CONTENTS To open a chapter, click on the chapter name. ELECTRICITY To turn pages within a Chapter 1: Electronic Components and Circuits chapter, use the keyboard Chapter 2: Properties of Capacitors and Inductors arrow keys. Chapter 3: Electrical Properties of Skin Chapter 4: Electrical Stimulation of Nerve and Muscle To navigate within a chapter, Chapter 5: Rectification and Amplification use the navigation buttons on each page. FIELDS Chapter 6: Electric and Magnetic Fields Chapter 7: Therapeutic Fields: Shortwave Diathermy Chapter 8: Non-Diathermic Fields WAVES Chapter 9: Sound and Electromagnetic Waves Chapter 10: Therapeutic Waves: Ultrasound Chapter 11: Electromagnetic Waves for Therapy GENERAL Chapter 12: Dosage and Safety Considerations Chapter 13: Electrical Safety Appendices: Quantities, Units and Prefixes
COMPONENTS AND CIRCUITS 1 1 Electronic Components and Circuits Although the central theme of this chapter is electronic Though the use of electricity in a medical context is no recent development. the last components and hardware, several decades have seen an unparalleled growth in electronic technology and its most of the concepts are application in medicine. Sophisticated electronic apparatus is now as commonplace in needed to understand the the hospital and clinical setting as it previously was in the research laboratory. It is relevant physiology and evident that all health professionals must become more and more accustomed to the electrotherapy covered in use of electronic instrumentation and its proper place in the practice of their profession. later chapters. The physiotherapist makes use of such apparatus for a variety of diagnostic procedures Apart from conductors and treatments, collectively described by the terms electrodiagnosis and electrotherapy. and insulators there is The aim here is to provide a physical basis for electrodiagnosis and electrotherapy. We a third, smaller group begin by discussing the 'nuts and bolts' of electronic apparatus, the basic components comprised of materials which go to make up an electronic circuit. displaying intermediate Electronic circuits and electronic components owe their existence to the fact that the conduction properties: this everyday materials with which we are familiar have quite dissimilar electrical properties. group includes the The vast majority of materials can be readily categorized as belonging to one of two semiconductors which form groups: either conductors or insulators. First let us consider conductors and insulators the basis of modern and some additional concepts which are fundamental to an understanding of electricity. electronics technology. We will talk more about SOME FUNDAMENTALS semiconductors in chapter two. Conductors are familiar to us in everyday life: the filament of a light bulb, the copper wire connected to car batteries and copper wire in the cables leading to most electrical appliances are a few examples. If we picture atoms as consisting of a positively charged nucleus together with electron 'shells' surrounding the nucleus then the atoms of conductors have a characteristic feature. Good conductors (iron, copper and other metals are examples) share the common feature that the outermost electrons of the atoms are only loosely bound to the nucleus. For this reason they can readily transmit electrons. A good way to picture this is to COMPONENTS AND CIRCUITS 2 imagine a fairly rigid lattice of atomic nuclei with a sea of electrons a metal consists of a rigid and contained within the boundaries of the piece of metal. regular, crystalline array of metal ions An electron entering one end of a length of wire causes a 'ripple in the sea ( ). A 'sea' or cloud of electrons ( ) of electrons' and, a split second later, an electron is ejected from the other end. fills the spaces between the ions. A flow of charges through any material is what we call an electric current and materials which readily transmit charges are termed conductors. The standard unit of current flow in the systeme internationale (Sl) is the ampere (abbreviated amp and given the symbol A). The current in amperes is a measure of the quantity of charge flowing through a conductor each second. The unit is named after the noted French physicist André Marie Ampère. Insulators have their electrons tightly bound in their shells so that the process of conduction described above can not take place under normal conditions. Plastics and ceramics are good insulators; thus we can use copper wires coated in plastic for the power leads to electric toasters, jugs, TV sets and so on. The telegraph poles in the street have the wires fastened not to the wood but around small white ceramic insulators. In order to cause a flow of current through a conductor we need a device which will produce an excess of electrons at one end of a wire and a deficiency at the other: such a device is a source of electrical potential energy and is said to produce a potential difference. A typical example of one of these devices is the battery found in a torch or transistor radio. The two terminals of a battery are at a different electrical potential - that is, there is a potential difference between the terminals. If a conductor is connected between the terminals electrons move from the terminal at a high potential to that at low potential. The Sl unit of potential difference is the volt (symbol V). Often the terms
COMPONENTS AND CIRCUITS 3 potential difference and voltage are used interchangeably. It is important to clearly understand the difference between voltage and current. Using Just as we can have a the analogy with water in pipes we can liken voltage to the water pressure and current substantial pressure in the to the volume rate of flow of water. water mains without any flow of water through a tap, so we If we connect a piece of material between the terminals of a battery the amount of can have a substantial current flow will depend on the battery voltage and the resistance to current flow which potential difference without the material offers. Good conductors offer little resistance to current flow while any flow of current. insulators offer substantial resistance. The Sl unit of resistance is the ohm (abbreviated Ω). We say something has a resistance of one ohm when a potential difference of one volt produces a current of one amp through it. A good conductor such as a metre length of household mains wire would have a resistance measured in milliohms while a good insulator, a block of ceramic, would have a resistance of several thousand megohms. Resistance values used in electronic circuits are typically measured in ohms (Ω), kilohms (kΩ) or megohms (MΩ). 1 kΩ = 103 Ω = 1000 Ω 1 MΩ = 106 Ω = 1 000 000 Ω COMPONENTS An electronic device such as a radio, TV set or CD player can be very complex but the complexity lies in the arrangement and total number of components. When we examine a typical circuit we find only a few different kinds of components. Resistors are the most common circuit components: they come in a variety of shapes and sizes and usually have their values coded in the form of three or four coloured stripes on the body. High power resistors are larger and usually have the resistance and power rating stamped on the body. Rheostats and potentiometers are variable resistors having two or three terminals respectively. COMPONENTS AND CIRCUITS 4 Capacitors come in a greater range of sizes and shapes than resistors but can readily be distinguished with a little practice. fixed value capacitors They consist of two metal plates (usually aluminium) and an insulator. The insulator may be air (in the case of variable capacitors), mica or a plastic film. The different kinds of capacitor are named after the insulator used, thus we have mica capacitors, polyester capacitors, ceramic capacitors and so on. Typical examples of fixed value capacitors are shown, together with a variable capacitor such as might be used in the tuning section of a radio. For typical values of capacitance large areas of metal are involved and it is convenient to save space by making the plates of thin aluminium foil and rolling them into a cylinder. Thus a tubular capacitor has the internal construction shown. We will return to discuss what a capacitor does in the next chapter. The unit of capacitance is the farad (F) but the values found in typical circuits are measured in microfarads (µF),
COMPONENTS AND CIRCUITS 5 nanofarads (nF) or picofarads (pF). 1 µF = 1 1 000 F = 10-6 F 1 nF = 1 1 µF = 10-9 F 1 pF = 1 1 000 µF = 10-12 F 000 000 000 Electrolytic capacitors are polarized; that is, they have definite positive and negative terminals and can only be connected one way around in a circuit. Ordinary capacitors (mica, polyester, etc.) are non-polarized and can be connected either way around. Inductors and transformers form our third category of components. An inductor is simply a coil of wire: the wire may be wound on various kinds of core, depending on the specific role for which it is intended. An example is shown. The unit of inductance is the henry (H). You will also see the terms millihenry (mH) and microhenry (µH) used frequently. 1 mH = 1 H = 10-3 H 1 µH = 1 mH = 10-6 H 1000 1000 When two inductors are wound on the same core in close proximity or overlapping we have a transformer. Valves, Transistors and Diodes are the 'workhorses' of any electronic circuit, the devices which have permitted the development of long distance voice communication, radio and television, computers and space exploration, guided missiles and the reproduction of music to name a few of the more obvious applications. Today valves are seldom used except for special applications: they are bulky and inefficient and have largely been superseded by transistors, their semiconductor equivalents. The transistor performs much the same job as a valve but is physically much smaller. COMPONENTS AND CIRCUITS 6 Most electronic equipment today uses Integrated Circuits (IC's), small devices having 8 or more pins and which contain complete circuits having many transistors, diodes, resistors and capacitors fabricated directly in a single package. The pins are used for access to various points in the circuit so that the designer can tailor the circuit to specific requirements by connection to external components. Integrated circuits have permitted a further reduction in the physical size of complex circuits comparable to the reduction that was achieved by the replacement of valve circuitry with transistors. The pocket calculator, desktop computer and space satellites have all been made possible by the miniaturization permitted with integrated circuits. ClRCUITS The electronic components we have met so far are typically found with a complex maze of interconnections between them. The particular way in which they are connected defines a circuit and the form that the circuit takes will depend on the job it is designed to do. If we consider, for example, a radio, the job it has to perform is exceedingly complex. It must pick up radio transmissions, convert the radio waves to electrical signals, amplify them and convert them to audible sound waves. Not only that but it must also be capable of selecting a particular frequency of radio wave (the one from the radio station you wish to tune-in on) and ignoring the remainder. Needless to say, the circuitry required to perform these tasks is very complex. In the next section we will consider extremely simple circuits in order to examine the characteristic behaviour of some of the components discussed so far. But first a few words on circuits themselves and their physical construction.
COMPONENTS AND CIRCUITS 7 The earliest circuits, those found for example in a valve radio receiver, were made up of a large number of bulky components - the transformer, valves, etc., and the circuit interconnections reflected this bulkiness. Large components such as the transformer were bolted to a metal frame or chassis along with sockets for the valves, and the circuit was built-up by interconnecting with smaller components (resistors and capacitors) and lengths of wire. With the advent of the transistor and the consequent trend to miniaturization which this permitted, it was found to be more convenient and simpler to replace most of the wiring with copper strips firmly attached to an insulating board having holes drilled in appropriate places. All of the miniature components (transistors, resistors, capacitors) could then be mounted on the board and soldered to the copper conductors. Only a limited number of bulky components (transformer, volume control and speaker in a mains-powered radio) need then to be mounted on a chassis or casing. Circuits assembled on these printed circuit hoards can be rapidly mass produced and easily put together. Consequently they are used almost exclusively by manufacturers. When integrated circuits are used rather than transistors a further reduction in size is achieved. The printed circuit boards can be made smaller as the l.C.'s themselves are small and relatively few external components are required. SIMPLE CIRCUITS WITH RESISTORS Perhaps the simplest kind of circuit which can be constructed comprises a battery (a source of electrical energy) and a single electrical component such as a resistor or a torch globe. First consider the simple circuit arrangement shown in figure 1.1. On the left is a small lamp (torch globe) connected by means of wires to a dry cell (battery). On the right is the circuit diagram of the arrangement: note the circuit symbols used for the lamp and dry cell. When connected this way current will flow from one terminal of the battery, through the lamp and back to the opposite terminal. It is common convention to say that current flows from the positive terminal of the battery to the negative terminal; although this is the opposite direction to the flow of COMPONENTS AND CIRCUITS 8 electrons. The difference between reality and convention is not important as it has no Figure 1.1 effect on the magnitudes of currents and voltages in a circuit. A very simple circuit and its Under normal circumstances representation using a circuit the flow of current through the lamp will cause it to glow - the diagram normal purpose of such a circuit. If a break occurs in the wire for example, or the wire is not secured to the battery, current will not flow and we have an open circuit. If the wire connected to one terminal of the lamp comes loose and touches the other wire, current can flow directly through the wires and bypass the lamp - this condition is known as a short circuit. The switch on a torch is just a means of introducing an open circuit condition at will: preventing the flow of current and thus conserving the electrical energy stored in the battery. Going one step further than the circuit shown in figure 1.1 we may add a resistor to produce the circuits shown in figure 1.2. Note the circuit symbol (zig-zag line) for a resistor. Figure 1.2(a) shows the resistor in series with the lamp, i.e. the resistor and lamp are connected end to end so that current must flow through both the lamp and the resistor. In 1.2(b) the resistor is connected in parallel with the lamp so that some current can flow through the lamp and some through the resistor. The resistor is thus allowing part of the current to bypass the lamp and return to the opposite terminal of the battery.
COMPONENTS AND CIRCUITS 9 RESISTORS IN SERIES Figure 1.2 Two less simple circuits. (a) a Two or more resistors may be connected in series or in parallel. When the resistors are connected in series as in figure 1.3 the current must pass through series and (b) a parallel each resistance in turn and the total resistance offered by the circuit is equal to the arrangement. sum of the individual resistances. Figure 1.3 In general for a circuit having n resistors connected in series the total resistance Resistors in series of the circuit, R, is equal to the sum of the individual resistances i.e. R = R1 + R2 + R3 ........ + Rn .... (1.1) As mentioned previously, the current flowing is a measure of the rate of flow of charges around the circuit and is measured in amperes, or amps for short. Since there can be no accumulation of charges within a resistor the current flowing into a certain resistor is equal to the current flowing out. Thus the same current must flow through each resistor when they are connected in series. Although the current in each resistor is the same, the potential difference across each resistor in a series circuit is, in general, different. The sum of the potential differences across the resistors is equal to the potential difference generated by the battery (the battery voltage). COMPONENTS AND CIRCUITS 10 RESISTORS IN PARALLEL When a number of resistors are connected in parallel as in figure 1.4 the current flowing in each resistor may be different. In this case the total resistance of the circuit, R, is given by the equation 1 = 1 + 1 + 1 R R1 R2 R3 or more generally, where there are n resistors in parallel in a circuit 1 = 1 + 1 + 1 ..... + 1 .... (1.2) R R1 R2 R3 Rn Although the current in each resistor may be quite different, the voltage or Figure 1.4 potential difference across each resistor is the same and is equal to the Resistors in parallel battery voltage. This must be so since the ends of each resistor are connected directly to the terminals of the battery. The largest amount of current will flow through the smallest resistance when resistors are connected in parallel. OHM'S LAW The relationship between potential difference (V), resistance (R) and current (I) through a resistor is summarized by Ohm's law which states that the current flowing through a resistor is proportional to the potential difference across the resistor and is inversely proportional to the resistance. The equivalent mathematical expression of Ohm's law is I = V .... (1.3) R
COMPONENTS AND CIRCUITS 11 where I is the current in amperes, V is the potential difference in volts and R is the resistance in ohms. Example 1. If, in figure 1.3, R1 is 50 ohms, R2 is 100 ohms and R3 is 150 ohms then the total resistance R is R = 50 + 100 + 150 = 300 Ω If these are connected to a 10 volt battery the current flow will be I = V = 10 = 0.033 amps or 33 milliamps. R 300 Here we have calculated the total current drawn from the battery which is equal to the current flowing through each resistor. The battery gives a total potential difference of 10 volts across the combination of resistors. If we wish to calculate the potential difference across each individual resistor we again use equation 1.3 but apply the equation to each resistor in turn. For example, for the 50 ohm resistor we know that I = 0.033 amps and R = 50 Ω so I = V becomes V = 0.033 and hence V = 50 x 0.033 = 1.7 volts. R R For the 100 ohm resistor V = 100 x 0.033 = 3.3 volts and for the 150 ohm resistor V = 150 x 0.033 = 5.0 volts. The total potential difference across the combination of resistors is V = 1.7 + 3.3 + 5.0 = 10 volts as expected. COMPONENTS AND CIRCUITS 12 Example 2. Now consider the same three resistors as in the previous example connected in parallel to a 10 volt battery as in figure 1.4. The total resistance, R, is now given by 1 = 1 + 1 + 1 = 6 + 3 + 2 = 11 R 50 100 150 300 300 300 300 so R = 300 = 27 ohms 11 The total current is I = V = 10 = 0.37 amps R 27 To calculate the current in each individual resistor we use the fact that the potential difference across each resistor is the full 10 volts. Then for the 50 ohm resistor we have I = V = 10 = 0.20 amps. R 50 For the 100 ohm resistor I = 10 = 0.10 amps 100 and for the 150 ohm resistor I = V = 10 = 0.07 amps R 150 So the total current drawn from the battery is I = 0.20 + 0.10 + 0.07 = 0.37 amps as calculated previously. The calculations are made relatively simple by remembering that when resistors are connected in series the current through each is the same and when resistors are connected in parallel the potential difference across each is the same.
