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The Catt Anomaly
I seem to remember reading an article by Ivor Catt in Wireless World, many years ago, considering what would happen when a sinewave signal travelled along a feeder, and encountered a termination at the far end, consisting of a perfect diode. This, of course, is quite a complicated scenario, since waveform distortion will occur, leading to the generation of harmonic frequencies. However, just recently, as a result of the reappearance of Ivor in the pages of Electronics World, I paid a visit to his website. You can find an animation demonstrating the Catt Anomaly (he prefers to call it the Catt Question or the E-M Question) at http://www.electromagnetism.demon.co.uk/catanoi.htm and it is in fact a very much simpler scenario than that mentioned above. It set me thinking, and some of these thoughts are set out below. As usual, my approach has been in terms of the fundamental properties of the circuit, rather than a mathematical description of the phenomena. I have always been surprised that so many engineers, including many of my colleagues when I worked in industry, seemed happy simply to turn the handle on the textbook maths and arrive at a working circuit, with no real understanding - or even curiosity - as to the details of what was actually going on, and just how the circuit in fact functioned. (Maths does not explain anything, it simply provides you with the tools to calculate the quantitative relationships between the variables. The maths itself is just arranged to mirror the facts, once they have been understood and explained.) Vector diagrams, Argand diagrams, Nyquist diagrams, Bode plots and the like can provide an insight into the functioning of a circuit, which an equation does not, at least not to the ordinary mortal. An example is the functioning of a second order phase lock loop: the treatment in my Radio Frequency Handbook (Ref 1) uses graphical methods to elucidate the workings, arriving at the result without resort to any complicated mathematics. In the same way, the analysis below is based firmly on the physical properties, the relationships between voltage, current, charge, capacitance, inductance, power, energy etc. in the circuit.
The Catt Anomaly
Figure 1 shows a battery connected to a two wire transmission line of finite length, and we may assume, from the position that the TEM wavefront has reached, that the battery was connected a (very!) short while previously. On the part of the line which has already been charged by the source, positive charge appears on the top conductor, and negative charge on the bottom conductor. Electric field lines are indicated, originating on the positive charges and terminating on the negative charges. The electric field strength, in volts per metre, will be equal to the potential difference between the upper and lower conductors, divided by the distance between them. At least, this is the case for the field lines shown, in this two dimensional representation. There will also be longer such lines, starting on the upper conductor and terminating on the lower, in the same plane as the lines shown, but bulging out in front of the paper, and likewise behind. There are even notional lines, of negligible importance, heading off vertically from the top conductor towards infinity, and likewise downwards from the lower, assuming the two conductors are located somewhere in free space. The electric force results in an electric flux, of strength depending upon the permitivity of the medium in which the line is immersed. This is analogous to the way magnetic field strength H results in a magnetic flux density B, depending upon the permeability of the medium. For simplicity, let’s assume the line is in air, or even in vacuo.
The Catt Anomaly, as I understand it, lies in the question as to where the charge at any point on the surface of the lower conductor comes from, bearing in mind that its greatest concentration will be where the lines of electric force landing on it are most concentrated, i.e. directly underneath the upper conductor. Ivor points out that it cannot be delivered by the displacement current flowing between the lines, as displacement current is not a flow of real current (Ref 2).
Ivor says on his website that he has asked well known academics where the negative charge on the lower conductor comes from. He divides those who could be induced to give any answer at all, into “Westerners” and “Southerners”. The former say the charge comes from the source at the left-hand end of the line, and the latter, from within the line itself at that point. But, he maintains, it cannot come from the left-hand end as that would suppose that it travels at the speed of light3. But the negative charge consists of electrons, each of which has a small but finite mass (9.109 ´ 10-31kg), and therefore cannot travel at the speed of light.
