Experimental and Analytical Studies on Electromagnetic Wave Propagation in DC Circuits

Abstract

It is well known that two pieces of electrical conductors behave as a waveguide when they are employed to transmit AC signals. Some experimental and analytical studies are reported in this paper to demonstrate that two pieces of electrical conductors also behave as a waveguide when they are employed to transmit DC signals in practice. Specifically, the speed of wave propagation is measured in the experiments, and the analytical studies are based on the theory of transmission line.

1. Introduction

It is well known that two pieces of electrical conductors behave as a waveguide when they are employed to transmit AC signals. For instance, the inner conductor and outer conductor of a co-axial cable jointly guide dynamic electromagnetic wave propagation inside the co-axial cable, as illustrated by numerous classic textbooks [1,2]. When two pieces of electrical conductors are employed to transmit a DC signal, the two pieces of conductor do not behave as a waveguide in theory. A DC voltage produces a static electric field, and a DC current produces a static magnetic field. According to Maxwell’s equations, a static electric field and a static magnetic field are de-coupled from each other, and consequently, they do not jointly support electromagnetic wave propagation. In theory, a DC signal starts at “time = −∞” and ends at “time = +∞”. Nevertheless, in practice, a DC signal must be turned on at a certain “turn-on moment” and turned off at a certain “turn-off moment”. Generally, a DC signal cannot be established instantaneously at the turn-on moment; rather, a transient state with a certain time span is necessary for the DC signal to be established. Similarly, a “turn-off transient state” is usually necessary for the DC signal to vanish gradually. Since the “turn-on transient state” and “turn-off transient state” both rely on electromagnetic wave propagation, the DC signal in between these two transient states ought to be sustained by electromagnetic wave propagation as well. Therefore, two pieces of electrical conductors behave as a waveguide when they are employed to transmit DC signals in practice. The electromagnetic wave that supports DC transmission is named “stationary wave” in [3].

In this paper, some experimental and analytical studies are reported to demonstrate the electromagnetic wave propagation in DC circuits. In the experiments of this paper, electromagnetic wave propagation is explicitly demonstrated when a DC signal is turned on and turned off. The analytical studies in this paper based on the theory of transmission line further demonstrate that the electromagnetic waves do not vanish in the DC steady state. In other words, the electric field and magnetic field are coupled with each other when a DC signal is established, and they remain coupled with each other after the DC signal is established. The experimental and analytical studies jointly indicate that two pieces of conductors behave as a waveguide or transmission line when they are employed to transmit a DC signal in practice. The practical applicability of the studies in this paper is vague. The state-of-the-art technology for designing, analyzing, fabricating, and testing DC circuits appears robust and reliable without taking electromagnetic wave propagation into account. The authors therefore have no intention to suggest incorporating electromagnetic waves into the design, analysis, or test of practical DC circuits. Although the practical applications are unclear, the authors hope that the studies in this paper may lend certain insights toward understanding the transition from the transient state to steady state in DC circuits.

The rest of this paper is organized as follows. In Section 2, some experimental results are presented to demonstrate the electromagnetic wave propagation when a DC signal is turned on and turned off. Analytical studies are conducted in Section 3 to demonstrate that the electromagnetic wave propagation does not vanish in the DC steady state. Finally, conclusions of our experimental and analytical studies are drawn in Section 4.

2. Experimental Study on Electromagnetic Wave Propagation in DC Circuits

A benchmark experiment was constructed, as shown in Figure 1. A signal generated by a waveform generator is split into two signals that are identical to each other. Afterwards, these two signals are transmitted to an oscilloscope via two paths, respectively. Over one path, one of the two signals is delivered to Channel 1 of the oscilloscope using a “through cable”, which is a BNT co-axial cable. The other path is made of one cable connected to the waveform generator and one cable connected to Channel 2 of the oscilloscope, as displayed in Figure 1. The signal generated by the waveform generator is a square wave, whose voltage level alternates between 0 and 1 Volt with a frequency of 50 Hz.