COMPONENTS AND CIRCUITS 13 POWER DISSIPATION IN RESISTORS When electric current flows through a conductor, whether the conductor is a resistor, copper wire, the filament of a light bulb or whatever, electrical energy is converted to heat energy in the conductor. An extreme example of this is seen with a lamp - the heat generated is so great that the filament wire glows brightly. The same heating effect takes place, though to a lesser extent, in a resistor and clearly there will be a limit to how much heat a resistor can generate per second before it starts glowing and disintegrates or melts. The maximum rate at which a resistor can safely dissipate electrical energy is called the power rating of the resistor and this depends on the size and physical construction. The actual rate of dissipation of electrical energy at any particular time is the power dissipated. This depends on both the current flowing through the resistor and the potential difference across it. The power (in watts) is equal to the product of current (in amps) and potential difference (in volts). If we denote power by the symbol P the relationship may be written P = V.I .... (1.4) This formula can be usefully used with Ohm's law to calculate the maximum values Starting with the expression for voltage or current for a particular resistor of known resistance and power rating. P = V.I and substituting V = I.R Alternatively if we know any two of the values of V, I or R for a given resistor, we can we have P = (I.R).I or P = I2R calculate the power dissipated in the resistor using equation 1.4 or the two expressions obtained by solving equations 1.3 and 1.4. These are: Starting with the expression P = V.I and substituting I = V/R P = V2 .... (1.5) we have P = V.(V/R) or P = V2 R R where we have eliminated I in solving the two expressions and P = I2.R .... (1.6) where we have eliminated V. COMPONENTS AND CIRCUITS 14 With these three alternative formulae for calculating power we would choose the Figure 1.5 equation which gives the required answer with a minimum of calculations. Resistors in a series/parallel Consider the three resistors connected in series as in example 1 previously. If we wish to calculate the power dissipated in each resistor we would first calculate the arrangement current (which is the same for each resistor) and then use equation 1.6. In example 2 previously the resistors are in parallel and the potential difference is the same for each resistor. Equation 1.5 is thus the choice for power calculation. RESISTORS IN A SERIES/PARALLEL COMBINATION The ideas we have considered previously can be applied to more complex circuits where resistors are combined in a series/parallel arrangement. Figure 1.5 shows one possible series/parallel combination. To calculate the total resistance in this circuit we would first need to calculate the resistance of R2 and R3 in parallel using equation 1.2. The total resistance is obtained by adding this to R 1. This is because R1 is in series with the parallel combination of R2 and R3 so equation 1.1 applies. To calculate voltages and currents in this circuit it is important to visualize current flowing from the positive terminal of the battery, through R1 and then through the combination of R2 and R3. The current flowing through R1 splits. Some goes through R2 and some through R3. The current though R2 plus the current through R3 must equal that through R1 since the moving charges cannot disappear nor can new ones appear. Another important idea is that current (charge) flows around the circuit, losing electrical potential energy in the process. Some energy is lost as charges flow through R1. The remainder is lost as charges flow either through R2 or R3. The potential difference across R2 is one-and-the-same as the potential difference
COMPONENTS AND CIRCUITS 15 across R3. The potential difference across R1 plus that across the R2/R3 combination must equal the battery voltage. Example 3. If we have a resistor combination like that shown in figure 1.5, with the resistance values and battery voltage are as shown alongside: (i) what is the current through each of the resistors? (ii) do the currents calculated add-up i.e. is the current through the 50 Ω resistor equal to the total current through the 100 Ω and 200 Ω combination? (iii) what is the potential difference across the 50 Ω resistor? (iv) what is the power dissipated in the 100 Ω resistor? Answers: (i) First calculate the total resistance. For the parallel combination, 1 = 1 + 1 = 3 so R = 200 = 67 Ω R 100 200 200 3 The total resistance is thus 67 + 50 = 117 Ω The battery voltage is 10 V so the total current is I = V = 10 = 0.085 A R 117 All of this current flows through the 50 Ω resistor so the current through it is 0.085 A or 85 mA. The potential difference across the 50 Ω resistor is V = I.R = 0.085 x 50 = 4.3 V, which answers question iii above. The potential difference across the 100 Ω/200 Ω parallel combination is 10 - 4.3 = 5.7 V. COMPONENTS AND CIRCUITS 16 The current through the 100 Ω resistor is I = V = 5.7 = 0.057 A R 100 The current through the 200 Ω resistor is I = V = 5.7 = 0.029 A R 200 (ii) The total current through the 100 Ω/200 Ω parallel combination is 0.057 + 0.029 = 0.086 A which (allowing for rounding errors as we have used two-figure accuracy) agrees with the value calculated for the 50 Ω resistor. (iii) See above. (iv) The power dissipated in the 100 Ω resistor can be calculated using either equation 1.4 or 1.5 or 1.6 as we know the potential difference, current and resistance. Using equation 1.4 we have P = V.I = 5.7 x 0.057 = 0.33 W. AC AND THE MAINS SUPPLY DC and AC If we connect a battery to a lamp, as in figure 1.1 previously, current will flow from one terminal of the battery, through the lamp and return to the opposite terminal. Such a flow of current is termed direct current (DC) because the flow is always in one direction. A graph of voltage against time or current against time would look like figure 1.6. DC is directly available from a battery and it is DC that we need to power most electronic circuits. The voltage available at the power points in the laboratory or home is not, Figure 1.6 however, constant or direct but fluctuates rapidly and regularly between Direct current as from a battery about + 340 and -340 volts. The repetition rate or frequency of the alternating voltage is 50 times per second. If we plotted a graph of voltage against time it would look like figure 1.7.
COMPONENTS AND CIRCUITS 17 The alternating voltage supplied by the mains gives rise to an alternating current (AC) in any device which is connected to a power point. During the positive half-cycles of the waveform shown in figure 1.7 the current flows in one direction through the appliance and during the negative half-cycles the current flows in the opposite direction. Because the waveform in figure 1.7 is symmetrical the current flow in one direction during the positive half-cycles will be exactly balanced by the current flow in the opposite direction during the negative half-cycles. Hence there is no net transfer of charge over a full cycle; electrons are simply moved back and forth. This is in contrast to direct current where electrons are moved in one direction only. The frequency or number of times an AC waveform repeats itself per Figure 1.7 second is expressed in units of hertz (abbreviated Hz). 1 Hz is 1 cycle per second. Alternating current as from the mains supply It is easy to calculate the frequency of an AC waveform from a graph like figure 1.7. First we determine the period of the waveform. The period is simply the time taken for one complete cycle which is read from the horizontal axis. In this example, the alternating current is from the mains supply and the period, the time for one complete cycle or oscillation, is 0.02 second or I /50th of a second. The frequency (f) is the reciprocal of the period (T). In symbols: f = 1 .... (1.7) T Hence for AC supplied by the mains the frequency is 1/0.02 = 50 Hz. During the first hundredth of a second the voltage in figure 1.7 has gone from zero to +340 volts and back to zero again. The effective (average) voltage will be some figure between these limits. The figure of 340 volts (the maximum value) is termed the peak voltage but we normally specify AC by an average rather than peak value. The average during the first hundredth of a second is 340/ or 240 volts in this example. COMPONENTS AND CIRCUITS 18 Thus the appliances we plug into the mains must have a rating of 240 volts AC. Figure 1.8 Since the voltage is rapidly oscillating between + 340 and -340 volts the current through a light globe connected to the mains will flow alternately in one direction and (a) Circuit symbol and (b) then the other and the globe will flicker in brightness according to the voltage. It is construction of a transformer only because the frequency of fluctuation is sufficiently rapid that this effect is not noticeable. TRANSFORMERS AND THE MAINS SUPPLY Transformers are the devices we use to step-up or reduce AC voltage. If we apply a certain AC voltage to one winding of a transformer (the primary winding) an AC signal of the same shape but different amplitude (peak voltage) will appear on the other winding (the secondary winding). Power Transformers are specifically designed to convert 240 volts AC to some other voltage: for example a stereo tuner/amplifier would require only 20 to 60 volts to power the circuitry comprising the radio and amplifier, it would thus require a power transformer to step down the mains voltage to an appropriate value. An example of a transformer was shown in section 1.1. Figure 1.8 illustrates the circuit symbol and construction of a transformer. A transformer consists of two coils of wire which are insulated from each other but in close proximity. In practice the two coils of wire are often wound on a common core, made of iron (see figure 1.8(b)). This allows the transformer to work more efficiently. When AC is applied to one coil of wire (the 'primary' coil), an alternating current is induced in the other (the secondary) and the voltage will, in general, be different. If the transformer is constructed with more turns on the primary winding than on the secondary then it will act as a step-down transformer. For example if there are 200 turns on the primary winding and 50 on the secondary the voltage from the secondary will be one quarter of that applied to the primary. If 240 volts is applied to the primary 60 volts will be produced at the secondary.
COMPONENTS AND CIRCUITS 19 If the transformer is constructed with fewer turns on the primary winding than on the The principle that 'you can't secondary then it will act as a step-up transformer. For example if there are 20 turns get something for nothing' on the primary winding and 100 on the secondary the voltage from the secondary will applies here. If the trans- be five times that applied to the primary. If 240 volts is applied to the primary 1200 former steps-up the voltage, volts will be produced at the secondary. the available current is Because of their physical construction - two inductors in close proximity but insulated reduced. If the transformer - they play a useful role in isolating the voltage applied to one winding (the primary) steps-down the voltage, from that available at the other winding (the secondary). Most appliances are isolated the current is increased. from the mains voltage in this way - by their transformers. The isolation given by a transformer is important because the voltage Figure 1.9 from the secondary is earth-free. Isolation with a power The power from the mains comes from two cables, the active cable (colour coded brown) and the neutral cable (colour-coded blue). transformer The neutral cable is earthed at the power station and at distribution Current can only flow along substations. That is, the neutral cable is physically connected to the 'active' wire if it can the ground through a metal conductor (for example, a thick length simultaneously return via of wire connected to a metal stake or water supply pipe buried in the 'neutral' wire. Because the ground). The neutral cable is thus kept at zero volts potential. the neutral wire is earthed, The problem with this arrangement is that if a connection is anything touching the earth inadvertently made between the active cable and earth, such as by is provided with a return a person accidentally touching the active cable, this will complete a pathway via the neutral wire. circuit and current will flow from the active wire through the person to earth, giving an 'earth' shock. To obtain a shock from the 'earth free' power supplied at the transformer secondary it would be necessary for the person to touch both terminals simultaneously. As long as there is no connection between the primary and secondary windings, touching only one secondary terminal and thus making a connection between the secondary circuit and earth will not complete a circuit and no current can flow, hence no shock. Mains supply and the mechanisms of electric shock are described in more detail in a later chapter. COMPONENTS AND CIRCUITS 20 EXERCISES 1 The lamp shown in figure 1.1 has a resistance of 3.0 Ω and is connected to a 1.5 V battery. Calculate the current flowing through the lamp and the power dissipated. 2 An electric jug draws a current of 1.2 amps when connected to the 240 volts mains supply. Calculate the resistance and power rating of the heating element. 3 The power dissipation of an electric heater is 1000 watts when connected to a 240 volt supply. Calculate its resistance, and the current it will draw. 4 Three resistors are connected in series as shown in figure 1.3. The battery voltage is 12 V. The resistors have values R1 = 100 Ω, R2 = 500 Ω and R3 = 600 Ω. Draw a diagram of the circuit and label the values of the components. What is: (a) the current through each resistor (b) the potential difference across each resistor (c) the power dissipated in each resistor Do the values calculated in (b) add to equal the battery voltage? 5 Three resistors are connected in parallel as shown in figure 1.4. The battery voltage is 5 V and the resistors have values R1 = 50 Ω, R2 = 100 Ω and R3 = 75 Ω. Calculate: (a) the current through each resistor (b) the potential difference across each resistor (c) the power dissipated in each resistor (d) the total resistance of the circuit and hence (e) the total current drawn from the battery.
COMPONENTS AND CIRCUITS 21 22 (f) does the value calculated in part (e) agree with the sum of the values found in part (a) above? 6 Three resistors are connected in the circuit arrangement shown. The current through the 100 Ω resistor is 50 mA. Determine: (a) the battery voltage and (b) the power dissipated in the 30 Ω resistor. 7 In the course of an experiment a student sets-up the circuit shown below using a torch globe, 2 resistors and a battery. A voltmeter connected across the 200 Ω resistor registers 8 volts. (a) What is the resistance of the torch globe? (b) What is the current through the globe? COMPONENTS AND CIRCUITS 8 Consider the circuit shown. The current through the 50 kΩ resistor is measured as 67 µA. Determine the: (a) value of the unknown resistor, R. (b) total current drawn from the battery. (c) power dissipated in the 50 kΩ resistor. (d) potential difference across the resistor, R. (e) current through the 100 kΩ resistor. 9 Consider the circuit shown. The total current drawn from the battery is 0.2 A. What is the: (a) current through the 30 Ω resistor. (b) value of the unknown resistor, R. (c) power dissipated in the 50 Ω resistor. (d) potential difference across the 30 Ω resistor. (e) power dissipated in the resistor, R. 10 (a) What are the two important functions of a transformer? (b) A transformer has 100 turns on the primary winding and 40 turns on the secondary. If an AC signal of amplitude 60 V is applied to the primary winding what voltage will appear at the secondary? (c) A transformer has 200 turns on the primary winding. An AC signal of amplitude 240 V is applied to the primary winding and a voltage of 12 V is produced by the secondary. How many turns are there on the secondary?
CAPACITORS AND INDUCTORS 23 2 Properties of Capacitors and Inductors Figure 2.1 shows the circuit symbol for a capacitor and an inductor. Note the parallel Figure 2.1 between the circuit symbol and the physical construction of the component. A Circuit symbol for (a) a capacitor consists of two metal plates (often sheets of aluminium foil) separated by capacitor and (b) an inductor. an insulator. An inductor is simply a coil of wire. Knowing the physical construction of each we can predict what resistance they offer when connected in a circuit to a battery. The capacitor, with an insulator between its plates, offers an extremely high (virtually infinite) resistance to DC. Current can not flow from one plate to the other. Inductors, being simply coils of conducting wire, have an extremely low resistance to DC (almost zero). STORAGE CAPABILITY OF CAPACITORS A special property of a capacitor is its ability to store charge and thus electrical energy. If we connect a battery to the capacitor current will flow for a very short time and, since the current can not flow continuously through the insulator, the capacitor will accumulate charge on the plates. When the charge has built up until the voltage across the capacitor is equal to the battery voltage, current will no longer flow. The capacitance tells us how much charge a capacitor will accumulate for a given battery voltage. Capacitance (C), charge (q) and voltage (V) are related by the formula C = q .... (2.1) V Once charged, a capacitor can store the charge indefinitely. If the battery is disconnected, the charge on the capacitor and can only leak away if we connect an external resistance. The circuit shown in figure 2.2 can be used to observe the discharge of a capacitor. CAPACITORS AND INDUCTORS 24 Suppose that the capacitor is first fully charged to 12 V, then connected into the circuit. Upon closing the switch, current flows through the 500 kΩ resistor, allowing the capacitor to discharge. The reading on the voltmeter progressively falls. Table 2.1 shows measured values of the voltage taken at 2 second intervals for 20 seconds using a 25 µF capacitor. The 25 µF capacitor was then replaced with a 50 µF capacitor and the second set of results was obtained. Time Voltage on the Voltage on the Figure 2.2 (s) 25 µF capacitor 50 µF capacitor A circuit for measuring the 0 2 12 12 discharge of a capacitor 4 10.2 11.1 6 10.2 Table 2.1 8 8.7 Measured voltages for the 10 7.4 9.4 discharge of capacitors through 12 6.3 8.7 14 5.4 8.0 a resistance of 500 kΩ 16 4.6 7.4 18 3.9 6.8 20 3.3 6.3 2.8 5.8 2.4 5.4 A graph of these voltages against time is shown in figure 2.3. Note that it takes about 8.5 seconds for the 25 µF capacitor to discharge to half of the original voltage and 17 seconds for the 50 µF capacitor. On this basis we predict that it would take about 85 seconds for a 250 µF capacitor to discharge to half the original voltage through the same resistance.
CAPACITORS AND INDUCTORS 25 What we observe is that the time to half discharge (τ1/2) Figure 2.3 Discharge of capacitors is directly proportional to the capacitance (C) of the capacitor. In symbols this is written through a resistance τ1/2 α C Other experiments, where the resistance through which the capacitor discharges is varied, show that the time for half-discharge is also proportional to the resistance (R) through which the capacitor discharges. That is, τ1/2 α R Combining these equations we have that the time for half discharge is directly proportional to the product of the resistance and the capacitance. In symbols τ1/2 α R.C .... (2.2) This is a general relationship between resistance, capacitance and discharge time for a resistor/capacitor combination. Sometimes the term 'half-discharge time' is used, sometimes 'RC time-constant'. They are related, but are not the same thing. The RC time- constant is simply the product R.C. The constant of proportionality in equation 2.2 is 0.693 (the natural logarithm of 2) so if we know the RC time constant, τ1/2 is easily calculated. Notice that for a short period of time, such as 1/50th of a second, the 25 µF capacitor in table 2.1 will not have discharged appreciably. If it discharges by 0.9 volt in the first second, in l/50th of a second it will discharge by less than 0.02 volt. The importance of this and the previous ideas will be considered later. CAPACITORS AND INDUCTORS 26 INDUCTORS AND MAGNETIC FIELDS Most of you would have made an electromagnet at school. When current flows though an inductor, a magnetic field is produced. In fact, Insulated copper wire is whenever current flows, there is an associated magnetic field so a straight piece of wound around, say, a nail to wire which is carrying current will have a magnetic field around it. The reason make an inductor. When the inductors are special is that the wire is wound in closely spaced loops and this ends of the wire are connect- results in a much more intense magnetic field. The field can be further intensified by ed to a battery, the nail wrapping the coil around a piece of iron or mild steel. Both the magnetic field becomes magnetized. intensity and the inductance are increased if the inductor is wound around a core of Infants can hear sound iron or other so-called ferromagnetic material. frequencies of 20 kHz or more. We will have more to say about magnetic fields (and their effects on tissues) in later With age, this maximum chapters. frequency drops so that most people over 35 can not hear IMPEDANCE OF CAPACITORS AND INDUCTORS frequencies above about 14 kHz. A major part of the field of electronics is concerned with processing alternating current of various frequencies and it is for this reason that capacitors and inductors are so important. Resistors are impartial as far as direct current and alternating current is concerned: they offer the same resistance to AC and DC. Capacitors and inductors display no such impartiality: they both react quite differently to AC and DC. Before describing the properties of capacitors it is useful to have some idea of the kind of frequencies we are dealing with. Sounds which are audible to humans lie in the range from 20 Hz to up to a maximum of about 20 kHz (1 kHz = 1000 Hz). Normal (AM) radio transmissions are made in the frequency range from 500 to 1500 kHz. FM radio transmissions are made in the range 88 to 108 MHz. (l MHz = 1000 kHz or 1000 000 Hz). Check-out the numbers on your tuner or pocket radio display panel. Television transmissions are made in the frequency range on either side of the FM range, i.e. between 40 and 80 MHz and above 108 MHz up to about 130 MHz. In each case the sound, radio or television frequency wave can be converted to alternating current of the same frequency which can be amplified, filtered and processed by an electronic circuit to produce the resulting sound or image.