The Southerners’ explanation entails charge moving from the inside of the material of the line, to the surface i.e. at right-angles to the direction of propagation, which he also he also finds problematic. On the face of it, these two positions seem to be irreconcilable, and perhaps for this reason, Ivor assumes they are both wrong. On the contrary, I maintain, they are in fact both partially right.
From the outset, I was puzzled by Ivor’s fixation on the negative charge on the lower conductor. Surely any complete explanation must account equally for the positive charge on the upper conductor. So that became my point of departure.
A different view.
Consider the line before the battery is connected. The metal - let’s assume it is 100% pure copper - consists of atoms which are in “fixed” positions, the material being a solid. Although fixed, they are in fact each vibrating about some mean position, assuming the temperature is not absolute zero. Nevertheless, for the purpose of this exercise, that is immaterial; they cannot actually wander around, and they will therefore be regarded as stationary. This applies to the protons and neutrons in the nucleus and also to most of the electrons, in each atom.
But there are also “free electrons” forming a sort of “conducting gas” within the conductor. These may be on a somewhat freer rein, but they too, for the purposes of this argument, can be considered as stationary. Thus at every point along the line there are equal numbers of protons and electrons per unit length. This has been plotted in a graph at the top of Figure 2, where the x axis is length, the same as the
line itself. Before the switch is closed, the equal number of protons and electrons per unit length is indicated by the black line labelled “p, e”. Thus there is no net charge at any point on either conductor, and the potential difference between them is everywhere zero.
An atom is not proprietorial about its free electron: if the latter departs to the left as part of an electric current, the positive charge on the nucleus is happy to let it go, provided that another electron comes along from the right to take its place, which will always be the case in a complete circuit.
Now consider the case when the switch is closed, Figure 3. Conventional current
flows into the top conductor of the line, but in fact a conventional current flowing to the right actually consists purely of electrons moving to the left (at least, in a metallic conductor, which - unlike a semiconductor - has no mobile positive charges or “holes”). Assuming the velocity of the wavefront on the line is the same as the speed of light (for an open air-line, it won’t be that much less), then after half a nanosecond it will have reached the point indicated by the line A A’, and after a further half nanosecond it will have reached B B’, as indicated in Figure 3. The instant the switch is closed, the leftmost electron in the upper conductor (the “first” electron) will start to move, eventually passing via the switch and into the matched source, the internal resistance and emf of which have been shown separately. The second electron now finds the third electron closer to it than the first. As like charges repel each other the more, the closer they are together, the second electron experiences a net force pushing it to the left and starts to move leftwards also. Now, the third electron finds the fourth electron closer to it than the second and also starts to move - - and so on. The electrons may be moving at a snail’s pace relative to the speed of light, but the disturbance just described propagates along the line at the speed of light, reaching B B’ in just a nanosecond.
To the right of B B’ there are still equal numbers of protons and electrons per unit length of line, so there is no net charge and the voltage on the line there is as yet zero. To the left of B B’ a line of electrons is moving to the left at a constant speed, the spacing between the moving electrons being everywhere equal, x + δ say; slightly greater than to the right of B B’. Therefore there is everywhere a slight deficit of electrons per unit length of the upper conductor, their number being represented by the green line “e” in Figure 3 (not to scale). The resultant constant net positive charge per unit length is responsible for the constant positive potential on the line, relative to the lower conductor, indicated in Figure 3 by the red line labelled “V”. With a line of electrons all moving at the same speed and with a constant spacing between them, the current in the line to the left of B B’ is everywhere constant, indicated by the blue line labelled “I” in Figure 3. The ratio of V to I gives the “characteristic impedance” or “surge impedance” of the line, which I will assume to be 300W, a typical value for a two-wire air-line, though it could be anywhere from about a third to five times that value, depending upon the thickness of the conductors, their spacing and the dielectric separating them.
