ELECTRICAL STIMULATION OF NERVE AND MUSCLE 98 Kots and co-workers measured the maximum force which could be elicited using AC in the frequency range 100 Hz to 5 kHz. Current was applied either using either two equal-sized electrodes placed over the muscle belly (referred-to as 'direct' stimulation) or using a small 'active' electrode over the nerve trunk supplying the muscle and a larger 'indifferent' electrode placed elsewhere, so as to avoid excitable tissue (referred-to as 'indirect' stimulation). They established that maximal force at the pain- tolerance threshold was obtained at 2.5 kHz if the muscle was stimulated directly or 1 kHz if the muscle was stimulated indirectly. Kots also advocated a '10/50/10' treatment regime i.e. 10 seconds of stimulation followed by a 50 second rest period, repeated 10 times. His argument was that to produce strengthening, the electrical stimulation should be non-fatiguing. He reported that with intense stimulation for periods over 10 sec, fatigue is evident, whereas no force decline is seen if the duration is 10 sec or less. To avoid a force decline from one 10 sec stimulation period to the next, a rest period of 50 sec is needed. If this rest period is allowed, no force decline is seen over the 10 repeats. The validity of Kots' argument for the '10/50/10' treatment regime is questionable. The quoted findings were obtained using low frequency monophasic pulsed current, not kHz frequency AC. With AC bursts, the nerve firing rates would be expected to be higher and, as a consequence, the rate of fatigue would be higher. The strength gains reported by Kots are supportive, but whether the '10/50/10' treatment regime is optimal with AC burst stimulation remains open to question. EXERCISES 1 Briefly explain what is meant by the following terms: (a) threshold potential (b) absolute refractory period (c) relative refractory period (d) hyperpolarization ELECTRICAL STIMULATION OF NERVE AND MUSCLE 99 2 What is meant by the term 'accommodation' as applied to nerve fibres? Explain the significance of accommodation as far as stimulation by very low frequency AC is concerned. 3 Consider the strength-duration curve shown in figure 4.6. Explain why: (a) greater stimulus amplitudes are needed to elicit a muscle contraction when very short pulses (less than about 0.1 ms) are used. (b) a muscle contraction is produced with the same stimulus voltage for all values of pulse duration above a few milliseconds. 4 It is observed that the sensitivity of nerve fibres to stimuli changes as the stimulus frequency is increased above about 100 Hz. (a) Describe the change and briefly explain why it occurs. (b) Why does the effect become more pronounced at frequencies above about 1 kHz? 5 (a) Compare figures 4.6 and 4.8 and explain why the chronaxie for denervated muscle is much greater than that of typical nerve fibres. (b) Does this also account for the rheobase of denervated muscle being lower than that of nerve? Explain. 6 Consider the strength-duration curves for sensory, motor and pain responses shown in figure 4.10(b). What range of pulse duration should be used for producing: (a) A maximum sensory response with minimum motor involvement? (b) A maximum motor response with minimum physical discomfort? (c) A pain response with a minimum motor response? What is a motor point and what relevance has it to (a), (b) and (c) above?
ELECTRICAL STIMULATION OF NERVE AND MUSCLE 100 7 Consider DC pulses used for stimulation of denervated muscle. (a) What range of pulse width and frequency is most useful? (b) What is meant by the term 'selective' stimulus? Give an example and explain why the waveform you have nominated is selective. (c) Apart from pulse width and frequency, what other characteristic of a 'selective' waveform should be adjustable by the therapist? 8 Consider rectangular pulsed waveforms used for stimulating nerve fibres. (a) What is the most useful range of pulse width and frequency? (b) How does the depth efficiency of stimulation vary with pulse width and why? (c) When is surging of a pulse train useful? Why? 9 (a) What is the principal disadvantage of 50 Hz sinusoidal AC when the objective is to stimulate muscle? (b) Describe one useful practical application of 50 Hz sinusoidal AC in therapy. 10 (a) Describe the result of combining two medium frequency (interferential) currents in tissue. List the characteristics (carrier frequency, beat frequency, period and shape) of the resulting waveform. (b) What are the differences in depth efficiency of short duration pulsed current and interferential currents? Draw diagrams to illustrate. Why do interferential currents have greater depth efficiency? 11 Consider interferential currents flowing through tissue as in figure 4.15. (a) Why is the stimulation intensity greater in the regions between lines connecting electrode pairs rather than along lines connecting the electrodes? (b) Assuming that the stimulus intensity is high enough to excite nerve fibres regardless of their orientation, what differences in fibre firing rates would ELECTRICAL STIMULATION OF NERVE AND MUSCLE 101 exist between fibres aligned parallel to the current pathways and those aligned along lines bisecting the angles between the current paths? 12 Describe the similarities and differences between Russian currents and premodulated interferential currents.
RECTIFICATION AND AMPLIFICATION 102 5 Rectification and Amplification Electric light bulbs are very inefficient. More than 80% of DIODES, TRANSISTORS AND VALVES the electrical energy is dissipated in the form of heat. The humble electric light bulb is such a commonplace item that we take it for granted. Less than 20% of the energy Yet its development marked a turning point in human societal development. No is released as light. longer were we constrained by daylight hours. Although fire, the candle and later, gas lighting had enabled people to extend their daytime activities, the light bulb (and the establishment of a statewide system for providing electricity) virtually eliminated human dependence on daylight. The development of the light bulb also signalled the birth of electronics, an area of science which has transformed the world in which we live. The radio, TV, mobile phone and desktop computer are but a few examples of developments in electronics. Electronic technology led to the development of the neuromuscular stimulators used by physiotherapists and also enabled scientists to better study the workings of the human body, in particular the nervous and neuromuscular systems. So what is so special about an electric light bulb, that it can lay claim to the birth of electronics? A common electric light bulb consists of a filament mounted inside an evacuated glass envelope. The filament is a coil of resistance wire, usually tungsten. The glass bulb has virtually no air inside as oxygen in the air would react with the tungsten at high temperatures, forming tungsten oxide, which is an insulator so the filament would no longer conduct. Oxidation is prevented by the vacuum and when current from the mains or a suitable power source is passed through the tungsten filament It heats up and glows. Heat and light energy are produced at the expense of electrical energy. Heat and light production by a metal are fundamentally related to the movement of electrons. The greater the movement energy (principally of the electrons: the atoms are more locked in place in the crystal structure), the greater the heat energy and the greater the emission of light. Light is emitted whenever electrons are accelerated and, in the case of a light globe, electrons are continually accelerated when a current RECTIFICATION AND AMPLIFICATION 103 flows through the filament. Some electrons are given so Acceleration of electrons at red-hot temperatures and above, results in emission of much energy as a result of electrons from the surface of the hot filament. The phenomenon of electron emission collision that they can escape accompanying heating is given the name thermionic emission. The relationship the confines of the metal between heat, light and movement will be considered further in later chapters. For surface. the moment consider the electrons emitted by a glowing filament. In the normal run of things these electrons, having left behind positively charged ions from the parent metal, will be attracted back to the filament. The filament of a light globe is thus continually emitting and recapturing electrons in the process of emitting light. If heat and light production were the only physical phenomena associated with light bulbs then this, in itself, would be justification for their importance. the history of electronics would, however, would be quite different to the one we know. Electronics has its origin in a device based upon a simple light globe and invented by the English physicist J. A. Fleming in 1904. The device is a valve diode. Fleming's diode resembled an electric light bulb with an extra part, a metal electrode called the plate, included inside the glass envelope. The construction is illustrated in figure 5.1(a). By including the plate Fleming was able to capture some of' the electrons emitted by the filament. This is achieved by making the plate positively charged so that electrons are attracted to the plate more strongly than to the filament. Thus the diode allows a flow of current with electrons moving from filament to plate, providing the plate is sufficiently positive with respect to the filament. Notice, however, that a diode is asymmetric. Electrons can move from filament to plate but not from plate to filament even when the filament is made positive with respect to the plate. This is because the plate is not heated and so does not spontaneously emit electrons. The diode allows current to flow in one direction but not the other. Valve diodes require the plate to be at a potential some tens of volts higher than the filament for conduction to occur. Semiconductor diodes (figure 5.1(b)), their modern day counterparts, require potential differences of only a few tenths of a volt tor
RECTIFICATION AND AMPLIFICATION 104 conduction to occur and their energy efficiency is much higher i.e. less electrical energy is dissipated as heat. Fleming's diode was an extremely important development but the next step was even more important. By adding a 'grid' (actually a fine wire grid or mesh) between the filament and metal plate of a diode a device which could amplify electrical signals was invented. The development of this device (the triode valve) is generally credited to an American, Lee de Forest. This was in 1906. The construction of a triode valve is shown in figure 5.2. The mechanism by which it amplifies electrical signals will be described shortly. The triode valve, because of its ability to amplify very weak signals from a microphone and then apply them to a transmitter, started a revolution in science and technology. Although transmissions of signals across the Atlantic Ocean had been made by Marconi in 1901 these transmissions strained to the limit the detection facilities available. With amplifying valves it became possible to transmit and receive over much greater distances and even to amplify the signals so that they could be clearly heard without headphones, through a loudspeaker! By 1920 the valve had transformed wireless transmission from an extension of electrical and telegraph practice into a new and fascinating technology. In the 1940's when valves were a familiar part of the fields of home entertainment, communications and science the growing pressures of developments in science and technology had pointed up the limitations of the valve - high power consumption, large size and excessive heat generation. These factors became more and more critical as scientists and engineers attempted to apply electronics to more sophisticated tasks. The first computers were constructed using hundreds of valves, fully occupying large rooms and requiring elaborate ventilation and cooling systems in addition to enormous quantities of electrical power. RECTIFICATION AND AMPLIFICATION 105 Fortunately science again came to the rescue with the development in 1947 of the first functional transistor. Developed by physicists Shockley, Brittain and Bardeen the device revolutionized future developments in technology. The computers could be drastically reduced in size and power consumption. Printed circuit boards became a possibility so resistors, capacitors and inductors were miniaturized to suit. The computer occupying a large room could now be assembled as a device the size of a filing cabinet. More was still to come - the pressure was on for smaller and lighter components to build the apparatus used in navigation, guidance and communication in aeroplanes, missiles and satellites - to perform increasingly complex scientific tasks. Engineers quickly realized that transistors could be scaled down in size to fit many devices in the same volume which one previously occupied - they could even be interconnected in the one device in the kind of circuit arrangement equipment engineers might require. The same fabrication techniques could be adapted to produce resistive or capacitive interconnections between the transistors - microscopic versions of the familiar resistors and capacitors used in everyday circuits. Thus was the first integrated circuit (IC) produced. Early IC's, introduced in 1964 contained up to about 10 components in one tiny package. By 1968 this figure had risen to about 500. Today it numbers in the millions. Nowadays, large scale integration (LSI) permits many hundreds of transistors and diodes together with the necessary interconnections of components to fit into a package not much bigger than a postage stamp. A good example of LSI technology is the pocket calculator. Because they are based upon one or at most a few large scale integrated circuits, they can be cheaply mass produced. Most of the fabrication, testing and assembly is performed by automated apparatus under the control of minicomputers, themselves the product of IC technology. Many more examples of the impact of modern electronics are to be found in everyday life - digital wrist watches and clocks, the desktop computer and video games are a few which spring to mind. Watch for others - they are evident in virtually all aspects of life, art and science.