CAPACITORS AND INDUCTORS 27 Capacitors and inductors are useful because their resistance to alternating current depends on the frequency of the AC. Next we examine this frequency dependence. Knowing the physical construction of capacitors and inductors we can predict what resistance they offer when connected in a circuit to a battery. The capacitor, with an insulator between its plates, offers an extremely high (virtually infinite) resistance to DC. Current can not flow from one plate to the other. Inductors, being simply coils of conducting wire, have an extremely low resistance to DC (almost zero). For alternating current the resistance of these components is not predicted quite so simply. Instead of attempting a prediction let us look at some experimental observations with AC. When we talk about resistance to the flow of alternating current we no longer use the term 'resistance' but impedance. The impedance is still measured in ohms but we reserve 'resistance' for use when talking about DC. The symbol Z is usually used for impedance. Impedance can be calculated from measurements using a circuit such as that shown in figure 2.4. The potential difference across R and that across C are found to depend on the AC frequency. The two values added equal the potential difference produced by the AC source. The voltage across the resistor can be used to calculate the current flowing around the circuit (using Ohm's Law, I = V/R). Since C and R are in series, this equals the current flowing to the capacitor. The impedance of the capacitor is then calculated using Z = V .... (2.3) Figure 2.4 I A circuit for measuring the impedance of a capacitor Which is Ohm's law in its more general form. If the capacitor in figure 2.4 is replaced by an inductor, impedance values for the inductor can be calculated in a similar manner. Table 2.2 gives the measured impedance of a 20 millihenry (mH) inductor and a 1 microfarad (µF) capacitor at various frequencies. CAPACITORS AND INDUCTORS 28 Table 2.2 Frequency Impedance of 1 µF Impedance of 20 mH Variation of impedance with f (kHz) capacitor (ohms) inductor (ohms) 0.2 25.1 frequency for a 1 µF 0.3 795 37.7 capacitor and a 20 mH 0.4 530 50.2 0.6 398 75.4 inductor 1.0 265 126 2.0 159 251 3.0 377 4.0 79.5 502 5.0 53.0 628 39.8 31.8 A graph of these values of impedance at each frequency is shown in figure 2.5. Notice that for the inductor the graph is a straight line passing through the origin. This tells us that the impedance of the inductor is directly proportional to the frequency of the AC. For the capacitor the curve suggests an inverse relationship between impedance and frequency: this is, in fact, the case. To check that the relationship is an inverse one rather than inverse square or inverse cube we would need to plot impedance versus 1/frequency and establish that this gives a straight line graph. This is left as an exercise for the interested reader. The mathematical expression for the dependence of impedance on frequency for each of these components is Zαf for the inductor and Z α 1/f for the capacitor. By substituting different values of inductance and capacitance, it is established that for inductors, Z α L and for capacitors, Z α 1/C.
CAPACITORS AND INDUCTORS 29 Since for inductors, impedance is proportional to the frequency and is also proportional to the inductance, we can combine these expressions and write Z α f.L. When the impedance is specified in Ohms, the inductance in Henrys and the frequency in Hz, the constant of proportionality is 2π. The relationship between Z and f for an inductor is thus Z = 2πfL .... (2.4) Similarly for the capacitor: since impedance is proportional to the reciprocal of frequency and the reciprocal of the capacitance we can combine the expressions and write Z a 1/f.C. When the impedance is specified in Ohms, the capacitance in Farads and the frequency in Hz, the constant of proportionality is 1/2p. The relationship between Z and f for a capacitor is thus Z = 1 .... (2.5) 2πfC Figure 2.5 From equations 2.4 and 2.5 we can see that it is predicted that doubling the Impedance versus frequency value of inductance would double the impedance at all frequencies. Doubling graph for (a) a 1 µF capacitor the value of capacitance would halve the impedance at all frequencies. These predictions are easily confirmed experimentally. and (b) a 20 mH inductor Two important points to note from the equations (or the graphs) are that capacitors have an infinite impedance to direct current (zero frequency) whereas inductors have zero impedance to DC. This is in agreement with our previous predictions based on the physical construction of each. CAPACITORS AND INDUCTORS 30 RESONANT CIRCUITS Figure 2.6 A parallel resonant circuit and how its A traditional and simple way of producing alternating impedance varies with AC frequency current relies on the properties of capacitors and inductors and their behaviour when connected together in a circuit. An inductor and capacitor combined in parallel as in figure 2.6 forms what is called a parallel resonant circuit. The impedance of the inductor increases with frequency while the impedance of the capacitor decreases with increasing frequency (figure 2.5). The parallel combination will have a low impedance at low frequencies (because of the inductor) and at high frequencies (because of the capacitor). At an intermediate frequency the combination has a maximum impedance. This is when the impedances of the components are equal. The peak in the graph shown in figure 2.6 occurs at this frequency. A formula for calculating the frequency at which the impedances of the capacitor and inductor are equal is obtained as follows. The impedance of the capacitor is given by equation 2.5, and the impedance of the inductor by equation 2.4. When the impedances of the components are equal 1 = 2πfL 2πfC which rearranges to give f2 = 1 4π2LC
CAPACITORS AND INDUCTORS 31 or f = 1 .... (2.6) Figure 2.7 Initial current flow in a 2π√LC resonant circuit This frequency is termed the resonant frequency of the combination, and for good reason. Imagine that the capacitor in figure 2.6 is disconnected from the inductor and connected to a battery to charge it. The battery is then removed and the capacitor connected to the inductor again. What subsequently happens? The capacitor immediately begins to discharge and current flows through the inductor. In so doing the current flow sets up a magnetic field around the inductor. Immediately the charge disappears the current ceases, causing the magnetic field to collapse. The collapsing magnetic field then induces a current in the inductor and the capacitor recharges, but with opposite polarity. The sequence of events is illustrated in figure 2.7 where the arrows show the direction of current flow. The capacitor next begins to discharge in the opposite direction (not shown in figure 2.7 - this would follow (d)) and the whole cycle is repeated again and again. What is happening is that the electrical energy stored in the capacitor is converted to magnetic field energy around the inductor. Because a magnetic field can only exist if current is flowing, the magnetic field must collapse as the capacitor becomes discharged. The magnetic field energy can not disappear, but is converted back into electrical energy which moves charge around the circuit and reverse-charges the capacitor. The reverse-charged capacitor (figure 2.7(d)) would then discharge through the inductor. Current would flow in the CAPACITORS AND INDUCTORS 32 reverse direction to that in figure 2.7, producing a magnetic field which would collapse Figure 2.8 and re-charge the capacitor to its original state (figure 2.7(a). The process would (a) undamped and repeat indefinitely. (b) damped sinewaves If this were the whole story the circuit would resonate, generating a continuous alternating flow of current through the inductor. A graph of current versus time would be an undamped sine wave as shown in figure 2.8(a). A graph of voltage across the capacitor against time would also resemble figure 2.8(a). In practice, capacitors and inductors are not ideal. The inductor will have some DC resistance and the capacitor will allow some leakage of current so that energy is lost during each oscillation. The oscillations will be damped and must eventually come to an end. The net result will be a damped oscillation as shown in figure 2.8(b). The natural frequency of oscillation is the same frequency indicated by the peak in the graph shown in figure 2.6: the resonant frequency. This is also the frequency at which the impedances of the capacitor and inductor are equal (where the graphs intersect in figure 2.5). [A graph similar to that in figure 2.6 could be arrived at simply by adding impedances graphically, say by using figure 2.5. The method gives a correct qualitative result. It does not give a correct quantitative prediction as no consideration is given to the phase relationship between the current in each component. The 'resonant frequency' obtained is correct but the height of the impedance peak is not. A rigorous mathematical treatment shows that, for ideal components, the impedance is infinite at the resonant frequency. For real components the impedance will be finite because of the resistance of the inductor and leakage of the capacitor].
CAPACITORS AND INDUCTORS 33 Although resonant circuits are most commonly encountered in the parallel combination shown in figure 2.6, they can occur as a series combination (figure 2.9). Again there is a definite resonant frequency at which the impedances of the two components are equal. For a series resonant circuit the impedance has a minimum value at the resonant frequency. Because of the non-ideal nature of the components the impedance is not zero at the resonant frequency but when component losses are low the impedance can suddenly drop to a very low value as this frequency is approached. COUPLING AND RESONANT CIRCUITS If an oscillator is designed to operate at a fixed frequency the usual way of coupling its signal to an amplifier or, more generally, between circuits is by use of a resonant circuit. This method is most commonly used at high frequencies (above 100 kHz). Figure 2.10 shows how a resonant circuit is used to transfer Figure 2.9 electrical energy from one circuit to another most efficiently. A series resonant circuit and how its In other words to 'couple' the circuits most efficiently. impedance varies with AC frequency The signal from the first circuit supplies energy for the resonant circuit to oscillate. The signal is induced in the second circuit by the transformer action of the two inductors. In order for transformer action to operate most efficiently the signal produced by circuit 1 should have the same frequency as the resonant circuit. This means either that the frequency of circuit 1 can be adjusted to match that of the resonant circuit or that the frequency of the resonant circuit can be adjusted to match that of circuit 1. The latter would require the inductor or capacitor in the resonant circuit to be variable so that the resonant frequency could be adjusted to suit. CAPACITORS AND INDUCTORS 34 Figure 2.10 Coupling with a resonant circuit. The combination of L and C will have a resonant frequency which is determined This is analogous to pushing by their inductance and capacitance values according to equation 2.6. The current a swing. The swing has its through the inductor connected to circuit 1 will induce a current of the same own, natural or resonant frequency in L. If the current induced in L does not have the same frequency as frequency. If you apply the resonant circuit, little energy will be transferred. If the resonant frequency of L pushes at the same and C in combination is the same as the frequency of the signal supplied by frequency, the swing circuit 1, then the circuit will resonate and maximum electrical energy will be oscillations are reinforced. transferred. Pushing at a different frequency would be SUSTAINED OSCILLATION counterproductive. By appropriate choice of the capacitor and inductor a resonant circuit can be made to generate any frequency of sine wave. A resonant circuit alone is not sufficient, however, to generate a sustained oscillation. To produce a continuous, steady supply of alternating current (as in figure 2.8a) we must arrange for the resonant circuit to be continuously supplied with energy to overcome the losses in the components and keep it oscillating. By use of an amplifying circuit, we can provide the energy to overcome the circuit losses and prevent the oscillations from dying. An amplifier provides positive feedback. Amplifiers and positive feedback are considered in chapter 5.
CAPACITORS AND INDUCTORS 35 PIEZOELECTRIC CRYSTALS So far our discussion of oscillators has been restricted to resonant circuits in the form of inductor and capacitor combinations. Unfortunately, such combinations tend to drift in frequency over a period of time and with changes in ambient temperature. The effect is only slight and not very important at low frequencies unless high accuracy is required. The effect is much more significant at radio frequencies (greater than 500 kHz) where it is very important to have good frequency stability - imagine switching on your radio and never knowing quite where to find your favourite radio station! The effect is really quite serious in TV and radio transmission and communications. The frequencies allocated to users are quite closely spaced and any drift in transmission frequency could result in overlap with adjacent transmitters. Similarly the shortwave diathermy equipment used in physiotherapy operates at radio frequencies and radiates a certain amount of energy as radio waves - for this reason only certain frequencies are permitted for their operation and little deviation or drift is permitted. Good frequency stability can be achieved by use of a piezoelectric crystal. These crystals have the special property that when squeezed or stretched, a potential difference is produced between each surface. This piezo-electric effect is illustrated in figure 2.11. The other side of the coin with piezoelectric crystals is that if a potential difference is applied to their opposite sides, they change in thickness. Thus in the example shown in figure 2.11, if a potential difference is applied to an unstressed crystal and the voltage is positive on top, the crystal will shrink in thickness. If the potential difference is applied negative on top in this example, the crystal will expand. An interesting thing happens if a potential difference is suddenly and briefly applied. The Figure 2.11 piezoelectric crystal reacts like a bell which is struck by a hammer and starts ringing. The A piezo-electric crystal thickness of the crystal changes when the voltage is applied but the molecules have a compressed and stretched momentum which causes them to overshoot their equilibrium positions. They are pulled back and overshoot in the opposite direction. The cycle continues and the result is that the crystal vibrates continuously. Figure 2.12 illustrates the process and what would be observed in practice. CAPACITORS AND INDUCTORS 36 The crystal resonates mechanically at a particular frequency. Its thickness oscillates about a mean which is its normal, unstressed thickness (the horizontal axis in figure 2.12a). As the circuit resonates mechanically, the surfaces become charged according to whether the crystal is stretched or compressed (figure 2.11). Hence, as the thickness oscillates, the charge on each surface will vary and a graph of charge versus time would resemble figure 2.12(a). A graph of potential difference versus time would look the same. Crystal resonators consist of a quartz wafer between two electrodes. The physical dimensions of the crystal determine the resonant frequency and if the crystal is maintained at a constant temperature a very high order of frequency stability can be obtained. When included in a circuit the crystal only permits current to flow when the frequency of the current is equal to the natural frequency of oscillation of the crystal. Quartz crystals can thus replace the resonant circuits of figure 2.6 and 2.9 and Figure 2.12 set the frequency of the oscillator more precisely. The frequency stability of (a) The change in thickness of a quartz crystals is exploited in applications where precise timekeeping is piezoelectric crystal in response required. Everyday applications include wristwatches and the clock which is the heart of every computer. to (b) a briefly applied voltage We will meet quartz crystals again in a later chapter describing the production of ultrasound waves. CAPACITORS AND DEPOLARISATION Earlier in this chapter we discussed the charge storing capability of capacitors and introduced the half-discharge time, τ1/2 which characterizes the rate of discharge of a capacitor through a resistor (equation 2.2). τ1/2 α R.C .... (2.2)
CAPACITORS AND INDUCTORS 37 When a capacitor is charged through a resistor, equation 2.2 also applies to the charging behaviour. Hence in a circuit such as that shown in figure 2.13, when the switch is closed the voltage across the capacitor will increase as shown. The voltage across the capacitor increases and approaches the battery voltage asymptotically. For the graph shown the battery voltage is 12 V. From the graph, τ1/2 is approximately 6 seconds. The reason that the graph has this shape is that the current flowing to charge the capacitor decreases as the capacitor charges. Initially, the capacitor is uncharged so when the switch is closed, the potential applied to the left side of the resistor is 12 V and the potential on the right side is zero. The potential difference across the resistor is maximum (in this circuit, 12 V) so the current flow is maximum. When the capacitor has charged to, say, 3 V, the potential difference across R is 9 volts, so the current CAPACITORS AND INDUCTORS 38 flow (calculated using Ohm's law or measured directly) is less. When the capacitor is charged to 9 V, the potential difference across R is only 3 V and the current is one- quarter of its original value. The closer the capacitor comes to fully charged, the smaller is the potential difference across R and the smaller is the charging current. Consequently, the rate of charging, determined by the current flow through the resistor, becomes smaller and smaller as the capacitor approaches fully charged and the capacitor voltage approaches the battery voltage but never quite gets there. Next consider what happens if a pulsed voltage is applied to the resistor-capacitor combination shown in figure 2.14. At the start of the pulse, the current flow through the resistor is high and the capacitor charges rapidly. As the capacitor charges, the rate of charge decreases and the potential difference apclarotesasuth(efigcaupreaci2to.1r 4(aV)C.) approaches a Figure 2.14 A graph of A pulsed voltage applied to a current flow through the resistor or resistor-capacitor combination. potential difference across the resistor (a) potential difference across the versus time mirrors the graph of VC capacitor and (b) current through (figure 2.14b). The initial charging current is high but reduces rapidly as the resistor versus time. the capacitor charges. Note that the current flow through the resistor is AC. Current flows through the resistor as the capacitor charges and an equal amount flows in the reverse (negative) direction as the capacitor discharges.
CAPACITORS AND INDUCTORS 39 The pulse duration in this example is significantly greater than the half- Figure 2.15 Response of the circuit shown in figure discharge/half-charge time, τ1/2 so the pulse is able to (almost) completely 2.14 when the pulse duration is (a) charge the capacitor much greater than R.C, (b) comparable before coming to an end and allowing the to R.C and (c) less than R.C. capacitor to discharge. If the RC time constant (and hence τ1/2) of the resistor-capacitor co- mbination was com- parable to the pulse width, the capacitor would not fully charge (figure 2.15b) and the current flow through the resistor would be more sustained. [The dashed lines in figure 2.15 indicate the voltage across the capacitor when it is fully charged (in this case, it is the pulse voltage).] If the RC time constant was greater than the pulse width, the capacitor would charge minimally (figure 2.15c) and the current flow through the resistor would be well sustained. The importance of the RC time constant will be apparent when we consider transcutaneous electrical nerve stimulation in subsequent chapters. CAPACITORS AND INDUCTORS 40 A final point which should be noted is that in each example in figure 2.15, the current through R is AC. The capacitor charges by a certain amount then discharges, so the net movement of charge is zero. Whatever charge flows through R during the pulse must flow back afterwards. EXERCISES 1 The table below shows readings of voltage obtained at 2 second intervals as a capacitor discharged through a resistance. The circuit arrangement used is shown in figure 2.2. In this case the value of the capacitor was known to be 50 µF but the resistance was unknown. _______________________________________________________ Time Voltage on the 50 µF Time Voltage on the 50 µF ____(_s_e_c_)______c_a_p_a_c_it_o_r___________(s_e_c_)______c_a_p_a_c_it_o_r _______ 0 12.0 12 1.8 2 8.7 14 1.3 4 6.3 16 0.9 6 4.6 18 0.7 8 3.3 20 0.5 _____1_0___________2_.4_____________________________________ (a) Plot a graph of voltage against time for comparison with figure 2.3. (b) How long does it take for the 50 µF capacitor to discharge to half the original voltage? (c) What would the unknown resistance need to be to give the results shown above?