Exactly the same mechanism which has been used to account for the appearance of a positive charge on the upper conductor, accounts equally well for the appearance of a negative charge on the lower conductor. Only now, as conventional current flows from the source into the upper conductor, it returns from the lower conductor, into the negative pole of the source. This implies that the negative pole forces electrons into the lefthand end of the lower line. What was the first electron there now finds one closer to it on its left than the second electron on its right and - - - I won’t bore you with a blow by blow account again, but to the left of B B’ the electrons are slightly closer together (x – δ) than to the right, all equally spaced and moving rightwards at a constant speed. They thus constitute a conventional negative current flow to the left, indicated by the blue line below the baseline in the lower graph in Figure 3. This is basically the Westerner view advocated by Dr. Neil MacEwan and similar to the analogy put forward by Dr. J. W. Mink of the IEEE. He points to the analogy of a droplet of water entering one end of a pipe, promptly forcing a drop out of the other end, but not of course the same drop.
Although I have described events on the upper and lower conductors separately, the boundary between the charged and uncharged sections of the line proceeds at the same rate on both conductors, and indeed this common boundary is the wavefront. Also, the foregoing might seem to imply a neat single line of free electrons, rather like peas in a peashooter. But the current of 1A flowing in the line consists of the passage past any point of a charge of one Coulomb (1C) per second. The charge on an electron is 1.602 ´ 10-19C, so there are 6.242 ´ 1018 electrons entering the lower and leaving the upper conductor per second. The speed with which the electrons are moving depends of how many of them are involved in carrying the current. In a line composed of very thin conductors, the current will presumably be carried by fewer electrons, travelling faster, than in one with very thick conductors. In either case, the velocity of the electrons is very much less than that of light. Assuming the same conductor spacing, these two lines would clearly have very different characteristic impedances, so the applied voltage needed to cause a current of 1A to flow would differ greatly. Nevertheless, the mechanism of propagation of the wavefront on the line is as described; but many electrons are involved rather than just the “first”, “second” etc. I can see no flaw in the Westerners’ explanation of how the negative charge appears on the lower conductor, resulting from the slight bunching up of the free electrons due to those entering the line. Professor Pepper’s contribution is to point out that the negative charge at any point on the lower conductor, past which the wavefront has travelled, will distribute itself over the circumference of that conductor, in proportion to the density of lines of electric flux terminating at each point around the circumference. MacEwan explains how the charge got there in the first place; Pepper explains, given that it is there, how it distributes itself over the surface of the conductor, as a result of the electric field between them. . We must assume that the deficit of electrons in the upper conductor (the positive charge, due to nuclei short of one attendant electron) distributes itself on the surface, around the circumference, in a mirror image of the distribution of the additional electrons on the lower conductor, there being no deficit of electrons within the upper conductor.
During the two nanoseconds the wavefront takes to traverse the line depicted in Figure 3, the source supplies 300nJ/ns of energy to the line - delivered by a current of 1A - and “thinks” it is connected to a 300W resistor. If the length of the line is infinite, the source will continue to “see” a 300W resistor indefinitely, and similarly if the end of the line in Figure 3 is terminated in a real 300W resistor. But the 300W dissipation in the resistor will not commence until 2ns after the switch is closed. During that time, 600nJ of energy is stored in the electric and magnetic fields of the line, and 300W will continue to be dissipated in the resistor for 2ns after the switch is subsequently opened.
It is worth considering in detail what happens at 2ns, when the wavefront reaches the open circuit end of the line in Figure 3. The conventional “explanation” is that an open circuit reflects the voltage in-phase and the current out of phase, while a short circuit reflects the voltage out of phase and the current in-phase. A more complete analysis follows. At time t = 2 ns the source has delivered 600 nJoules of energy to the line (up to this instant the termination, if any, is immaterial).
So the source has supplied 2 nCoulombs of charge to the line.
There is a uniform PD of 300 Volts along the line.
Q = CV, therefore 2 nCoulombs = C ´ 300 where C is the capacitance between the conductors. (It’s just unfortunate that C stands both for capacitance and Coulomb.)