RECTIFICATION AND AMPLIFICATION 106 DIODES AND RECTIFICATION Valve diodes are rarely used nowadays. They have been superseded by semiconductor diodes, their solid-state equivalent. Semiconductor diodes are smaller, more efficient and generate less heat. The first semiconductors were fabricated from crystals of germanium. Nowadays silicon is the semiconductor of choice because of its superior physical properties. Both germanium and silicon are elements which fall midway between the good conductors and the good insulators. We now consider the properties of a silicon diode but you should bear in mind that other kinds of diode (selenium or germanium diodes, vacuum tube diodes) have similar though not identical properties. In addition there are diodes specifically tailored for somewhat different roles to those we will examine in this section (zener diodes and light emitting diodes are examples). These have distinct circuit symbols of their own which are variations on the normal diode symbol. Figure 5.3 shows the circuit symbol of a simple diode. Note the naming of the different sides of the diode. The circuit shown in Figure 5.4 can be used to demonstrate the conduction properties of a diode. With the circuit arrangement shown we would find that the potential difference across the diode is about 0.7 volt and the current flowing about 110 mA (0.11 amp). The resistance of the diode, calculated using Ohm's law, is thus R= V = 0.7 = 6.4 ohms I 0.11 Figure 5.4 Circuit for demonstrating the conduction properties of a diode RECTIFICATION AND AMPLIFICATION 107 This arrangement, with the anode (the arrow in the circuit symbol) connected to the positive terminal of the power supply and the cathode (the straight line) to the negative terminal, is called forward biasing of the diode. When forward biased, the diode has a very low resistance. With the diode reversed in the circuit, that is anode and cathode reversed, the diode is reverse biased. The voltage across the diode would be measured as 12 volts and the current about 0.5 microamp or 5 x 10-7 amp - too small to register on most ammeters. For all intents and purposes, the reverse current flow is negligible, meaning that the resistance is close to infinite. The diode resistance calculated from the measurements quoted is R= V = 12 = 24 x 106 Ω = 2.4 x 107 Ω or 24 MΩ. I 5 x 10-7 An ideal diode has zero resistance in the forward direction (when forward biased) and an infinite resistance in the reverse direction (when reverse biased). Silicon diodes come reasonably close to his ideal. Half Wave Rectification This unique conduction property of diodes makes them well suited to the task of' rectifying alternating current, that is converting alternating current to direct current. The circuit shown in figure 5.5 illustrates one way of doing this. Figure 5.5 Rectification by a diode
RECTIFICATION AND AMPLIFICATION 108 The AC source produces a sinusoidal alternating voltage, as shown in figure 5.6(a). This means that the diode will be forward biased when the voltage is positive i.e. during the positive half-cycles then reverse biased during the negative half-cycles. Only when the diode is forward biased will current flow through the resistor so a graph of current in the resistor will resemble figure 5.6(b). A graph of potential difference across the resistor would also resemble 5.6(b). Figure 5.6 (a) sinusoidal alternating voltage produced by the AC source in figure 5.5 and (b) the resulting current flow through the resistor, R. The current through the resistor is DC: not like the DC produced by a battery to be sure, but DC nonetheless because the current flow is only in one direction. A more accurate description of this half-wave rectified current would be pulsed DC. By placing a capacitor in parallel with the resistor as shown in figure 5.7 we can 'smooth' the pulsed DC to provide a more even flow of current. Figure 5.8 shows the waveforms which are obtained when different size capacitors are connected in parallel with the resistor, R. Figure 5.7 Smoothing rectified AC with a capacitor RECTIFICATION AND AMPLIFICATION 109 As the capacitor is made Figure 5.8 larger the waveform Smoothing using a capacitor. becomes more like the straight line graph The effect of different size obtained with a battery. capacitors. The voltage becomes more nearly constant. The reason that the capacitor has this effect is that the capacitor stores electrical energy. While the diode is conducting, current will flow through the resistor and at the same time the capacitor will charge to the peak voltage of the waveform. When the diode ceases to conduct the capacitor will discharge through the resistor (we discussed capacitor discharging in chapter 2). The rate at which the capacitor discharges depends on the size of the capacitor and also the resistance through which it discharges. The larger the capacitance and the larger the resistance, the slower the rate of discharge will be and the smoother will be the waveform. If we replaced the resistor in figure 5.7 with a higher value resistor then the waveforms shown in figure 5.8 would be obtained with capacitors of lower value. Conversely, if the capacitors remained the same, smoother waveforms would result. The amount of smoothing is determined by the RC time constant (chapter 2). For mains supplied electricity, where the AC frequency is 50 Hz, the time between pulses in figure 5.8(a) is 1/50th sec or 20 ms. For efficient smoothing, the RC time constant should be greater than 20 ms. Thus if R is large, meaning that the amount of current
RECTIFICATION AND AMPLIFICATION 110 drawn from the supply is small, C can be relatively small. If R is small and a lot of Figure 5.9 current is drawn from the supply, C would need to be relatively large to produce full-wave rectification using adequate smoothing. a diode bridge. Full Wave Rectification The half wave rectifier circuit is very simple but rather inefficient. Only half of' the original AC waveform is being used. The power supply is virtually switched off for half the time as no current flows during negative half cycles. The full wave rectifier circuit shown in figure 5.9 is a more efficient rectifier. The arrangement is called a bridge rectifier. The positioning of diodes ensures that current will always flow through the resistor in one direction and both halves of the AC waveform are used. Consider what happens when terminal A of the AC supply is positive - that is, on the positive half-cycle of the waveform. Current flows through diode 2 but not diode 1 (because of their polarity). It then flows through the resistor through diode 3 and back to terminal B of the transformer. It can not flow through diodes 1 or 4 because their opposite terminals are at a higher potential - remember there is a relatively large potential difference across the resistor. On the negative half cycle of the waveform, terminal A is at a lower potential than terminal B. Current flows from B through diode 4 through the resistor in the same direction as before, through diode 1 to terminal A. Current can not flow through diodes 2 or 3 as the voltage is higher at the opposite terminal. The net result is that the current through the resistor or potential difference across the resistor resembles figure 5.10(a). RECTIFICATION AND AMPLIFICATION 111 Full-wave rectified AC can be smoothed with capacitors in the same way as the Figure 5.10 half-wave rectified AC. A smooth waveform is more easily obtained with the full-wave Current flow through the rectifier because the capacitor is recharged 100 times per second rather than 50 resistor in figure 5.9 (a) without times per second with half-wave rectified AC. Compare figure 5.10(b) with the a capacitor and (b) with a half-wave rectified waveform of figure 5.6(c). The R and C values are the same but capacitor as in figure 5.9(c) the waveform in 5.10(b) is smoother because the pulses of rectified current are Athough the grid, when closer together. positive, attracts electrons, most shoot straigh through AMPLIFICATION the empty spaces in the grid The Triode Valve and add to the current flowing between filament and plate. The triode valve (figure 5.2) was the first electronic device capable of amplification. If a relatively high voltage is applied to the plate, electrons emitted by the filament will be attracted to, and accelerated towards, the plate, so a current will flow. The grid can control the flow of current. If a (relatively small) negative voltage is applied to the grid, electrons will be repelled and the flow of current will be reduced. If a small positive voltage is applied to the grid, electrons will be attracted and the flow of current will be increased. The grid of the valve is placed closer to the filament than to the plate, with the result that very small changes in grid voltage produce very large changes in the current through the valve. The valve thus functions as an amplifier. If a small alternating voltage is applied to the grid, the result is a large fluctuation in the current flowing through the valve. Again the small voltage applied to the grid
RECTIFICATION AND AMPLIFICATION 112 results in a large change in the current flowing through the triode. If a resistor is Figure 5.11 connected in series with the triode, the large fluctuations in current through the triode (a) a transistor and (b) its produce a large change in the potential difference across the resistor. A small alternating voltage applied to the grid produces large fluctuations in the voltage circuit symbol across the resistor. Thus the small signal is amplified. The Transistor The transistor is the semiconductor equivalent of the triode valve. It is used today in preference to valves in almost all electrical equipment, either in the form of a discrete component or as a part of an integrated circuit. A transistor has no filament and hence no heating requirements, is much smaller than a valve and consumes less power. It is more suited to low power applications, and so is used almost exclusively in electronic stimulators and many other pieces of apparatus. Figure 5.11 shows a transistor and its circuit symbol. The transistor (like the triode valve) has three terminals - called the collector, base and emitter (abbreviated c, b and e in figure 5.11). The names were given to indicate that the emitter 'emits' electrons which are 'received' or collected by the collector. The base controls the flow of current between emitter and collector. The arrow in the circuit symbol for the transistor is used in the same way as for a diode (figure 5.3). It points in the direction of easy current flow. The base-emitter junction in fact behaves just like a diode: the resistance to current flow in the direction of the arrow is very low, the resistance in the opposite direction is extremely high. The resistance between collector and emitter can vary from very low to very high depending on the current flowing between base and emitter. This 'variable resistance' property gives the transistor its name. Transistor is an abbreviation of trans-resistor. The current flowing between the collector and the emitter is directly proportional to the base-emitter current. Thus if the base to emitter current is zero, the collector to emitter current is also zero. If a small current flows from base to emitter, a larger current can flow between the collector and the emitter. The collector current is always RECTIFICATION AND AMPLIFICATION 113 many multiples of the base current. The ratio (collector current/base current) is called the current gain (or amplification) of the transistor. The amplification of the transistor depends on how the transistor has been made, its size and other factors. Typical values of current gain lie in the range 50 to 500. In other words the resistance of' the transistor between collector and emitter decreases in proportion to the base current. As the base current is made greater the collector to emitter resistance decreases so that the collector current increases in proportion to the base current. The transistor is thus a very good current amplifier - if we pass a certain amount of current through the base-emitter junction a much larger current will flow from collector to emitter. Operational Amplifiers It would be unusual nowadays to find a piece of electronic equipment built entirely from discrete components. Integrated circuits are now produced in huge numbers using automated fabrication techniques and this has reduced their cost to a point where it is, more often than not, cheaper to use one integrated circuit in applications where previously several individual transistors were used. One of the most common integrated circuits is the operational Figure 5.12 amplifier or op-amp for short. The operational amplifier is (a) an integrated circuit containing comprised of many transistors, resistors and capacitors fabricated in one tiny package with the components four operational amplifiers and interconnected to produce an amplifier of very high gain. By (b) the circuit symbol for an adding a few external components the op-amp can be adapted operational amplifier. to suit a variety of particular applications. Figure 5.12 shows an integrated circuit which contains four, independent operational amplifiers alongside the circuit symbol for a single operational amplifier.
RECTIFICATION AND AMPLIFICATION 114 All operational amplifiers require a power supply. For simplicity, this is not shown in figure 5.12. Of the 14 pins on the IC shown in figure 5.12, twelve are used for connection to the four op-amps and the remaining two are used for connection to a power supply. The operational amplifier has two inputs, the inverting input (labelled -) and the non- An op-amp amplifies the inverting input (labelled +). When a signal is applied to the inverting input the output potential difference between is a much amplified and inverted version of the input signal. Signals applied to the the inverting and non-inverting non-inverting input are amplified without being inverted. If the same signal is applied inputs. to both inputs, the output is zero. Figure 5.13 The voltage amplification or gain of an op-amp is very high, typically in the range 106 An inverting amplifier. to 1014. More often than not such high gains are not required in practical electronic circuits. The gain is easily reduced by adding a few external resistors. Figure 5.13 shows a practical op-amp circuit which acts as an inverting amplifier. Notice that we have again omitted the power supply connections for simplicity. The circuit is arranged so that signals are applied to the inverting input via resistor R1. The non-inverting input is connected to ground (earthed). A resistor (R2) connects from the output, back to the inverting input. This resistor will allow some of the output signal to feed back into the input. Note that the output is inverted with respect to the input. This means that the signal fed back will tend to cancel the input and so reduce both the input and output of the op-amp. The principle being used here is that of negative feedback. The gain (G) of the amplifier shown in figure 5.13 is given by the formula G = R2 R1 For example, if R2 is 20 kΩ and R1 is 4 kΩ, the gain would be 20/4 = 5. Negative feedback can be used to reduce the gain of an amplifier to any desired value. For the circuit shown in figure 5.13, if the feedback resistor (R2) is one hundred times as big as the input resistor (R1) the gain is set at 100 times. If, instead, the feedback resistor was ten times as big as the input RECTIFICATION AND AMPLIFICATION 115 resistor (R1) the gain would be 10 times. Figure 5.14 The advantage of using external resistors and negative feedback is that the op-amp A non-inverting amplifier. is then a general purpose device. Instead of having to produce a multiplicity of different op-amps, each with a particular gain, manufacturers need only produce one device which can be tailored to suit any particular application. An alternative amplifier arrangement is shown in figure 5.14. This shows a non-inverting amplifier. In this circuit negative feedback is once again used to set the gain. The input signal is applied to the non-inverting input and the inverting input is connected to ground by a resistor (R1). The inverting input cannot be connected directly to ground otherwise all the feedback current would flow to ground and not into the inverting input: there would be no feedback. The gain (G) of this circuit is given by the formula G = 1 + R2 R1 Thus if R2 is 50 kΩ and R1 is 2 kΩ, the gain is 1 + 50/2 = 26. If the 50 kΩ resistor was decreased to 10 kΩ the gain would be reduced to 1 + 10/2 = 6. Notice that although different inputs are used for the signal to be amplified with inverting and non-inverting amplifiers, feedback is always applied to the inverting input to produce negative feedback and reduce the gain to a value determined by the ratio of two resistors. If the feedback was applied to the non-inverting input we would have positive feedback and the amplifier would be unstable. Most of us have experienced the effect of positive feedback when a microphone is moved too close to the loudspeaker of a public address system or the volume control is advanced too far. The circuit becomes unstable and an unpleasant howl is generated which can quickly damage the loudspeaker and/or amplifier and/or listener's ears! Used properly and carefully positive feedback can be of advantage in electronic
RECTIFICATION AND AMPLIFICATION 116 circuits. We will consider an example later. As a general rule, however, positive feedback is not used in circuits whose sole purpose is to amplify. PRODUCTION OF ALTERNATING CURRENT In chapter 2 we saw that a resonant circuit, consisting of an inductor and a capacitor connected in parallel, has a natural (resonant) frequency which can be calculated using the formula If electrical energy (i.e. a pulse of current) is applied to the circuit, it resonates. That is, current flows backwards and forwards around the circuit and this alternating current has a particular frequency, the resonant frequency. As noted in chapter 2, 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 alternating current 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 amplifier and positive feedback we can provide this energy. The circuit alongside shows one way of generating sustained oscillations using an operational amplifier with positive feedback. RECTIFICATION AND AMPLIFICATION 117 L1 and C1 form the resonant circuit and L2 is an extra inductor in close proximity to L1. The combination of L1 and L2 is, of course, a transformer. The oscillating current in L1 will induce a current in L2. The current induced in L2 produces an AC potential difference between the two input terminals of the op-amp. The output of the op-amp, which is an AC signal which is in synchronization with the AC in the resonant circuit, is fed back to the resonant circuit through R and this compensates for the natural energy loss and so keeps the resonant circuit oscillating. A problem with this circuit is that it is unstable. If the amount of feedback is too small, the oscillations will die-out. If the amount of feedback is too large, the oscillations will increase out of control. In practice it is impossible to have precisely the right amount of feedback to generate a steady, sustained oscillation. The problem is overcome by using a voltage controlled amplifier whose gain is controlled by negative feedback. Figure 5.15 shows how this is achieved. The AC potential difference across the Figure 5.15 resonant circuit is rectified and A circuit for producing steady smoothed to produce a DC voltage which is directly proportional to the AC continuous AC signal. This DC voltage is used to control the gain of the op-amp. If the AC signal increases, the DC voltage applied to the op-amp increases and its gain is reduced. This reduces the amount of feedback and the AC signal is reduced. If the AC signal decreases, the DC voltage applied to the op-amp decreases and its gain is increased. This increases the amount of feedback and the AC signal is increased. In this
RECTIFICATION AND AMPLIFICATION 118 way the AC signal is prevented from either decreasing or increasing appreciably. Piezoelectric Crystal Oscillators As indicated in chapter 2, LC resonant positive feedback piezoelectric output circuits tend to drift slightly in frequency crystal due to factors such as ageing and R temperature change. When extreme rectifier frequency stability is needed, voltage with piezoelectric crystals are used. These controlled crystals have the property that when a amplifier smoothing potential difference is applied to their opposite sides, the crystal resonates mechanically. When included in the 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. In practice crystal resonators consist of negative feedback a quartz wafer between two electrodes. The physical dimensions of the crystal Figure 5.16 determine the resonant frequency and A circuit for producing stable if the crystal is maintained at a constant high frequency AC using a temperature a very high order of frequency stability can be obtained. piezoelectric crystal Figure 5.16 shows a suitable circuit for the production of stable, high frequency AC using a piezoelectric crystal. The LC resonant circuit of figure 5.15 is replaced by a piezoelectric crystal which is connected directly to the voltage controlled amplifier. A circuit using a piezoelectric crystal can also be made to produce rectangular pulses over an extremely wide range of pulse durations and repetition rates. RECTIFICATION AND AMPLIFICATION 119 Extremely short pulse durations are required for computing and other applications. For reaction testing in electrotherapy, the shortest duration in use would not normally be less than 10 microseconds. For muscle therapy the pulse widths in use might lie in the range 20 microseconds up to a few seconds. EXERCISES 1 (a) Draw the circuit symbol for a semiconductor diode. Include arrows to show the directions of high current flow (low resistance) and low current flow (high resistance). (b) What is meant by the terms 'forward bias' and 'reverse bias' of a diode? State typical values for the resistance of a semiconductor diode when forward biased and reverse biased. 2 The circuit below is used to convert AC from the mains to DC of lower voltage. (a) Draw a graph of the potential difference across the 1 kΩ resistor versus time. (b) Why does this graph represent DC and not AC? (c) How would the graph be changed if the diode was connected into the circuit with its terminals reversed? (d) Describe (with the aid of graphs) the effect of connecting different size capacitors in parallel with the resistor.
RECTIFICATION AND AMPLIFICATION 120 3 The circuit shown below is used to convert AC from the mains to DC of lower voltage. (a) Draw a graph of potential difference across the I kΩ resistor versus time. (b) In what way is this circuit more efficient than that shown in question 2? 4 Consider the circuit shown in question 3 above. (a) Draw graphs to show the effect of different size capacitors connected in parallel to the 1 kΩ resistor. (b) What is the advantage of full-wave rectification compared to half wave rectification as regards the size of capacitor needed to smooth the rectified waveform? (c) Draw graphs to show the effect of removing one of the diodes from the circuit. 5 Consider an operational amplifier integrated circuit. (a) Draw a circuit diagram for an inverting amplifier with a gain of 20. (b) describe two ways by which the gain of the amplifier in (a) above could be increased to 50. RECTIFICATION AND AMPLIFICATION 121 6 Figure 5.14 shows a circuit diagram for a non-inverting amplifier. (a) What would be the gain of this circuit if resistor R2 is 50 kΩ and R1 is 10 kΩ? (b) What value resistor would need to be used instead of the 50 kΩ resistor tor a gain of 8? 7 What is meant by the term 'positive feedback'? Why does positive feedback produce instability? 8 Consider the following amplifier circuits. (a) Which is the inverting amplifier? Which is the non-inverting amplifier? (b) What is the gain (amplification) of each circuit? (c) If the 4 kΩ resistor was replaced by a 1 kΩ resistor, what would be the new value of the gain of each circuit? (d) Explain, in terms of positive and negative feedback, the arrangement of resistors in each circuit. 9 The circuit shown in figure 5.15 can be used to produce continuous sinusoidal
RECTIFICATION AND AMPLIFICATION 122 alternating current. (a) Explain why positive feedback is needed in this circuit. How is positive feedback produced? (b) Why is negative feedback needed? Explain how negative feedback is achieved. 10 Under what circumstances would the circuit shown in figure 5.16 be preferred to that shown in figure 5.15?