CAPACITORS AND INDUCTORS 41 2 A capacitor is charged to a potential of 15 V then connected to a 250 kΩ resistor and allowed to discharge. The time taken to half discharge is 2.4 seconds. How long would the capacitor take to half discharge if: (a) the resistor was replaced by one of value 500 kΩ (b) the resistance was halved in value (c) the capacitance was doubled in value (d) the resistance was halved in value and the capacitance doubled (e) the resistance was doubled in value and the capacitance also doubled. 3 What is the impedance of: (a) a 10 µF capacitor (b) a 0.047 µF capacitor (c) a 10 µH inductor (d) a 2 mH inductor at a frequency of 1 kHz? How would the impedance change if the frequency was doubled? 4 A capacitor is connected directly to a 12 volt AC supply (of frequency 50 Hz). The current flowing through the capacitor is found to be 15 mA. What is the capacitance? 5 A capacitor has an impedance of 600 ohms at a frequency of 2 kHz. What is its impedance at a frequency of: (a) 100 Hz (b) 1 MHz. 6 Figure 2.5 includes a graph of impedance versus frequency for a 20 mH inductor. The graph is obtained using the values of impedance and frequency listed in table 2.2. Use the values listed to plot your own graph of impedance CAPACITORS AND INDUCTORS 42 versus frequency for a 20 mH inductor. On this graph draw lines showing the relationship between impedance and frequency for: (a) a 10 mH inductor (b) a 50 mH inductor. 7 Use the values in table 2.2 to plot a graph of impedance versus frequency for a 1 µF capacitor. On the same graph draw curves relating impedance and frequency for: (a) a 0.5 µF capacitor (b) a 2 µF capacitor. 8 Figures 2.6 and 2.9 show the variation of impedance with frequency for series and parallel resonant circuits. Explain the shape of each graph in terms of the variation of impedance with frequency of the individual inductors and capacitors. 9 A resonant circuit is made by combining a 0.1 µF capacitor with a 3.0 mH inductor. What is the resonant frequency of such a combination? 10 A 50 mH inductor in combination with a capacitor is found to have a resonant frequency of 1 kHz. What is the value of the capacitor? 11 When a pulse of current is applied to the circuit shown in figure 2.22 it 'resonates' generating an alternating voltage as shown in figure 2.24(b). (a) explain what is meant by resonance and why the circuit resonates (b) explain why the AC is damped. 12 Two circuits are coupled together as shown in figure 2.10. The values of the components are C = 100 pF and L = 1 mH. (a) Under what circumstances will power be transferred most efficiently between the circuits?
CAPACITORS AND INDUCTORS 43 (b) At what frequency will maximum power transfer occur? (c) If C was increased to 500 pF, what would be the new frequency for maximum power transfer? 13 An AC signal of frequency 100 kHz is generated by circuit 1 in figure 2.10. The value of C is 0.01 µF. (a) What value of L is needed for maximum power transfer to circuit 2? (b) If L is 3 mH, at what frequency will maximum power be transferred? 14 A DC pulse is applied to a resistor/capacitor combination. The result is a potential difference and flow of current through the resistor as shown. (a) why does the graph of current flow through the resistor droop so rapidly? (b) why must the current flow through R be purely AC? (c) if the capacitance was increased, how would this affect the shape of the graph of current through R? Why? (d) under what circumstances will the graph of current flow through the resistor resemble that of the applied voltage? CAPACITORS AND INDUCTORS 44 15 DC pulses as shown below are applied to a resistor/capacitor combination. (a) what can you conclude about the RC time constant for this circuit? (b) if the RC time constant was reduced (by decreasing C or R) how would this affect the shape of the graph of potential difference or current flow through the resistor? Give an explanation of your reasoning.
ELECTRICAL PROPERTIES OF SKIN 45 3 Electrical Properties of Skin In most areas of the body, the subcutis is Physiologists study the bioelectric properties of nerve and muscle using invasive predominantly adipose surgical techniques where the nerve trunk is exposed and so rendered accessible to (fat storing) tissue. It is, direct electrical stimulation. Sometimes nerve fibres are teased-out and stimulated nonetheless, quite individually. In this way much has been learned about the electrical characteristics of conductive because of nerve. This knowledge is used by the physiotherapist to choose electrical stimulus its water (and ion) content characteristics (such as the intensity, treatment time, pulse width and frequency) to and the presence of an invoke the desired physiological response during patient treatment. A complication is extensive network of that electrical stimulation for therapy is almost invariably applied non-invasively. It is blood vessels. normally achieved by applying an electrical stimulus through the skin using surface mounted electrodes. The skin has complex electrical characteristics and these must be taken into account when transcutaneous electrical nerve stimulation (TENS) is used. SKIN AND APPENDAGES The biological makeup of skin and its physiological state determine its electrical properties so it is worthwhile to consider some aspects of its structure which are relevant to electrical stimulation. Examined microscopically, the skin is found to have two distinct layers, the dermis and epidermis, as illustrated in figure 3.1. The dermis and epidermis together constitute the skin. The epidermis is punctured by the skin appendages: the sweat gland ducts and hair follicles. Beneath the skin is the subcutis, also referred to as the superficial fascia or simply subcutaneous tissue. Blood vessels, lymph vessels and nerves infiltrate the subcutis and dermis but not the epidermis. There are other fundamental differences between the dermis and epidermis. The dermis is well hydrated. It consists of a matrix of collagen and elastin fibres embedded in a 'ground substance' rich in proteoglycans and hyaluronic acid. Fibroblasts are the predominant cells in this layer. The high degree of hydration makes the dermis electrically conductive. The epidermis is less hydrated and consists of a matrix of keratin fibres. The cells present in this layer are predominantly keratinocytes which ELECTRICAL PROPERTIES OF SKIN 46 receive nutrients from capillaries in Figure 3.1 the underlying dermis. The basal Some important features of skin. layer of the epidermis is metabolically very active, with the cells regularly undergoing mitosis. Keratinocytes, formed and pushed upwards from this layer, synthesise keratin and store it within the cytoplasm. In their life cycle, the keratinocytes move towards the skin surface, becoming less metabolically active as diffusion limits the rate of nutrient supply. Near the surface the cells die and shrivel, turning into little sacs of (mostly) keratin. The dead cells form a scaly shell called the stratum corneum. The stratum corneum is thus the withered, dehydrated remains of keratinocytes packaged full of keratin. From formation in the basal layer to desquamation (flaking off as tiny scales from the surface) takes the keratinocyte approximately 40 days. The balance between desquamation and mitosis in the basal layer keeps the thickness of the epidermis constant. The stratum corneum is a dry, insulating but very thin shell which separates and isolates the highly hydrated soft tissues of the human body from the drier and far more changeable external environment. If the stratum corneum is removed, the body loses water and if too great an area of the stratum corneum is lost or damaged, the resulting water loss can be fatal. As noted above, the fibrous protein, keratin forms the bulk of this dead, insulating layer which is so essential for water homeostasis. From the point of view of transcutaneous electrical nerve stimulation, the presence of the stratum corneum is significant
ELECTRICAL PROPERTIES OF SKIN 47 because it is an electrical insulator. Figure 3.2 The skin appendages, the hair follicles and sweat gland ducts, are important both An electrical model for physiologically and electrically. The keratinous structures which we call hairs serve transcutaneous stimulation. two important physiological roles; one sensory, one thermal. Hairs are relatively rigid structures which are anchored in the skin. Sensory receptors close to the hair root detect and respond to movement of the hair, so providing sensitive touch receptors. A sufficient drop in temperature will signal a reflex response from receptors which triggers contraction of smooth muscle fibres connected to the hair root. The myofibril contraction causes the hairs to 'stand on end', trapping a volume of warmer air which separates the living tissue from the external environment. This effect is pronounced in humans and is also seen in other species which have greater hair covering. An elevated temperature triggers an increased blood flow to the dermis, thus increasing the rate of heat transfer from deeper structures and also activates the sweat glands to release water on the skin surface and provide evaporative cooling. Electrically, the hair follicles and sweat gland ducts are conductive pathways through the insulating layer, the stratum corneum. AN ELECTRICAL MODEL FOR TRANSCUTANEOUS STIMULATION When an electrode is placed on the skin surface we create a situation where two conductors are separated by an insulator, in other words, a capacitor. The electrode and the tissue beneath the stratum corneum are the two conductors. The stratum corneum is the insulator although, as we have seen, the stratum corneum is a 'leaky' insulator as it is punctured by conductive channels created by the skin appendages. The combination of electrodes and tissue can thus be modelled by a resistor capacitor combination. Figure 3.2 shows an electrical model for two electrodes placed on the skin surface. ELECTRICAL PROPERTIES OF SKIN 48 C is the capacitance of the stratum corneum. Rp, the resistor in parallel with C, In this model, the resistance represents the conductive pathway created by the skin appendages. Rs, the series of the electrodes is resistance, represents the resistance of the tissue volume under the stratum assumed to be negligible, corneum between the two electrodes. The resistance of the electrodes is assumed but could be included to be negligible. The actual values of Rp and C depend on factors such as the putting a resistor at each electrode size, ambient temperature and skin condition. The quantity Rs varies little. end of the model or including An important point which we will return to is that nerve fibres, whether sensory motor the electrode resistance or pain, are located beneath the stratum corneum. In terms of the electrical model, In Rs . this means that the potential difference across Rs determines the stimulation intensity experienced by nerve fibres. The potential difference across C and Rp represents lost electrical energy as far as nerve fibres are concerned. RESPONSE OF THE MODEL TO STEADY DIRECT CURRENT If a source of direct current, such as a battery is connected to the circuit shown in figure 3.2, some current will flow through the three resistors in series and some will flow on to the capacitors, so charging them. The initial flow of current is large as it includes charging current flowing to the capacitors (figure 3.3a) which is limited only by Rs. After the capacitors have charged, the current flow is lower as there is only current flow through the resistors (figure 3.3b). If Rp, the parallel resistance of the stratum corneum, is high (as it is, under normal circumstances), the 'steady state' current flow (figure 3.3b) will be low - appreciably lower than the initial current flow which includes charging current for the capacitors. At the extreme, if the stratum corneum was not 'leaky' i.e. if the parallel resistance, Rp, in figure 2 was infinite, no steady direct current could flow. There would be a brief flow of current as the capacitors charged, then no further current flow. In reality, Rp allows current flow, so after C is charged the only current flow is through Rs and the two resistances Rp which are connected in series.
ELECTRICAL PROPERTIES OF SKIN 49 Question 1: Figure 3.3 How much steady direct current would flow in a Response of the electrical model typical 'real life' situation if the applied DC to direct current (a) initially and (b) voltage is 50 V and electrodes with an area of 10 after the capacitors have charged. cm2 are used? Information: Skin capacitance is proportional to the electrode area. A typical value for the capacitance per unit area is 0.05 µF.cm-2, so for a 10 cm2 electrode area, the capacitance, C, is 0.5 µF. Skin resistance is inversely proportional to the electrode area. The larger the area, the greater the number of conductive channels through the stratum corneum and the lower the total resistance. The parallel resistance x unit area might typically be 10 kΩ.cm2, so for a 10 cm2 electrode area, the cpma2ra=lle1lkΩre.siTsthaencrees,isRtapn, cies about 10 kΩ.cm2/10 of the tissue volume beneath the stratum corneum is typically about 200 Ω. Answer: Under steady-state conditions, the capacitors are charged so no current flows to the capacitors and the value of C is irrelevant. The combination of electrodes and tissue behaves as three resistors in series. The total resistance is (1 kΩ + 200 Ω + 1 kΩ) = 2200 Ω. The applied potential difference is 50 V so the resulting current flow is, from Ohm's Law, I = V/R = 50/2200 = 0.0227 A = 22.7 mA. ELECTRICAL PROPERTIES OF SKIN 50 Question 2: How big is the initial current flow for the 'real life' situation described in question 1? Information: Initially, the capacitors act as a 'short circuit'. They are uncharged and offer no resistance to current flow. As each capacitor charges, the potential difference across the capacitor increases and this opposes further flow of current. When fully charged, an ideal capacitor has infinite resistance. Uncharged, the 'resistance' is zero. Answer: The initial flow of current is resisted only by Rs. I The capacitors offer zero resistance when they are uncharged so the current flow is = V/R = 50/200 = 0.25 A = 250 mA. This is more than ten times larger than the steady-state current (question 1). Question 3: What is the steady-state potential difference across Rp and Rs for an applied potential difference of 50 V? Answer: Since the three resistors are in series, the same current (22.7 mA) flows through each. The potential difference across each resistor can be calculated using Ohm's Law. The resistance Rs is 200 Ω so the potential difference across it is V = I.Rs = 0.0227x200 = 4.5 Volts. For each parallel resistance, Rp, V = I.Rp = 0.0227x1000 = 22.7 Volts. THE USE OF STEADY DC The previous calculations demonstrate an important idea concerning the use of steady direct current for patient treatment. Namely, that steady DC is of little practical use for stimulation of nerve fibres. For steady, continuous flow of direct current
ELECTRICAL PROPERTIES OF SKIN 51 through the skin, the tissue impedance is high (Rs + 2Rp) and the potential difference Notice that in the answers to across Rs is small. question 3, for an applied Nerve fibres are located in the tissues underlying the stratum corneum (represented potential difference of 50 V, by Rs) so the potential difference across them is small and their stimulation intensity only 4.5 V is produced is low. Most of the applied voltage drop occurs across the stratum corneum (Rp). across the tissue beneath This is one reason why steady direct current is not used for transcutaneous nerve the stratum corneum. stimulation. A second, more important, reason is that nerves are relatively insensitive to steady DC because they accommodate. The firing threshold progressively increases with a constantly applied DC stimulus, so the nerve will cease firing once the threshold has risen above the applied DC stimulus. Accommodation is described in more detail in chapter 4. IONTOPHORESIS It is for the reasons outlined above that steady direct current is not used for transcutaneous nerve stimulation. Steady DC does, however, have an important clinical role in iontophoresis, where drugs or other chemical agents are driven through the skin to the underlying tissue. Iontophoresis only works if the chemical to be driven is charged. It uses steady DC so that the chemical ions are driven continually in one direction while no nerve stimulation (in particular, no stimulation of pain or motor fibres) occurs. An example is acetate ion iontophoresis for decalcification of connective tissue. A soluble acetate, such as sodium acetate is dissolved in conductive gel. This is applied to the skin under the positive electrode (anode). Steady DC is applied at a current level of about 10 mA over a treatment period of 10 to 20 minutes. The negatively charged acetate ions are driven through the skin into deeper tissue. The calcium in calcified tissue is in the form of calcium phosphate crystals, which react with the acetate ions to form the soluble substance, calcium acetate. In this way the calcium phosphate crystals are removed. If the ion to be driven through the skin was positively charged, it would be applied under the negative electrode (the cathode). ELECTRICAL PROPERTIES OF SKIN 52 RESPONSE OF THE MODEL TO PULSED DIRECT CURRENT If a pulse of direct current is applied to the circuit shown in figure 3.2, the initial current flow through Rs will be large, but will quickly drop as the capacitors charge. The time constant for the charging process is proportional to Rs.C. Figure 3.4 shows the effect of stimulus pulse width and time constant on the current flow. In (a) the pulse duration is long Figure 3.4 compared to Rs.C so the capacitors charge during the pulse and the Response of the electrical model to pulsed current quickly drops to resistive current flow only. This means that if direct current (a) when the pulse duration Rp is large, the current will drop to a very low value. At the end of the is long compared to R s.C and (b) when the applied voltage pulse the capacitors pulse duration is short compared to R s.C. discharge, producing a negative- going current spike. Nerve fibres, located in deeper tissue beneath the stratum corneum (represented by Rs), would thus experience two short- duration pulses of stimulating current, one at the beginning and one at the end of the applied voltage pulse. Most of the applied voltage is ineffective as far as stimulation of nerve fibres is concerned. In (b) the pulse duration is short compared to Rs.C. The current decreases only fractionally during the pulse and most of the resulting current is due to the capacitors charging. The resulting current through Rs is high and does not decrease much during the pulse. Most of the applied voltage pulse is effective.