Therefore capacitance of line = 2 ´ 10-9 / 300 = 6.667 ´ 10-12 Farads.
Energy stored = ½CV2 = ½ ´ 6.667 ´ 10 -12 ´ 3002 = 300 nJoules.
This is half the energy put in, so the rest must be in the magnetic field.
Energy stored = ½LI2.
Therefore ½ L ´12 = 300 nJoules.
Therefore L= 600 nHenries.
It may seem strange to talk of the inductance of a
600mm line which is open circuit, but it is quite logical if attention is
restricted purely to the instant t = 2ns, when there is current flowing in the
whole length of each conductor.
Now for a lossless line, as assumed,
Line impedance = (L/C)0.5 = (600 ´ 10-9 / 6.667 ´ 10-12)0.5 = 300W, which all seems to tie up. Half of the total energy supplied is stored in the electric field, and half in the magnetic.
In the open circuit case, after t = 2ns, what actually happens is this:-
The electron at the extreme righthand end of the lower conductor (the “last” electron) cannot move substantially to the right; it is at an open circuit. But the potential between the conductors at the input is still 300V, so the source continues to feed electrons into the lefthand end of the line. The “train” of electrons thus continues on its route, forcing the “second but last” electron, closer to the stationary last electron. When the spacing between them falls from (x - δ) to (x - 2δ), the force of repulsion between them becomes so great as to force the second but last electron to a halt. Successively, the third but last and other electrons also grind to a halt, so the current at the righthand end of the line is zero. The boundary between the moving and stationary electrons propagates to the left at the speed of light, even thought the speed of movement of individual electrons is comparatively very slow.
Simultaneously, when the right-most electron in the upper conductor starts to move to the left, becoming the last “truck” in a “goods train” of electrons steaming to the left, the nucleus of the right-most atom finds no electron arriving from the right to replace it. It is therefore unwilling to relinquish its electron entirely. The electron truck is therefore promptly uncoupled again from the train. The next atom in from the left is similarly unwilling to relinquish its electron, since no replacement is available from the right. The second from right electron is thus also uncoupled from the train, but the spacing between these two now stationary electrons is (x + 2δ). The train continues to the left, surrendering 1nC/ns of charge to the source, but getting steadily shorter as more and more electron trucks are abandoned on the rails, all at a spacing of (x + 2δ), until after three nanoseconds, the situation is as depicted in Figure 4.
Whereas electron spacing of (x ± δ) on the conductors equated to a potential difference between them of 300V, (x ± 2δ) equates to a potential difference 600V.
To the right of B B’, the deficit of electrons per unit length of the upper conductor is twice as great as to the left of B B’, indicated by the green line labelled e on the graph. Therefore the potential on the upper conductor, relative to the lower, is now 600V, as per the red line labelled V, while the current on the line to the right of B B’ is zero, since the electrons there are stationary. At four nanoseconds after switch S1 was closed, the length of the “goods train” has reduced to zero, all the “trucks” having been abandoned at a spacing of (x + 2δ), the electrons on the lower line are all at (x - 2δ) and the potential on the line is everywhere 600V. There is thus no potential difference between the two ends of the 300W source resistor, so the flow of conventional current being supplied by the source EMF ceases abruptly, leaving as many extra electrons per unit length on the lower conductor as the deficit of electrons per unit length on the upper conductor.
During the period t = 2ns to t = 4ns, the length of line in which current is flowing reduces steadily from 600mm to zero, and the inductance reduces similarly from 600nH to zero, converting the 300nJ of energy stored in the magnetic field to energy stored in the electric field. Together with the 600nJ supplied by the source during this period, this brings the total energy stored in the line to 1200nJ, all stored in the electric field.
