4.4.1 How Do Transmission Lines Work?
Heaviside Came Close, But Missed An Important Point
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The transatlantic telegraph cable was “coaxial.” Insulation (a kind of rubber called gutta-percha) enveloped the center conductor and was surrounded in turn by a braided conducting ground sheath. It may be a bit easier to understand transmission lines by considering a pair of wires.
Telephone and Ethernet signals travel along twisted pairs of wires to better confine the signals so they won’t couple to adjacent twisted pairs. Let’s untwist a pair and consider a voltage sine wave signal moving along the transmission line, as in Figure 4.40. The signal propagates at the speed of light. At a particular planar cross-section, the electrons are a bit compressed in the top wire resulting in a net negative charge. In the bottom wire, the electrons are a bit rarefied or stretched out, resulting in a net positive charge.
Consider the fields in this cross-section of the twin-lead transmission line. The electric fields originate on the positively charged wire and terminate on the negatively charged wire (just as we saw back in Figure 3.18). At the same time, the opposing currents in the wires give rise to magnetic field loops around the wires (just as we saw in Figures 3.23 and 3.24). In the twin-lead transmission line of Figure 4.40, the electric and magnetic fields act at right angles to each other, conveying energy along the line. The fields reach out from the line and can couple to adjacent lines. Twisting the wires of each transmission line together, however, minimizes the interference because it randomizes whether the signal is adding constructively or destructively to adjacent wires.
Oliver Heaviside discussed how voltage (V) and current (I) work in transmission lines in the second volume of his Electrical Papers.
The electric and magnetic forces are perpendicular, and so are V and I regarded as vectors, [i.e., their elements E and H are perpendicular]. The product VI is the energy current ; their ratio is the important quantity, the impedance [[i]].
The units of electric field are volts per meter; magnetic field is amps per meter. Their product is volt amps per meter squared or power per area.
Provided electric and magnetic fields act orthogonally or at right angles to each other, the Poynting-Heaviside theory is identical to the traditional result that power is the product of voltage (V) and current (I), P = VI, but on a per area basis.
Maxwell published his “Dynamical Theory of the Electromagnetic Field” in 1865 [[ii]], and the first edition of his Treatise appeared in 1873. Maxwell recognized that an electromagnetic wave has electric and magnetic energy in equal amounts [[iii]]. Heaviside made the same point [[iv]]. In August 1876, the twenty-six-year-old Heaviside published a paper with the intriguing title, “On the Extra Current,” explaining how signals propagate on transmission lines [[v]].
Oliver Lodge (1851–1941) explained the crux of Heaviside’s brilliant approach in a moving obituary for his deceased colleague. For any true wave, there is an equality of potential and kinetic energy. In the case of an electromagnetic wave:
…the electric and magnetic energies are required to be equal. If one overpowers the other we have distortion, varying rates of transmission, or no rate of transmission at all, nothing but diffusion if the overpowering is excessive. In Kelvin’s theory it was excessive, because the inductance was considered zero, that is to say, the magnetic field was neglected ; the electric field alone was taken into account. Heaviside pointed out that the magnetic field was really doing the work, though greatly overpowered by the electric, especially in cables of large capacity. By adding self-inductance, magnetic energy could be increased and made more equal to the electric energy ; by leakage the overplus of electric energy could be still further reduced, though less economically. If the two energies could be made quite equal we should have true distortionless waves, with every feature of the signal reproduced at the far end, subject to nothing but attenuation [[vi]].
That was the justification for Heaviside’s “Telegrapher’s Equation.” Wave propagation requires electric and magnetic energy in equal amounts. Since electric energy is stored in capacitance and magnetic energy in inductance, getting their ratio correct is crucial to good cable behavior.
Control the balance of electric and magnetic energy with a careful selection of inductance and capacitance, and all is well. Ignore this principle, and dispersion distorts the signal, rendering a transmission line unusable for voice communication after a relatively short distance. Even more narrowband signals (like the dots and dashes of telegraphy) become difficult to resolve if the cable is long enough, as the Atlantic cable pioneers discovered.
Where Thomson’s model described a diffusion of charge into a cable and ignored inductance, Heaviside showed a true distortionless wave propagation was possible, in most cases, by adding some extra inductance to a transmission line. If we consider four parameters, the resistance R, the inductance L, the capacitance C, and the conductance (or shunt inverse resistance) G of a unit length (dx) of transmission line, then if the Heaviside condition, G/C = R/L, is satisfied, we eliminate dispersion and get propagation without distortion. The mathematical details will take us too far off the trail, but are available elsewhere [[viii], [ix]].
Heaviside thought deeply about power and energy in transmission lines. Elsewhere in his Electrical Papers, he says:
When oppositely travelling waves meet, the resultant V is the sum of the two V’s, and the resultant I the sum of the two I’s.