ELECTRICAL PROPERTIES OF SKIN 53 Figure 3.4 shows examples close to the range of extremes for stimulation with rectangular pulsed current. Note that the horizontal (time) axis scale is different in each example. In figure 3.4(a) the time scale is relatively long compared to Rs.C and sharp spikes are seen in the graphs of current through Rs versus time. In figure 3.4(b) the time scale is relatively short compared to Rs.C and the current through Rs shows only a small decrease during the stimulus pulse. During the spikes in 3.4(a), appreciable current flows through Rs, meaning that electrical energy is dissipated in the tissue volume beneath the stratum corneum. Between spikes, little current flows through the deeper tissue (Rs), and most of the electrical energy is dissipated in the stratum corneum. The short duration pulse in 3.4(b) results in appreciable and sustained current flows through Rs, so most of the electrical energy is available for nerve stimulation. The above is one of the reasons why modern stimulators use short duration pulses for transcutaneous nerve stimulation. A second, perhaps more important reason is that better discrimination between sensory, motor and pain fibre stimulation is achieved with narrower stimulus pulses. This will be considered further in chapter 4. RESPONSE OF THE MODEL TO ALTERNATING CURRENT The response of the model to alternating current is particularly interesting as The term 'low frequency' alternating current (at kilohertz frequencies) is commonly used in physiotherapy for currents refer to those in the patient treatment, either in the form of 'interferential currents' or 'Russian stimulation'. 'biological' frequency range Interferential currents are 'medium frequency' AC currents normally applied at a from 0 to approximately 100 frequency of 4 kHz. Russian current has a frequency of 2.5 kHz. Part of the reason Hz. 'Medium frequencies' are that these frequencies are used is that at kHz frequencies, the stratum corneum has in the kHz to tens of kHz range. a low electrical impedance. 'High frequencies' are in the The impedance of the stratum corneum is the total impedance of Rp and C in the 100 kHz and upward range. electrical model (figure 3.2). It is strongly dependent on the alternating current frequency because of the capacitance, C. The impedance of C is given by the formula: ELECTRICAL PROPERTIES OF SKIN 54 Zc = 1 .... (2.5) In most descriptions of 2πfC interferential or Russian currents, the choice of a where Zc is the impedance and f is the alternating current frequency. kilohertz frequency is justified as being due to the low skin Question 3: impedance at kHz What is the capacitative impedance of the stratum corneum, Zc, at frequencies of (a) frequencies. The low skin 50 Hz, (b) 500 Hz and (c) 5 kHz? impedance is due to the low impedance of both the Information: stratum corneum and the A typical value for the skin capacitance per unit area is 0.05 µF.cm-2. A typical underlying skin layers. electrode area is 10 cm2 so the capacitance, C, in the model is about 0.5 µF. Answer: (a) Using equation 2.5, the impedance at 50 Hz is Zc = 1 = 1 = 106 = 6400 Ω 2πfC 2 x 3.142 x 50 x 0.5 x 10-6 50 x 3.142 (b) Using equation 2.5, the impedance at 500 Hz is Zc = 1 = 1 = 106 = 640 Ω 2πfC 2 x 3.142 x 500 x 0.5 x 10-6 500 x 3.142 (c) Using equation 2.5, the impedance at 5 kHz is Zc = 1 = 1 = 103 = 64 Ω 2πfC 2 x 3.142 x 5000 x 0.5 x 10-6 5 x 3.142 If the stratum corneum was purely capacitative, its impedance would vary as calculated above. In fact the skin appendages provide a parallel resistive path for current flow (Rp in figure 3.2) so the total impedance of the stratum corneum is lower,
ELECTRICAL PROPERTIES OF SKIN 55 significantly so at low frequencies. The parallel resistance x unit area of the stratum corneum is typically about k1Ω0.ckmΩ2./c1m0 2c,ms2o for a 10 cm2 electrode area, the parallel resistance, Rp, is about 10 =1 kΩ. At a frequency of 5 kHz, the capacitative impedance of the skin is about 64 Ω and placing a 1 kΩ resistance in parallel makes virtually no difference to the total impedance. At a frequency of 500 Hz, the capacitative impedance of the skin is about 640 Ω and placing a 1 kΩ resistance in parallel reduces the total impedance to appreciably less than 640 Ω. At a frequency of 50 Hz, the capacitative impedance of the skin is about 6400 Ω and placing a 1 kΩ resistance in parallel reduces the total impedance to less than 1 kΩ. Despite the effect of skin appendages offering a conductive pathway (Rp) at low frequencies, the total impedance of the stratum corneum (C and Rp in parallel) still shows a dramatic decrease with increasing frequency. Thus, for a given stimulus voltage, greater current flows at higher frequencies. This means that the current through Rs, representing the underlying tissues, is higher at higher frequencies and the potential difference across Rs is correspondingly higher. Question 4: What is the impedance of the stratum corneum at frequencies of (a) 50 Hz, (b) 500 Hz Strictly speaking, capacitative and (c) 5 kHz? and resistive impedances must be added using Use the model in figure 3.2 and values of C and Rp used previously assuming an electrode area of 10 cm2. complex number algebra. If a resistor, R p and a capacitor, Use the formula for two resistances in parallel to add the two impedances. C, are connected in parallel, Information: the total impedance, Z, is As indicated previously, a typical value for the skin capacitance for an electrode area given by the formula: of 10 cm2 is about 0.5 µF. The capacitative impedance is thus 6400 Ω, 640 Ω and 64 Ω at frequencies of 50 Hz, 500 Hz and 5 kHz respectively. The parallel resistance, Rp, Z= Rp is about 1 kΩ. (1+(2πfRp C)2 )1/2 ELECTRICAL PROPERTIES OF SKIN 56 Answer: The impedance of the stratum corneum at each frequency is the total impedance of C and Rp in parallel. Two resistances in parallel can be added using the formula 1 = 1 + 1 RT R1 R2 Adding the impedances of C and Rp in this way we use 1 = 1 + 1 Z Rp Zc where Zc is the impedance of C at the frequency concerned. (a) Using the above formula, the impedance at 50 Hz is calculated as follows: 1 = 1 + 1 = 1000 + 6400 = 6500 Z 6400 1000 1000 x 6400 1000 x 6400 so Z = 6400 x 1000 = 985 Ω 6500 (b) The impedance at 500 Hz is given by 1 = 1 + 1 = 1000 + 640 = 1640 so Z = 640 x 1000 = 390 Ω Z 640 1000 1000 x 640 1000 x 640 1640 (c) The impedance at 5 kHz is given by 1 = 1 + 1 = 1000 + 64 = 1064 so Z = 64 x 1000 = 60 Ω Z 64 1000 1000 x 64 1000 x 64 1064 The impedance of the stratum corneum (C and Rp in parallel) thus shows a dramatic decrease with increasing frequency: from about 1 kΩ at 50 Hz to about 60 Ω at 5 kHz.
ELECTRICAL PROPERTIES OF SKIN 57 It is the potential difference across Rs which determines the stimulation intensity experienced by nerve fibres, which are located under the stratum corneum. This depends on both Z and Rs. Rs is about 200 Ω and does not vary appreciably with frequency. Since Z decreases appreciably with increasing frequency, the potential difference across Rs will correspondingly increase. Question 5: What is the potential difference across Rs at frequencies of (a) 50 Hz, (b) 500 Hz and (c) 5 kHz for an applied potential difference of 50 V? Use the values of Z and Rs calculated previously assuming an electrode area of 10 cm2. Information: Accurate calculation of the potential difference across Rs requires complex number algebra, but a good first-approximation can be made by treating the impedances as simple resistances in series. Answer: (a) The impedance of the stratum corneum at 50 Hz is 985 Ω and Rs is 200 Ω . Thus the total impedance is 985+200+985 = 2170 Ω. For an applied potential difference of 50 V, the current through the series combination is I = V/Z = 50/2170 = 0.023 A = 23 mA. This is also the current through I.RRss, so the potential difference across Rs is V = = 0.023 x 200 = 4.6 volts. (b) The impedance of the stratum corneum at 500 Hz is 390 Ω and Rs is 200 Ω. Thus the total impedance is 980 Ω. For an applied potential difference of 50 V, the current through the series combination is I = V/Z = 50/980 = 0.051 A = 51 mA. This is also the current through Rs, so the potential difference across Rs is V = I.Rs = 0.051 x 200 = 10.2 volts. ELECTRICAL PROPERTIES OF SKIN 58 (c) The impedance of the stratum corneum at 5 kHz is 60 Ω and Rs is 200 Ω. Thus the total impedance is 320 Ω. For an applied potential difference of 50 V, the current through the series combination is I = V/Z = 50/320 = 0.156 A = 156 mA. Thus the potential difference across Rs is V = I.Rs = 0.156 x 200 = 31.2 volts. Notice that at 5 kHz, the potential difference across Rs is 31.2 volts, which is more than 60% of the applied (50 V) stimulus. At 500 Hz, the potential difference across Rs is about 20% of the applied stimulus and at 50 Hz, about 9%. If the objective is to stimulate nerve fibres which are located beneath the stratum corneum, the higher the alternating current frequency, the less the energy wasted in the stratum corneum and the greater the current through, and voltage across, the deeper tissues. If skin impedance was the only factor, then the higher the alternating current frequency, the lower the skin impedance and the more efficiently nerves could be stimulated. Following this logic, if 5 kHz stimulation is more efficient than 500 Hz, then 50 kHz should be even more so. In fact this is not the case. The reason is that the impedance of the nerve fibre also varies with frequency. NERVE FIBRE IMPEDANCE The electrical properties of nerve fibres are analogous to those of skin and underlying tissue. The nerve fibre membrane, like the membrane of all cells, is a phospholipid bilayer with an embedded patchwork or mosaic of protein molecules. This fluid-mosaic structure, illustrated alongside, explains the measured electrical properties. Orange shapes are the polar, high-water-content, protein molecules. Blue circles with two dangling chains represent the phospholipid molecules which have two long, non-polar 'tails'. The phospholipid bilayer is an insulator. The hydrocarbon tails of the phospholipid molecules separate the conductive intracellular fluid from the equally conductive extracellular fluid. Thus the nerve-fibre membrane acts as a capacitor. Protein molecules which span the width of the
ELECTRICAL PROPERTIES OF SKIN 59 membrane provide conductive channels, allowing Figure 3.5 some ions to cross the membrane continuously. For An electrical model for a this reason the membrane behaves electrically like a capacitor, C, with a parallel resistance Rp. nerve fibre Electrical stimulation of the nerve fibre involves current flow across the membrane, through the intracellular fluid and out across the membrane at some other point (usually the adjacent node of Ranvier). The internal contents of the nerve fibre, the intracellular fluid, is purely resistive so the nerve fibre can be electrically modelled as shown in figure 3.5. C in figure 3.5 is the capacitance of the nerve-fibre membrane: an insulating phospholipid bilayer. Rp is the resistance of conductive channels through the bilayer (the protein molecules which span the full width of the membrane). Rs is the resistance of the intracellular fluid. To excite a nerve fibre, that is, to cause the nerve to fire and produce an action potential, the membrane potential must be changed from its resting value to the excitation threshold. For nerve excitation it is not the potential difference across the fibre which determines this but the potential difference across the fibre membrane (C and Rp in figure 3.5). Since the membrane acts as a capacitor, if alternating current is used for nerve stimulation, the potential difference across the fibre membrane will be less at higher alternating current frequencies. The decrease in membrane impedance with increasing frequency means that a greater potential difference will be produced across Rs and a correspondingly lower potential difference will be produced across C and Rp. At high frequencies, nerve fibres are less excitable because the potential difference produced across the fibre membrane is reduced. In practical terms this means that electrical stimulation with alternating current does not become more and more efficient at higher alternating current frequencies. the decrease in impedance of the stratum corneum, which results in a higher potential difference across the ELECTRICAL PROPERTIES OF SKIN 60 underlying tissues at kilohertz frequencies is countered by the decrease in impedance of the nerve fibre membrane. In theory, the decreased nerve fibre sensitivity could be compensated-for by increasing the stimulus intensity but in practice this would result in a high current flow through the skin and consequently a high electrical power dissipation and the risk of tissue damage. The result is that frequencies above about 10 kHz are of little use for eliciting a nerve response. A similar argument applies to Figure 3.6 stimulation with rectangular pulses. As indicated previously, long duration Change in potential difference across the pulses are relatively ineffective for transcutaneous nerve stimulation as nerve fibre membrane (voltage across C and most of the electrical energy is dissipated in the stratum corneum and Rp in the electrical model shown in figure the current through the deeper tissues 3.5) when the pulse duration is (a) much is in the form of two spikes, at the onset and cessation of the stimulus longer than R s.C and (b) shorter than R s.C. (figure 3.4(a)). With short duration pulses (figure 3.4(b)), little energy is dissipated in the stratum corneum and the current through the deeper tissues is sustained during the pulse. However, if the pulse duration is extremely short (tens of microseconds), there will be insufficient time during the pulse for the nerve membrane capacitance to charge. Figure 3.6 illustrates the response of the electrical model shown in figure 3.5 to short and longer duration current pulses. With a sufficiently long duration pulse, there is time for the membrane capacitance to charge (figure 3.6(a)) and the change in membrane potential is maximum. With a very short duration pulse, there is insufficient time for the membrane capacitance to fully charge and the change in membrane potential is small (figure 3.6(b)). The use of a
ELECTRICAL PROPERTIES OF SKIN 61 higher stimulus intensity can compensate for the short pulse duration, but if the intensity required is too large, there is the risk of skin damage. Specifically, at stimulus intensities of several hundred volts, there is the risk of skin electrical breakdown where tiny regions of skin under the electrodes suddenly become highly conductive, allowing an extremely high current to flow and causing tissue damage. The net effect is that for transcutaneous stimulation using rectangular pulses, there is a 'window' of pulse widths between tens of microseconds and about 1 ms, outside which stimuli are either ineffective or inefficient and potentially dangerous. EXERCISES 1 Explain, in terms of the structure of skin, why the electrical model shown in figure 3.2 is an appropriate model for a description of transcutaneous electrical nerve stimulation. 2 When a constant DC stimulus is applied, via electrodes, to the skin surface, there is a large transient flow of current through the tissue. Explain why the initial flow of current is high and why the steady-state current is lower. 3 (a) What is iontophoresis? (b) Why must the active agent be charged (i.e. in the form of an ion)? (c) What is the advantage of iontophoresis for application of medication? 4 Figure 3.4, page 52, shows graphs of the current flow through tissue in response to stimulus pulses of different duration. (a) explain why long duration pulses produce only short-duration current flows, as in figure 3.4(a). (b) Why is the current flow sustained when short duration pulses, as in figure 3.4(b), are applied transcutaneously? 5 For transcutaneous nerve stimulation, body tissue can be modelled by the resistor/ capacitor combination shown below. Skin capacitance per unit area is ELECTRICAL PROPERTIES OF SKIN 62 approximately 0.05 µF.cm-2 and the parallel skin resistance x unit area, approx- imately 10 kΩ.cm2. The subcutaneous resistance is typically around 200 Ω. Sinusoidal AC with an amplitude of 20V is applied between A and B using 1 cm2 electrodes. (a) Calculate the impedance of the capacitor at AC frequencies of: * 50 Hz * 500 Hz * 5 kHz. (b) Calculate the impedance of the stratum corneum; i.e. the impedance of the parallel (Rp and C) resistor capacitor combination, Z||, at each of the three frequencies. (c) What is the potential difference across the deep tissue, modelled as Rs, at each of the frequencies? 6 If the electrode area in question 5 above was 50 cm2, what would be: (a) the impedance of the capacitor, C, at the same AC frequencies? (b) the impedance of the stratum corneum; i.e. the impedance of the (Rp and C) resistor-capacitor combination at each of the three frequencies?
ELECTRICAL PROPERTIES OF SKIN 63 (c) the potential difference across Rs (representing the deeper tissues) for the same AC frequencies? 7 A rectangular pulse of amplitude 40 V is applied to the circuit shown in question 5. What is: (a) the peak current through Rs (b) the minimum current through Rs when a long duration pulse is used? 8 (a) Draw graphs of the voltage across Rs versus time when rectangular pulsed current is applied to the circuit in question 1. Consider three situations (i) the pulse width is small compared to Rs.C, (ii) the pulse width is comparable to Rs.C and (iii) the pulse width is large compared to Rs.C. (b) explain, using graphs to illustrate, the effect of Rp on the voltage measured across Rs. 9 The skin acts as a capacitative barrier to the flow of current, meaning that the higher the AC frequency, the lower the skin impedance. For this reason kHz frequency sinusoidal AC is used for transcutaneous electrical nerve stimulation. (a) why are higher frequencies (tens or hundreds of kHz) not used clinically to reduce skin impedance further? (b) what is the clinically useful frequency range for AC stimulation? 10 Pulsed current with widths between about 50 µs and 1 ms are normally used for transcutaneous electrical nerve stimulation. Why are pulses of duration (a) less than 50 µs and (b) greater than about 1 ms not normally used?
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 64 4 Electrical Stimulation of Nerve and Muscle The resting membrane potential is also affected by Both low frequency pulsed current and kHz frequency alternating current are used by the movement of other ions, the physiotherapist for the stimulation of nerve and muscle. Low frequency including sodium ions, but stimulation using short duration pulses has most often been used by physiologists because their permeability is for studying nerve and muscle. Consequently, the physiological basis for electrical much lower than that of stimulation with these currents is reasonably well understood. Less is known of the potassium, their effect on the effects of kHz frequency alternating currents, particularly when applied trans- resting membrane potential cutaneously. The aim of this chapter is to present some of the important is less. observations that have been made concerning the response of nerve and muscle to electrical stimulation using both kinds of currents. When a nerve fibre is in its resting state there is a potential difference of some 70 millivolts between the interior and exterior of the fibre. This is called the 'resting membrane potential'. The inside of the fibre is negative with respect to the outside. The resting membrane potential originates from the difference in concentration of different ions inside and outside the cell and the permeability of the fibre membrane to particular ions. Potassium ions contribute most to the resting membrane potential. The intracellular and extracellular concentrations of potassium ions differ markedly, with the result that they diffuse down their concentration gradient, producing a difference in electrical potential across the nerve fibre membrane. The origin of the resting membrane potential is described in most physiology text books and so will not be elaborated here. If a current of sufficient intensity is passed through tissue containing a nerve fibre the potential difference set-up across the fibre may be sufficient to cause depolarization of the fibre membrane and the nerve is stimulated. The depolarization of the membrane, once induced, is transmitted along the length of the nerve fibre and is indistinguishable from a normal nerve impulse (sometimes referred to as an action potential). The important idea is that the potential difference across the membrane must be changed by a critical amount to produce the transient, but large, membrane depolarization which is known as an action potential. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 65 STIMULATION OF NERVE FIBRES Figure 4.1 Response of a nerve fibre to In order to elicit an action potential the potential difference across a nerve fibre must stimuli of increasing intensity. be changed to more than some critical value known as the threshold potential. An electrical stimulus resulting in less than the threshold potential across the nerve fibre will not trigger any response. The process is illustrated in figure 4.1. The threshold potential of most excitable membranes is between 5 and 15 mV more positive than the resting potential. Thus if the resting potential is -70 mV the threshold potential may be -60 mV. To generate a nerve impulse the potential must be changed by more than 10 mV in this case. Once the potential is increased above threshold the nerve fibre is 'fired'. The response is 'all or none'. That is, any stimulus above the threshold value produces the same size of response. The membrane potential rapidly changes to around +30 mV then decreases to the resting value, typically in about one millisecond. Refractory Periods After a nerve has been stimulated there is a short period of time, typically around 10 milliseconds for sensory and motor neurones, during which the sensitivity of the nerve to stimuli is decreased. During this time the nerve membrane is said to be refractory to a second stimulus. The threshold potential is increased above the normal value as shown in figure 4.2.