ELECTRIC AND MAGNETIC FIELDS 123 6 Electric and Magnetic Fields The French scientist, Arsenne d'Arsonval described, in 1893, Pulses or bursts of electric current, delivered at relatively low frequency, have a major passing a current of 1 Amp AC effect on nerve and muscle because the current causes depolarization of the nerve- at a frequency of 500 kHz, fibre membrane and production of action potentials. When sinusoidal alternating through two human volunteers current (AC) is used, the physiological effects depend on the AC frequency. This is, at and an electric light bulb least in part, because each AC sinewave produces a positive and negative pulse of connected in series. The light current. At higher frequencies (several tens or hundreds of kHz), the pulses of current glowed brilliantly but the in each sinewave are of short duration and the nerve fibre does not have time to volunteers felt nothing. Only respond to the positive pulse before the negative pulse cancels out its effects. In when the current was increased other words, the pulse duration is too short to produce an appreciable change in the to 3 Amperes did the subjects nerve fibre membrane potential before the effects are reversed. Thus high frequency complain of a disagreeable alternating current flow has no direct excitatory effect on nerve fibres. Current flow sensation of heat. does, however, result in heating of tissue because electrical energy is converted to heat energy according to Joule's Law: P = V.I. Any flow of current will result in heat production. In this and the following chapter we consider how AC current flow induced in tissue by high-frequency alternating electric and magnetic fields produces heating of tissue. We also ask why certain tissues are heated more rapidly by high-frequency alternating electric and magnetic fields. Chapters 6 and 7 are both important for an understanding of the principles of diathermy. Diathermy (which literally translates as 'through-heating') refers to the heating of deeply located tissues; that is, the heating of tissues lying below both the skin and superficial fatty tissue. The practical difficulty involved in this form of therapy is that of minimizing the heating of superficial tissue while raising the temperature of the deeper structures by a therapeutically useful amount. The objective, then, is to produce a greater increase in temperature in the tissue layer to be treated. By varying the dosage, the desired amount of tissue heating can be chosen. We begin our coverage of diathermy using electric and magnetic fields with some basic physical principles associated with these fields. ELECTRIC AND MAGNETIC FIELDS 124 STATIC ELECTRIC FIELDS It is common knowledge that there are two kinds of electric charge: the negative charge associated with electrons and the positive charge associated with the nucleus of an atom. We also know that like charges repel each other and unlike charges attract. The quantitative relationship between the forces between charges and their magnitude and distance apart was first determined by Charles Augustin de Coulomb in 1785. The relationship is summarized in what we now call Coulomb's law which states that the force between two charges is directly proportional to the magnitude of each charge and inversely proportional to the square of their separation. Coulomb's law is expressed mathematically as: F α q1.q2 .... (6.1) r2 where F is the force experienced by two charges q1 and q2 and r is their distance apart. If we wish to calculate the force on a charge q due to a large number of charges q1, .... qn, we could achieve this by using equation 6.1 and calculating the force on q due to q1, the force on q due to q2 and so on and then summing the forces to obtain the resultant force. Since forces are vector quantities the summation must be by vector addition. In practice we do not often come across situations which approximate to two point charges nor even to simple distributions of charges. More often we encounter charges spread over surfaces of different shapes where the actual charge distribution
ELECTRIC AND MAGNETIC FIELDS 125 is unknown (and in practice difficult to measure). Under these circumstances it is By moving a small positive difficult to calculate the force on another charged object. It is usually more convenient test charge around in an electric and easier experimentally to measure the magnitude and direction of the force on a field and measuring the force at 'test charge' placed in the field of influence of the charged object. numerous points, the electric field can be mapped-out. The concept of a field of influence, the electric field, is an extremely useful one because it is possible to measure forces in the field without any prior knowledge of the arrangement of charges causing the field. We define the intensity at any point in an electric field, E, to be the force per unit positive charge at that point. This defines the magnitude of the field and its direction. If a small test charge is placed in a field the magnitude of the force it experiences together with the direction of the force determine the electric field intensity at that point. The field intensity is thus a vector quantity given by the relationship E = F .... (6.2) q' Where q' is the magnitude of the test charge placed in the field. Since the units of force are Newtons and the units of charge are Coulombs, the field intensity has units of Newtons per Coulomb (written N.C-1). The field surrounding a single positive charge q can be obtained from equations 6.1 and 6.2 as E = F = q .... (6.3) q' r2 Where E is the magnitude of the field and its direction is the same as that of the force (in a straight line between the charges in this case). ELECTRIC AND MAGNETIC FIELDS 126 Figure 6.1 shows the pattern of the electric field around point charges and pairs of charges. Figure 6.1 Fields of point charges (a) single positive charge (b) single negative charge (c) unlike pair of charges (d) like pair of charges
ELECTRIC AND MAGNETIC FIELDS 127 The lines drawn in figure 6.1 indicate the direction of the field and are called lines of Positive charge will move force because they indicate the direction along which the electric force acts at any along the lines of force in the point. same direction as the field, An important point to note is that electric fields are always produced by separation of negative charge along lines charges. Electrons are removed from their parent atoms to produce the charge in the opposite direction. separation so that for every positive charge that is produced, there is a corresponding negative charge somewhere. The electric field lines thus have their origin on charges and can only terminate on opposite charges. By convention we indicate the relative strengths of electric fields by the number of lines of force going through a unit area perpendicular to them. In figure 6.1 the field lines are closest together near the point charges so the field intensity is greatest there. In figures 6.1(a) and (b) it is apparent that at greater distances from the point charges the field line density is reduced meaning that the field intensity is correspondingly decreased. Another point worthy of note is that lines of force represent vector directions. It follows that no two lines of force can ever cross each other - the fields at the point would simply add by vector addition. This is just another way of saying that a point charge placed in a field will only move in one direction it cannot be pushed in either of two directions. THE FIELD BETWEEN CAPACITOR PLATES Consider the case of two large, flat metal surfaces each carrying opposite charges and parallel to each other. The metal plates are good conductors so charges are able to move freely in response to electric forces. On one plate we have an accumulation of positive charges and on the other an accumulation of negative charges. ELECTRIC AND MAGNETIC FIELDS 128 In this situation electric field lines will originate on the positive plate Figure 6.2 and terminate on the negative plate. The field lines so produced will Electric field between parallel, originate perpendicular to the conductor surfaces (this is illustrated in figure 6.2). It is generally true that at the surface of a conductor equal size capacitor plates. through which no charge is moving, the field lines are perpendicular to the conductor surface. If this were not so the lines of force would have a component tangential to the conductor surface. The effect of the tangential component would be to move charges, so changing the field. The charges would continue to move and readjust the field until there were no longer any field line components tangential to the surface causing them to move. In insulators, where the charges are not free to move, this does not apply and the field lines may be at any angle to the insulator surface. Figure 6.2 shows the pattern of electric field lines between two capacitor plates. The field lines originate perpendicular to one plate and terminate perpendicular to the opposite plate. If the surfaces of each metal plate were infinitely large, the field lines would all be straight and uniformly spaced - in other words the field intensity would be the same at all points, not growing weaker at large distances from the centre of the plates. In practice the plates cannot be infinitely large and the field lines curve away from each plate at the edges. When the plates are close together as in figure 6.2(a) the field is relatively uniform - only the outermost areas (near the edges of each plate) show any curvature, or weakening, of the field pattern. In 6.2(b) the plates are more widely spaced and more weakening of the field is evident. Only in the central region is the field uniform. Clearly, to obtain the most uniform field intensity it is desirable to have large capacitor plates separated by a relatively small distance.
ELECTRIC AND MAGNETIC FIELDS 129 Figure 6.3 shows the field patterns expected for different sizes and arrangements of electrodes. ELECTRIC AND MAGNETIC FIELDS 130 When interpreting electric field patterns it is important to remember that not only is the Saying that the electric field field direction shown by the lines of force but that the field intensity is by convention intensity is 100 newtons per indicated as the number of lines per unit cross section area. coulomb is the same as Imagine now a small positive charge placed in the field between two capacitor plates. saying the intensity is 100 The charge will experience a force in the direction along the field lines and hence will volts per metre. accelerate in this direction, thus gaining kinetic energy. The gain in kinetic energy is offset by a loss of (electrical) potential energy. In other words the charge will lose potential energy as it moves along field lines. The change in potential energy per unit positive charge is called the electric potential difference between the points. This quantity is the same potential difference we have already met in chapter 1 and the unit is, of course, the volt. A region of positive charge is a region of high electric potential and a negative charge region is at a low potential so that going from a positive to a negative region involves a potential drop, i.e. a negative potential difference. The electric field also goes from positive to negative so that moving along a field line in the direction of the field involves travelling across a potential drop. The idea of charges losing potential energy as they move in an electric field gives us another way of talking about electric field intensity. Clearly an intense electric field will be associated with a large potential difference between two points a certain distance apart. A weak field has a lower potential difference for the same distance. The field intensity can thus be specified in units of potential difference per unit distance: in other words in volts per metre (V.m-1). The units of volts per metre are identical to the units of field intensity mentioned earlier, namely newtons per coulomb. STATIC MAGNETIC FIELDS The basic groundwork for our understanding of magnetic phenomena was laid around 1820 by André Marie Ampère when he first studied the forces of attraction and repulsion between wires which are carrying current. Before these pioneering
ELECTRIC AND MAGNETIC FIELDS 131 experiments magnetism was regarded as an isolated and puzzling phenomena associated with a few naturally occurring (magnetic) materials. What Ampère did was to establish the first links between electricity and magnetism: essential groundwork which culminated in the development of modern electromagnetic theory. Ampère observed that if two loops of wire were mounted so that each had one side in close proximity to a side of the other loop, a force was produced when a current was made to flow in each loop. The arrangement is shown in figure 6.4. Ampère found that when current was passed through each loop in the same direction the loops were attracted to each other - this could be seen by movement cf the pivoted loop. If the current in one loop was reversed the force was also reversed - a repulsive force was produced. He also noted that if current was made to flow in only one loop, no force was experienced by either loop. Experiments can be performed to ascertain what effect the size of the current in each loop has on the force and what effect have the loop separation, r, and the length of wire, L. The results are summarized in what we now know as Ampere's law for the force between two parallel conductors (equation 6.4). ELECTRIC AND MAGNETIC FIELDS 132 F α I1.I2.L .... (6.4) r In comparing this with Coulomb's law of electrostatics, (equation 6.1) we can see some similarities and some differences. The table below summarizes the corresponding features of each kind of force. Electrostatic Force Magnetic Force * between two charges. * between two currents * repulsion of Iike charges. * attraction of 'like' currents. * depends on size of charges. * depends on size of currents. * is proportional to 1/r2. * is proportional to 1/r. * force in a straight line between * force at right angles to direction of charges. current. The two most striking properties of the magnetic force which contrast with the electrostatic case are the attraction of 'like' currents, i.e. currents in the same direction (figure 6.5) and the line of action of the magnetic force - perpendicular to the direction of current flow. Figure 6.5 Forces between two current- carrying wires.
ELECTRIC AND MAGNETIC FIELDS 133 Inspection of figure 6.5 leads to the prediction that two current carrying Figure 6.6 conductors placed at right angles will experience no force. This is borne out Magnetic field lines around a experimentally. Remember that the electric field near charged bodies could be mapped by wire carrying current. placing a test charge in the field and observing the direction of the force on the charge - let us now consider the relationship between magnetic fields and forces. The use of a compass or iron filings to visualize the magnetic field around a bar magnet is an experiment which most of us have carried-out. The compass or iron filings line up in the direction of the magnetic field and the field lines are readily apparent. The same technique can be used to map the field lines around a wire carrying an electric current. Figure 6.6 shows the sort of pattern which is obtained. It was Ampère himself who first thought of obtaining a more intense magnetic field by passing current through a wire wound in a closely spaced spiral. Figure 6.7 shows the magnetic field expected for a single loop of wire and a number of loops wound as a solenoid. Figure 6.7 Magnetic field lines around a loop and a solenoid. ELECTRIC AND MAGNETIC FIELDS 134 Now let us return to consider what is happening to a small element of current in a wire placed near to the straight wire in figure 6.6. Figure 6.8 shows the main features of such an arrangement. Here we have the most striking feature of magnetism: that the force on a moving charge is perpendicular to both the current and magnetic field directions. This observation stands in stark contrast to our experience with gravitational and electrostatic fields, where the field direction is also the direction of the force experienced by the mass or charge. The scope of this book precludes our delving into Figure 6.8 theories of the origin of magnetic force: its explanation Forces on a current element in was a triumph for relativity theory in modern physics, when the magnetic force was shown to be simply a result of the movement of a magnetic field (B). charges with their associated electric field. The theory indicates that the force between charges in motion is slightly larger than that between stationary charges. This slight increase in the electric force is, in fact, the 'magnetic' influence of moving charges on one another. Although we have, so far, only discussed electric current in the form of charges flowing through wires, what we have described also applies to isolated charges moving in a vacuum. A good practical example of this is in the television set. Here a stream of electrons emitted from a glowing filament (the cathode) is accelerated by an electric field and impinges on the screen. The phosphor coating on the screen is utilized to convert the kinetic energy of the electrons to light energy and a visible spot is produced on the screen. The electron beam is deflected rapidly both horizontally and vertically by magnetic fields produced by coils or solenoids attached to the neck of the picture tube. By applying rapidly alternating pulses of current to the coil, impulsive forces are applied to the stream of electrons and the spot is moved accordingly.
ELECTRIC AND MAGNETIC FIELDS 135 In summary it must be remembered that the movement of charges is central to our The size of the magnetic force understanding of magnetic phenomena. The movement of charges results in a depends on sin(θ ) where θ is magnetic field and a static magnetic field can only have an influence on a charge the angle between the field when that charge is in motion. Motion at right angles to the field results in a maximum line and the direction of force. Movement of charges along a magnetic field line will result in no force being charge movement. exerted. Since we know that movement of charge in a static magnetic field is necessary for a force to be exerted on the charge, we might logically ask such questions as whether the charges in a conductor will experience a force if the conductor is moved in a magnetic field, or whether a force is exerted on a stationary charge by a moving or changing magnetic field. We will discuss the answers to these questions later in this chapter. ELECTRIC FIELDS AND DIELECTRICS In talking about the electric field between charged bodies and the magnetic field associated with moving charges we have not considered what effect, if any, the surrounding medium has on the fields in each case. Let us consider the electrostatic case first. When a material is given an electric charge, the charge may remain localized in the region of generation for some considerable time or it may spread over the surface almost instantaneously. In the first case the material is called an insulator or dielectric, and in the latter case a conductor. An ideal insulator would retain a charge indefinitely and materials such as glass, plastics and waxes come very close to this ideal. Pure metals such as gold and copper are close to being ideal conductors. Of course the line of demarcation between conductors and insulators is not sharp and many substances, particularly ELECTRIC AND MAGNETIC FIELDS 136 those of biological origin, are neither good conductors nor good insulators. Dielectric Constant It was found by Henry Cavendish, and later independently by Michael Faraday, that the capacitance of a capacitor - its ability to store charge - can be increased by placing a dielectric material between the plates. If Co is the capacitance when measured in a vacuum and C the capacitance when the region between the plates is filled with a dielectric, the ratio of C to Co, is found to be independent of the shape and size of the capacitor. This ratio is dependent only on the dielectric medium itself and is called the dielectric constant (ε) of the material. We thus have ε = C .... (6.5) Co Capacitance is defined as the amount of charge the capacitor will acquire (and store) for each volt of applied potential difference. Mathematically this is written: C = q .... (6.6) V where the symbols have their usual meaning. This equation tells us that if the capacitance is changed by inserting a dielectric between two capacitor plates either the charge will be increased or the potential will be decreased. If the capacitor plates are connected to a battery or other power source capable of maintaining a constant potential difference then the charge on the plates will be increased. The increase in capacitance is very small in the case of gases at normal pressure, but for materials such as oil the capacitance is doubled. Values of ε determined in this way range from very close to 1 for gases up to 81 for pure water. Table 6.1 gives values of ε for a representative range of materials.