A snapshot of the oscilloscope’s screen is shown in Figure 2, with “5 ms per horizontal division” as the temporal scale of the display. Unsurprisingly, the two waveforms, detected by Channel 1 and Channel 2 of the oscilloscope, respectively, appear almost identical to each other. When the temporal scale of the display is reduced to “100 ns per horizontal division” in Figure 3, it is observed that the rising edges of two waveforms look different from each other. When the temporal scale of the display is further reduced to “10 ns per horizontal division” in Figure 4, a time delay of about 10 ns is visible between the two rising edges. This time delay is apparently due to the difference between the physical lengths of two transmission paths, which is approximately 1.8 m. The speed of “$(1.8\mathrm{m})/(10\mathrm{ns})=1.8\times {10}^8\mathrm{m}/\mathrm{s}$” must be associated with electromagnetic wave propagation. Since the cables in Figure 1 include dielectric materials, it is reasonable that the speed of electromagnetic wave propagation over these cables is lower than $3\times {10}^8\mathrm{m}/\mathrm{s}$, the speed of light in free space. Figure 5 and Figure 6 are the counterparts of Figure 3 and Figure 4. The falling edges of Figure 2 are visualized in Figure 5 and Figure 6, with “100 ns per horizontal division” and “10 ns per horizontal division”, respectively. A time delay of about 10 ns is obvious between the two falling edges in Figure 6. Therefore, the falling edges in Figure 2 travel from the source (i.e., the waveform generator) to the load (i.e., the oscilloscope) via electromagnetic wave propagation.

The square waveform in Figure 2 can be considered as a DC signal being turned on and off periodically. Each square pulse consists of three parts: a rising edge, a DC signal with a voltage level of 1 Volt, and a falling edge. The rising edge embodies a transient state for the DC signal to be established. The falling edge embodies a transient state for the DC signal to vanish gradually. Because the two transient states both rely on electromagnetic wave propagation, it is reasonable to hypothesize that the steady state (which resides in between the two transient states in time) also relies on electromagnetic wave propagation. In practice, a DC signal must be turned on at a certain “turn-on moment” and turned off at a certain “turn-off moment”. Consequently, a DC signal must be associated with a certain “turn-on transient state” and a certain “turn-off transient state” in practice, generally speaking. In Figure 2, the temporal length of the DC steady state (in between the “turn-on transient state” and “turn-off transient state” in time) is as short as 10 ms, whereas a practical DC signal’s steady state usually lasts for a much longer time (one hour, one day, or even one month, for instance). Nevertheless, the temporal length of steady state does not prevent one square pulse in Figure 2 from exemplifying a practical DC signal. Therefore, it is a rational statement that the three cables in Figure 1 behave as waveguides when they are employed to transmit DC signals as well as AC signals.

On the basis of the benchmark experiment in Figure 1, two long wires (one in red color and one in yellow color) are placed between the waveform generator’s cable and oscilloscope’s cable. Each wire has a physical length of about 2.5 m. The two wires are placed approximately parallel to each other, as shown in Figure 7. All the other experimental parameters remain unchanged with respect to the benchmark experiment in Figure 1.

The two waveforms detected by the oscilloscope can hardly be differentiated from each other with “5 ms per horizontal division”, as displayed in Figure 8. With “10 ns per horizontal division”, the rising edges and falling edges of the waveforms are demonstrated in Figure 9 and Figure 10, respectively. A time delay of about 20 ns is obvious between the two rising edges in Figure 9 as well as between the two falling edges in Figure 10. Compared with the experimental results of Figure 1, the extra time delay of about 10 ns is undoubtedly due to the two 2.5 m-long wires. Thus, the two long wires construct a waveguide as the three cables in Figure 1 do. It appears that the rising edges, falling edges, and DC signals travel along the two wires in the form of electromagnetic wave propagation and the propagation speed is fairly close to the speed of light in free space.

According to the experimental results in Figure 8, Figure 9 and Figure 10, the two long wires in Figure 7 construct an excellent TEM waveguide in the frequency range of [0, 10 MHz]. Thus, as far as the spectrum of [0, 10 MHz] is concerned, the two wires in Figure 7 can be modeled as a transmission line.

The transient state represented by the rising edges in Figure 9 can be considered as “a process of charging” during which the source (i.e., the waveform generator) injects power into the waveguides. The transient state represented by the falling edges in Figure 10, in contrast, can be considered as “a process of discharging” during which the energy stored in the waveguides is released to the load (i.e., the oscilloscope). During the steady state (which resides in between the two transient states in time), neither the process of charging nor the process of discharging stops. Instead, these two processes balance each other such that the electric field and magnetic field over the waveguides appear static. “The process of charging” and “the process of discharging” are demonstrated by several analytical examples in Section 3.

During the steady state, there is a DC voltage of 1 Volt between the red wire and yellow wire in Figure 7. The two channels of the oscilloscope have an equivalent input impedance of a 1 MΩ resistor in parallel with a 16-pF capacitor. Therefore, there is a DC current of about 1 μA over the red wire and yellow wire during the steady state. The DC current distribution and charge distribution over the two wires establish the boundary conditions required by the electromagnetic wave propagation, consistent with the fact that the AC current distribution and charge distribution over the two wires establish the boundary conditions required by the electromagnetic wave propagation when the two wires are employed to transmit AC signals.