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 66 For the first millisecond or so after initiation of a Figure 4.2 nerve impulse no stimulus, no matter how large, Refractory periods for a will produce a second impulse. This is the absolute refractory period. This is followed by a nerve fibre. further period, the relative refractory period, during which only a larger than normal stimulus will produce a response. Full recovery is normally complete in about 10 to 15 milliseconds, when the threshold potential has returned to its original value. Termination of the action potential involves the membrane potential returning to its resting value. The membrane potential typically decreases towards resting, overshoots, and returns to baseline during the relative refractory period. During the overshoot, the magnitude of the membrane potential is somewhat greater than the normal 70 mV and the membrane is described as being hyperpolarized. Hyperpolarization and refractoriness following an action potential lasts for 10 to 15 milliseconds in large diameter sensory and motor neurones. For smaller diameter fibres, the refractory period is longer. This is associated with the observation that the firing rates of smaller diameter fibres are typically less than those of larger diameter fibres. Accommodation Three characteristics of an electrical impulse influence its ability to stimulate nerve fibres: * the size or amplitude of the pulse, * the width or duration of the pulse, and * the rate of change (or rise) of the pulse. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 67 The size or amplitude of the pulse is clearly important in that the larger the pulse, the more rapidly the nerve fibre will reach threshold. The width or duration of the pulse is also important in that the longer the pulse duration, the more time is available for the fibre to reach threshold. Figure 4.3 The effect of stimulus rise-time on action potential production. (a) short stimulus rise-time, (b) lower rate of rise of the stimulus and (c) very low rise time. The rate of change (or rise) of the pulse is important because, in general, a stimulus pulse which rises slowly to its maximum value is less effective than a sudden sharp pulse, other things being equal. If a slow rising pulse is used then the minimum amplitude needed to elicit an action potential will be greater. This happens because the nerve fibre is able to accommodate to a
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 68 slow change in potential. Indeed, if the pulse rises at a sufficiently low rate, no nerve Hille (Hille B. Ion Channels impulse will be generated. The effect of accommodation is illustrated in figure 4.3. In of Excitable Membranes. figure 4.3(a) the pulse rise-time is short and threshold is reached before any 2nd Edn. Sinauer Associates. appreciable change in threshold potential occurs. In 4.3(b) the pulse increases at a 1992) provides a simple, faster rate than the threshold potential so threshold is reached after several yet detailed explanation milliseconds. In 4.3(c) the pulse rises at the same rate as the threshold potential and of the molecular dynamics so does not 'catch up' and cannot generate an action potential: the nerve fibre has of an action potential. 'accommodated' to the rising stimulus intensity by becoming insensitive to electrical Transcutaneous electrical stimulation. stimulation then, particularly The refractory period and accommodation both stem from the same basic molecular with short duration pulses process by which the nerve impulse, once generated, is terminated and the (durations less than the time membrane potential returns to the resting value. An explanation involves voltage- constant for muscle fibres), sensitive 'gates' which control opening and closing of conduction channels in the preferentially recruits nerve nerve fibre membrane. fibres. STIMULATION OF NORMALLY INNERVATED MUSCLE When the nerve supply to a muscle or group is intact, transcutaneous electrical stimulation will normally evoke a motor response - not by a direct effect on the muscle fibres, but indirectly via excitation of the motor nerve fibres (α-motoneurons). The reasons for this are twofold. The first is that many muscle fibres are deeply located and so less likely to be stimulated than those closer to the stimulating electrodes. The second, and perhaps more important reason, is that the time constant for depolarization of a muscle fibre is much greater than that of a nerve fibre. If short duration pulses are used, muscle fibres do not have sufficient time to respond. Nerve fibres, which have a smaller time constant and depolarize more rapidly, are more likely to have sufficient time to reach threshold and fire. Where to Stimulate? When the aim of transcutaneous stimulation is to produce a motor response, the electrodes are normally placed either over the nerve trunk or directly over the muscle to be stimulated. Stimulation of the nerve trunk is described as 'indirect' and will ELECTRICAL STIMULATION OF NERVE AND MUSCLE 69 evoke a response from all of the muscles innervated by fibres in the trunk. Stimulation Note that the terms 'direct' over the muscle ('direct' stimulation) will preferentially activate just that muscle. For and 'indirect' stimulation refer example, if the femoral nerve is stimulated at the level of the groin, the quadriceps to proximity to the muscle. femoris group will be activated. Alternatively, electrodes may be positioned over an Direct stimulation does not individual muscle to activate just one member of the quadriceps group. mean that muscle fibres are The best response from an individual muscle is obtained if the stimulus is applied at stimulated directly but rather a motor point. This is often the region of skin which is over the point where the main that stimulation is via nerve nerve enters the muscle. In the case of deeply placed muscles the motor point is fibres entering the muscle. usually where the muscle emerges from under cover of the more superficial ones. Note, however, that motor point locations are defined as points where a motor response is most easily produced and such points are determined experimentally. Thus a particular motor point may not fit either of the above descriptions - for example it may be simply an area where the nerve is located more superficially. Electrode Orientation and Size To stimulate nerve, current must flow through tissue between two electrodes which are normally positioned so that current flows parallel to the nerve fibres. If the current flow is at right angles to the fibres, much higher stimulus intensities are required. The reason is that in order to produce an action potential, current must flow in across the fibre membrane at one Node of Ranvier, along the fibre and out at an adjacent node. The amount of current flow depends on the applied potential difference and the greatest potential difference will be produced between adjacent nodes if the current flow direction is parallel to the nerve fibre. A related idea is that action potentials will more readily be generated under the negative electrode (the cathode) than under the positive electrode (the anode). The reason for this is as follows. The nerve fibre membrane, in its resting state, is polarized. The outside is positively charged and the inside, negatively charged. A potential difference of about 70 mV exists across the membrane. If a stimulus is applied using two electrodes, the resulting current flow will depolarize the membrane at one Node of Ranvier while hyperpolarizing the membrane at its neighbour.
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 70 Figure 4.4 illustrates the effect. The resting membrane Figure 4.4 is positively charged on the outside and negatively Current flow between anode and charged on the inside. The membrane also acts as a cathode produces depolarization under capacitor, which can store charge. The flow of current the cathode and hyperpolarization between anode and cathode increases the positive charge on the membrane closer to the anode, while under the anode. reducing the charge on the membrane closer to the cathode. The result is that the potential difference across the membrane near the anode is increased and the membrane becomes hyperpolarized. The potential difference across the membrane near the cathode is decreased and the membrane becomes less polarized. When the reduced polarization (i.e. depolarization) is sufficient, an action potential is generated. To enhance the efficiency of stimulation under the cathode, a small electrode size can be used. A larger anode helps to ensure that the current density beneath the electrode is low while the smaller cathode concentrates the current flow and more specifically targets Nodes of Ranvier closest to this electrode. Under these conditions, the location of the anode is relatively unimportant as the cathode acts as the 'active' electrode. The anode acts as an 'indifferent' or 'dispersive' electrode ('dispersive' referring to spreading of the current over a larger area). The main criterion for the location of the anode is that it should not be located over an electrically sensitive area such as a motor point or muscle belly. Sometimes equal size electrodes are used for transcutaneous electrical stimulation. This is often the case when electrodes are placed over a muscle belly. The cathode is positioned distally and the anode proximally. The reason for the 'cathode distal' arrangement is that an action potential generated near the cathode may not propagate through the region under the anode. The phenomenon is referred-to as anodal block. The idea is ELECTRICAL STIMULATION OF NERVE AND MUSCLE 71 that if an action potential is generated at a particular node of Ranvier, it will normally Two-way action potential trigger an action potential at the nodes immediately adjacent. This means that when propagation does not the nerve is stimulated electrically, action potentials can propagate in both directions happen physiologically as along the fibre from the site of stimulation. Propagation in the direction towards the action potentials are always anode, however, will not occur if the adjacent node is kept hyperpolarized by the initiated either at a synapse stimulus pulse. If the objective is to elicit a motor response, a cathode distal or a nerve ending. arrangement should be used, so that action potential propagation towards the muscle The term 'tetanus' refers to a fibres is not blocked. Conversely, if the objective is stimulation of sensory fibres, tetanic muscle contraction. It where the aim is sensory input to the central nervous system, a cathode proximal is also used to describe a arrangement would be appropriate. pathalogical condition produced by the toxin of a Recruitment and Summation bacillus which causes tetanic muscle contraction. When muscle fibres are stimulated indirectly, via their nerve supply, the muscle fibres are activated synchronously because their motoneurons are activated simultan-eously. Each stimulus pulse activates a proportion of the fibres in the nerve trunk and the activated fibres evoke a twitch response in the muscle fibres which they innervate. The number of fibres recruited, and hence the force of the muscle contraction depends on the stimulus intensity. At low intensities, only a small proportion of the fibres are recruited. At higher intensities, a greater proportion of the nerve fibres are activated. Whatever the intensity, the muscle response to transcutaneous electrical nerve stimulation is critically dependent on the stimulus frequency. At low frequencies (a few Hz or less), isolated twitches are produced in response to each stimulus pulse. There is time for the muscle to relax before the next contraction. If the frequency is more than a few Hz, the muscle fibres do not have time to completely relax between pulses. Each successive contraction occurs on the tail of the previous one and the peak force is greater. With a further increase in the frequency it becomes more difficult to distinguish the effects of individual stimuli. The twitch responses fuse and the contraction becomes stronger still. With most human muscles, at a stimulus frequency of about 20 Hz, only small ripples are seen in the force record. This is described as partial tetany. Between 20 Hz and 50 Hz, the ripples disappear, the contractile force reaches a plateau and the contraction is described as tetanic.
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 72 Figure 4.5 illustrates the effect of a progressive Figure 4.5 increase in stimulus frequency on the evoked Muscular force produced in muscle response. In this diagram, the frequency response to short duration has been ramped from 2 Hz to 50 Hz. Note that at rectangular pulsed current about 20 Hz, the response is almost tetanic. with frequencies in the range Above this frequency complete tetany occurs. Note also that the tetanic force is about four times 2 Hz to 50 Hz. greater than the isolated twitch force. In the example considered, the fusion frequency is about 30 Hz. The fusion frequency varies between muscles and depends on the muscle fibre types present. In terms of twitch times, two groups of fibres are distinguished: fast- and slow-twitch. The contraction time, defined as the time from the start of the contraction to peak force, is about 40 ms for human fast- twitch muscle fibres and about 120 ms for slow-twitch fibres. Muscles such as soleus contain mostly (80%) slow twitch fibres. The twitch contraction time is long and consequently the fusion frequency is low. At the opposite extreme, orbicularis oculi, an eye movement muscle, contains mostly (85%) fast-twitch fibres and the fusion frequency is high. Fusion frequencies can thus vary from less than 20 Hz to close to 80 Hz. Many human skeletal muscles have roughly equal proportions of slow and fast-twitch fibres. For example, biceps and triceps brachii are comprised of about 60% fast-twitch fibres, wile the figure for quadriceps is close to 50%. Effect Of Pulse Duration Earlier we saw that the characteristics of an electrical impulse which determine its effectiveness in stimulating nerve fibres were pulse amplitude, duration and rate of rise. We now examine the first two factors in more detail. For the stimulation of normally innervated muscle it is customary to use rectangular pulses of short duration. The reasons for this are as follows: ELECTRICAL STIMULATION OF NERVE AND MUSCLE 73 * the short rise-time of a rectangular pulse overcomes the problem of accomm- Figure 4.6 odation of the nerve fibre membrane. A strength-duration curve for normally innervated muscle * some sensory nerves will invariably be stimulated. The sensation associated with pulses of short duration (less than 1 ms) is less unpleasant than that associated with a longer pulse duration (above 1 ms). * for long pulses, only the early part is effective in stimulating nerve. If the pulse duration is a few multiples of the skin RC time constant, significant subcutaneous current flow will only occur at the beginning and end of the pulse (chapter 3, figure 3.4). A graph of stimulus strength needed to produce a minimal muscle contraction against pulse duration is shown in figure 4.6. The method of obtaining these results is as follows. Pulses of long duration (usually 100 ms) are applied to a muscle and the stimulus intensity is increased until a minimal contraction is obtained. The stimulus intensity (voltage or current) is then recorded. The pulse duration is then decreased and the intensity needed for minimal contraction is again determined. The process is repeated until enough results are obtained to give a graph like that shown in figure 4.6. The graph (the strength-duration curve) is normally plotted using a logarithmic axis for pulse duration. This magnifies the region of the curve showing the effect of short pulse durations and makes interpretation easier. It is found that pulses of long duration (about 10 ms or more) produce a muscle contraction with the same voltage for all durations. When the duration is decreased below a certain point (in this example, a little over 1 ms) the
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 74 stimulus intensity needed to produce contraction is increased. The increase in intensity with decreasing pulse duration can be explained in terms of the electrical properties of the nerve fibre: specifically the capacitance of the nerve fibre membrane (see chapter 3 previously). To generate an action potential, the membrane must be depolarized to threshold. That is, the potential difference across the membrane, the resting potential, must be changed by a certain amount by charging the capacitative membrane. As figure 3.6 shows, if the pulse duration is long, the capacitor charges fully. With short duration pulses, the membrane does not have time to fully charge. However, if a higher voltage is used, the overall current flow is higher and the membrane potential changes more rapidly. Threshold is approached more rapidly so a shorter charging period (short duration) is compensated by a greater charging force (higher voltage pulse). The plateau above 1 ms in this example relies on two factors: the membrane capacitance and the skin capacitance. When the pulse duration is long, the skin capacitance will fully charge and the flow of current through the deeper tissues will decrease to a negligible amount during the pulse (figure 3.4a). A long duration pulse also allows time for the membrane capacitor to fully charge, and any additional time does not produce further charging (figure 3.6a). Once the time has elapsed for both of these processes to reach a steady state, any further increase in pulse width has a negligible effect. Chronaxie and Rheobase Two important quantities are obtained from the strength-duration curve, the chronaxie and the rheobase: * The rheobase is the minimum voltage (or current) which will produce a response if the stimulus is of infinite duration. In practice a pulse width of 100 ms duration is used, quite satisfactorily, to assess this. * The chronaxie is the minimum duration of impulse which will produce a response with a voltage (or current) of double the rheobase. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 75 In figure 4.6 the rheobase is 25 volts so the chronaxie is the minimum duration required with a 50 volt stimulus. In this case the chronaxie is 0.03 ms (see alongside). Strength-duration curves and their chronaxie and rheobase values can be used clinically to assess and monitor muscle which may have suffered damage to its nerve supply. Strength-duration graphs for denervated muscle are quite different to those of normally innervated muscle, as are the chronaxie and rheobase values - but more of this later. Effect of Pulse Frequency We have already described the response of typical skeletal muscles to nerve impulses of different frequencies. Single muscle twitches are produced with low frequency stimuli (less than about 5 per second) and as the frequency approaches 20 Hz, the twitches summate to produce partial tetany (see figure 4.5). At some frequency above 20 Hz, a tetanic contraction results. Once a fused, tetanic contraction is induced, any further increase in stimulus frequency does not induce any increase in muscle force. For typical human muscles (which have mixed fibre types in roughly equal proportions) the fusion frequency is around 40 Hz. For muscles with a high proportion of fast twitch fibres, the fusion frequency is higher. For muscles with a high proportion of slow twitch fibres, the fusion frequency is lower. What of stimulation at frequencies above the fusion frequency? At frequencies which are high enough that successive stimuli arrive within the refractory period, the nerve fibre response depends on the intensity of the stimulus. Just at threshold, one stimulus pulse will produce an action potential, the next will not, as the nerve fibre will
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 76 be refractory. Well above threshold, when fibres are stimulated at multiples of their McComas (McComas AJ. threshold intensities, firing will occur within the refractory period and the firing rate will Skeletal Muscle: Form and equal the stimulus frequency. It has been demonstrated experimentally that nerve Function. Human Kinetics. fibre firing rates up to the limit determined by the absolute refractory period can be 1996) provides a wealth of produced by stimulus intensities of only a few times threshold. Thus if the absolute fascinating information on refractory period is 1 ms, the maximum firing rate would be every millisecond so the nerve firing frequencies in frequency would be 1 kHz. The experimentally determined maximum firing rate of α- different activities. motoneurons is 800 Hz, a value in close agreement with measured absolute refractory periods. Nerve fibre firing rates with electrical stimulation can thus be much higher than those produced physiologically. In a sustained, weak voluntary contraction, firing rates of 8 - 12 Hz are typical. Lower firing rates are found with repetitive weak contractions. For a steady, sustained forceful contraction, an upper limit to the firing rate seems to be about 30 Hz in human skeletal muscle. These are firing frequencies which result in a partially fused contraction. How, then, are smooth, controlled voluntary movements possible when low forces are involved and the firing rates are very low? Why is it that no twitching or fluttering is seen when the firing rates are below the fusion frequency? The answer is that the activity of different motor units is asynchronous. Although individual motor units may be firing at low frequency and producing a fluttering, partly fused contraction in individual muscle fibres, there is no synchronization between different motor units. At the level of the whole muscle, the total force is the sum of the contributions of all active motor units so the ripples in force output from each motor unit are smoothed i.e. lost in the total. By contrast, when muscles are activated electrically, all of the activated fibres are synchronously activated so smooth contractions are only possible when the induced firing frequencies are greater than, or equal to, the fusion frequency. The very large range of force output of which human muscles are capable is only partly due to variation in nerve firing rates. A second factor which is at least as important is recruitment. In a weak contraction only a few motor units may be active. In a stronger ELECTRICAL STIMULATION OF NERVE AND MUSCLE 77 contraction, more motor units are recruited. The gradation in force which all skeletal Larger muscles such as muscles exhibit is achieved by a combination of increase in firing rate and increase in biceps brachii and deltoid, number of motor units recruited. Different skeletal muscles rely to different extents on which contain a large these two strategies. number of motor units, rely With rapid, forceful contractions, initial nerve fibre firing rates can be as high as 100 Hz more on recruitment than or so, but this is never sustained. Such rates are only observed at the start of a smaller muscles, such as contraction and drop to much lower 'steady' values within a few seconds. With adductor pollicis and the first prolonged effort and fatigue, the maximum steady firing rate might typically drop from dorsal interosseous muscle. 30 Hz to about half this figure. It is interesting to note that as muscle fibres fatigue, their twitch duration increases so the associated decrease in firing rate does not result in a partially fused contraction becoming unfused. Were this to occur, a very large drop in force would result (figure 4.5). Rather the decrease in firing rate seems to be balanced by the increase in contraction time. Fatigue considerations An observation made very early in the history of electrically induced muscle contraction is that the rate of fatigue is much greater with electrically induced contractions than with voluntary contractions of the same magnitude. Two factors contribute to the difference: the firing rates of the motor units and the number and nature of the motor units which are recruited. As discussed previously, in order to produce a smooth, non-twitching motor response, the frequency of electrical stimuli must be higher than the fusion frequency of the excited muscle fibres. 50 Hz is a 'ball-park' figure for most skeletal muscles. A voluntary contraction of the same magnitude would involve lower firing frequencies and, to compensate, greater recruitment of motor units. The difference is that physiologically, the load is spread over more motor units which, individually, do not have to work as hard. The result is a lower rate of fatigue. The second difference involves different muscle fibre types. Muscle fibres are typed, as described previously, as fast- or slow-twitch. They are further categorized according to fatigue resistance where slow, fast-resistant and fast-fatigable fibres are