ELECTRIC AND MAGNETIC FIELDS 137 Notice that the dielectric constant of petroleum oil is similar to that of polythene. Although these materials are quite different in some properties, most notably physical appearance, they have the same basic molecular composition. Each is made up of a long hydrocarbon chain. The chains in polythene are longer and the subunits have many chain crosslinks but the elemental compositions of the two materials are very similar. Compare these with ethyl alcohol and then water. The dielectric constant is closely correlated with an intrinsic property of the molecules making up a substance - the polarity. Of the substances in table 6.1 the most polar is water, then ethyl alcohol. The principal constituents of air, namely oxygen and nitrogen, are non-polar. The remaining substances in the table are of intermediate polarity. We consider the effect of dielectrics on an electric field next but it is Table 6.1 worthwhile to pause and note at this point the distinction between Dielectric constants of materials insulating properties and dielectric properties. The terms insulator and dielectric are used almost synonymously to describe a particular group of materials - those with poor conduction properties. The terms 'good insulator' and 'good dielectric', however, mean two different things. * A good insulator is a substance which offers a very high resistance to the flow of current. This refers to how easily charges can move through a material and says nothing about the dielectric constant. * A good dielectric is a substance with a high dielectric constant. This depends on the polarity of the molecules in the substance, not on its conduction properties. Though good dielectrics must be reasonably good insulators the best dielectrics are not necessarily the best insulators. For example water in table 6.1 is the substance ELECTRIC AND MAGNETIC FIELDS 138 with the highest dielectric constant. Although pure water is a reasonably good Figure 6.9 insulator the other substances in this table are all better insulators. (a) Unpolarized dielectric and Dielectrics In An Electric Field (b) polarized dielectric Consider what happens when we put a material made up of polar molecules between two charged capacitor plates; that is, in an electric field. For simplicity we assume that the polar molecules are dipoles having a single 'slightly negative' region and a corresponding 'slightly positive' region. In the absence of a field the molecules will be randomly aligned as shown in figure 6.9(a). In an electric field the dipolar molecules will-align themselves as shown in figure 6.9(b) with the positive and negative ends pointing in directions along the field lines. The material is then said to be polarized. The alignment of the molecules has an effect on the applied electric field. This is because each dipole has its own field comprised of field lines which spread from the positive end of the molecule to its negative end. The field of a single dipole is shown in figure 6.10. Although the field around a single dipolar molecule is extremely weak, when a dipolar material is polarized and the molecules are aligned as in figure 6.9(b) the net effect of all of the dipoles can have a significant effect on the externally applied field.
ELECTRIC AND MAGNETIC FIELDS 139 Consider first the effect of a single dipole on the external field. The central field line in Figure 6.10 figure 6.10 points in the direction of the externally applied field (from left to right). The Electric field of a dipole remaining field lines, however, curve around above and below the dipole and the arrows point in a direction opposing the external field (from right to left). The net result Figure 6.11 is that these field lines cancel part of the externally applied field and so weaken it. Polarization of a molecule When we take into account the fields surrounding all of the dipoles in a polarized material the net result is a weakening of the externally applied field within a dielectric. in an electric field The field inside a dielectric will be less than the externally applied field: this is because each of the dipoles aligns to produce a field opposing the externally applied field. Why then should there be any effect with 'non-polar' molecules such as oil or bakelite? The origins of the effect with these materials can best be understood by considering a single non-polar molecule such as the one shown in figure 6.11. In the absence of a field the electron 'cloud' around the molecule is symmetrical: the molecule is non-polar. If now an electric field is applied the electron cloud can distort giving rise to an induced dipole. The molecule becomes polarized and will remain so as long as the external electric field is maintained. Non-polar molecules will polarize and so have the same effect on an externally applied field as polar molecules. The dipole field produced will oppose the external field and so weaken it. Generally the effect of non-polar molecules on an electric field is not as great as those of polar molecules. This is because the induced dipoles are not as strong as those of naturally polar substances. We thus have the general result that the field within any dielectric is less than the field that would exist without the dielectric being present. The next question we must ask is how the pattern of field lines is affected. ELECTRIC AND MAGNETIC FIELDS 140 For simplicity we take the single dipole shown in figure 6.10 and Figure 6.12 consider what happens when it is placed in a uniform field. The The effect of a dipole on a uniform electric field original field must be added (vectorially) to that of the dipole. The result is the field pattern shown in figure 6.12. Figure 6.13 The most significant thing to note from figure 6.12 is that the field Effect of a dielectric cube on a uniform electric field lines converge towards the dipole. The field is made more intense immediately in front of the faces perpendicular to the field direction. Above and below the dipole the field lines are more spread out; that is, the field here is weaker. If now we wish to consider a large object of high dielectric constant in an electric field we can apply the same ideas. Take the case of a simple cube of dielectric. We can add the contribution to the field from each dipole using the arrangement shown in figure 6.9(b). The result will be similar to figure 6.12 but now we have many dipoles joined head to tail and stacked above and below each other. The net result is shown in figure 6.13. Again we observe the field lines converging towards the dielectric, so increasing the field intensity near faces perpendicular to the field. The field intensity near the remaining faces of the cube is correspondingly decreased. Figure 6.13 also shows why, even though field lines converge on the cube, the field within the dielectric is weak. It is because some field lines terminate on charges on the surface of the dielectric. The origin of the surface charge can be seen by inspection of figure 6.9(b). Within the polarized dielectric, dipoles are aligned in a head to tail pattern so that for each positive 'head' there is an
ELECTRIC AND MAGNETIC FIELDS 141 adjacent negative 'tail'. The opposite charges so cancel each other. On the faces perpendicular to the field, however, there is a charge imbalance. One surface has an excess positive charge and the opposite surface has an excess negative charge. It is on this surface charge of polarization that field lines will terminate. The amount of surface charge produced in an electric field will depend on the dielectric constant of the material. For a material with a high dielectric constant the surface charge of polarization will be high and a large proportion of field lines will terminate on the dielectric surface. When the dielectric constant is low the proportion of field lines which terminate will be small. This means that the field within a dielectric is reduced in proportion to the dielectric constant. The higher the dielectric constant, the lower is the field within the material. In the limiting case where the dielectric constant is extremely high the field within the material is close to zero. Figure 6.14 shows what happens when either a spherical or Figure 6.14 cylindrical dielectric is placed in an electric field. Note the regions Effect of a dielectric sphere on a in which the field is intensified (more field lines per unit area) and where the field is reduced. uniform electric field REFRACTION OF FIELD LINES Inspection of figures 6.13 and 6.14 reveals certain discontinuities. Two things are apparent. First we notice that some field lines terminate at the dielectric interface. Second we notice that the lines which do not terminate impinge on the interface at one angle and leave it at a different angle. The change in direction at an interface is called refraction. Field lines are refracted at a dielectric interface. The phenomenon is not restricted to dielectrics but also applies to conductors. In this section we consider how the dielectric constant and conductivity of materials determines the extent of refraction at an interface. ELECTRIC AND MAGNETIC FIELDS 142 Refraction and Dielectrics First let us consider the refraction of field lines as they pass from one dielectric medium to another. We ask the question 'what are the constraints on field lines crossing a dielectric boundary?' There are two constrains, or boundary conditions, on the electric field at an interface. The first is that the tangential component of the field (the component parallel to the boundary) should be continuous. That is, the tangential component of the field must be the same on each side of the boundary. This is illustrated in figure 6.15. The tangential component of E1 is E1.sinθ1 and the tangential component of E2 is E2.sinθ2. These are equal so we have E1.sinθ1 = E2.sinθ2 .... (6.7) The second boundary condition is that the normal component of the field in each dielectric (the component at right-angles to the boundary) is decreased in proportion to the dielectric constant. We have seen previously that the field within a dielectric is decreased. The second boundary condition states this precisely. Since the normal component of the field is decreased in proportion to the dielectric constant the product of dielectric constant and the normal component of the field is the same in each medium. Mathematically this is written Figure 6.15 Refraction of an electric field line at a ε1.E1.cosθ1 = ε2.E2.cosθ2 .... (6.8) boundary between two dielectrics
ELECTRIC AND MAGNETIC FIELDS 143 Dividing equation 6.8 by equation 6.7 and cancelling the E1s and E2s, we obtain ε1.csoinsθθ11 = ε2.scionsθθ22 .... (6.9) or ε1.cotθ1 = ε2.cotθ2 An Example. Suppose that in figure 6.15, medium 1 is air with a dielectric constant of A similar calculation for an 1.0 and medium 2 is water with a dielectric constant of 81 (table 6.1). The field line incident angle θ1 of 10o gives shown for E1 strikes the boundary at an angle θ1 of 20o. We wish to calculate the an angle of refraction θ2 of angle θ2 at which the field line leaves the boundary. 8 6 o. Rearranging equation 6.9 we have cotθ2 = εε21.cotθ1 = 18.10.cot20o = 18.10.2.75 = 0.0399 which gives θ2 = 88o For an angle of incidence θ1 of 20o, the angle of refraction, θ2, is predicted to be 88o. In this example the angle of refraction is always considerably greater than the angle of incidence. This is because medium 2 has a much higher dielectric constant than medium 1. The implication is that field lines are refracted greatly on entering a medium of high dielectric constant. For a moderate angle of refraction the angle of incidence must be very small: in other words the field lines must enter the medium of high dielectric constant almost at right angles. ELECTRIC AND MAGNETIC FIELDS 144 Dielectrics Between Capacitor Plates Now consider what happens when we place a material of high dielectric constant between two capacitor plates. The result is shown in figure 6.16. To explain the effects shown here we must remember the main points of the preceding discussion. Two important effects are evident in figure 6. 16. An increase in field strength between each plate and the dielectric surface. This is predicted by equation 6.6. The capacitance is increased in the presence of a dielectric. This means that the charge on the plates must increase and consequently the field strength between the plates and the dielectric must increase if the potential difference between the plates is kept constant. A decrease in field strength within the dielectric. Figure 6.16 Two factors contribute to this effect. The first is Change in the electric field pattern the surface charge of polarization which allows field lines to terminate on the dielectric surface, in the presence of a dielectric. so reducing the field intensity within. The second is the refraction of field lines at the dielectric interface (equation 6.9). Notice that the field lines in figure 6.16 enter the dielectric with a small angle of incidence and are refracted greatly. Refraction spreads the field lines and so reduces the field in the dielectric.
ELECTRIC AND MAGNETIC FIELDS 145 Refraction and Conductors The field within a non- ideal conductor is not zero, nor do So far we have talked about dielectrics in an electric field. We dealt with dielectric all the field lines terminate on properties alone and made no mention of the effect of conductivity on the electric field the conductor surface. pattern. In the case of good insulators such as plastics, glass and oil it is reasonable to treat them as ideal, non-conducting dielectrics. In the case of water and materials containing water, this approximation cannot be made. Hence for biological materials which generally have a high water (and ion) content we must also consider the effect of conductivity on the electric field pattern. Here then we consider the effect of conductors on an electric field. We will not be so concerned with the ideal or perfect conductors, which must have field lines perpendicular to the conductor surface, but with non-ideal conductors which are much more commonplace in biological and in everyday situations. For ideal conductors two conditions apply: * the field lines must be perpendicular to the conductor surface. * all field lines must terminate on the surface of the conductor. In this sense an ideal conductor can be considered to be 'infinitely polarizable'. This means that the field within a perfect conductor is zero. For non-ideal conductors these constraints do not apply. What then are the properties of a non-ideal conductor and what are the laws governing its behaviour? We have already met one of the properties of a conducting medium in a previous chapter. This is Ohm's law which states that the current in a conductor is proportional to the potential difference between the conductor ends. The constant of proportionality is 1/R where R is the resistance. Ohm's law was covered in chapter 1. ELECTRIC AND MAGNETIC FIELDS 146 Mathematically, Ohm's Law is written: Figure 6.17 Electric field and current in a I = V .... (1.3) R conductor When we are interested in the bulk flow of current within a material it is more convenient to use a different expression for Ohm's law. The alternative expression is i = E or i = σ.E .... (6.10) ρ Where E is the electric field intensity in volts per metre and i is the current density in amps per square metre passing through a surface perpendicular to E (figure 6.17). For an area A, with a current I flowing through it, the current density is I/A amperes per square metre (A.m-2). The resistivity, ρ, is defined as the resistance of a one metre length of conductor with a cross sectional area of one square metre. The unit of resistivity is the ohm.metre. The conductivity, σ, is simply the reciprocal of resistivity and thus has units of (ohm.m-1) which are called siemens per metre (S.m-1). We now ask what happens to the electric field lines (and hence the direction of current flow) on passing from one conductive medium to another. Figure 6.18 shows the refraction which occurs when a field line crosses to a medium of higher conductivity. The angle of refraction is greater than the angle of incidence in this case. The conditions applying to the electric field lines are analogous to those which apply in dielectric media. When a field line crosses into a medium of higher conductivity, the angle of refraction is greater than the angle of incidence. Conversely, when a field line crosses into a medium of lower conductivity, the angle of refraction is less than the angle of incidence.
ELECTRIC AND MAGNETIC FIELDS 147 Compare figure 6.18 with figure 6.15. The first condition on the electric field is that the tangential components of the field be equal on both sides of the boundary i.e. E1.sinθ1 = E2.sinθ2 .... (6.11) The second condition is that the flow of current normal to one side of the boundary should be equal to that normal to the other side i.e. i1.cosθ1 = i2cosθ2 This makes sense since the net current flow into the boundary from side 1 must equal that entering side 2 from the boundary. Using equation 6.10 to eliminate i1 and i2 from this last equation we Figure 6.18 have: Refraction of electric field line at a boundary between two conductors. σ1.E1.cosθ1 = σ2.E2.cosθ2 .... (6.12) Combining 6.11 and 6.12 we obtain: σ1.cotθ1 = σ2.cotθ2 .... (6.13) Compare these equations with 6.7, 6.8 and 6.9. The close analogy between current flow in conductors and polarization in a dielectric is quite evident from these equations. The implications of equation 6.13 for conductors are similar to those we found using equation 6.9 which applies to dielectrics. Electric field lines are refracted significantly on passing from a poor conductor to a better conductor. ELECTRIC AND MAGNETIC FIELDS 148 Equation 6.13 predicts that when the conductivity of medium 2 is extremely high (when The reason that high σ values medium 2 comes close to being an ideal conductor) the field lines in medium 1 will accompany high ε values is that enter the boundary between the two conductors almost normally (almost at right angles). In other words, if material 2 is an ideal conductor the field lines in medium 1 tissues of high water content will impinge on the boundary at an angle of 90o. have a high ion content. A high The similarity of equations 6.9 and 6.13 lead to the general conclusion that field lines are refracted greatly when entering a medium of high dielectric constant or water content results in a high ε, conductivity. To obtain an equation which takes account of both factors simultaneously the dielectric constant and conductivity of the media must be a high ion content results in a combined to calculate their electrical impedances. The amount of refraction then depends on the impedance of each medium and the refraction equation has a similar high σ. form to equations 6.9 and 6.13. It turns out that for biological materials there is a good correlation between the insulating properties (hence the conductivity) and the dielectric properties (dielectric constant). Tissues with high dielectric constant are poor insulators. In general, the higher the value of ε, the higher the value of σ. Hence if both of ε and σ increase by a factor of 10, say, on going from one medium to another equations 6.9 and 6.14 predict the same relationship between angle of incidence and angle of refraction. In this case both equations correctly predict the angle of refraction and the complexity of combining ε and σ in a single equation is avoided. Electric Fields in Tissue Table 6.2 gives values of dielectric constant and conductivity for some body tissues measured at 37oC. Also included for comparison are data for water, oil and metals. The fields used in therapy are not static but alternate at very high frequencies (normally 27.12 MHz). At high frequencies the dielectric constant is lower and the conductivity higher than in the electrostatic case.