The experimental setup in Figure 11 has only one difference from the experimental setup of Figure 7. In Figure 7, the two long wires are placed over the surface of a desk; in Figure 11, the two long wires go through multiple “tunnels”, which are copper pipes with circular cross-section. Since the wires are covered by a certain plastic material, there is no electrical contact between the wires and copper pipes. The experimental results obtained by the setup in Figure 11 reinforce the conclusions drawn above.

The screen snapshots in Figure 12, Figure 13 and Figure 14 portray the two waveforms detected by the oscilloscope, their rising edges, and their falling edges, respectively. The phenomena exhibited by Figure 12, Figure 13 and Figure 14 have little difference from those exhibited by Figure 8, Figure 9 and Figure 10. As demonstrated by the experiments in Figure 1 and Figure 7, the two long wires can be modeled as a transmission line. In Figure 11, the two wires loaded by copper pipes still behave as a transmission line. Compared with the transmission line in Figure 7, the loading of copper pipes in Figure 11 changes the characteristics of the transmission line, such as $C^{\prime }$ (the capacitance per unit length) and $L^{\prime }$ (the inductance per unit length) along the transmission line. However, from Figure 13 and Figure 14, the value of $1/\sqrt{L^{\prime }{C}^{\prime }}$, which determines the propagation speed, remains unchanged with respect to the propagation speed in Figure 9 and Figure 10. This is reasonable as the propagation speed of a TEM waveguide solely depends on the propagation medium. The wave propagation between the two wires occurs primarily in the air in both Figure 7 and Figure 11 (it seems that the dielectric material of desk does not have any strong impacts). It is therefore not surprising that the speed of light in free space is demonstrated in both Figure 7 and Figure 11.

3. Analytical Studies on Electromagnetic Wave Propagation in DC Circuits

In the experiments of Section 2, electromagnetic wave propagation is explicitly demonstrated when a DC signal is turned on and turned off. It is therefore hypothesized that the electromagnetic wave propagation does not vanish in the DC steady state (which resides in between the “turn-on transient state” and “turn-off transient state” in time). In this section, some analytical studies are conducted to demonstrate the electromagnetic wave propagation in the DC steady state.

In theory, electromagnetic wave propagation does not exist in DC circuits. A DC voltage produces a static electric field, and a DC current produces a static magnetic field. According to Maxwell’s equations, a static electric field and a static magnetic field are de-coupled from each other, and consequently, they do not jointly support electromagnetic wave propagation. Nevertheless, the Poynting’s Theorem has been proved to hold true in DC circuits by many researchers [4,5,6,7,8,9]. The Poynting vector, which is the cross product between the electric field vector and magnetic field vector, is directly associated with the propagation of electromagnetic energy density with a speed close to the speed of light [10]. Thus, the validity of Poynting’s Theorem implies that the electrical power is transported by electromagnetic wave propagation in DC circuits (indeed, while it is undoubted that the transportation of electrical power in AC circuits is completely characterized by the Poynting vector, there are still controversies regarding how the electrical power is transported in DC circuits [11,12,13]). The contradiction laid out above (i.e., the contradiction between the de-coupling of static fields and the validity of Poynting’s Theorem) might be addressed by the hypothesis in Section 2: The electromagnetic wave propagation in the transient state remains active after the circuit reaches the DC steady state. Apparently, solving Maxwell’s equations merely in the DC steady state (such as the work in [4,5,6,7]) is insufficient to portray the transition from the transient state to DC steady state. Rather, modeling the transition from the transient state to DC steady state calls for the solutions to Maxwell’s equations in a broad frequency range with DC covered, which is a challenging task. In this section, the transition from the transient state to DC steady state is analyzed by resorting to the theory of transmission line, in order to avoid the heavy-duty computations required to solve Maxwell’s equations in a wide frequency range. Though the analysis based on the theory of transmission line is not as rigorous as the analysis based on Maxwell’s equations (such as the analyses in [4,5,6,7]), it is much simpler and thus is capable of clearly unveiling the power propagation in the transient state as well as the steady state. The primary objective of this section is to demonstrate that the power and energy of a DC circuit could be logically interpreted by electromagnetic waves. We are endeavoring to apply FDTD [14] and FEM [15] to conduct the full-wave analysis based on solving Maxwell’s equations.