Thus, if the waves be so shaped as to fit, then, on coincidence, V is doubled and I is annulled. The energy is then all electric. But if the electrifications be opposite, V is annulled and I is doubled, on coincidence. The energy is then all magnetic. On emergence, however, the two waves are unaltered, save in the attenuation that is always going on [[xi]].
In this description, Heaviside comes so very close to capturing an important but overlooked aspect of how electromagnetism behaves. In the interaction Heaviside describes, the two opposite traveling waves exchange their energy. When the energy becomes entirely electric or entirely magnetic, it becomes static or stationary. The voltage and current waves continue propagating at the speed of light (which may be different from the free space speed of light, depending on the line impedance). Heaviside failed to note, however, that after the waves emerge from their superposition, they have swapped their energy: the forward traveling wave conveys the energy originally in the reverse wave and vice versa. The details of this process were not fully explained until over a hundred years later [[xii], [xiii], [xiv]].
An imbalance in electric and magnetic energy on a transmission line distorts a signal propagating in a particular direction as some of the energy slows down and falls behind the wavefront. That is the foundation of Heaviside’s Telegrapher’s Equation. The next step, the step Heaviside failed to take, is that the imbalances resulting from an interference between two waves propagating in different directions similarly slow down energy, leading to energy being exchanged from one wave to the other. And that behavior demonstrates unequivocally that fields and energy are two different phenomena. They cooperate yet follow different trajectories. That fields guide energy is only now beginning to be understood. We will pick up our discussion of this simple yet subtle principle in Chapter 6.
Next time: 4.4.2 The Catt Question: How Does a Net Charge Get On A Wire?
Full Table of Contents [click here]
Chapter 4 Electromagnetism Comes of Age
4.5 An Introduction to Electromagnetic Models
4.5.1 Potentials and Actions at a Distance
4.5.2 Jefimenko & Lorentz
4.5.3 A Synthesis
4.6 Hertz & Radiation Fields
4.7 How Does Radiation Work?
4.8 Summary & Conclusions
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References
[[i]] Heaviside, Oliver, Electrical Papers, vol. 2, New York: Macmillan and Company, 1894, p. 126. Note, I used substituted the modern “I” for current instead of the “C” Heaviside used to denote current, and omitted his Lv for impedance. We’ll get into such technical details in chapter 6.
[ii] James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, §73 (Edinburgh: Scottish Academic Press, 1982), p. 70. Originally published as “A Dynamical Theory of the Electromagnetic Field,” Philosophical Transactions of the Royal Society of London CLV (1865).
[[iii]] Maxwell, James Clerk, A Treatise on Electricity and Magnetism, London: Oxford University Press, vol. II, 3rd ed., §792, 1892, pp. 440-441.
[[iv]] Heaviside, Oliver, Electrical Papers, vol. 1, London: The Electrician Publishing Company, 1892, p. 379.
[[v]] Heaviside, Oliver, “On the Extra Current,” Electrical Papers, vol. 1, London: The Electrician Publishing Company, 1892, pp. 53-61.
[[vi]] Lodge, Oliver, “Oliver Heaviside. An Appreciation : The Personal Equation : The Work of a Genius Elucidated,” The Electrician, February 13, 1925, p. 174.
[[vii]] See: https://en.wikipedia.org/wiki/Telegrapher's_equations#/media/File:Telegrapher_equation.gif; see also: https://twitter.com/j_bertolotti/status/1172517281374572551
[[viii]] Schantz, Hans G., The Art and Science of Ultrawideband Antennas, 2nd ed., Boston: Artech House, 2015, pp. 129-133.
[[ix]] Williams, Al, “Loading Coils, The Heaviside Condition, and Pupin Coils, Hackaday, June 19, 2020. See: https://hackaday.com/2020/06/19/loading-coils-the-heaviside-condition-and-pupin-coils/
[[x]] See: https://en.wikipedia.org/wiki/Telegrapher's_equations
[[xi]] Heaviside, Oliver, Electrical Papers, vol. 2, London: The Electrician Publishing Company, 1892, p. 126. Note, I used substituted the modern “I” for current instead of the “C” Heaviside used to denote current, and omitted his Lv for impedance. We’ll get into such technical details in chapter 6.
[[xii]] Schantz, Hans G., “On the Superposition and Elastic Recoil of Electromagnetic Waves,” FERMAT, Vol 4 No 2, Jul-Aug, 2014. See also: https://arxiv.org/abs/1407.1800.
[[xiii]] Schantz, Hans G., “On Energy Flow in Standing Waves,” 2014 Antenna Applications Symposium, Allerton Park, IL, September 25, 2014. See: https://www.researchgate.net/publication/265849380_On_Energy_Flow_in_Standing_Waves
[xiv] Schantz, Hans G., (2018) Energy velocity and reactive fields Phil. Trans. R. Soc. A. 3762017045320170453 http://doi.org/10.1098/rsta.2017.0453 See: https://royalsocietypublishing.org/doi/10.1098/rsta.2017.0453