ELECTRIC AND MAGNETIC FIELDS 149 The results in table 6.2 were obtained with an Table 6.2 alternating electric field of frequency 50 MHz - Dielectric constants and conductivities the values of conductivity would be about 20% lower at 27 MHz (the frequency normally used at 37oC and 50 MHz. for therapy) and the dielectric constant some 20% higher. Both quantities depend quite strongly on the frequency of alternation of the field. From table 6.2 it is apparent that at this frequency tissues with high water content (muscle, spleen and kidney) have very similar dielectric and conduction properties to saline solution. This is not surprising as the tissues contain about 80% by weight of water with the remaining 20% being tissue protein. The lower water content of fatty tissue and bone marrow is also reflected in their dielectric constant and conductivity values. Now let us apply what we have found so far to a more practical situation and ask what the pattern of field lines would look like for a patient's arm or leg placed between circular capacitor plates. Figure 6.19 shows a highly simplified representation of the arm or leg in (a) longitudinal and (b) perpendicular cross-section. It can be seen that the field intensity is highest in the air space and decreases markedly on entering the fatty tissue. A further reduction occurs when the field lines enter the muscle. ELECTRIC AND MAGNETIC FIELDS 150 In figure 6.19(a) the reduction in field intensity within the arm or leg is due to two factors: (a) refraction of field lines at the air/fatty tissue and fatty tissue/muscle boundaries which spreads the lines apart, so reducing the field intensity and (b) termination of some of the field lines on surface charge of polarization of the fat and muscle.
ELECTRIC AND MAGNETIC FIELDS 151 Note that in figure 6.19(a), in bone there are more field lines than in muscle. This is Iron-cored electromagnets because bone has a lower dielectric constant and conductivity than muscle and so can generate fields up to does not polarize to the same extent. Its electrical properties are similar to fatty tissue. several thousand times as Figure 6.19(b) also shows the effects of refraction and termination of field lines. strong as would be produced We will return to consider the implications of these field patterns. For the moment it is without the iron core present. sufficient to note the diminished field intensity in materials of high dielectric constant due to the polarization of the material and consequent termination of some field lines. Note also the refraction of field lines on entering a material of high dielectric constant or conductivity which can result in a focussing of the field (figure 6.19(b)) or defocussing (figure 6.19(a)). MAGNETIC FIELDS IN MATERIALS We saw previously that the dielectric constant of a material could be measured by the increase in capacitance which occurs when the material is placed between the plates of a parallel plate capacitor. An analogous situation is found with magnetism. When different materials are placed along the axis of a solenoid or inductor (figure 6.7), the inductance changes. There is also a proportionate change in the strength of the magnetic field around the inductor. The effect can be seen in the dramatic increase in magnetic field strength which occurs when a bar of iron is placed in a solenoid to make an electromagnet. Iron is, however, a member of a small group of elements which show such an effect on the magnetic field. Nickel and cobalt are two others. The materials in this group are said to be ferromagnetic. An extremely small effect on the magnetic field is found with other substances. ELECTRIC AND MAGNETIC FIELDS 152 By analogy with the electrostatic case we define the permeability of a material by its ability to change the inductance of a solenoid. The permeability, µ, is the ratio of the inductance L, to the inductance without the material being present, Lo, i.e. µ = L .... (6.14) Substances with µ less than Lo one are termed diamagnetic while substances with µ The permeability of most materials is found to be extremely close to unity. Unlike the greater than one are called paramagnetic. electrostatic case (where ε is always greater than unity) µ may be less than or greater than unity. Table 6.3 lists values of µ for a number of different materials. Certainly for biological materials we can consider µ to be sufficiently close to one to make no difference. In other words the magnetic field lines will be virtually unchanged on passing through biological materials: tissues are 'transparent' as far as the magnetic field is concerned. Ferromagnetic materials will influence a magnetic field in the same way Table 6.3 that dielectrics influence an electric field. While an electric field will Permeability of various materials. polarize a dielectric a magnetic field will magnetise a ferromagnetic material. The analogy could be carried further with discussion of the field in and around a ferromagnetic material. Figures 6.13 and 6.14 would be appropriate if we replaced the electric field E with the magnetic field B and substituted µ for ε. However since the effects are negligible for biological materials we will not pursue this topic further. Magnetic Fields and Induced EMF We now return to the questions left unanswered earlier and further examine the effect of magnetic fields on charges. We have already considered electric charge moving in a magnetic field and found that the
ELECTRIC AND MAGNETIC FIELDS 153 charge experiences a force in a direction perpendicular to both the magnetic field and the current direction (figure 6.8). This applies both to current flowing along a wire, as in figure 6.8 and to current flow in a vacuum or near vacuum, as in a television picture tube. Consider the following experiments which can be performed with two loops of wire. Figure 6.20 shows the arrangement of the apparatus which is needed. Loop 2 is connected, through a switch, to a battery. Loop 1 is connected to a sensitive current meter or galvanometer. If the switch were in the closed position, current is flowing through loop 2 and a magnetic field exists. Since the electrons in loop 1 are not moving, we know from Ampere's law that there is no force on them and hence no force on the loop. What happens if now we move loop 1 away from loop 2? In moving the loop upwards we find that the galvanometer deflects indicating a flow of current in loop 1. As soon as we stop moving loop 1, the current flow ceases. If we use the three dimensional axes of Figure 6.20 figure 6.8 and regard the direction of An experiment with moving wire loops. movement of the loop as the current direction we predict a force on the electrons in loop 1 in a direction along the wire. This then is the explanation for the induced current in loop 1. ELECTRIC AND MAGNETIC FIELDS 154 If now we fix loop 1 in figure 6.20 and instead move loop 2 downwards what is the v in the diagram above is the effect? In this instance instead of moving electrons through a magnetic field we have direction of movement - left the electrons alone and moved the field. The net result is the same as we found in either of the field, the wire or the first example: the galvanometer deflects indicating a flow of current in loop 1. electrons (current). Clearly it is only the relative motion of the conductor and field which is important in determining whether current is induced. When moving the conductor in a fixed field or moving the field with the conductor fixed, the essential process occurring is that charges are crossing magnetic field lines. Whenever this happens the charges experience a force. What happens then if we keep both loops fixed and suddenly switch off the current in loop 2? As far as loop 1 is concerned one of two things could have happened. Either the magnetic field disappeared because the current causing it was stopped or the loop responsible for the field was suddenly accelerated away from the vicinity. The net effect is the same - current is induced in loop 1 because of the changing magnetic field. The observation can be explained by picturing the magnetic field collapsing on loop 2. With current flowing in loop 2 a magnetic field, represented by concentric circles around the wire, is present (figure 6.7). When the current is switched off the circular field lines can be visualized as shrinking; converging on the wire and disappearing into it. Thus the field direction is in concentric circles but the direction of movement of the field is radially inwards towards the wire of loop 2. This is illustrated in figure 6.21. Figure 6.21 Magnetic force acting on charges when current in in loop 2 is switched off.
ELECTRIC AND MAGNETIC FIELDS 155 Closing the switch in figure 6.20 causes current to flow in loop 2 and a field The experiments described builds up around the wire. The increasing field is represented by a series demonstrate the principle of of expanding concentric rings emanating from the wire. The direction of electromagnetic induction. A current increase of the field is radially outwards (opposite to v in figure 6.21). is induced in loop 1 either by moving Experimentally we observe a flow of current in loop 1, in the opposite the loops or by switching the current direction to that when the switch was opened (i.e. F is reversed when v is and so causing the field to change. In reversed in figure 6.21) each instance the charges in loop 1 Once the current flow is steady, the magnetic field is constant and current is are crossing magnetic field lines. no longer induced in loop 1. This results in a force on the charges Now consider what happens if, instead of switching the current in loop 2 of and hence charge movement. The figure 6.20 on or off, we pass an alternating current through it. By the direction of the induced current is at principles outlined above we would expect to find an alternating current right angles to both the field direction produced in loop 1. This process, as you may have realized, forms the and the direction of movement. basis of transformer action: a process described in chapter 1. When an alternating current flows in the primary winding of a transformer, an alternating magnetic field is produced around the primary and an alternating current is thus induced in the secondary. The secondary circuit need not be closed. If no ammeter or other components are connected to the secondary, current will still be able to flow to the ends of the secondary winding. This will result in a difference in charge between each end of the winding and thus a potential difference between the endings. A major point that should be noted here is that although we have talked about the current induced in a conducting wire loop, we need not have restricted the discussion to conductors. Although insulators have their electrons tightly bound to the molecule and current will not flow, the charges will still experience a force and this force will polarize the molecules. ELECTRIC AND MAGNETIC FIELDS 156 Since electric field strength is defined as the force per unit charge we could include both insulators and conductors in our discussion by referring to the induced electric field or electromotive force (EMF) arising as a result of changing magnetic field. The principle of electromagnetic induction applies to any material placed in a changing magnetic field. An electric field is always produced as a result of the change in a magnetic field. If a conductor is in the changing magnetic field, a current will be induced whereas if an insulator is in the changing magnetic field only polarization will result. When an alternating current is induced in a slab of conducting material rather than a wire the currents are given the special name 'eddy currents'. The term arises because the most common geometry, a conducting cylinder placed in a solenoid as in figure 6.22, gives rise to circular current paths at right angles to the magnetic field. Provided that the magnetic field of the solenoid is changing i.e. the field lines are moving, force will be produced on charges in the conductor, resulting in current flow. An alternating current in the solenoid will result in an induced alternating current flow in the conductor. To understand why the induced current follows circular pathways we need Figure 6.22 to think about the direction of the magnetic field lines and their direction of Current induced in a conductive movement. A force will be produced with a direction at right-angles to each material placed in a solenoid. of these. Figure 6.7 shows the magnetic field pattern around a solenoid. Magnetic field lines inside the solenoid run parallel along the central axis. If alternating current flows though the solenoid, the magnetic field will build-up then collapse, build-up in the reverse direction then collapse in repetitive cycles. As the field builds-up, the field line loops in figure 6.7 will grow larger as new loops form. This is illustrated in figure 6.23.
ELECTRIC AND MAGNETIC FIELDS 157 Note the movement of the field lines. One field line is highlighted in red to Figure 6.23 show how the line moves towards the central axis as the current flow (and Movement of field lines as the current field intensity) increases. For field lines inside the solenoid, the field line movement is always radially inwards as the field increases and radially through a solenoid increases. outward as the field decreases. Figure 6.24 shows the conductive cylindrical object in figure 6.22, viewed end- on. The magnetic field lie (B), direction of movement (v) and resulting force (F) vectors are shown at different points. The B arrows point out of the page, directly towards you and are shown as blue circles. Because the field, B, is always pointing out of the Figure 6.24 page (along the cylinder axis) and v is always radially Direction of force and induced inwards, the resulting force (EMF) always acts current as a result of an increasing around the circumference of a circle. This is why the induced current follows circular pathways. magnetic field intensity. ELECTRIC AND MAGNETIC FIELDS 158 The force, F, acts clockwise when v points inwards i.e. when the current and, When the current through the consequently the magnetic field is increasing, When the current decreases, the coil is increasing, the induced magnetic field collapses and loops shrink towards the coil (the reverse of that shown current flows in an anti- in figure 6.23) so the direction of v in figure 6.24 is reversed. The consequence is that clockwise direction. When the the direction of F is reversed and the induced current flow reverses direction. Thus an current flow is decreasing, the alternating current is induced in the conductor as a result of the alternating magnetic induced current flows in a field. clockwise direction. The important conclusion to draw from figures 6.22 to 6.24 is that an alternating current in the solenoid gives rise to an alternating magnetic field. This, in turn, gives rise to an alternating EMF in the material within the solenoid. If the material is a conductor a current will be induced which follows a circular path parallel to the current in the solenoid. If the material is an insulator the molecules will polarize in alternating directions along arcs parallel to the solenoid loops. In either case an induced electric field is produced with the field direction parallel to the wires in the solenoid. The actual amount of induced current flow will depend on the dielectric constant and conductivity of the material. Magnetic Fields in Tissue From the previous discussion it should now be apparent that three factors determine the effect of an alternating magnetic field on a material: the permeability, µ, which is a measure of the 'magnetizability' of a material. This determines the magnetic field strength around and within a material placed in the field. The permeability is very close to unity for most biological materials: only ferromagnetic substances have a significant effect on the field strength. the conductivity, σ, which determines the amount of current flow in response to the applied (alternating) magnetic field. The higher the conductivity the greater will be the
ELECTRIC AND MAGNETIC FIELDS 159 induced current. the dielectric constant, ε, which measures the 'polarizability' of a material. The higher the dielectric constant the greater will be the amount of charge movement as a result of polarization of the material. A significant difference between electric and magnetic fields in tissue is that biological tissue is 'transparent' in a magnetic field. The permeability, µ, is close to 1.000 meaning that the magnetic field is virtually unaffected by the presence of biological tissue. This contrasts with biological tissue in an electric field, where the field intensity varies according to the electrical properties of different tissues. While the electrical properties of fat, muscle and bone are quite different, the magnetic properties are almost identical. This means that for a body segment in an electric field, the field within tissue will vary according to tissue type. The field in muscle is lower than in the fatty tissue or bone (figure 6.19). In a magnetic field, no such variation occurs. The field intensities in fat, muscle and bone are virtually identical. Thus if, for example, a limb segment is exposed to a magnetic field by a surrounding coil as in figure 6.22, the magnetic field intensity (and consequently the induced EMF) within fatty tissue, muscle and bone will be the same. The differences between current induced by an alternating electric field and that induced by an alternating magnetic field will be discussed further in chapter 7. ELECTRIC AND MAGNETIC FIELDS 160 EXERCISES 1 Two oppositely charged spheres are separated by a distance d as shown. The force of attraction between the spheres is measured as 50 N. What would this force be if: (a) the distance d was halved. (b) the distance d was doubled. (c) the charge q1 was reduced by one half. (d) both charges were halved. (e) both charges were halved and the distance doubled. 2 The quantitative relationship between charge and electrostatic force is: F = k.q1r2.q2 .... (6.1) where k has the value 9 x 109 N.m2.C-2. Calculate the magnitude of the force of attraction between: (a) one coulomb of negative charge and one coulomb of positive charge separated by a distance of 1 m. (b) two opposite charges, both of magnitude 1 microcoulomb separated by a distance of 1 mm. (c) a sodium ion (Na+) and a chloride ion (Cl-) in crystalline NaCI. The distance between ions in the crystal is 0.3 nm and the charge of each ion is 1.6 x 10-19 coulomb.