As argued in Section 2, two pieces of conductor can be modeled as a transmission line when they are employed to transmit AC signals as well as DC signals. This fact motivates the circuit diagram in Figure 15. In Figure 15, a load resistor $R_L$ is connected to a battery via a piece of transmission line deployed along the z axis. The battery has a DC voltage of $v_{DC}=1.5\mathrm{Volt}$ and an internal resistance of $R_b$. The transmission line is assumed to be lossless, its physical length is assumed to be l, its characteristic impedance (which is purely resistive for a lossless transmission line) is assumed to be $Z_0=50\mathsf{\Omega }$, and the phase velocity of wave propagation over the transmission line is assumed to be c, the speed of light in free space. In order to facilitate the analysis, the transmission line in Figure 15 is assumed to exhibit no dispersion; in other words, its characteristic impedance and phase velocity are constants without any frequency dependence. It is observed that the transmission lines in Section 2 have little dispersion or loss in the frequency range of [0, 10 MHz]. As far as the signals studied in Section 2 are concerned, most of the energy is held in the frequency range of [0, 10 MHz]. Thus, the ideal model in Figure 15 ought to be capable of demonstrating the fundamental physical phenomena, though it does not precisely characterize the loss or dispersion of the transmission lines in Section 2.

When the load in Figure 15 is a matched load $Z_0=50\mathsf{\Omega }$ and the internal resistance of battery is $Z_0=50\mathsf{\Omega }$ as well, the circuit problem is analyzed in Figure 16. At t = 0 (that is, time is zero), the two switches are turned on, and a wave termed as “Wave 0” is launched. Specifically, the voltage and current of “Wave 0” are

$$\left\{\begin{array}{l}{v}_0(z,t)={\displaystyle \frac{v_{DC}}{2}}H\left(t-{\displaystyle \frac{z+l}{c}}\right)\\ i_0(z,t)={\displaystyle \frac{v_{DC}}{2Z_0}}H\left(t-{\displaystyle \frac{z+l}{c}}\right)\end{array}\right.$$

(1)

where $H(\cdot )$ is the Heaviside step function

$$H(t)=\left\{\begin{array}{ll}0& \mathrm{when}t<0\\ 1& \mathrm{when}t>0\end{array}\right.$$

(2)

Following the classic theory of transmission line [16], the polarity of voltage drop $v_0$ is defined to be from the top line to the bottom line, the polarity of current flow $i_0$ is defined to be toward +z over the top line, and the polarity of current flow $i_0$ is defined to be toward −z over the bottom line, as denoted in Figure 16. At $t=0$, the voltage and current in Equation (1) satisfy the Telegrapher’s equations as well as the boundary conditions. Over the transmission line, the voltage represents the electric field, the current represents the magnetic field, and the voltage–current product represents the magnitude of the Poynthing vector. In other words, though the waves in this section are characterized by voltage and current, they are essentially electromagnetic waves. “Wave 0” propagates toward the +z direction, as evidenced by the experimental results in Section 2. When “Wave 0” arrives at the load at $t=\delta =l/c$, the transient state is over, and the entire circuit reaches the steady state. The battery starts supplying a constant power of $\left(v_{DC}/2\right)\left(v_{DC}/(2Z_0)\right)={(v_{DC})}^2/(4Z_0)$ at $t=0$, whereas the load starts consuming a constant power of ${(v_{DC})}^2/(4Z_0)$ at $t=\delta$. During the temporal range of $t\in [0,\delta ]$, the battery “charges” the transmission line, i.e., the power supplied by the battery during the period of $t\in [0,\delta ]$ is stored over the transmission line. To be specific, the amount of energy stored over the transmission line, defined as $E_0$, is the amount of energy supplied by the battery in the transient state, that is, when $t\in [0,\delta ]$:

$$E_0={\displaystyle \frac{{(v_{DC})}^2}{4Z_0}}\delta .$$

(3)

The battery continues charging the transmission line after t = δ, because the matched load does not provide any feedback to the battery and the battery has no reason to alter the process of charging. Although the voltage appears to be static and the current appears be static in the steady state, electrical power is transported from the battery to the load after $t=\delta$ via electromagnetic wave propagation. After t = δ, the amount of energy stored over the transmission line stays unchanged, since the battery supplies energy to the electromagnetic wave with the same rate as the load accepts energy from the electromagnetic wave.