The apparent flow of conventional current from the upper to the lower conductor is what is meant by displacement current, a concept introduced by Maxwell to retain consistency with Kirchoff’s First Law, in cases where there is no obvious circuit. Note that the notional displacement current only flows where the electric field strength (voltage gradient) is changing. In Figure 3, at one nanosecond after the switch S1 is closed, the voltage on the line to the left of B B’ is everywhere constant at 300V, and to the right of B B’ is everywhere zero. So the displacement current of 1A flows only at the point where the wavefront is. This implies that if the voltage step is instantaneous, the length of conductors between which the displacement current flows is infinitesimal, and the displacement current density therefore infinite. This is a major difficulty, and one reason why the concept of displacement current can be consigned to history.
Is there crucial difference between the 300W characteristic impedance of the line, and a capacitor, or a 300W resistor? At twice the voltage, a capacitor stores four times as much energy, and a resistor dissipates energy at four times the rate. In the case of the open circuit line, at 2ns, the energy stored at 300V is 600nJ, and after 4ns, at twice the voltage, the stored energy only doubles to 1200nJ. But after 2ns, only 300nJ is stored in the capacitance of the line, the other half of the energy supplied being stored in its inductance. Thus the energy stored in the line’s capacitance quadruples when the voltage doubles to 600V. During the four nanoseconds, 1200nJ has been dissipated in the internal resistance of the source, so the total energy supplied is 2400nJ.
At four nanoseconds, the energy stored in the line is ½CV2 = ½ ´ 6.667 ´ 10-12 ´ 6002 = 1200nJ, exactly the same as for a discrete capacitor of the same capacitance as the line. The crucial difference is that if the discrete capacitor is connected to a 300W resistor, it will deliver an initial 2A, decaying exponentially whereas the charged line will deliver a constant 1A for 4ns, ceasing abruptly thereafter. This is why a delay line rather than a capacitor is used to supply anode power to a magnetron, to form a pulse flat-top pulse in a radar transmitter.
An explanation of the appearance of negative charge on the lower conductor of a line, on which a Transverse Electric Magnetic wavefront from a matched voltage source, is propagating, has been presented. This explanation has been extended to the case of an open circuit line of finite length. An explanation, in similar terms, of what happens when a TEM wave from a matched ideal current source propagates along a short-circuited line, can also be developed. However, the situation looks, at first sight, a little more complicated, and I am still thinking about it. So, in common with all the best textbooks, I can only say that this question “is left as an exercise for the reader".
The discussion above applies in the case of an ideal lossless line, but applies in broad principle to practical lines. However, there are two unwanted characteristics of real lines which have been ignored: attenuation and dispersion. These are interesting, important and well worth studying and becoming familiar with. But there is enough meat in this article for now, so I hope to look at these topics later. For the present then, here endeth the first lesson.
1 Newnes Practical RF Handbook, 4th Edition 2007, ISBN-13: 978-0-7506-8039-4, ISBN-10: 0-7506-8039-3 Butterworth-Heinemann
2 Displacement current, and Maxwell’s Equations generally, seem to have fallen into disfavour since I was at college: at one company, where I was working as a contract engineer on the development of epirbs (emergency position-indicating radio beacons), the manager to whom I reported (younger than me) stoutly maintained that he had never heard of displacement current, and didn’t believe such a thing existed. In a sense he is right of course; it certainly does not account for the negative charge on the lower conductor. Displacement current is a notional flow of current which is used to account for an apparent flow of current where there is no conducting circuit (e.g. “through” a capacitor when the voltage gradient in the dielectric changes). Displacement current is proportional to the rate of change of the electric field strength D in volts/m:- displacement current = k dD/dt in the one dimensional case, it’s rather more complicated in the general three dimensional case - see Maxwell’s Equations.
3 (or very nearly at the speed of light, in an open air-line. In a coax the speed is only about two thirds that of light, in delay cable used to feed the Y plates of an oscilloscope, it is much lower still, while in a loaded telephone cable it is only about one twentieth of the speed of light).
The author gratefully acknowledges helpful discussion and comment, in the preparation of this article, from his old colleague and longtime friend M. H. Gross, C.Eng., MIEE.
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