# How to Calculate a Transfer Function From S-Parameters

### Key Takeaways

• S-parameter measurements are commonly used to characterize high-speed and high-frequency circuits in the frequency domain.

• An alternative tool in the frequency domain is the transfer function, which defines how a circuit network can act like an amplifier or filter.

• When you need to relate a launched signal to the value received at a load, you can use some basic matrix manipulations to calculate the transfer function from S-parameters. Once you have S-parameter measurements, you can use them to calculate a transfer function from S-parameters.

Just like the tools in your toolbox, different mathematical tools have different uses for circuit and signal analysis. A circuit network can have a complex structure that is difficult to analyze with Ohm’s law and Kirchoff’s laws. Enter S-parameters, which nicely summarize bidirectional signal behavior, including reflection and transmission into an N-port network.

Another important tool in your engineering toolbox is a transfer function, which defines how a circuit or network responds to signals with different frequencies. If you want to easily analyze the frequency response of a circuit in order to examine signal distortion and impulse response, one option is t use the transfer function for the circuit.

Why worry about getting a transfer function from S-parameters? When evaluating circuit and transmission line responses, the S-parameters are normally measured with a vector network analyzer or extracted from simulations. However, the causal response in the network cannot be simulated without the impulse response function, which is calculated with the transfer function. This simplifies analysis of broadband circuit responses in the time domain for high speed digital systems.

In the guide below, we’ll show you how to calculate a transfer function from 2-port S-parameters. The equation shown below is derived by solving a simple system of linear equations, which can be generalized to N-port S-parameters with programs like MATLAB. Once you have a transfer function for your system, you have the important function you need for time-domain modeling of your circuits, giving you everything you need to know about your circuit’s electrical behavior.

## Transfer Function from S-parameters: Theory for 2-port Networks

To get started, we’ll look at a 2-port network, and the procedure shown here can be generalized to an N-port network. The term “2-port” refers to a circuit network with 2 physical ports (input and output), both of which are referenced to ground. A higher-order network would be applicable to a circuit with multiple inputs and outputs. An example might be a system with two single-ended or differential inputs referenced to ground (4-port) or other systems involving multiple inputs.

### Definition of S-parameters

The image below shows the general form of S-parameters for a 2-port network. The important point in the below graphic, which is poorly explained in nearly every S-parameter tutorial, is that the “network” can be anything. It could be a complicated circuit with multiple elements and well-defined input and output ports. Another example of a network is a transmission line; it has a source end and a load end. The S-parameters define what would be measured at the load (source) end of the network if a signal is injected at the source (load) end of the network.

S-parameters are useful as they combine signal levels at different ports (transmission lines use a combination of source voltage and current) into a single matrix calculation. They also define directionality; a signal a1 or a2 can be input to the network from the left or right, respectively, both of which will produce the output signals b1 and b2. This can then be easily extended to an N-port network.

Before determining the transfer function for a circuit network or device under test (DUT), you need to extract the S-parameters for the DUT. This is normally done with a vector network analyzer, which sweeps a source signal into a DUT and measures the reflected/transmitted waves. The relationship between the S-parameters and the input/output signals measured from the DUT is shown below. After de-embedding the S-parameters for the connection fixtures and transmission line leading to the DUT, you now have the S-parameters for the DUT itself. Relationship between S-parameters, input signals, and output signals.

To get to the transfer function from S-parameters, we don’t need the definition of a and b terms here. Instead, we can work with current and voltage amplitude values at the signal source and network input. By using these values, we can construct a new matrix, called the ABCD matrix. This important matrix will provide the link for calculating a transfer function from S-parameters.

### ABCD Matrix Definition

The ABCD matrix relates the voltage and current seen at the load back to the voltage and current provided to a network from a source. Be careful when looking at different definitions of the ABCD matrix; some definitions place the ABCD matrix on the opposite side of the equation. The most common definition of the ABCD matrix is: Definition of the ABCD matrix in a 2-port network.

The V and I terms here play a similar role as the a and b terms in the S-parameter definition. These are values measured at the source end (S) and the load end (L). Now, we can use a standard definition that relates the S-parameters for a network to its ABCD parameters using the network’s characteristic impedance Z: Relationship between S-parameters and ABCD parameters in a 2-port network.

Now that we have ABCD parameters, we can more easily calculate the transfer function for the network.

### Transfer Function Definition from ABCD Parameters

To get the transfer function from the ABCD parameters, we can use the equation shown below. In this equation, we consider the impedance from the source side of the network (S) and the load side (L). If the network is terminated to the characteristic impedance on each side, then the two values are equal to the characteristic impedance Z. Relationship between ABCD parameters and transfer function.

Note that the ABCD parameters are complex quantities and are frequency-dependent, thus the transfer function will have a complex phase and magnitude as a function of frequency, as one would expect. This gives a simple way to get a transfer function by simply looking at the input impedance and S-parameters, both of which can be measured for a given DUT.

The equation shown above is defined for a 2-port network. As examples, this is applicable to a highly isolated antenna feedline, SerDes channel, or other circuit network that does not couple to any other electrical network. In more complex N-port networks, you can derive a transfer function using the ABCD matrix and S-parameter matrix by solving a system of linear matrix equations. These problems can be solved in principle, but they become intractable for large networks. For this reason, programs like MATLAB include a function to convert between S-parameters and a transfer function matrix for any Ni-port network. You could also use Mathematica to derive analytical equations.

### Example: Isolated Lossy Transmission Line

A transmission line transfer function is easy to take out of context because there are different formulas found in different references. These formulas correspond to different systems, so it is important to look at the general case for a transmission line with known characteristic impedance. The matrix below shows the ABCD parameters for a lossy transmission line with characteristic impedance Z0 and length l: ABCD parameters for a lossy transmission line.

From here, you can simply place the ABCD parameters into the transfer function equation. This gives you everything you need to know about the transmission line’s frequency response, which can then be used to calculate the causal response in the time domain. The impulse response function in the time domain or the transfer function itself can be generalized to determine the time-domain response for any input; you only need to know the frequency-domain voltage waveform being injected into the line from the driver.

Looking at the time-domain response to a specific stimulus is done by taking the convolution of the transmission line’s impulse response function (weighted by a sign(0) function) and the time-domain function for the input signal. The impulse response function can be calculated by taking the Fourier transform of the transfer function. This aspect of modeling is critical for examining intersymbol interference, ringing due to broadband impedance mismatch, and superimposed random noise, especially in multi-level signaling schemes (e.g., PAM-4 as specified in the IEEE 802.3 Task Force Proposal for 100G Ethernet). Relationship between the transfer function (H), impulse response function (h), and the input and output signals in the time domain.

While most transfer functions are working pretty automatedly in your analysis and simulation tools these days, speed, efficiency, and accuracy are still important and viable models to consider when looking into your tools. After all, no point in waiting 72 hours for an analysis process that could only take 10 hours and be ready for when you jump into your design the next morning.

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