ELECTRIC AND MAGNETIC FIELDS 161 3 The magnitude of the electric field around a single positive charge q is: E = k.rq2 .... (6.3) where k has the value 9 x 109 N.m2.C-2 (see equation 6.3). Calculate the field of: (a) one coulomb of positive charge at a distance of 1 m. (b) one coulomb of positive charge at a distance of 1 km. (c) one microcoulomb of positive charge at a distance of I m. (d) a sodium ion (charge 1.6 x 10-19 C) at a distance of 0.3 nm. 4 The electric field of a single positive charge is measured as 1.5 V.m-1 at a distance of 0.3 m. Use equations 6.1 and 6.3 to calculate: (a) the magnitude of the charge (b) the force which would be experienced by a charge of 2 µC placed at a distance 0.3 m from the first charge. The constant of proportionality in equations 6.1 and 6.3 is 9 x 109 N.m2.C-2. 5 Consider the electric field patterns shown in figure 6.3. Draw diagrams to show the effect on the field when: (a) the small plate in figure 6.3(a) is made smaller (b) the angle between the plates in 6.3(b) is made greater (c) the plates in 6.3(c) are offset further. 6 Two current carrying wires are separated by a distance, d, as shown. The force of attraction between the wires is 50 N. ELECTRIC AND MAGNETIC FIELDS 162 What would be the force if: (a) the distance d was halved (b) the distance d was doubled (c) the current, I1, was reduced by one half (d) both currents were halved (e) both currents were halved and the distance doubled (f) I1 was reduced to zero. (g) I1 was reversed in direction. 7 The quantitative relationship between current and magnetic force is: F α k.I1.Ir2.L .... (6.4) where k has the value of 10-7 N.C-2.s2 (see equation 6.4). Calculate the force between two parallel wires of length 0.8 m. The wires each carry a current of 0.5 ampere and are separated by a distance of 2 cm. 8 Two metal plates separated by a thin air space are found to have a capacitance of 15 pF. The plates are connected to a power supply and charged to a potential difference of 200 V. (a) Use equation 6.6 to calculate the resulting charge on each plate. (b) The power supply remains connected and the space between the plates is filled with petroleum oil. What is the charge on each plate? What is the potential difference between the plates? 9 If the power supply in question 8(b) above was disconnected before the oil was introduced what would be: (a) the charge on each plate
ELECTRIC AND MAGNETIC FIELDS 163 (b) the potential difference between the plates after introduction of the oil? 10 (a) Define the terms 'good insulator' and 'good dielectric'. (b) Which of the substances listed in table 6.2 are good insulators and which are good dielectrics? 11 Briefly describe how the dielectric constant and conductivity of a material will change if there is a change in proportion of: (a) polar molecules (b) non- polar molecules (c) ions. 12 (a) Draw a diagram similar to figure 6.13 to show the effect of a rectangular dielectric on a uniform electric field. Label your diagram to show clearly regions where the field intensity is increased and where it is decreased. (b) Explain the origin of the charges shown on the dielectric surface in figure 6.13. What effect do the charges have on the field intensity within the dielectric? 13 An electric field line crosses from medium I (air) to medium 2 (water) as shown in figure 4.15. Use equation 4.9 and values of s from table 4.1 to calculate the angle of refraction for an incident angle of: (a) 1o (b) 5o (c) 15o What general conclusion can be drawn about field lines crossing into a medium of high dielectric constant? ELECTRIC AND MAGNETIC FIELDS 164 14 Draw a diagram similar to figure 6.16 to show the effect of a dielectric on the field between capacitor plates. (a) Label the diagram to show clearly regions where the field intensity is increased or decreased. (b) Briefly explain the origin of the changes in field intensity. 15 A cylindrical block of material of conductivity 0.8 S.m-1 carries a current of 0.2 amps along its length. The block has faces each of area 10 cm2 and a length of 25 cm. (a) What is the current density in the material? (b) Use equation 6.11 to calculate the electric field intensity in the material. (c) What is the potential difference between the two faces? 16 A rectangular length of material of resistivity 25 Ω.m has a potential difference of 5 V between its ends. The material has a length of 30 cm and a cross-sectional area of 2 cm2. (a) What is the field intensity within the material? (b) What is the current density within the material? (c) What is the total current in the material? 17 An electric field line crosses from medium 1 (conductivity 0.05 S.m-1) to medium 2 (conductivity 0.8 S.m-1) as shown in figure 6.18. Use equation 6.14 to calculate the angle of incidence for a refraction angle of: (a) 40o (b) 80o (c) 89o
ELECTRIC AND MAGNETIC FIELDS 165 18 Figure 6.19 shows the pattern of field lines in a model for an arm or leg (longitudinal section) placed between capacitor plates. Explain the pronounced difference in field intensity in fat and muscle tissue in terms of refraction of field lines and surface charge of polarization in each tissue. 19 Figure 6.20 shows the pattern of field lines in a model for an arm or leg (perpendicular cross-section) between two capacitor plates. Explain why: (a) the field intensity in air close to the fatty tissue is greater than that without the limb present. (b) the field in muscle tissue is lower than without the limb present and lower than that in fatty tissue. 20 Consider a length of wire wound as a solenoid as shown in figure 6.7. Draw a diagram of the solenoid and the associated magnetic field when a current flows through the wire. Is the magnetic field changed by the introduction of biological materials (for example, an arm or leg) within the coil? Explain. 21 (a) What is meant by the term 'electromagnetic induction'? (b) Describe the effects of electromagnetic induction on conductors placed in a magnetic field. 22 Consider a material placed within a coil of wire wound as a solenoid as shown in figure 6.24(a). Describe the effect of an alternating magnetic field on the material if the material is: (a) a good conductor (b) a good dielectric. 23 Figure 6.22 shows the pathway of induced current when a conductive material is placed in a solenoid through which alternating current is flowing. Explain why the ELECTRIC AND MAGNETIC FIELDS 166 induced current follows circular paths, parallel to the coil. 24 Consider a loop of wire positioned parallel to the surface of a conductive material as shown. (a) Draw a diagram showing the magnetic field produced by the loop. (b) Suppose the current through the loop is alternating so that current is induced in the conductor. Draw a diagram showing the pathways of the induced current. Explain why the current flows follows these particular pathways. (c) where, in the conductive material, is the induced current greatest? Why?
SHORTWAVE DIATHERMY 167 7 Therapeutic Fields: Shortwave Diathermy While shortwave diathermy units do radiate waves with a 'Shortwave diathermy' refers to heating of deeply located tissue using electric or frequency of 27.12 MHz, this magnetic fields which alternate at high frequency. The term 'shortwave diathermy', is is a side-effect. The something of a misnomer as the contribution of waves, as such, to the treatment is physiological effects are due negligible. The physiological effects are a result of electric and magnetic fields to the powerful electric or generated by the shortwave diathermy apparatus. Shortwave radiation plays little or magnetic fields generated by no role in the therapy. the apparatus. The apparatus used by physiotherapists generates alternating electric and magnetic fields with a frequency of 27.12 MHz. Since radio waves with frequencies in the range 10 MHz to 100 MHz are termed short waves the term has been, rather inappropriately, applied to this therapeutic modality. PRODUCTION OF THE FIELD Shortwave diathermy apparatus consists of a sinewave generator circuit which produces alternating current with a frequency of 27.12 MHz and a resonant circuit which can be tuned to exactly the same frequency. The sinewave generator supplies energy to the resonant circuit by transformer action. Figure 7.1 illustrates the arrangement. Figure 7.1 Shortwave diathermy apparatus (schematic). SHORTWAVE DIATHERMY 168 The sinewave generator consists of a power supply (chapter 5), an oscillator with Any mains-frequency AC good frequency stability (chapters 2 and 5) and a power amplifier (chapter 5). The produced by the apparatus is power supply converts AC from the mains (of frequency 50 Hz) to DC which is needed also not conducted to power the equipment. It consists of a transformer (to convert the 240 V AC from the appreciably to the patient mains to the voltage needed by the rest of the circuitry), and a rectifier to convert the AC circuit as the resonant to DC. The DC is used to power a sinewave generator; a resonant circuit which frequency (27.12 MHz) is oscillates at 27.12 MHz and an amplifier, which boosts the current produced by the vastly different to the mains resonant circuit to higher levels, as needed for patient treatment. frequency (50 Hz). Electrical energy produced by the sinewave generator is coupled to the patient tuning circuit by transformer action (figure 7.1). Two inductors are placed close together so that energy produced by the power amplifier is transferred to the patient circuit. This method of coupling ensures that DC in the apparatus is unable to reach the patient and the risk of electric shock is minimized. A variable capacitor, C, is included in the patient circuit so that the resonant frequency of the patient circuit can be made equal to the frequency of the oscillator. This ensures maximum efficiency of energy transfer (chapter 2) and reliable operation of the apparatus. A power meter or indicator lamp shows when resonance is achieved and maximum power is transferred. In older machines, the variable capacitor, C, was manually adjusted with the operator adjusting a knob while observing the power meter and adjusting for maximum power. Modern machines use electronic control of the variable capacitor and are described as 'auto-tuning'. The principal advantage of automatic tuning is that if the patient should move during treatment the machine will adjust to keep the patient circuit in resonance. With manual tuning machines, movement of the patient or electrodes can result in de-tuning and a drop in output of the machine. The output of the apparatus is coupled to the patient via electrodes (in the capacitor field technique represented in figure 7.1) or via an induction coil. The coil or electrodes are connected directly to the output of the machine and the part of the
SHORTWAVE DIATHERMY 169 patient to be treated is positioned in the electric or magnetic field. In figure 7.1, the When an induction coil is area highlighted in yellow is circuitry inside the machine. used, the presence of The part of the patient to be treated would be positioned between the external biological tissue in the field is capacitor plates shown in figure 7.1. The plates are normally in the form of two metal irrelevant but the tissue disks, each inside a clear plastic container or envelope. The electrical characteristics volume to be treated will of the patient's tissue affects the capacitance of the patient circuit, as does the influence the number of turns electrode size and spacing. For this reason it is necessary that the apparatus be of the coil and their radius. tuned (by adjusting C in figure 7.1) with the patient positioned in the field. Similarly, if an induction coil is used rather than capacitor plates, tuning will be necessary. This is because when the coil is wrapped around the part of the patient to be treated, the inductance of the coil will depend on the number of turns of the coil and their radius. MOLECULES IN AN ELECTRIC FIELD In shortwave diathermy treatment a high frequency AC electrical signal is produced and applied to the patient via an induction coil or electrodes. The high frequency signal will produce a corresponding high frequency alternating electric or magnetic field in the patient's tissue. We now consider what effect this has on the tissue. Since an alternating magnetic field gives rise to an induced alternating electrical field (as described in chapter 6) we first examine the effects of an alternating electric field on the different molecules found in human tissue. Charged Molecules The conductivity of tissue is determined by the number of free ions in the tissue fluid. In the presence of an electric field these ions will migrate along field lines and so constitute an electric current. The process is not unlike electrical conduction in metals. Metallic conduction results from the movement of free electrons. In electrolytes the charge carriers are not electrons but ions; these are tens of SHORTWAVE DIATHERMY 170 thousands of times more massive than electrons. Figure 7.2 Under the influence of the electric field ions will be Response of molecules to a high accelerated along field lines - but they will not travel far before frequency alternating electric field. colliding with other molecules and losing their acquired kinetic energy. The repeated sequence of accelerations and collisions is the way in which electrical energy is converted to heat energy, which is the random-motion energy of the molecules. At the frequencies associated with shortwave diathermy the field alternations are so rapid that the ions oscillate about a mean position rather than undergoing any large scale movement, but the alternations are not so rapid that movement is prevented and heat generation is not impaired. Dipolar Molecules Dipolar molecules such as water will orient themselves in an electrical field and if the field is alternating this will result in backwards and forwards rotation of the dipoles. In a liquid the molecules are continually in motion (due to their thermal energy) and are loosely associated with each other (coupled); thus some of the rotational energy of the molecules will be converted to heat energy by what can be thought of as a frictional drag between adjacent molecules. Non Polar Molecules Though not normally polar these molecules will undergo a distortion of their electron 'clouds'; that is, they will polarize in an electric field. In an alternating field the electron clouds will
SHORTWAVE DIATHERMY 171 oscillate back and forth to each end of the molecule. Since this kind of motion does not involve transport or rotation of the molecule as a whole it can only be coupled indirectly with the gross molecular movement associated with heat energy. Figure 7.2 summarizes, by illustration, the response of ions, polar molecules and non-polar molecules to a high frequency alternating electric field. In each case there is a net back and forth movement of charge: in other words, an alternating flow of current. REAL AND DISPLACEMENT CURRENT From the previous discussion it is apparent that the different kinds of molecule in a material will each respond differently to an applied electric field. The back and forth movement of ions and the consequent collisions will result in a very efficient conversion of electrical energy into heat energy. The rotational movement of polar molecules provides a less efficient mechanism of energy conversion. The electron cloud distortion of non-polar molecules represents the least efficient means of heat production. Nonetheless each kind of molecule responds to an alternating electric field in a way which results in movement of charges and hence an alternating current. The difference is in the proportion of electrical energy converted to heat energy when the alternating current is produced. With this in mind we distinguish real and displacement current. * Real current is that associated with heat production. When real current flows through a material the rate at which electrical energy is converted to heat energy is given by Joule's law: P = V.I .... (1.4) where V is the potential difference and I is the real current flowing through the material. P is the power dissipated (in watts), in other words the amount of electrical energy dissipated per second (1 watt (W) = 1 joule per second (J.s-1)). SHORTWAVE DIATHERMY 172 * Displacement current is current flow which does not produce any heating. Figure 7.3 In this case the power dissipated, and hence the heat generated, is zero. Real and displacement current Ionic materials are associated principally with real current and hence substantial in an AC circuit. heat production. Polar substances are associated with both real and displacement current and hence less heat production. Non-polar materials are principally associated with displacement current and hence minimal heat production. An example which serves to illustrate the distinction between real and displacement current is given in figure 7.3. Here we have a resistor and a capacitor connected in series to a source of alternating current. In this case we suppose that the capacitor is ideal - it comprises two metal plates separated by a perfect insulator which can polarize and depolarize with no loss of electrical energy to heat energy. The magnitude of the current flowing in this circuit will depend on the voltage of the AC source and the total impedance of the resistor/capacitor combination. The actual impedance of the capacitor is calculated using equation 2.5. The real current (Ir) flowing through the resistor will result in power dissipation according to equation 1.4 and hence heat production in the resistor. The displacement current (Id) flowing through the capacitor (assumed ideal) gives no power dissipation and hence no heat production as the material between the plates is able to polarize and depolarize with no energy loss. In this case, then, the current flowing from the AC source appears as real current in the resistor R and displacement current in the (ideal) capacitor C. Charges move and heat is produced in the resistor while the charge movement (displacement current) in the capacitor produces no heating. The two currents, which are different forms of the same thing, are necessarily the same size.