Figure 17 has only one difference in comparison with Figure 16: The load is a matched load (i.e., $R_L=Z_0=50\mathsf{\Omega }$) in Figure 16 and is a short (i.e., $R_L=0$) in Figure 17. The circuit analysis for Figure 17 is fundamentally the same as the circuit analysis for Figure 16, though more complicated. At t = 0, “Wave 0” is launched, which is specified by blue color in Figure 17. When $t\in [0,\delta ]$, “Wave 0” propagates in the same manner as in Figure 16. When “Wave 0” arrives at the load (which is a short in Figure 17) at $t=\delta$, “Wave 1⁻” (specified by brown color in Figure 17) emerges in order to satisfy the boundary condition at $z=0$. The sum of Wave 0’s voltage and Wave 1⁻’s voltage is enforced to be zero at $z=0$. “Wave 1⁻” propagates toward the −z direction and arrives at the battery at $t=2\delta$. The battery supplies power to the transmission line during the period of $t\in [0,2\delta ]$, and it stops supplying power after $t=2\delta$. The amount of energy stored over the transmission line is $2E_0$ at $t=2\delta$, where $E_0$ is defined in Equation (3). “Wave 0” is supported by the battery during the period of $t\in [0,2\delta ]$, and it loses the battery’s support at $t=2\delta$. Therefore, the complete mathematical expression of “Wave 0” in Figure 17 is

$$\left\{\begin{array}{l}{v}_0(z,t)={\displaystyle \frac{v_{DC}}{2}}\left\{H\left(t-{\displaystyle \frac{z+l}{c}}\right)-H\left(t-{\displaystyle \frac{z+3l}{c}}\right)\right\}\\ i_0(z,t)={\displaystyle \frac{v_{DC}}{2Z_0}}\left\{H\left(t-{\displaystyle \frac{z+l}{c}}\right)-H\left(t-{\displaystyle \frac{z+3l}{c}}\right)\right\}\end{array}\right.$$

(4)

Please note that the mathematical expression of the blue-colored “Wave 0” in Figure 17 is incomplete (it is the correct expression when $0<t<2\delta$). Because “Wave 1⁻” results from “Wave 0”, “Wave 1⁻” vanishes soon after “Wave 0” vanishes. Specifically, the voltage and current associated with “Wave 1⁻” are

$$\left\{\begin{array}{l}{v}_1^{-}(z,t)=-{\displaystyle \frac{v_{DC}}{2}}\left\{H\left(t+{\displaystyle \frac{z-l}{c}}\right)-H\left(t+{\displaystyle \frac{z-3l}{c}}\right)\right\}\\ i_1^{-}(z,t)={\displaystyle \frac{v_{DC}}{2Z_0}}\left\{H\left(t+{\displaystyle \frac{z-l}{c}}\right)-H\left(t+{\displaystyle \frac{z-3l}{c}}\right)\right\}\end{array}\right.$$

(5)

Similarly, the mathematical expression of the brown-colored “Wave 1⁻” is incomplete in Figure 17 (it is the correct expression when $\delta <t<3\delta$). After $t=2\delta$, the boundary conditions at $z=-l$ include “voltage being 0” and “current being $v_{DC}/Z_0$.” To satisfy the boundary conditions at $z=-l$, a new wave termed as “Wave 1⁺” is generated at $t=2\delta$; in Figure 17, “Wave 1⁺” is in red color. Although “Wave 1⁺” and “Wave 0” appear alike, they are actually quite different from each other: “Wave 0” is supported by the battery, whereas “Wave 1⁺” is supported by “Wave 1⁻”. At $t=2\delta$, the transient state is over and the circuit reaches the steady state. In the steady state, the voltage is 0 and the current is $v_{DC}/Z_0$ throughout the transmission line. However, electromagnetic waves do not vanish in the steady state. When “Wave 1⁺” arrives at the load at $t=3\delta$, “Wave 2⁻” (in green color) is generated. Furthermore, when the green-colored “Wave 2⁻” arrives at the battery at $t=4\delta$, it generates “Wave 2⁺” that travels toward the +z direction. With the marching of time, “new waves” keep being generated sequentially by “old waves” in the steady state. In the steady state, the amount of energy over the transmission line stays unchanged as $2E_0$, stored in the form of two groups of waves: One group consists of two (rather than one) waves traveling toward the +z direction and the other group consists of two (rather than one) waves traveling toward the −z direction. The two groups of waves sustain each other, as illustrated in Figure 17. It is worth noting that there is one +z traveling wave but no −z traveling wave in Figure 16. Thus, unsurprisingly, the energy stored over the transmission line in Figure 16 is $E_0$ and the energy store over the transmission line in Figure 17 is $2E_0$.

In Figure 18, the load of the transmission line is an open (i.e., $R_L=\infty$). Everything else in Figure 18 remains unchanged compared with Figure 16 and Figure 17. The circuit analysis in Figure 18 is fundamentally the same as the circuit analysis in Figure 17. When the circuit reaches the steady state at $t=2\delta$ in Figure 17, the DC voltage over the transmission line is $v_{DC}$ and the DC current over the transmission line is zero. In contrast, the DC voltage over the transmission line is zero and the DC current over the transmission line is $v_{DC}/Z_0$ in Figure 16. Nevertheless, Figure 16 and Figure 17 share the same amount of stored energy in the steady state; we are currently endeavoring to verify it experimentally.

