How to calculate a transfer function from S-parameters (2023)

the central theses

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

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

• If you need to relate an enabled signal to the value received from a load, you can use some basic matrix manipulation to calculate the S-parameter transfer function.

  (Video) S-Parameter & Transfer Function Measurement

Once you have S-parameter measurements, you can use them to calculate a transfer function from S-parameters.

Like the tools in your toolbox, different math tools have different uses for circuit and signal analysis. A network of circuits can have a complex structure that is difficult to analyze using Ohm's law and Kirchoff's laws. Enter the S-parameters that summarize bidirectional signal behavior well, including reflection and transmission in 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 at different frequencies. If you simply want to analyze a circuit's frequency response to examine signal distortion and impulse response, one way is to use the circuit's transfer function.

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

In the following guide we will show you how to calculate a transfer function from 2-port S-parameters. The equation shown below is obtained by solving a simple system of linear equations that can be generalized to N-port S-parameters using programs such as MATLAB. Once you have a transfer function for your system, you have the important function you need for time domain modeling of your circuits and gives you everything you need to know about the electrical behavior of your circuit.

Parameter transfer function S: Theory for 2-port networks

First we consider a 2-port network and the procedure shown here can be generalized to an N-port network. The term "2 ports" 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 could be a system with two differential or single-ended inputs referenced to ground (4 ports) or other multi-input systems.

Definition of parameters p

The image below shows the general form of the S-parameters for a 2-port network. The important point in the graph below, which is poorly explained in almost every S-parameter tutorial, is that the "mesh" 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 when a signal is injected into the source (load) end of the network.

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

Before you determine the transfer function for a circuit network or device under test (DUT), you must extract the S-parameters for the DUT. This is typically done with a vector network analyzer that samples a source signal in a DUT and measures the reflected/transmitted waves. The relationship between the S-parameters and the input/output signals measured by the DUT is shown below. After thisDescaling of S-parameters for connection fittings and transfer linelead to the DUT, you now have the S-parameters for the DUT itself.

Relationship between S parameters, input signals and output signals.

In order to get the transfer function from the parameters S, we do not need the definition of the terms a and b here. Instead we can work with current and voltage amplitude values ​​at the signal source and mains input. With these values ​​we can create a new matrix called the ABCD matrix. This important matrix provides the link to calculate a transfer function from S-parameters.

ABCD-Matrixdefinition

The ABCD matrix relates the voltage and current seen across the load to the voltage and current supplied to a network by 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 play a similar role here as the a and b terms do in defining the parameter S. These are values ​​measured at the source end (S) and load end (L). Now we can use a standard definition that relates the S parameters for a network to its ABCD parameters using the networkWellenwiderstand Z:

Relationship between S parameters and ABCD parameters in a 2-port network.

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

Definition of the transfer function from ABCD parameters

To get the transfer function of the ABCD parameters we can use the equation shown below. In this equation weConsider impedancefrom the source side of the network (S) and the load side (L). If the network ends with the characteristic impedance on each side, then the two values ​​are equal to the characteristic impedance Z.

Relationship between the ABCD parameters and the transfer function.

Note that the ABCD parameters are complex quantities and depend on frequency, so the transfer function is expected to have complex phase and magnitude as a function of frequency. This provides an easy way to obtain a transfer function simply by 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. For example, this applies to a heavily isolated antenna feed line, a SerDes channel, or any other circuit network that is not coupled to another electrical network. In more complex N-port networks, you can derive a transfer function using the ABCD matrix and the S-parameter matrix by solving a system of linear matrix equations. These problems are solvable in principle, but become unsolvable for large networks. For this reason, programs like MATLAB include a function to convert between S-parameters and a transfer function matrix for each Ni-port network. You could also use Mathematica to derive analytic equations.

Example: transmission line with isolated loss

It is easy to take a transmission line transfer function out of context as different formulas can be found in different references. These formulas correspond to different systems, so it is important to consider the general case of a transmission line with a known characteristic impedance. The following matrix 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 plug the ABCD parameters into the transfer function equation. This gives you everything you need to know about the frequency response of the transmission line, which can then be used to calculate the causal response in the time domain. The time domain impulse response function, or the transfer function itself, can be generalized to determine the time domain response for any input; You just need to know the frequency domain voltage waveform that is injected into the line by the controller.

With view onTime domain response to a specific stimulusthis is done by taking the convolution of the transmission line 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 essential for studying intersymbol interference, ringing due to broadband impedance mismatch, and superimposed random noise, particularly in multi-level signaling schemes (e.g.,

Relationship between the transfer function (H), the impulse response function (h) and the input and output signals in the time domain.

While most transfer functions work fairly automatically in your analysis and simulation tools these days, speed, efficiency, and accuracy are still important and viable models to consider when looking for your tools. After all, there's no point waiting 72 hours for an analysis process that might only take 10 hours and be ready when you start your design the next morning.

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