SHORTWAVE DIATHERMY 173 For a capacitor to be ideal the material between the plates must be an ideal dielectric Most gases come close to - a substance capable of polarizing in an electric field and depolarizing on its removal being ideal dielectrics, as do without any dielectric absorption. In other words, with no conversion of electrical some oils. Water being a energy to heat energy. highly polar molecule, falls Biological materials, particularly those with high water and ion content are far from short of this ideal and being ideal dielectrics. When placed in an electric field the induced current will be a dielectric absorption results combination of real and displacement current. The proportions of each kind of current in significant heating at any will depend on the proportions of ionic, polar and non-polar molecules. frequency below about 1010 We now consider biological tissue exposed to an electric or magnetic field which Hz. alternates at a frequency of 27.12 MHz, the frequency licensed for use in shortwave diathermy. As we have seen, shortwave diathermy may be applied using capacitor plates (which produce an electric field) or an inductive coil (which generates a magnetic field). CAPACITOR FIELD TREATMENT Consider first the situation depicted in figure 6.19(a), where an arm or leg is positioned between two capacitor plates. Figure 6.19(a) shows the electric field pattern, which is affected by refraction and termination of field lines. The total current flowing through the tissue will be determined by the total impedance of the tissue plus the air space between tissue and capacitor plates. Current will flow in the direction of the field lines and the proportions of real and displacement current will depend on the electrical properties of the particular tissue. The amount of heating in any tissue layer will be determined by two factors: the field intensity within the layer and the amount of real, rather than displacement, current. Calculation of the proportions of real and displacement current in a particular tissue is not difficult. Measured values of dielectric constant and conductivity are all that are SHORTWAVE DIATHERMY 174 needed. Calculation of the field pattern is much more difficult and has only been done The conductivity determines using simplified models: even simpler than the somewhat idealized geometries the amount of real current shown in figure 6.19. flow, the dielectric constant Useful qualitative pictures are nonetheless obtained by combining diagrams such as determines the amount of those shown in figure 6.19, with calculated values of real and displacement current in displacement current. each tissue layer. At a frequency of 27.12 MHz the current flow in fatty tissue and bone is approximately 50% displacement. In muscle and tissues of high water content the proportions are approximately 80% real current to 20% displacement current. Figure 7.4 shows a revised view of figure 6.19(a) which takes into account the two kinds of current flow which occur. In the air spaces the current flow is entirely displacement current. In fatty tissue and bone the current is assumed to be one half real current and one half displacement current. For simplicity, muscle is shown as having entirely real current. Figure 7.4 Current type and directions in a model for an arm or leg.
SHORTWAVE DIATHERMY 175 When viewing diagrams such as these, bear in mind the simplifications made. The pictures can be misleading if interpreted too literally. You should also bear in mind that even a single tissue layer may be inhomogeneous at both the microscopic and macroscopic level. An example of the complications introduced by tissue inhomogeneity is seen with fatty tissue in the shortwave field. Fatty Tissue A practical limitation on the amount of heat which can be produced in deeply located tissue is the heat production in fatty tissue. When using capacitor plates the rate of heating of fatty tissue is always greater than that of the underlying muscle tissue. Part of the reason is that fatty tissue is inhomogeneous. The tissue is not a uniform distribution of cells but a complex structure incorporating regions of high conductivity and dielectric constant: the lymphatic and blood vessels. The high conductivity and dielectric constant of the vessels will result in field lines being focussed or channelled into them with a resulting high local field intensity and corresponding high rate of heating in and near the vessels. The phenomenon is illustrated in figure 7.5. The localized high heat production will result in greater temperature Figure 7.5 elevation of the vessels than the fatty tissue as a whole and a greater Focussing of electric field lines in blood sensation of heat than would be expected if the tissue layer was homogenous. and lymphatic vessels in fatty tissue. INDUCTIVE COIL TREATMENT We now consider application of the shortwave field with an induction coil. The objective is to induce an electric field and hence a flow of current as a result of the alternating magnetic field produced by the coil. In the example illustrated in figure 7.6 a cable carrying the shortwave frequency current is wrapped around a patient's lower SHORTWAVE DIATHERMY 176 limb. Figure 7.6(a) shows the inductive coil wound as a solenoid around the patient's lower limb and figure 7.6(b) shows the current pathways in the different tissues. The current pathways shown are predicted assuming that the alternating magnetic field gives rise to an induced EMF in the patient's tissue. In this case the current will follow circular paths parallel to the turns of the coil in figure 7.6(a). Note that in figure 7.6(b) the current through the fatty tissue is shown as half displacement current and half real current while muscle is assumed to have real current only. As indicated previously, this is only an approximation: while the proportion of real current in fatty tissue is about 50%. in muscle it is about 80%. If the coil in figure 7.6 had a large number of closely spaced turns and the coil Figure 7.6 diameter was small compared to its length, then the magnetic field inside the Current flow induced in a limb by coil would be uniform and the induced EMF would be the same throughout the tissue volume. Were this the case, the relative amounts of current flow in each inductive coil treatment. tissue would simply depend on the tissue impedance (which is determined by the dielectric constant and conductivity).
SHORTWAVE DIATHERMY 177 A complication is that with more widely spaced turns and a relatively large diameter, This means thaqt a greater the magnetic field inside the induction coil will be non-uniform. In an arrangement EMF will be induced in the like that shown in figure 7.6(a), the magnetic field would be strongest close to the coil superficial tissues and and decreasing in intensity towards the centre. The highest field intensity is thus in consequently there will be a the superficial tissues of the limb. greater current flow. Another Kind of Coil Most manufacturers of shortwave diathermy apparatus offer accessories which include a compact coil mounted in a plastic housing. This device is called a monode. The monode is pointed at the part of the patient to be treated so that the coil is in a plane parallel to the skin surface. With this arrangement (figure 7.7), currents are induced which flow in circular paths parallel to the skin surface. The cable supplied with the shortwave machine can, of course, also be wound into a spiral and positioned to produce a similar distribution of induced current. The spiral coil placed parallel to the skin produces more Figure 7.7 superficial heating than the solenoidal coil (figure 7.6). This is Induced currents with a spiral coil mounted because the magnetic field intensity decreases rapidly with distance from the coil, as the field lines diverge, spreading parallel to the skin surface. apart and looping round to the opposite side of the coil. The field spreading is similar to that which occurs at the ends of the coils in figures 6.7 and 7.6. Magnetic field lines become more separated, indicating a weaker magnetic field further from the coil and consequently less induced EMF and less induced current. Hence although the current induced in muscle is mostly real current, the amount of current at depth is much less than with a solenoid (figure 7.6). SHORTWAVE DIATHERMY 178 Capacitative Effects Figure 7.8 Electric field pattern (blue lines) between A practical complication which occurs with inductive coil treatment, whether with a solenoid or a spiral coil (monode), is that in addition adjacent turns of an inductive coil. to the currents induced by the magnetic field there is also a pronounced electrostatic effect. There is a certain capacitance between the loops of the coil. In fact whenever a cable or wire is folded back on itself or coiled we have produced a situation where there are two conductors separated by a space; thus we have produced a capacitor. Although in the case of a cable wound as a coil the capacitance is very small, the effect is quite significant at MHz frequencies. The inductive coil behaves as an inductor in parallel with a capacitor. At the high frequencies used for shortwave diathermy the inductance of the coil results in a high impedance to current flow in the cable (equation 2.4). The capacitance associated with the coil presents a lower impedance pathway for current to take (equation 2.5). In consequence the induced current patterns are not as simple as those shown in figure 7.6(b). The electric field between adjacent turns (Figure 7.8(a)) results in current flow along the field lines shown in blue. Because the electric field is stronger closer to the coils, greater current flows and this adds to the current induced by the magnetic field. The consequence is greater current flow in, and greater heating of, superficial tissue (figure 7.8(b)). The electric field between adjacent loops is similar to that between two small electrodes (figure 6.1(c)). The field is most intense close to the cable. A consequence is that there is a risk of burning the superficial tissues with the electric field of the coil rather than
SHORTWAVE DIATHERMY 179 heating deeper tissue with current induced by the alternating magnetic field. When the adjacent turns are A similar argument applies for a spiral coil. An electric field is produced between closer together, the electric adjacent turns within the loop. Close to the coil, the electric field is intense and field is actually greater, but it greater current flows. This adds to the current induced by the magnetic field so there is also more localized to the is greater current flow in, and greater heating of, superficial tissue. space between the turns of Superficial heating due to the electric field can be minimized in three ways: (a) by the coil. winding the turns of the coil close together, (b) by keeping the cable away from the patient's skin using towelling and/or rubber spacer designed for this purpose and (c) by using an electrostatic shield. Electric field heating effects can also be minimized, in the case of a solenoid, by positioning an earthed metal cylinder between the coil and the patient's limb. If a monode is used, a flat metal plate between the monode and the patient's tissue would be needed. The plate will screen-out the electric field while having little effect on the magnetic field of the coil. The electric field inside the metal cylinder or behind the metal plate would be almost nil because the metal is a good conductor and field lines will terminate on its surface. Most metals are, however, transparent as far as magnetic fields are concerned so the magnetic field is virtually unchanged. Some, but not all, inductive coil applicators are supplied with an inbuilt electric field screen. Screening is an important feature when depth efficient heating is required. In summary, the options with inductive coil treatment are a coil wound around the part of the patient to be treated or a flat coil (monode) positioned over the body part. The difference is the depth efficiency of treatment. A solenoidal coil (figure 7.6) has greater depth efficiency as far as tissue within the coil s concerned. A flat spiral coil (figure 7.7) has greater effect on superficial tissues. With either method of application, there is the risk of excessive superficial heating due to the electric field between adjacent turns of the coil or spiral. the risk is minimized by spacing the coil or spiral away from the patient's superficial tissues. SHORTWAVE DIATHERMY 180 ELECTRODE PLACEMENT - CAPACITOR FIELDS With capacitor field treatment, the therapist has more control over the field intensity in different areas than with inductive coil treatment. We have discussed previously how the combination of tissue layers in the part of the patient being treated alters the shape of the electric field. The other factors influencing the field pattern involve the placement of the electrodes. Each factor listed below must be taken into account in the practical application of shortwave diathermy using capacitor field treatment. * The shape of the part of patient in the field. Compare Figure 6.19(a) with 6.19(b). In addition, if the electrodes are placed over any prominence an undesirable concentration of the field can result. * The arrangement of tissues layers in the treated structure. As discussed previously, this factor plays a significant role in determining the final shape of the field. * The size, spacing and orientation of the electrodes. Some examples of the electric field in the absence of any object were shown in figures 6.2 and 6.3. We consider below the effect when the patient is in the field. Electrode Size In general, it is preferable to use electrodes which are somewhat larger than the structure to be treated. This results in the central, more uniform, part of the field being used (figure 6.2). The dielectric constant and conductivity of tissue are much higher than those of air (table 4.2). Thus, with large electrodes, the field lines are bent towards the limb and spreading of the field is minimised. The effect is illustrated in figure 7.9 where the effect of the different tissue layers is ignored for simplicity.
SHORTWAVE DIATHERMY 181 Compare these with figures 4.19 and 4.20. The use of Figure 7.9 small electrodes results in an undesirably high field Effect of electrode size: (a) correct intensity in the superficial tissues. electrode size (b) electrodes too small Unequal size electrodes (figure 5.10(c)) can be used to (c) arrangement for selective heating. selectively heat tissue located closer to one surface of a limb. Large differences in electrode size, however, can sometimes lead to difficulty in tuning or instability in machine operation. Electrode Spacing The electrode spacing should normally be as wide as possible. In this way the problems associated with a non-uniform field pattern are minimised. The machine itself, however, sets the limit on the maximum spacing which can be used. As the electrodes are moved further apart the capacitance of the two plates decreases. In addition the field intensity (and consequently the rate of heating) will decrease. A point will be reached where the machine can no longer be tuned or insufficient power is available for adequate heating: this sets the limit on the separation of the electrodes. By use of a wide spacing the electrical properties of the tissue have a smaller effect on the overall field pattern and the electrical properties of air play a greater role. Thus the field pattern is more uniform and less subject to variation with movement of the patient in the field. Figure 7.10 illustrates the effect of electrode spacing. In 7.10(a) the electrode to surface distance varies considerably resulting in a local high field intensity SHORTWAVE DIATHERMY 182 in the limb. In 7.10(b) the electrode to surface Figure 7.10 distance does not vary greatly and the field within Effect of electrode spacing: (a) narrow the limb is more uniform. Clearly, if a relatively uniform field pattern is required the arrangement spacing, (b) wide spacing and (c) shown in 7.10(b) is to be preferred. The unequal spacing. arrangements shown in 7.10(c) and 7.11(c) are both suitable if we wish to selectively heat one surface of a limb. They would also be suitable for heating a structure which is located close to one surface of a limb or trunk - for example, the hip joint. Electrode Orientation In the examples considered previously the electrodes were placed parallel to each other in order to obtain a relatively uniform heating pattern. However if one part of the surface of a structure is closer to an electrode, the field lines will be concentrated in that region. Figure 7.11 shows electrodes applied to the shoulder. Compare this with figure 6.16. Electrodes which are parallel to each other as in figure 7.11(a) do not give a uniform field because the air spacing varies considerably. The dielectric constant and conductivity of each field-line pathway varies considerably, resulting in variation in the field intensity. In figure 7.11(b) the distance between the plates varies but the electrical characteristics of each pathway are similar: thus the field is relatively uniform. Clearly the arrangement shown in figure 7.11(b) is preferred when uniform heating is the objective.
