RF Isolator S-Matrix Explained

This page describes the RF Isolator S-Matrix (scattering matrix). It also covers the basics of the RF Isolator.

What is an RF Isolator?

As we know, an RF Isolator is a 2-port microwave device that forwards the signal in one direction and blocks the signal in the other direction.

It is used for a wide variety of applications such as the protection of test and measurement equipment, and delicate devices such as MMIC amplifiers and attenuators.

Figure 1: Working of RF Isolator

As shown in Figure 1, the RF signal can pass from port-1 to port-2 only with port-3 terminated.

Following are the typical specifications of an isolator device:

• Operating frequency range
• Isolation from port-2 to port-1
• Insertion Loss from port-1 to port-2
• VSWR (Voltage Standing Wave Ratio)

RF Isolator S-Matrix | RF Isolator Scattering Matrix

Figure 2: 2-Port Scattering Parameters

As an Isolator is a 2-port device, let’s consider a simple 2-port network as shown in Figure 2. Waves traveling towards the n-port are $a_i = (a_1, a_2, ...., a_n)$ and waves traveling away from the n-port are $b_i = (b_1, b_2, ...., b_n)$.

The relation between $a_i$ and $b_i$ can be expressed by the following equations:

$b_1 = S_{11}*a_1 + S_{12}*a_2$

$b_2 = S_{21}*a_1 + S_{22}*a_2$

Figure 3: RF Isolator S-Matrix

From the above, the S-matrix can be written as follows:

Isolator S-matrix = $\begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix}$

Here:

• $S_{11}$ is the input reflection coefficient with the output terminated by a matched load.
• $S_{21}$ is the forward transmission from port-1 (input) to port-2 (output).
• $S_{12}$ is the reverse transmission from port-2 to port-1.
• $S_{22}$ is the output reflection coefficient.

Here,

$S_{11} = \frac{b_1}{a_1}$, $S_{12} = \frac{b_1}{a_2}$, $S_{21} = \frac{b_2}{a_1}$, $S_{22} = \frac{b_2}{a_2}$

Figure 3 depicts the RF Isolator S-matrix, i.e., the Scattering matrix.