### Introduction

Transmission line theory has been around for more than a century. However, the distributed nature of it has kept it interesting for all these years. One interesting subject is the step response in a hybrid lumped-distributed network. While for a purely lumped network it is quite easy to analyze and derive the response equation, for a distributed or a hybrid network, it may not be. This short post covers this topic briefly . It is assumed the reader has the fundamental knowledge of transmission lines.

### A Review of Transmission Line

Transmission lines are mainly defined with the following factors:

- Characteristic impedance where are the resistance, inductance, conductance and capacitance per unit length. In a lossless transmission line .
- Propagation constant where is the loss (this post considers a lossless transmission line in which =0) per unit length and is the wave number.
- The wave velocity (phase velocity) where is the speed of the light and is he permitivity of the medium. For an IC designers =4.2 as the signal travels in silicon-doixide.
- The signal wavelength where is the wave velocity and is the frequency of the signal.

In a circuit where the signal wavelength is comparable to the size of the elements, the lumped analysis of the voltage and current (Kirchhoff circuit laws) is not valid anymore and should be treated with wave equation. In

this case, the signal travels like a wave on a transmission line and could have reflection if not terminated with a matched impedance load. If a transmission line with characteristic impedance of is terminated with a load impedance of , then the reflection coefficient is defined .

### Step Response Equation

We now can use the terminologies we defined earlier to derive the step response of a distributed network. Consider the circuit illustrated in the following figure.

The source has an impedance of driving a transmission line with a length of and characteristic impedance of . The transmission line is then terminated with . and are the reflection coefficients at the source and load respectively. Assume at the source voltage is launched. Then there is a voltage divide ratio between the source impedance, , and of the line. This defines the forward voltage travelling from source to the load at . We call this . It takes for to get to the other side of the line where it sees . The reflected wave is now . The same analysis is applied for the travelling signal at each end point of the transmission

To derive the equation of the voltage at each node one need to usethe superposition of each reflected waves. Let's start with .Looking at Figure 1 it is shown this node will have transition at time . So:

\begin{align}

V_{I}(t)&=V_{1}u(t)+(V_{2}+V_{3})u(t-2t_{0})+(V_{4}+V_{5})u(t-4t_{0})\\ \nonumber

&=+...+(V_{2n}+V_{2n+1})u(t-2nt_{0})

\label{eq:VI}

\end{align}

Similarly, is derived as:

\begin{align}

V_{L}(t)&=(V_{1}+V_{2})u(t-t_{0})+(V_{3}+V_{4})u(t-3t_{0})\\ \nonumber

&=+...+(V_{2n-1}+V_{2n})u(t-(2n-1)t_{0})

\label{eq:VL}

\end{align}

To complete the discussion in this section the dual circuit of Figure 1 is also analyzed. Figure 2 shows the Norton equivalent of and . Since the reflection coefficient for current always has a negative sign, the sign of each reflected current is toggled every other time. This has been shown in Figure 2.

Similar to equation 2 one can derive with the same exact format. For a purely resistive network, both and are real numbers. This makes ploting the step response quite easy. For a complex source and load impedance such as RC, RL and RLC networks, the reflection coefficients are not real anymore and further consideration is required to derive the step response.

### Example: Purely Resistive Load

Consider a , a 2mm transmission line with that is terminated with a in silicon-dioxide medium. Plot and .

By plugging the above numbers into the equations in Figure 2 one can derive and . Figure 3 shows the waveforms of and .

**Reference:**

[1]. Brian C. Wadell, *Transmission Line Design Handbook*, Artech House, 1991.

[2]. Stuart M. Wentworth, *Applied Electromagnetics : Early Transmission Lines Approach*, Wiley, 2007.

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