SHORTWAVE DIATHERMY 183 In all of the examples discussed previously the arrangement Figure 7.11 of electrodes is contraplanar: that is, electrodes are placed Effect of electrode orientation. (a) and over opposite sides of a structure. Such an arrangement is needed if deeply located tissue is to be heated. (c): incorrect orientation (b) correct In some circumstances it is preferable to use a coplanar orientation. electrode arrangement. For example superficial structures, such as the spine, which are too extensive for contraplanar treatment may be treated in this way. Figure 7.12 shows a coplanar arrangement of electrodes. When using a coplanar arrangement it is very important to ensure that the spacing between electrodes is greater than double the skin to electrode distance. This results in the majority of field lines passing through tissue rather than the air space between the electrodes. Figure 7.12 A coplanar arrangement of electrodes. SHORTWAVE DIATHERMY 184 Even when using a contraplanar arrangement of electrodes considerable heating occurs in the superficial tissues closest to the electrodes. This effect can be minimized by using the cross-fire technique of treatment shown in figure 7.13. Half of the treatment is given with electrodes in one position (figure Figure 7.13 7.13(a)). The electrodes are then moved so that the new electric field The cross-fire technique. is at right angles to the old one (figure 7.13(b)) and the treatment is continued. In this way deeply located tissue receives treatment for Figure 7.14 twice as long as the skin. The cross-fire treatment may be used, for A hollow dielectric between example, on the knee joint or pelvic organs. It is also particularly useful for treating the walls of cavities within a structure, for example capacitor plates. the sinuses. Figure 7.14 shows the field pattern obtained with an object of high dielectric constant which has an air-filled hollow at its centre. The field lines are concentrated in the dielectric resulting in uneven heating of the walls of the cavity. Cross-fire treatment ensures that all of the cavity wall area is treated. HEATING OF TISSUE Earlier we discussed qualitatively and in molecular terms, the heating effect of a high frequency alternating electric field. We now consider heat production and temperature rise and take a larger scale view of matter: a view at the level of tissue rather than molecules. We saw in chapter 1 that the power dissipated by a resistor, the rate at which electrical energy is converted to heat energy, is given by equation 1.4: P = V.I .... (1.4)
SHORTWAVE DIATHERMY 185 This expression relates the current, I, flowing through a resistor to the total power, P, dissipated in the resistor. For resistors the current, I, is entirely real current and thus produces heat. When we consider biological tissues we must distinguish between real current and displacement current since only the real current results in heat production. In additional, we are usually more interested in the rate of heating at a particular point in the tissue rather than in the tissue as a whole. In this case a more useful expression of equation 1.4 is equation 7.1. Pv = E.ir .... (7.1) Here Pv is the power dissipated per unit volume of tissue at a particular point. The units of Pv are thus watts per cubic metre. E is the field strength (in volts per metre) and ir is the real component of current density (in amps per square metre) at that point. The power dissipated, Pv is equal to the rate of heat production. Hence, in order to determine the rate of heating at a particular point in tissue we need to know the electric field strength and the real current density. We begin by considering fields and currents produced using capacitor field treatment. Capacitor Field Treatment Whether electrodes are positioned in a coplanar arrangement (figure 7.12) or in a contraplanar arrangement (figures 7.9 to 7.11) the current flow in muscle will be determined by the total impedance of the tissue combination plus the air space between the tissue and capacitor plates. Figure 7.15 shows electrical equivalent circuits for the two electrode/tissue arrangements. The quantities Za, Zf, and Zm refer to the electrical impedances of air, fat and muscle respectively. SHORTWAVE DIATHERMY 186 Figure 7.15 Electrode/tissue configurations and their electrical equivalent circuits. (a) coplanar arrangement, (b) contraplanar arrangement. In figure 7.15(a) we ignore (displacement) current flow through the air directly between the electrodes. We also ignore current flowing directly through the fatty tissue and bypassing the muscle. If the electrode spacing is at least twice the electrode to tissue spacing this will be a reasonable approximation. The impedance presented by each alternate pathway will be sufficiently high to make these currents negligible. In figure 7.15(b) we ignore current flow through the bone, directly around the fatty tissue or through the air around the tissue. Again this is because these pathways have very high impedance compared to the ones shown. With these approximations the electrical equivalent circuits in 5.16(a) and (b) are the
SHORTWAVE DIATHERMY 187 same. Each of the electrode/tissue arrangements are equivalent to a series combination of impedances. Just as with resistive circuits (described in chapter 1), when impedances are connected in series the current in each impedance is the same. We thus have the following relationship: displacement displacement displacement current = + real current = + real current in air in fatty tissue in muscle As mentioned earlier, the proportion of real current in fatty tissue is approximately 50% The real current density in while in muscle the proportion is about 80%. Thus the amount of real current flow in muscle may be increased or muscle is 80/50 or about one and one half times greater than in fatty tissue. decreased if the electric field lines converge or diverge. Let us take the simple case where current spreading is minimal and estimate the This depends on the relative rate of heating in fatty tissue and muscle. We need to know both the real tissue/electrode geometry - current density and field strength in each tissue. The field strength is estimated see figure 6.19 for example. below. When resistors are connected in series the current flow in each is the same but the voltage across each resistor will, in general, be different. The largest resistor will have across it the greatest potential difference. The equivalent statement for tissues of different impedance is as follows: When tissues are arranged in series the field intensity will be greatest in the tissue with highest impedance. Inspection of table 6.2 shows that muscle has a higher conductivity and dielectric constant than fatty tissue: both figures are several times higher. Now a high conductivity and dielectric constant means a low impedance. Combining the two SHORTWAVE DIATHERMY 188 figures from table 6.2 we calculate that fatty tissue has an electrical impedance some ten times larger than muscle. The rate of heating of each tissue is given by equation 7.1: Pv = E.ir .... (7.1) The real current density in muscle is as we have seen, about one and a half times greater than in fatty tissue, however the field intensity in fatty tissue is approximately ten times higher. Hence the rate of heating of fatty tissue is predicted to be approximately 10/1.5 times higher than muscle. We thus have the general conclusion that if spreading or converging of the field is minimal the rate of heat production in fatty tissue will be about seven times higher than in muscle. If the electrode/tissue configuration permits spreading of the field in muscle the current density will be reduced and the rate of heating of muscle correspondingly reduced. Conversely if the geometry produces convergence of the field lines in muscle the current density will be increased and the relative rate of heating will be increased accordingly. Inductive Coil Treatment With capacitor field treatment tissues are effectively in a series electrical arrangement. The current flow in muscle is thus limited by the impedance of the fatty tissue layers. When inductive coil treatment is used such is not the case. Consider the coil and tissue arrangement shown in figure 7.7. Currents are induced in the plane of the fatty tissue and in the plane of the muscle. The current loops are complete electrical pathways in the one tissue. For this reason the current flow in
SHORTWAVE DIATHERMY 189 muscle is not limited by the fatty tissue but depends only on the strength of the induced electric field and the electrical characteristics of the muscle tissue. In other words the induced currents flowing in each tissue layer are independent of each other. The real component of the current density, the current density which determines heat production, is given by equation 6.10, which can be written: ir = σ.E .... (6.10) Substituting this formula into equation 7.1 we obtain an alternate expression for the power dissipated per unit volume: Pv = σ.E2 .... (7.2) Table 6.2 shows that the conductivity, σ, of muscle is some sixteen times greater than that of fatty tissue. Hence, for the same induced electric field strength, both the real current density and the power dissipated in muscle will be sixteen times greater than in fatty tissue. How large is the magnetically induced electric field? The intensity of the induced field The intensity of the induced is determined by the rate of change of the magnetic field and the permeability, µ, of the electric field is determined by material. The permeability is close to one for biological materials (see table 6.3) so the rate of change of the fatty tissue and muscle are alike in this regard. magnetic field and the permeability, µ, of the For the same strength of alternating magnetic field then, both fatty tissue and muscle material. It does not depend will have the same strength of induced electric field. Thus the rate of heating of on the electrical properties, σ muscle in this situation will be about sixteen times greater than that of fatty tissue. and ε, of the tissue. In practice such a degree of selective heating is difficult to achieve. This is for two reasons: SHORTWAVE DIATHERMY 190 * Muscle is located beneath fatty tissue and so is further from the induction coil. Thus the magnetic field is weaker in muscle and the strength of the induced electric field is correspondingly smaller. * Fatty tissue, being closer to the induction coil may also experience an appreciable electric field due to the capacitance between adjacent turns of the coil. This effect was described earlier (see figure 7.8). These two factors combine to increase the heating of fatty tissue relative to muscle so For more information on that a sixteen to one advantage is rarely obtained. Nonetheless efficient selective relative heating rates see the heating is achieved with close spacing of the turns of the coil and a sufficiently large book 'Therapeutic Heat and coil to patient distance. One would also expect good discrimination with applicators Cold' J F Lehmann (ed) which incorporate an electric field screen in front of the inductive coil. (1982) chapters 6 and 10. HEAT AND TEMPERATURE RISE Having described the factors determining the rate of heating of tissue we now consider the relationship between rate of heating and rate of increase of temperature. The rate of heating per unit volume is given in terms of electric field intensity and real current density by equation 7.1. Hence the amount of heat produced per unit volume, ∆Qv, in a time interval ∆t is given by equation 7.3. ∆Qv = E.ir.∆t .... (7.3) ∆Qv has units of joules per cubic meter (J.m-3). In considering the therapeutic effects of diathermy it is not the heat produced, as such, which determines the physiological response but the resulting temperature rise. Temperature is a key factor in determining the rates of chemical reactions and hence
SHORTWAVE DIATHERMY 191 physiological processes. The SI unit of temperature is the kelvin (symbol K). It is related to the perhaps more To convert from degrees familiar degree Celsius (oC) by the expression Celsius to Kelvin's, simply oC = K - 273.15 add 273.15 to the Celsius temperature. Notice that from this definition the size of the degree Celsius is the same as the kelvin. Alternatively, when the In other words a change in temperature of five degrees Celsius is precisely the same specific heat capacity of a as a change of five Kelvin's. When we are talking about increases in temperature substance is known, equation brought about by diathermy treatment the terms kelvin and degrees Celsius can be 7.4 can be used to calculate used interchangeably to describe the increase. the temperature increase resulting from the heat When a fixed amount of heat is supplied to different substances the increase in supplied. temperature of each will, in general, be quite different. The factor which determines the resulting temperature increase is the specific heat capacity of the substance. The specific heat capacity is defined as the amount of heat required to raise 1 kg of a substance through one kelvin. The units of specific heat capacity are thus joules per kilogram per kelvin. This can be measured experimentally by supplying a certain amount of heat (∆Q) to a known mass (m) of the substance an measuring the resulting temperature increase (∆T). The experiment must be arranged so that all of the heat supplied is used to increase the temperature of the substance. If the loss of heat is negligible then the specific heat capacity (c) can be calculated using equation 7.4: ∆Q c = m.∆T .... (7.4) When we consider the heating of tissues by diathermy, heat transfer between tissues and to the bloodstream will have a large effect on the temperature distribution during treatment. SHORTWAVE DIATHERMY 192 Prior to the start of treatment the body tissues are in a state of dynamic equilibrium. Cellular activity, metabolism and muscle contraction result in the steady production of heat and the circulation of blood and tissue fluids provide an efficient means of heat transfer. The net production of heat is balanced by net transfer of heat from the tissue and a stable temperature is maintained. Once treatment is started heat is produced in the tissue according to equation 7.3 and the temperature starts to increase. An expression for the initial rate of increase in temperature is obtained below. Rearranging 7.4 we have ∆Q = m.c.∆T .... (7.5) Dividing this expression by volume we obtain: ∆Qv = ρ.C.∆T where ρ is the mass per unit volume or density of the tissue. Dividing 7.5 by ∆t gives: ∆Qv = ρ.c.∆∆Tt .... (7.6) ∆t where ∆Qv/∆t is the volume rate of heating (in Joules per cubic metre per second) and ∆T/∆t is the rate of increase in temperature (in Kelvin's per second). This equation can be used to compare the initial rate of temperature increase in fatty Note that this conclusion is a tissue with that of muscle. The densities of the two tissues are similar but the heat general one. It applies not capacity of muscle is some 50% greater than that of fatty tissue. Thus if the rate of just to shortwave diathermy heating of each tissue is the same, the initial rate of temperature increase in muscle but to any diathermic will be only two thirds of that of fatty tissue. To produce the same initial rate of modality. increase in temperature in each tissue the rate at which heat energy is produced in muscle must be 50% greater.
SHORTWAVE DIATHERMY 193 An equation specifically applicable to shortwave diathermy is obtained by solving equations 7.5 and 7.3. We then have: ρ.c.∆T = E.ir.∆t, which on rearranging gives: ∆T = E.ir .... (7.7) ∆t ρc Equation 7.7 shows that the initial rate of increase in temperature (∆T/∆t) in shortwave diathermy depends on four factors: * E, the field intensity at the point * ir, the magnitude of the real current density at the point * ρ, the density of the tissue * c, the specific heat capacity of the tissue Once the temperature of any tissue has increased appreciably two things happen: * The body's temperature regulation mechanism responds. Blood vessels dilate, circulation is increased and more heat is transferred from the tissue. * Heat is transferred by the blood and tissue fluids to adjacent cooler tissues. Both of these effects lower the rate of increase in temperature. Eventually, the stage is reached where the temperature ceases to increase. A new dynamic equilibrium is achieved where the net production of heat is once again balanced by the net transfer from the tissue. Figure 7.17 illustrates the temperature variation during treatment. There is a transient period during which the tissue temperature increases, followed by a steady state where a constant (elevated) temperature is produced. The transient period for tissue volumes of interest in physiotherapy is typically of the order of twenty to thirty minutes SHORTWAVE DIATHERMY 194 (see Lehmann (1982), chapter 10). Thus for treatment times of up to several minutes, equation 7.7 gives a reasonable approximation to the real physical situation. Application of equations 7.1 and 7.7 to quantitative prediction of the rate of heating and rate of temperature increase in different parts of tissue is difficult. The difficulty arises in the calculation of the field intensity in a particular area. For a review of results obtained using various approximations see A. W. Guy in J F Lehmann (1982). In patient treatment, shortwave diathermy remains something of Figure 7.16 an art as well as a science. The physiotherapist must use a A simple model for tissue temperature knowledge of anatomy together with knowledge of the electrical properties of tissues to determine the optimum placement of variation during treatment. electrodes or coil to give the required field pattern. Once the field pattern is selected, the physiotherapist uses a knowledge of the relative heating of the tissues and the patient's report of a sensation of warmth to adjust the intensity of the applied field to an appropriate level. With this procedure it is not possible to accurately monitor dose or dose rate for the individual tissues. Since this is a problem common to all diathermic modalities we will defer further discussion of dosage until chapter eleven. Physiological Effects The therapeutic value of shortwave diathermy arises from the physiological response of tissues to an increase in temperature. A number of physiological responses are found: * at the cellular level an increase in temperature increases the rate of biochemical reactions. Thus cellular metabolism is increased - there is an increased demand for oxygen and nutrients and the output of waste products is increased.
SHORTWAVE DIATHERMY 195 * blood supply is increased. A number of factors determine this response. The increased output of cellular waste products triggers dilation of the capillaries and arterioles. The temperature increase itself causes some dilation, mainly in the superficial tissues where heating is greatest. In addition, stimulation of sensory nerve endings (again mainly in the superficial tissues) can cause a reflex dilation. * a rise in temperature can induce relaxation of muscles. If there is abnormal muscle activity caused by pain, for example, repeated treatment with shortwave diathermy can be beneficial. The treatment helps to interrupt the vicious circle of pain producing muscle activity which in turn produces more pain and so on. A number of factors may contribute to relaxation: the direct effect of heat on muscle tissue, the removal of any accumulated metabolites due to increased circulation and the sedative effect of heat on sensory nerves. * the response of sensory nerves to heat is useful for the relief of pain generally. Mild heating appears to inhibit the transmission of sensory impulses via nerve fibres. In addition, when pain results from inflammation of tissue an increase in the rate of absorption of exudate with increase in temperature can result in a secondary pain-relief effect. Some claims have been made that additional non-thermal effects can be produced under the conditions used for therapy. As yet there is no clinical evidence for these claims. Non-thermal effects seem to have been demonstrated using pulsed shortwave treatment when the peak power level is significantly higher than used for diathermy. The few published comparative studies indicate little or no nonthermal effect at the low continuous power levels of conventional shortwave field treatment. These points are considered further in chapter 8 following. SHORTWAVE DIATHERMY 196 EXERCISES 1 Figure 7.1 shows a schematic diagram of shortwave diathermy apparatus. (a) Briefly explain how the apparatus produces a high-frequency alternating electric or magnetic field. (b) What is the function of inductors L1 and L2? c) Why is the capacitor in the patient tuning circuit a variable one? 2 (a) Why is it necessary to tune shortwave diathermy apparatus with the patient coupled to the machine? (b) What is the advantage of automatic versus manual tuning of shortwave diathermy apparatus? 3 Figure 7.2 illustrates the response of ions, polar molecules and non-polar molecules to a high-frequency alternating electric field. (a) Briefly describe the movement of each kind of molecule in the field. (b) How is the movement related to heat production in a material? (c) Which kind of movement is associated with greatest heat production and which with least heat production? 4 (a) What is meant by the terms 'real current' and 'displacement current'? (b) Consider the movement of ions, polar molecules and non-polar molecules in an alternating electric field. Describe the relationship between each kind of movement and real and displacement current. 5 (a) Consider each of fatty tissue, muscle and bone in the shortwave diathermy field. Is current flow in each tissue best described as real or displacement current?
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