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 78 distinguished. Fatigue-resistance depends on cellular metabolism. Slow fibres have a Henemann et al, in 1965, long twitch-force duration and are fatigue-resistant. They rely on aerobic glycolysis for proposed the size principle of energy production. Fast-fatigable fibres are designed to produce very high peak motoneuron recruitment forces for a very short time (from a fraction of a second to a few seconds). They rely on based upon their own anaerobic metabolism for peak energy production: an oxygen supply from the experimental work and that of bloodstream is not an immediate concern. Fast-resistant fibres are relatively fatigue others. The principle states resistant but also have a short twitch-force duration and moderately high peak force. that with increasing contractile In a steady or repetitive voluntary contraction it is the slow, highly fatigue-resistant force, recruitment proceeds in motor units which are recruited first. For contractile forces up to about 20% of an orderly fashion from maximum, slow motor units dominate. Above this level, the contribution of fast- smallest to largest resistant units increases. Fast-fatigable units are the last to be recruited. motoneuron diameter. Slow motor units are the smallest in terms of the number of fibres innervated by an individual motoneuron and also in terms of the motoneuron diameter. Fast-resistant motor units are larger and the motoneuron diameters are larger. Fast-fatigable units are the largest on both counts. Clearly it is optimal to have small motor units with long twitch times and low fusion frequencies used for weak contractions. In this way, an unsteady twitching contraction is avoided. The order of recruitment described above applies to steady or repetitive contractions. In sudden movements, fast-fatigable units are activated at the beginning of the movement. These units fire very few action potentials in a single high-frequency burst. This produces a high peak force with a rapid initial rate of increase, such as would be needed to produce a sudden, brief acceleration of a limb segment. With electrical stimulation, the pattern of recruitment is very different to that which occurs physiologically. Two factors determine the order of recruitment: proximity to the stimulating electrode and nerve fibre diameter. Fibres closer to the stimulating electrode will experience a higher stimulation intensity than those further away. This is because current spreads within the tissue, resulting in a decrease in intensity. Close to the electrodes, spreading is minimal and the current density is highest. With increasing distance, the current density decreases. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 79 The current density, i, is given by the formula Figure 4.7 The spreading of current i = I ...... (4.1) within a volume conductor A where I is the current (in amperes) and A the area through which the current passes (in square metres). The units of current density are thus amperes per square metre (A.m-2). Close to the electrodes the current density will be greatest: approximately I/Ao, where Ao is the area of the electrode (figure 4.7). Further from the electrodes the area A through which the current passes is larger than Ao so the current density is less. Fibre diameter is important because the distance between adjacent nodes of Ranvier is greater for larger diameter fibres. Histological measurements show that the distance between nodes is directly proportional to nerve fibre diameter. As stated previously, initiation of an action potential relies on producing a potential difference between adjacent nodes. The greater the distance between the nodes, the greater will be the potential difference for a given stimulus intensity applied to the tissue. Larger diameter nerve fibres then, are more easily recruited than those of smaller diameter. This means that for nerve fibres at a certain distance from the stimulating electrode, the order of recruitment will be the reverse of that which occurs physiologically. The largest diameter fibres, which innervate fast-fatigable motor units which have the highest fusion frequencies, will be recruited first. The effect of electrode-to-nerve-fibre distance means that the reverse recruitment order will not be followed exactly, but fast-fatigable motor units will contribute
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 80 disproportionately to an electrically induced contraction. A consequence is that a high stimulus frequency is needed to achieve a smooth, fused response and this inevitably induces a high rate of fatigue. STIMULATION OF DENERVATED MUSCLE Even though a muscle may have lost its motor nerve supply it is possible to stimulate the fibres directly. The effect of electrical stimuli on muscle fibre and on nerve is similar: the potential difference across the muscle fibre membrane is reduced and this results in a wave of excitation which propagates along the fibre and is transmitted into the interior of the muscle fibre via the transverse tubule (T-tubule) system. Depolarization of the T-tubules triggers the calcium ion release which results in contraction of the fibre. There are three main differences between the response of innervated and denervated muscle to electrical stimulation. One difference is in the type of contraction produced and the others are in the effects of pulse shape and duration: * The contraction and subsequent relaxation of denervated muscle is more sluggish than innervated muscle. This is mainly due to the absence of synchronization in stimulation of the muscle fibres. * Denervated muscle shows a much less marked accommodation effect than nerve. Thus it is not necessary to use short duration, rectangular pulses for stimulation. An impulse which rises slowly in intensity can depolarize the muscle fibre membrane. For this reason impulses having, for example, sawtooth, trapezoidal or triangular shape and long duration are effective in stimulating denervated muscle. Such pulses are termed selective because it is possible to adjust the pulse duration and intensity for adequate stimulation of denervated muscle with minimal stimulation of nearby intact nerve fibres. * Denervated muscle is relatively insensitive to short duration stimuli. This important point is discussed next. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 81 Effect of Pulse Duration Figure 4.8 A strength-duration curve for The much smaller accommodation effect with denervated completely denervated muscle muscle compared to nerve results in longer pulses being more effective for stimulating denervated muscle. However, short duration pulses are considerably less effective. These experimental observations are illustrated in figure 4.8, which shows a strength-duration curve typical of completely denervated muscle. The graph shows that painfully high voltages must be used to stimulate denervated muscle if the duration is short. In contrast, if long duration pulses are used only low voltages are needed. In this particular example, comparison with figure 4.6 shows a greater sensitivity for denervated muscle to pulses of duration above 50 ms. The increased sensitivity to long pulse duration stimuli is typical of recently denervated muscle. The lack of neural input apparently causes an increased sensitivity of the muscle fibre membrane. The increase may be the result of normal control mechanisms whereby the muscle fibre adapts-to or compensates-for changes in neural activity. Effect of Pulse Frequency We saw previously that for frequencies above about 100 Hz, the higher the frequency the less efficient pulses are for direct stimulation of nerve fibres. Similar behaviour is observed when muscle is stimulated directly, though the frequencies concerned are different, as is the explanation of the effect. For stimulus frequencies above about 10 Hz the sensitivity of denervated muscle decreases with the effect becoming quite marked at frequencies above 50 Hz. A simple explanation for this behaviour is evident from the strength-duration graph of
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 82 denervated muscle. From figure 4.8 we see that for pulse durations below 100 ms The effect of stimulus the stimulus intensity needed for contraction begins to increase. For a pulse duration frequency then is somewhat of 100 ms the pulse frequency cannot exceed 1/(100 ms) = 1/(0.1 s) = 10 Hz. This simpler to explain than for would allow no 'rest' time between stimuli. As the frequency is increased above 10 Hz nerve fibre stimulation, where the pulse duration must inevitably decrease if there is to be any separation between the effects of accommodation the pulses. One pulse must finish before the next one is applied. and pulses applied within the To charge the muscle fibre membrane enough to depolarize it, the stimulus intensity relative refractory period are needs to be increased if the pulse duration is decreased. The membrane capacitor needed to account for the must be charged by a certain amount to trigger depolarization. This can be achieved observed variation in by a long duration pulse of relatively low intensity or a shorter duration pulse of higher sensitivity. intensity. Charge movement is the critical factor, as with nerve fibres, but the time- α-motoneurons are frames are very different. intrinsically more sensitive to electrical stimulation than the SENSORY, MOTOR AND PAIN RESPONSES smaller diameter A-δ and C fibres. So far the focus has been on the motor response to electrical stimulation. In reality, the forcefulness of the motor response will be limited by pain. Pain can be a direct result of the muscle contraction or can be due to stimulation of pain fibres (nocioceptive afferent nerve fibres) by the electrical stimulus. Clearly, if pain is produced as a result of the forcefulness of the muscle contraction, more efficient electrical stimulation will not enhance the motor response. On the other hand, if noxious electrical stimulation is the limiting factor, stimuli which preferentially recruit motor (A-α) fibres ahead of pain (A-δ and C) fibres will be more effective. Fortunately, a degree of selectivity can be achieved by appropriate choice of pulse duration. The reason is that α-motoneurons and pain fibres have a different range of diameters and different strength-duration behaviour. α-motoneurons have the largest diameters (range 12-20 µm), the largest internodal spacing (distance between adjacent nodes of Ranvier) and consequently, the lowest thresholds for electrical stimulation. The range of diameters of sensory fibres (A-α afferents, diameters 6-17 µm) overlaps with that of α-motoneurons, making it virtually impossible to elicit a motor response without also electrically activating sensory fibres. When stimulation is applied transcutaneously, a sensory response is, more ELECTRICAL STIMULATION OF NERVE AND MUSCLE 83 often than not, elicited before a motor response. The reason is that although motor A motor response is elicited fibres have, on average, larger diameters, they are located more deeply. Sensory first in some individuals. This fibres are in abundance near the skin surface and so will inevitably be closer to the can happen when the person electrodes. In other words, the effect of current spreading with depth tips the balance has a low skinfold thickness to favour sensory fibre activation before motoneuron activation in most individuals. i.e. a very thin layer of Pain fibres are also found in abundance near the skin surface. If this was the whole subcutaneous adipose story then one would expect a painful sensation before a motor response. That this tissue. The motor response seldom occurs is due to the fact that A-δ and C fibres are less sensitive to electrical inevitably produces a sensory stimulation than the larger diameter sensory and motor nerve fibres. response, but this is a result Suppose then that skin surface electrodes are attached to a subject and stimuli of of the muscle activity rather increasing intensity are applied. Three distinct responses may be obtained, each than electrical activation of response having a different threshold for its onset. As the stimulus intensity is sensory fibres. increased the first response normally noticed is sensory. The subject perceives the electrical stimulation before any muscular response is elicited. A further increase in intensity is needed for the onset of a motor response. This is followed, at higher intensities, by the subject reporting a sensation of pain. The sequence of responses and their separation in terms of the intensity required, depend on four factors: the placement of electrodes, the electrode size, the stimulus pulse width and the stimulus frequency. * Electrode placement is important in that to obtain a pronounced motor response without pain the electrodes should be over a motor point or region where the motor nerve is located superficially. Conversely, if the aim is to produce sensory fibre stimulation with no motor response (as would be appropriate for pain control), motor points and nerve trunks should be avoided. * The electrode size should be as large as possible to avoid concentrating the current in a small superficial region. Current spreads as it enters tissue and the greatest spreading is produced near the edges of the electrodes. Near the electrode centre the spreading is less. By using a large electrode the central part, where current spreading is least, covers a larger deep tissue area (see figure 4.9).
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 84 * The stimulus pulse width should be sufficiently small. Experimentally it has been shown that best discrimination between sensory, motor and pain responses is achieved using relatively narrow stimulus pulses. This point is discussed further below. * The stimulus frequency would be expected to influence discrimination because the nerve fibres associated with sensory, motor and pain responses have different refractory periods. To date, no studies of discrimination as a function of frequency appear to have been published. Effect of Pulse Duration Figure 4.9 The effect of electrode size on The differences in electrical characteristics of nerve fibre types and their different current density in tissue. (a) small depths of location in tissue results in separate strength-duration curves for electrodes, greater spreading of sensory, motor and pain responses. current, (b) larger electrodes, more Consider first the effect of fibre type and consequently fibre diameter. Other things being equal, the observation is that the strength-duration curve is shifted to the uniform current density right (to longer pulse durations) for smaller diameter fibres. The smaller the fibre The graphs shown in figure diameter, the larger is the chronaxie. This is another way of saying that the 4.10(a) are based on animal smaller the fibre diameter, the larger is the associated RC time-constant. Figure studies. There are greater 4.10(a) illustrates the differences in strength-duration curves of different diameter ethical problems associated nerve fibres. The results apply to nerve which is stimulated directly using surgical with human experimentation intervention. On this basis we would predict a recruitment order of motor then using the same design, so sensory then pain fibres, with the sensory fibres recruited almost as soon as the such studies have not been motor fibres. carried-out. As noted previously, with transcutaneous stimulation, depth in tissue also affects ELECTRICAL STIMULATION OF NERVE AND MUSCLE 85 the threshold for nerve excitation. This is because of current spreading and a consequent reduction in the local stimulus intensity. Superficially located fibres are therefore recruited at lower stimulus intensities. Figure 4.10(b) shows measurements obtained with human subjects and transcutaneous electrical stimulation. Note the horizontal axis (time) scale. In this figure the pulse widths are Figure 4.10 measured in microseconds (µs) and not milliseconds as have been Strength-duration curves for (a) different previously used to describe action potentials and the subsequent refractory period. Here we are dealing with pulse widths which are nerve fibre types, with the nerve trunk small compared to the time-course of an action potential. exposed and stimulated directly and (b) sensory, motor and pain thresholds measured using transcutaneous stimulation.
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 86 Two things are apparent from figure 4.10(b). First, that in reality the order of The results shown in figure recruitment is usually sensory, then motor, then pain at all pulse durations when 4.10(a) indicate that, with current is applied transcutaneously. Second, that as we go to shorter pulse durations surgically implanted the separation between the curves increases. electrodes, very good The separation due to fibre diameter is most marked in figure 4.10(a) and indicates discrimination between nerve that with direct nerve stimulation, by using sufficiently short pulse durations (around fibre types can be achieved 500 µs), the small diameter C fibres will not be stimulated at intensities which very by choice of an optimal pulse effectively recruit the larger A-δ, A-β and A-α fibres. A shorter pulse durations (around width. 50 µs), neither C nor A-δ fibres will be stimulated at intensities which efficiently recruit The pain threshold graph in A-β and A-α fibres. This indicates that as one goes to smaller pulse widths, the ease figure 4.10(b) should not be of discrimination between sensory and motor responses on the one hand, and pain identified with the A-δ and C responses on the other, is increased. fibre graphs. Very little The extent of discrimination evident with transcutaneous stimulation is less. As figure contribution would be made 4.10(b) shows, the sensory, motor and pain threshold graphs are more overlapped by C fibres. Rather, A-δ fibre and the variation occurs at smaller pulse widths. This is because the measured activity and the pain response depends not just on the fibre type (and the associated diameter) but also associated with a forceful two other factors: the depth of the fibres within tissue and the electrical characteristics muscle contraction would of the skin and underlying tissues. The capacitative nature of the stratum corneum determine pain thresholds. means that longer duration pulses are not more effective for nerve stimulation (whatever the fibre type) as the current flow in tissue beneath the stratum corneum is transient (figure 3.4). Spikes in the current flow are produced at the start and end of long duration pulses and increasing the pulse width does not result in a longer duration flow of current in tissue. Thus C fibres are not as more readily recruited at longer pulse durations as would be expected from figure 4.10(a). Nor are A-δ fibres, though the effect is less. The result is a plateau in the transcutaneous sensory, motor and pain threshold graphs at a pulse width much less than in figure 4.10(a). The observations regarding the effect of pulse width have important practical implication for therapy, and the results shown in figure 4.10(b) are most relevant. If long duration pulse widths are used then only small changes in intensity will be needed to change from a sensory response to a motor or pain response. By contrast, ELECTRICAL STIMULATION OF NERVE AND MUSCLE 87 if short duration pulses are used, much larger changes in intensity will be needed to On the basis of his work, recruit motor and pain fibres. If the objective is to produce a sensory response with d'Arsonval described the minimal motor or pain responses then short duration pulses are preferred (less than electric chair, recently 50 µs from figure 4.10a or perhaps 'the shorter, the better' from figure 4.10b). Short adopted in New York state for duration pulses will also be capable of producing an effective motor response with criminal executions, as minimum pain sensation. It is for this reason that modern electronic stimulators 'barbarous and unholy' as the produce higher voltage, shorter duration pulses than their predecessors. voltage chosen was too low A question arising from the foregoing discussion is whether very short pulses, around (1500 V) and death is slow. 2 to 10 µs duration, will give better discrimination with transcutaneous stimulation than, say, 20 µs pulses. The evidence certainly indicates that pulses of duration in the range 20 to 50 µs will more effectively discriminate than pulses with duration greater than 100 µs. It is not known whether this trend continues to very short pulse durations. Further research is needed before any firm conclusions can be drawn. STIMULATION USING SINUSOIDAL AC Sinusoidal alternating current has been used for patient treatment almost since devices for producing AC were first marketed in the late 1800s. It was soon established that low frequency AC produced noxious stimulation while AC in the kHz frequency range could produce strong muscle contractions without the degree of noxious stimulation associated with lower frequencies. The French scientist Arsène d'Arsonval studied the effect of AC stimulation on nerve and muscle, both using dissected animals and by transcutaneous stimulation of human subjects. He used an alternator, the first device built for generating AC and the one which is used in every modern-day motor car. An alternator works on the principle that if a coil of wire is made to spin in a stationary magnetic field, alternating current is produced in the coil. d'Arsonval reported in 1891 that with increasing frequency, the neuromuscular response to sinusoidal AC becomes stronger up to about 1400 Hz, is constant between 1500 and 2500 Hz and decreases to 5,000 Hz. He also reported that a current of 1500 Hz is more painful than 5,000 Hz but much less painful than currents of 75 and 20 Hz.
