# What is the ABCD matrix of an impedance step?

What is the ABCD matrix of the following configuration, where a transmission line with characteristic impedance $$\Z_0\$$ is connected to another line with characteristic impedance $$\Z_1\$$ (if we consider $$\Z_0\$$ as port 1 and $$\Z_1\$$ as port 2)? The $$\S\$$ matrix given in this reference is

$$S = \begin{pmatrix} \frac{Z_1 - Z_0}{Z_1 + Z_0} & \frac{2\sqrt{Z_0Z_1}}{Z_0 + Z_1}\\ \frac{2\sqrt{Z_0Z_1}}{Z_0 + Z_1} & \frac{Z_0 - Z_1}{Z_0 + Z_1} \end{pmatrix},$$

which I converted to the ABCD matrix as

$$\text{ABCD} = \begin{pmatrix} \sqrt{Z_1/Z_0} & 0\\ 0 & \sqrt{Z_0/Z_1} \end{pmatrix}$$

by the formulas from for example here. I wonder if there is a way to see the form of the ABCD matrix directly.

• Where are you wanting to measure the impedance and what does this actually mean: "There is no further electrical length on either side" - please be clearer. Jul 4, 2021 at 17:53
• The questioner has obviously an idea: He uses ABCD presentations for chained parts and calculates the total effect as a matrix product. Only the ABCD matrix of one element is missing: That part is the zero length piece of transmission line which has only an impedance step. He already has ABCD matrices for parts in both sides of the step. Jul 4, 2021 at 18:11

The ABCD matrix is defined by the equation

$$\begin{bmatrix}V_1\\I_1\end{bmatrix}=\begin{bmatrix}A & B\\C & D\end{bmatrix} \begin{bmatrix}V_2\\-I_2\end{bmatrix}$$

What equations apply at infinitesimal point of transition between transmission lines with different characteristic impedance?

$$V_1 = V_2$$ $$I_1 = -I_2$$

So now all you have to do is work out what values of $$\A\$$, $$\B\$$, $$\C\$$ and $$\D\$$ make the ABCD equation match the equation for your network.

• If $V_1 = V_2$, $I_1 = -I_2$, isn't the ABCD matrix just the identity matrix? But I think the answer should be $$\begin{pmatrix}\sqrt{Z_1/Z_0} & 0 \\ 0 & \sqrt{Z_0/Z_1}\end{pmatrix}$$ Jul 4, 2021 at 20:39
• How is KCL satisfied in that case? Jul 4, 2021 at 20:44
• Actually I think the answer should be $$\begin{pmatrix} 1 & 0 \\ 0 & Z_0 / Z_1 \end{pmatrix}.$$ I am getting this answer by converting the $S$ matrix shown in the reference to the ABCD matrix. Jul 4, 2021 at 21:27
• That's exactly where I'm confused about. I understand that by KCL $I_1 = -I_2$ should hold. But if $V_1 = V_2, I_1 = -I_2$, doesn't it imply both sides have the same impedance, which is not necessarily the case? Jul 4, 2021 at 21:30
• Because the thing you're characterizing with this ABCD matrix is not the transmission lines, but the connection between them. Which is essentially just an ideal wire in circuit theory terms. Jul 4, 2021 at 21:53