The scenario with $R_b=Z_0=50\mathsf{\Omega }$ and $R_L=2Z_0=100\mathsf{\Omega }$ is analyzed in Figure 19. “Wave 0” is the only wave propagating over the transmission line when $0<t<\delta$ in Figure 19. At $t=\delta$, “Wave 1⁻” emerges, and “Wave 1⁻” carries a smaller power than “Wave 0” does. The difference between Wave 1⁻’s power and Wave 0’s power, which is $2{(v_{DC})}^2/(9Z_0)$, is consumed by the load resistance, obviously. When “Wave 1⁻” arrives at the battery at $t=2\delta$, “Wave 1⁺” is generated, and “Wave 1⁺” carries a larger power than “Wave 1⁻” does. Apparently, the difference between Wave 1⁺’s power and Wave 1⁻’s power, which is $2{(v_{DC})}^2/(9Z_0)$, is supplied by the battery. The entire circuit reaches the steady state when $t=2\delta$. After $t=2\delta$, the battery supplies a power of $2{(v_{DC})}^2/(9Z_0)$ steadily and the load consumes a power of $2{(v_{DC})}^2/(9Z_0)$ steadily. The DC power supplied by the battery is transported to the load via two groups of electromagnetic waves, one traveling toward the +z direction and the other traveling toward the −z direction. During the period of $t\in [0,2\delta ]$, the amount of energy supplied by the battery is $2E_0$, and the amount of energy consumed by the load is $(8/9)E_0$. Thus, the energy stored over the transmission line in the steady state is $2E_0-(8/9)E_0=(10/9)E_0$. The stored energy of $(10/9)E_0$ can be decomposed into $E_0+E_0/9$: $E_0$ is stored in the form of +z traveling waves and $E_0/9$ is stored in the form of −z traveling waves.

In Figure 16, Figure 17, Figure 18 and Figure 19, the internal resistance of the battery, $R_b$, is assumed to be $Z_0$. As a result, the transient state is over at $t=2\delta$ in Figure 16. If $R_b\ne Z_0$ and $R_L\ne Z_0$, the transient state is infinitely long in time. One example is depicted in Figure 20, with $R_b=2Z_0=100\mathsf{\Omega }$ and $R_L=\infty$. When the switches are turned on at $t=0$ in Figure 20, “Wave 0” is launched over the transmission line with the following expression.

$$\left\{\begin{array}{l}{v}_0(z,t)={\displaystyle \frac{v_{DC}}{3}}\left\{H\left(t-{\displaystyle \frac{z+l}{c}}\right)-H\left(t-{\displaystyle \frac{z+3l}{c}}\right)\right\}\\ i_0(z,t)={\displaystyle \frac{v_{DC}}{3Z_0}}\left\{H\left(t-{\displaystyle \frac{z+l}{c}}\right)-H\left(t-{\displaystyle \frac{z+3l}{c}}\right)\right\}\end{array}\right.$$

(6)

The subsequent waves (i.e., “Wave 1⁻”, “Wave 1⁺”, “Wave 2⁻”, …) can be analyzed in the same manner as in Figure 16, Figure 17, Figure 18 and Figure 19. In order to simplify the notations, Figure 20 does not include the complete mathematical expression of the waves; instead, only the amplitude value of voltage and the amplitude value of current are shown for each wave in Figure 20. Differing from Figure 16, Figure 17, Figure 18 and Figure 19, the mismatching between $R_b=2Z_0$ and $Z_0$ at $z=-l$ in Figure 20 needs to be taken into account by the reflection coefficient below.

$${\mathsf{\Gamma }}_g={\displaystyle \frac{R_b-Z_0}{R_b+Z_0}}={\displaystyle \frac{2Z_0-Z_0}{2Z_0+Z_0}}={\displaystyle \frac{1}{3}}.$$

(7)

In theory, the circuit in Figure 20 reaches a steady state when $t\to \infty$. When $t\to \infty$, the voltage’s amplitude of the +z traveling wave over the transmission line evolves to

$${\displaystyle \frac{v_{DC}}{3}}\left[1+{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^3+\cdots \right]={\displaystyle \frac{v_{DC}}{2}},$$