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 88 d'Arsonval's observations of transcutaneous stimulation using human subjects and The Nemectrodyne sinusoidal AC laid the foundations for the use of kHz frequency AC in clinical practice. interferential stimulator was Sinusoidal AC stimulation has figured in clinical practice since the 1950s when a the first on the market and German physician, Hans Nemec, began advocating 'Interferential Currents' as a the company continues to means of producing comfortable, pain-free, muscle contractions and 'promoting sucessfully market tissue healing'. More recently, the use of interferential currents for pain control has interferential units. been advocated. Interferential currents used clinically are sinusoidal AC with Diadynamic currents have all frequencies around 4 or 5 kHz. but disappeared from clinical More than two decades after the introduction of interferential currents, 'Russian practice and will not be currents' became popular, principally due to the claims made by a Russian physician, discussed here. Details can Yakov Kots, in the late 1970s. Kots claimed that kHz frequency AC, modulated at 50 be found in older texts e.g. Hz with a 1:1 duty cycle, could produce large strength gains in stimulated muscle. He the 2nd (1986) edition of based his claims on studies made with young Russian athletes as subjects: athletes Electricity, Fields and Waves who were hoping to qualify for the Olympic games. Russia's success in the Olympics in Therapy'. and the intense competitiveness which existed at the time seems to have given weight to Kots' claims. Kots argued that an optimal AC frequency for muscle strengthening, one which produced maximal force at the pain-tolerance threshold, was 2.5 kHz if the muscle was stimulated directly (with the active electrode over the muscle) or 1 kHz if the muscle was stimulated indirectly (with the active electrode over the nerve trunk supplying the muscle). Both Russian currents and Interferential currents continue to be used in clinical practice. Interferential currents are popular in England, Europe and Australia. Russian currents are, somewhat paradoxically in the light of political relations post world war two, more popular in the USA. Stimulation with low frequency AC is seldom used nowadays. It is particularly painful. Nonetheless, it did experience some popularity in Europe in earlier decades. A particular form of low-frequency AC stimulation, called 'Diadynamic current' was popularized in Europe. The argument seems to have been that the discomfort associated with the stimulation had therapeutic benefits resulting from a counter- irritant effect. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 89 Low Frequency Alternating Current The term low frequency AC as applied in therapy relates to frequencies between about 1 Hz and 100 Hz. A sinusoidal current is, in effect, a continuous train of current pulses. For example, 50 Hz AC has one complete cycle every 1/50th of a second or 20 milliseconds. The 20 ms sinewave has a rounded 10 ms positive pulse followed by a rounded 10 ms negative pulse. The stimulus is therefore a series of 10 ms pulses. As noted previously, a pulse width of 10 ms results in little discrimination between sensory, motor and pain thresholds. Smaller diameter pain fibres are recruited at thresholds not much above those of the larger sensory and motor fibres (figure 4.10b). As the waveform does not have an abrupt rise, some nerve fibre accommodation will occur meaning that a greater current intensity will be needed to produce the same response as a 10 millisecond rectangular pulse. The effect of accommodation is greater in large diameter nerve fibres so there is dropout of their contribution if sinewaves rather than square waves are used. This means that there will be less stimulation of large (A-α and A-β) fibres with low frequency AC and, relatively, more contribution of smaller (A-δ) fibres. The pulse duration will result in stimulation which is both superficial and relatively non-discriminatory between sensory, motor and pain responses. Thus if the aim of therapy is to stimulate superficially and to produce, say, modest muscle contraction together with stimulation of pain fibres, or simply painful stimulation, then 50 Hz sinusoidal AC or one of its variants is a logical choice. Unsurged 50 Hz AC is sometimes used for a counter irritant effect. Counter-irritation has been dismissed as treating a patient with a sore right thumb by hitting the opposite, left thumb and producing more pain. Suddenly, the patient finds the right thumb more comfortable! The point which is ignored in this simplistic argument is
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 90 whether the pain relief persists, in which case counter-irritation stimulation is Alternating current at high vindicated. Unfortunately, no properly documented studies seem to have addressed (>100 kHz) frequencies does this question. Low frequency AC might thus have some potential in clinical practice. have a part to play in The evidence base has yet to be established. physiotherapy, but not transcutaneous electrical Medium Frequency Alternating Current nerve stimulation. Rather the tissue-heating effects are Medium frequency alternating currents are defined as currents in the frequency range exploited. This is discussed 1 kHz to 100 kHz. Above 100 kHz, alternating current is not able to excite nerve fibres in the later topic 'Fields'. and the only effect is one of tissue heating. Currents above 100 kHz are classified as 'high frequency'. In clinical practice, currents with frequencies between 1 kHz and 10 kHz are commonly used. Frequencies above 10 kHz are not. The reason is that above 10 kHz or so, the nerve fibre response diminishes while the power dissipated in tissue increases. At frequencies above 10 kHz, nerve sensitivity becomes lower while the electrical energy dissipated in tissue, and consequently the heating rate, increases. 10 kHz to 100 kHz is evidently the transition zone between direct electrical stimulation and tissue heating. As noted previously, nerve-fibre firing rates are well below 100 Hz during most voluntary activities, including strenuous exercise and generally less than a few tens of Hz on a sustained basis. With electrical stimulation at higher frequencies and sufficiently high intensity, firing rates approaching 1 kHz can be produced. The absolute refractory period places the limit on the maximum firing rate. If nerve is stimulated with AC at frequencies above 1 kHz, action potentials are produced with every second, third or fourth succeeding AC pulse. The fibre firing rate will thus be a sub-multiple of the AC frequency. If, for example, 4 kHz AC is used, the induced firing rate might be 100 Hz at intensities just above threshold. In this case the firing rate is determined by the relative refractory period. At higher intensities, higher firing rates are induced as action potentials are produced during the relative refractory period. At the highest intensities the firing rate might approach 1 kHz i.e. fibres firing immediately after the absolute refractory period. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 91 With AC stimulus frequencies above 10 kHz or so, the physiological response of nerve Higher frequency AC (above fibres become less and less while the power dissipated, and heating rate, become 10 kHz) thus has less direct larger and larger. The decreased nerve fibre response is because the membrane effect on the nerve-fibre capacitor has less and less time to charge during a pulse, so less depolarization is membrane and more effect produced. The higher tissue heating rate is because skin impedance decreases with on the sensory receptors increasing frequency (chapter 3 previously) so the current flow is higher for a given which detect heat. stimulus voltage and the power dissipation is correspondingly higher. INTERFERENTIAL CURRENTS Hans Nemec popularized interferential currents in the 1950s. Although Nemec published a number of articles describing and reporting on the effect of interferential currents, these were in German. Only one English language article exists. It was translated from German and published in the British Journal of Physiotherapy in 1959. In it, Nemec described interferential currents and made claims of therapeutic benefits. The claims, judged in terms of modern criteria, were inappropriately speculative i.e. were not adequately documented. They are, however, intriguing and not without some credence. Here we focus on the less speculative aspects. An interferential stimulator has two separate, Figure 4.11 electrically isolated circuits for applying current to the Application of interferential currents patient. The currents are applied using two diagonally opposed pairs of electrodes as shown in figure 4.11. The idea is that the two currents 'interfere' within the tissue volume, reinforcing each other and producing a greater effect at depth than would be possible using a single circuit. In the region of intersection (the cross- hatched area in figure 4.11), the resultant intensity is
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 92 high as it is the sum of the contributions of each current. Each circuit (A and B) supplies an AC signal of constant amplitude to the patient. If current spreading is not great, as is assumed in figure 4.11, the region of maximum stimulation is the cross-hatched area (the region of diamond shapes) in figure 4.11. This contrasts with the regions of maximum stimulation when only one circuit is used. In this case maximum stimulation is produced immediately under the electrodes. Figure 4.12 illustrates the difference. In practice, current spreading will make the difference shown in figure 4.12 less marked. The superimposition of the two currents will, however, help to counteract the reduction in stimulus intensity with depth, thus increasing the depth efficiency of stimulation. The original interferential machines Figure 4.12 produced a sinusoidal waveform Depth efficiency of (a) bipolar with a frequency around 4 kHz: thus and (b) quadripolar stimulation. the stimulus pulse width was 1/8000 sec or 125 µs. Some modern machines offer a choice of AC frequencies and use a rectangular pulsed AC waveform, rather than a sinewave. There is some evidence that a rectangular pulsed waveform is more comfortable than its sinusoidal counterpart and also evidence that optimal comfortable stimulation is achieved at a frequency of about 9 kHz. Beat or Modulation Frequencies A key feature of interferential currents is that the two circuits produce current of slightly different frequency. The difference is normally between 1 Hz and 150 Hz. When ELECTRICAL STIMULATION OF NERVE AND MUSCLE 93 applied to the patient the two currents interfere and produce a 'beating' effect in the patient's tissue. The interference or 'beat frequency' effect is illustrated in figure 4.13. Figures 4.13(a) and (b) show two sinusoidal waveforms applied to the patient via diagonally opposing pairs of electrodes as shown in figure 4.11. The total current at a particular point in the patient's tissue is the sum of the currents from each pair of electrodes. At points where the two currents are of equal amplitude the sum of the two signals will be an AC waveform which is amplitude modulated as shown in 4.13(c). The surge or modulation frequency is equal to the difference in frequency of the two Figure 4.13 currents. The frequency, f, of waveform (a) might be 4000 Hz and the frequency (f-δ) Interference of two sinusoidal of waveform (b) might be 4000-10 = 3990 Hz. In this case the value of δ, the currents of different frequency. modulation frequency is 10 Hz.
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 94 To make clear which frequency we are talking about, the Figure 4.14 terms 'carrier frequency' and 'beat frequency' are used. In Interference of two sinusoidal this example, the carrier frequency is 3995 Hz and the beat currents of different frequency frequency is 10 Hz. In figure 4.13, the currents are assumed to be of equal and different amplitude. amplitude. In regions of tissue where the two currents are not the same size, an interference effect will still be produced, but the resulting waveform will not drop to zero midway between the maxima. Figure 4.14 shows the effect of adding two currents of slightly different frequency when one current is twice as big as the other. An interference effect is still produced but the depth of modulation of the waveform is less. Depth Efficiency and Localization As noted earlier in this chapter, for maximum stimulation efficiency, current should flow parallel to the nerve fibres when there is a single current flow through tissue. When there are two intersecting currents of equal amplitude, maximum stimulation occurs along lines midway between the current paths. The reason is that the net current flow is the vector sum of the two currents. Consider first the situation where two current pathways are at right angles and the currents are equal. Nerve fibres aligned parallel to one of the current pathways will experience an unmodulated AC stimulus as shown in figure 4.13(a) or (b). Fibres aligned along lines midway between the current paths will experience a modulated stimulus (figure 4.13(c)) of higher intensity. Those fibres aligned in other directions will experience a partially modulated stimulus, similar to figure 4.14, with a depth of modulation which depends on the fibre orientation. Figure 4.15 shows the net current flow in different directions for the simple configuration in figure 4.11. The length of the black arrows is proportional to the ELECTRICAL STIMULATION OF NERVE AND MUSCLE 95 current intensity. In the horizontal and Figure 4.15 vertical directions, the net current is The variation in current intensity maximum and the modulation is 100%. In directions at 45o , there is no and amount of modulation modulation and the intensity is some with direction when using 30% lower. The pattern of stimulation is clearly more interferential currents. complex with interferential currents than with current applied using a single pair of electrodes. We can, however, draw some important conclusions: * nerve fibres aligned in directions which bisect the angle between the current pathways (horizontally and vertically in figure 4.15) will experience the greatest stimulation intensity and the stimulus will be a modulated AC signal. * fibres aligned parallel to the direction of the individual current flows will experience a lower, but still relatively high, stimulation intensity. The stimulating current will not be modulated. * nerve fibre firing rates will be much higher than with stimulation using single pulses applied at low frequency. Fibres aligned parallel to the direction of the individual current flows will fire at a rate determined by how far above threshold is the local stimulation intensity. * Fibres aligned in directions which bisect the angle between the current pathways will fire in bursts. The bursts of activity will be at the beat frequency and the number of action potentials per burst will depend on how far above threshold is the local stimulation intensity.
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 96 A widespread misconception is that with interferential currents, the nerve fibre firing frequency is equal to the beat frequency. This would only be the case for fibres stimulated at, or just above, their threshold. As noted previously, for stimulation intensities above threshold, nerve fibres will fire at much higher rates. When the stimulus intensity is modulated at low frequency, nerve fibres will fire in bursts, with each 'beat' of the current intensity. The beat frequency only determines the burst frequency of the action potentials. The number of action potentials per burst depends on how far the stimulus intensity is above threshold. Thus if a beat frequency of 50 Hz is chosen to produce repetitive, forceful muscle contractions, the rate of fatigue will be higher than if 50 Hz single-pulses were used as the average firing rate will be much higher. Another widespread misconception about interferential currents is that the pattern of stimulation is in the shape of a clover-leaf (a four-leafed clover) rather than the rounded-diamond shape shown in figures 4.11, 4.12 and 4.15. The misconception seems to have originated from the idea that nerve fibres are insensitive to an unmodulated AC stimulus i.e. that modulation at low ('biological') frequencies is necessary to produce a physiological response. Were this true, then fibres aligned parallel to the current paths (figure 4.15) would not be excited while those aligned along lines bisecting the angle between the current paths would be excited maximally. The pattern of stimulation would have four lobes, each lobe pointing to a corner of the rounded diamond shape. In fact, the clover-leaf pattern shows the areas of maximum interference, not maximum stimulation. The pattern applies to every small diamond shaped segment in the region of interference. It indicates the direction in which the stimulus intensity is greatest. It does not, in any way, represent the area of maximum stimulation. Within each diamond-shaped segment, a clover-leaf pattern can be drawn, showing the directions of maximum interference: in other words, the directions in which nerve fibres must be aligned to experience maximum stimulation. A misleading implication of the pattern is that no stimulation is produced if the nerve fibres are aligned along either of the current paths. ELECTRICAL STIMULATION OF NERVE AND MUSCLE 97 Premodulated Interferential Current Figure 4.16 Most interferential machines make provision for stimulation using either two pairs or a Russian currents: 2.5 kHz single pair of electrodes. Two pairs are needed for true interferential stimulation. The term 'premodulated interferential current' refers to a current waveform as shown in sinusoidal AC, burst figure 4.13(c), which is produced inside the interferential machine and applied to the modulated at 50 Hz i.e. 10 ms patient using a single pair of electrodes. 'Premodulated interferential' is thus something of a misnomer, as there are no currents interfering in tissue. 'on' and 10 ms 'off'. Premodulated current has the advantage that it is easier to apply, as only two electrodes are needed. The disadvantage is that there is no reinforcing at depth so maximum stimulation is produced immediately beneath the electrodes (figure 4.12(a)). RUSSIAN CURRENTS Russian currents are a particular form of electrical stimulation which became popular as a result of a talk given by Dr Y M Kots of the Central Institute of Physical Culture, Moscow, at a conference hosted by Concordia University, Montreal in 1977. He claimed strength gains of up to 40% in elite athletes as a result of this form of electrical stimulation. The term 'Russian currents' refers to sinusoidal AC of frequency 2.5 kHz which is burst-modulated at 50 Hz. The wave- form is shown in figure 4.16. It consists of 10 ms bursts of AC separated by 10 ms 'off' periods. The waveform repeats every 20 ms (1/50th sec) so the burst or modulation frequency is 50 Hz.
Search
Read the Text Version
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 10
- 11
- 12
- 13
- 14
- 15
- 16
- 17
- 18
- 19
- 20
- 21
- 22
- 23
- 24
- 25
- 26
- 27
- 28
- 29
- 30
- 31
- 32
- 33
- 34
- 35
- 36
- 37
- 38
- 39
- 40
- 41
- 42
- 43
- 44
- 45
- 46
- 47
- 48
- 49
- 50
- 51
- 52
- 53
- 54
- 55
- 56
- 57
- 58
- 59
- 60
- 61
- 62
- 63
- 64
- 65
- 66
- 67
- 68
- 69
- 70
- 71
- 72
- 73
- 74
- 75
- 76
- 77
- 78
- 79
- 80
- 81
- 82
- 83
- 84
- 85
- 86
- 87
- 88
- 89
- 90
- 91
- 92
- 93
- 94
- 95
- 96
- 97
- 98
- 99
- 100
- 101
- 102
- 103
- 104
- 105
- 106
- 107
- 108
- 109
- 110
- 111
- 112
- 113
- 114
- 115
- 116
- 117
- 118
- 119
- 120
- 121
- 122
- 123
- 124
- 125
- 126
- 127
- 128
- 129
- 130
- 131
- 132
- 133
- 134
- 135
- 136
- 137
- 138
- 139
- 140
- 141
- 142
- 143
- 144
- 145
- 146
- 147
- 148
- 149
- 150
- 151
- 152
- 153
- 154
- 155
- 156
- 157
- 158
- 159
- 160
- 161
- 162
- 163
- 164
- 165
- 166
- 167
- 168
- 169
- 170
- 171
- 172
- 173
- 174
- 175
- 176
- 177
- 178
- 179
- 180
- 181
- 182
- 183
- 184