# Formulate the equations that describe this transmission-line circuit (Veritasium's circuit)

## My question

Consider the following circuit. Two non-resistive long transmission lines ($$R' = G' = 0 \text{ } \Omega$$) , each of the same length $$L$$ and same inductance per unit length $$L'$$ and same capacitance per unit length $$C'$$, each with one of their ports short-circuited by ideal wires. In the other port of both TLs, between one of their terminals we connect an ideal constant voltage source (battery) of voltage $$V_\text{S}$$ in series with an ideal normally-open switch; between the other two terminals we connect an incandescent light bulb that is modelled as an ideal constant-resistance resistor $$R$$. The switch is closed at $$t = 0 \text{ s}$$.

Before that instant, the circuit has been for a long time in the configuration shown, meaning it is operating in steady-state (the charge distribution and electromagnetic field are steady), so the inductors and capacitors act as short-circuits and open-circuits, respectively.

The two short-circuits at one of the ports of the TLs, and the region where the light bulb, the battery and the switch are, are modelled as lumped circuit elements. The two TLs are modelled as distributed circuit elements.

The circuit is shown in the following figure.

Figure 1. Image source: own.

Given the above assumptions, I'd like to determine the currents $$i_1(z,t)$$ and $$i_2(z,t)$$ flowing through each TL as a function of space $$z$$ and time $$t$$, the voltages $$v_1(z,t)$$ and $$v_2(z,t)$$ across each TL as a function of space and time, and the voltage $$v_R(t)$$ across the incandescent light as a function of time. There's no need to solve them by hand, we can use a software such as Mathematica; I'd like the analytical solution if possible. Actually, in this post I'm not even interested in solving the equations, only in obtaining the correct equations and boundary conditions for the assumptions.

This is not homework. I'm trying to solve a circuit mentioned in a recent video of Veritasium's channel, where I'm modelling the conductors as transmission lines. Whether such model is correct or not is irrelevant to me; I just want to obtain the equations, given such model and the previous assumptions.

I wasn't sure of whether asking this here or in the Electrical & Electronics Stack Exchange.

Note that I just want to obtain the equations and boundary conditions that describe the circuit. Unlike what Veritasium's video asks, I don't want to know how much it takes for the light bulb to turn ON, assuming it turns ON as soon as any amount of current passes through it; also, I don't want to know the energy transfer direction.

I never had an actual class on PDE, only on ODE. As for transmission lines, I had one, but my friends and I agree that the teacher didn't really explain the theory. So, I'm not very good on those two topics.

## My attempt

As we know, the uncoupled telegrapher's equations for a lossless long transmission line in the time domain (which are pretty much the one-dimensional wave equation, a hyperbolic PDE) are:

\left\{ \begin{align} \dfrac{\partial^2 v}{\partial z^2} &= L' C' \dfrac{\partial^2 v}{\partial t^2} \\ \dfrac{\partial^2 i}{\partial z^2} &= L' C' \dfrac{\partial^2 i}{\partial t^2} \end{align} \right. \tag*{}

With the previous assumptions, the circuit of figure 1 becomes the following.

Figure 2. Image source: own.

I have the following questions regarding the previous figure and model:

1. Is it correct to consider that the end of TL1 is in the same space as the beginning of TL2 ($$z = + L$$)? I mean, sure, in reality, they aren't. But because we're considering the region where the light bulb, battery and switch are as lumped elements, I think it is correct.
2. When formulating the equations for both TLs, is it correct to use the same independent spatial variable $$z$$ for both sets of equations? I'm not sure. For example, maybe we must use two different spatial variables, like $$z_1$$ for TL1 and $$z_2$$ for TL2.
3. Should we restrict the values that $$z$$ can take in each TL? That is, for TL1, $$0 \text{ m} \le z \le L$$; for TL2, $$L \le z \le 2L$$.

If the answer to the three previous questions is yes, then I think the sets of equations for both TLs are:

\text{TL1} \left\{ \begin{align} \dfrac{\partial^2 v_1(z, t)}{\partial z^2} &= L' C' \dfrac{\partial^2 v_1(z, t)}{\partial t^2} \\ \dfrac{\partial^2 i_1(z, t)}{\partial z^2} &= L' C' \dfrac{\partial^2 i_1(z, t)}{\partial t^2} \end{align} \right. \quad , \, 0 \text{ m} \le z \le L, \, t \ge 0 \text{ s} \tag 1

\text{TL2} \left\{ \begin{align} \dfrac{\partial^2 v_2(z, t)}{\partial z^2} &= L' C' \dfrac{\partial^2 v_2(z, t)}{\partial t^2} \\ \dfrac{\partial^2 i_2(z, t)}{\partial z^2} &= L' C' \dfrac{\partial^2 i_2(z, t)}{\partial t^2} \end{align} \right. \quad , \, L \le z \le 2 L, \, t \ge 0 \text{ s} \tag 2

And applying KVL around the center of the circuit:

$$-v_1(L, t) + R \, i_1(L, t) + v_2(L, t) - V_\text{S} = 0 \text{ V} \tag 3$$

Are the above equations correct?

As for the initial and boundary conditions:

• At the short-circuited ports of the TLs:

$$v_1(0 \text{ m}, t) = 0 \text{ V} \tag 4$$

$$v_2(2 L, t) = 0 \text{ V} \tag 5$$

• At the center of the circuit:

$$i_1(L, t) = i_2(L, t) \tag 6$$

• Before the switch is closed, all voltages and currents are zero:

$$v_1(z, 0 \text{ s}) = 0 \text{ V} \tag 7$$

$$i_1(z, 0 \text{ s}) = 0 \text{ V} \tag 8$$

$$v_2(z, 0 \text{ s}) = 0 \text{ A} \tag 9$$

$$i_2(z, 0 \text{ s}) = 0 \text{ A} \tag {10}$$

Do we need more boundary conditions? Are the above correct?

• just an engineering suggestion: ignore the PDEs. Instead, knowing the length of the TL's you can replace them by lumped reactances between the load and the battery, one on the left and one on the right. Now calculate the current through the load and the branch voltages. Now that you know the voltage and current at the input of each TL, you are in business. Commented Dec 16, 2021 at 17:01
• @hyportnex Thanks for your suggestion. I’ve seen an equivalent pi and tee circuit for a long transmission line, but sinusoidal steady-state is assumed and phasors and complex impedances are used. Is there a way to obtain the eq. circuit in the time domain? Commented Dec 16, 2021 at 18:36
• forget about tee or pi, a short circuited TL is just a simple reactive impedance (inductive or capacitive depending on $\ell < or > \lambda/4$: $X_{in}=Z_0 tan (\beta \ell)$, see section Input impedance. In the time domain you will have to deal with a convolution! That is why Fourier transform was invented instead... Commented Dec 16, 2021 at 19:02