(8)

the current’s amplitude of the +z traveling wave over the transmission line evolves to

$${\displaystyle \frac{v_{DC}}{3Z_0}}\left[1+{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^3+\cdots \right]={\displaystyle \frac{v_{DC}}{2Z_0}},$$

(9)

the voltage’s amplitude of the −z traveling wave over the transmission line evolves to

$${\displaystyle \frac{v_{DC}}{3}}\left[1+{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^3+\cdots \right]={\displaystyle \frac{v_{DC}}{2}},$$

(10)

and the current’s amplitude of the −z traveling wave over the transmission line evolves to

$$-{\displaystyle \frac{v_{DC}}{3Z_0}}\left[1+{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^3+\cdots \right]=-{\displaystyle \frac{v_{DC}}{2Z_0}}.$$

(11)

The steady-state status in (8), (9), (10), and (11) is identical to the steady-state status in Figure 18, unsurprisingly. In Figure 18, the battery stops supplying power after $t=2\delta$. In Figure 20, however, the battery never stops supplying power. The values of voltage, current, and power with respect to time at the battery (i.e., at $z=-l$) are articulated in

Table 1. Voltage, current, and power values at z = −l in Figure 20.

	Voltage @ z = −l	Current @ z = −l	Power @ z = −l
t ∈ [0, 2δ]	${\displaystyle \frac{v_{DC}}{3}}$	${\displaystyle \frac{v_{DC}}{3Z_0}}$	${\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}$
t ∈ [2δ, 4δ]	${\displaystyle \frac{v_{DC}}{3}}\left[2+{\mathsf{\Gamma }}_g\right]$	${\displaystyle \frac{v_{DC}}{3Z_0}}({\mathsf{\Gamma }}_g)$	${\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[2{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2\right]$
t ∈ [4δ, 6δ]	${\displaystyle \frac{v_{DC}}{3}}\left[2+2{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2\right]$	${\displaystyle \frac{v_{DC}}{3Z_0}}{({\mathsf{\Gamma }}_g)}^2$	${\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[2{({\mathsf{\Gamma }}_g)}^2+2{({\mathsf{\Gamma }}_g)}^3+{({\mathsf{\Gamma }}_g)}^4\right]$
t ∈ [6δ, 8δ]	${\displaystyle \frac{v_{DC}}{3}}\left[2+2{\mathsf{\Gamma }}_g+2{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^3\right]$	${\displaystyle \frac{v_{DC}}{3Z_0}}{({\mathsf{\Gamma }}_g)}^3$	${\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[2{({\mathsf{\Gamma }}_g)}^3+2{({\mathsf{\Gamma }}_g)}^4+2{({\mathsf{\Gamma }}_g)}^5+{({\mathsf{\Gamma }}_g)}^6\right]$
⁞	⁞	⁞	⁞

. When $t\to \infty$, the total amount of energy supplied by the battery can be calculated using the power values in the last column of Table 1:

$$\begin{array}{l}2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\\ +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[2{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2\right]\\ +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[2{({\mathsf{\Gamma }}_g)}^2+2{({\mathsf{\Gamma }}_g)}^3+{({\mathsf{\Gamma }}_g)}^4\right]\\ +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[2{({\mathsf{\Gamma }}_g)}^3+2{({\mathsf{\Gamma }}_g)}^4+2{({\mathsf{\Gamma }}_g)}^5+{({\mathsf{\Gamma }}_g)}^6\right]\\ +\cdots \end{array}$$

(12)

Because

$$\begin{array}{l}2{\mathsf{\Gamma }}_g={\displaystyle \frac{2{\mathsf{\Gamma }}_g}{1-{\mathsf{\Gamma }}_g}}-{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^2}{1-{\mathsf{\Gamma }}_g}}\\ 2{({\mathsf{\Gamma }}_g)}^2+2{({\mathsf{\Gamma }}_g)}^3={\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^2}{1-{\mathsf{\Gamma }}_g}}-{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^4}{1-{\mathsf{\Gamma }}_g}}\\ 2{({\mathsf{\Gamma }}_g)}^3+2{({\mathsf{\Gamma }}_g)}^4+2{({\mathsf{\Gamma }}_g)}^5={\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^3}{1-{\mathsf{\Gamma }}_g}}-{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^6}{1-{\mathsf{\Gamma }}_g}}\\ \cdots \end{array}$$

(13)

Equation (12) can be re-arranged to be

$$\begin{array}{ll}& 2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\\ & +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{2{\mathsf{\Gamma }}_g}{1-{\mathsf{\Gamma }}_g}}-{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^2}{1-{\mathsf{\Gamma }}_g}}+{({\mathsf{\Gamma }}_g)}^2\right]\\ & +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^2}{1-{\mathsf{\Gamma }}_g}}-{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^4}{1-{\mathsf{\Gamma }}_g}}+{({\mathsf{\Gamma }}_g)}^4\right]\\ & +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^3}{1-{\mathsf{\Gamma }}_g}}-{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^6}{1-{\mathsf{\Gamma }}_g}}+{({\mathsf{\Gamma }}_g)}^6\right]\\ & +\cdots \\ =& 2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{2{\mathsf{\Gamma }}_g}{1-{\mathsf{\Gamma }}_g}}+{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^2}{1-{\mathsf{\Gamma }}_g}}+{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^3}{1-{\mathsf{\Gamma }}_g}}+\cdots \right]\\ & -2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^2}{1-{\mathsf{\Gamma }}_g}}+{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^4}{1-{\mathsf{\Gamma }}_g}}+{\displaystyle \frac{2{({\mathsf{\Gamma }}_g)}^6}{1-{\mathsf{\Gamma }}_g}}+\cdots \right]\\ & +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[1+{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^4+{({\mathsf{\Gamma }}_g)}^6+\cdots \right]\end{array}$$

(14)

Further, due to the following two identities

$$\begin{array}{l}{\mathsf{\Gamma }}_g+{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^3+\cdots ={\displaystyle \frac{{\mathsf{\Gamma }}_g}{1-{\mathsf{\Gamma }}_g}}\\ 1+{({\mathsf{\Gamma }}_g)}^2+{({\mathsf{\Gamma }}_g)}^4+{({\mathsf{\Gamma }}_g)}^6+\cdots ={\displaystyle \frac{1}{1-{({\mathsf{\Gamma }}_g)}^2}}\end{array}$$

(15)

the amount of energy in (14) can be evaluated to be

$$\begin{array}{l}2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{2}{1-{\mathsf{\Gamma }}_g}}{\displaystyle \frac{{\mathsf{\Gamma }}_g}{1-{\mathsf{\Gamma }}_g}}\right]\\ -2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{2}{1-{\mathsf{\Gamma }}_g}}{\displaystyle \frac{{({\mathsf{\Gamma }}_g)}^2}{1-{({\mathsf{\Gamma }}_g)}^2}}\right]\\ +2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{1}{1-{({\mathsf{\Gamma }}_g)}^2}}\right]\\ =2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{1}{{(1-{\mathsf{\Gamma }}_g)}^2}}\right]\\ =2\delta {\displaystyle \frac{{(v_{DC})}^2}{9Z_0}}\left[{\displaystyle \frac{1}{{\left(1-\frac{1}{3}\right)}^2}}\right]\\ =2\delta {\displaystyle \frac{{(v_{DC})}^2}{4Z_0}}\\ =2E_0\end{array}$$

(16)

When $t\to \infty$, the amount of energy stored over the transmission line is $2E_0$, which is identical to the amount of steady-state energy stored over the transmission line in Figure 18. Although the transient state never ends until $t\to \infty$ in theory, the circuit can be considered to have reached the steady state when the power level in the last column of Table 1 is negligibly low in practice. Obviously, the “steady state in a practical sense” relies on electromagnetic wave propagation.

Figure 20 exemplifies the scenario of “$R_b\ne Z_0$ and $R_L\ne Z_0$.” When $R_b$ and/or $R_L$ have other values, the circuit can be analyzed following the same procedure as in Figure 20. To be specific, a load voltage reflection coefficient ${\mathsf{\Gamma }}_L=(R_L-Z_0)/(R_L+Z_0)$ needs to be incorporated into the analysis when $R_L$ is not infinity (${\mathsf{\Gamma }}_L$ is 1 in Figure 20, which facilitates the evaluation of power and energy).

The analytical studies in this section resemble the analysis of time domain reflectometer [17,18]. Compared with the analyses of time domain reflectometers in the literature, the analytical studies in this section focus on analyzing power and energy. To be specific, the analysis in this section reveals that the power and energy in a practical DC circuit could be reasonably interpreted by electromagnetic waves.

4. Conclusions

Some experimental and analytical studies are conducted on the electromagnetic wave propagation in DC circuits. In the experiments of this paper, electromagnetic wave propagation is explicitly demonstrated when a DC signal is turned on and turned off. The analytical studies of this paper based on the theory of transmission line further demonstrate that the electromagnetic waves do not vanish in the DC steady state. In other words, the electric field and magnetic field are coupled with each other when a DC signal is established, and they remain coupled with each other after the DC signal is established. The experimental and analytical studies jointly indicate that two pieces of conductors behave as a waveguide or transmission line when they are employed to transmit a DC signal in practice.