# Understanding a Circuit Transfer Function from a Bode Plot

February 9, 2021

### Key Takeaways

• A Bode plot shows the magnitude and phase of a transfer function in a pair of graphs.

• You can convert between a Bode plot and a transfer function with some simple arithmetic involving complex numbers.

• A Bode plot conversion applies to any transfer function, including network parameter matrices.

Transfer functions describe the relationship between input and output signals

The transfer function provides important information regarding signal transformation through a circuit. It relies on a simple concept: any circuit will transform an input voltage or current into some measurable output voltage or current. However, transfer functions are used for more than just describing the relationship between input and output signals—they can verify causality and track signal transformation throughout a complex electrical network.

However, most engineers do not work with the transfer function directly. Instead, they work with a Bode plot. In fact, instruments used to measure signal transfer through an electrical network (such as a VNA) will provide Bode plot data, rather than transfer function data. When you need to get back to a transfer function from a Bode plot, there are simple methods you can use for conversion.

## Calculating a Transfer Function from a Bode Plot

A Bode plot simply shows the magnitude and phase of a transfer function, so the two are directly related. The magnitude of the transfer function is shown on a logarithmic scale and the phase is shown in radians or degrees. The magnitude and phase are normally shown together to communicate everything about the transfer function in a pair of corresponding graphs. A simple example of a filter with a secant transfer function is shown below.

A Bode plot example: in the left panel, the magnitude plot is shown on a logarithmic scale (red) and linear scale (blue). The right panel shows the phase of the transfer function.

It’s important to remember that a transfer function is generally a complex function of frequency. Since the Bode plot measures the magnitude and phase, a conversion back to the transfer function will need to include both pieces of information. Reconstructing the transfer function from a Bode plot is a simple matter of taking the phase and magnitude at each frequency and using the following process:

Conversion from Bode plot data to a transfer function

The final H function is the transfer function. This can then be plotted in terms of its real and imaginary parts, magnitude in phase on linear scales, or for use in other analyses. How you use the transfer function depends on how the circuit being examined is used in a larger electrical network.

## Using a Transfer Function from Your Bode Plot

A Bode plot is a universal way to visualize a transfer function on a logarithmic scale, but it is important to remember that there are many types of transfer functions. In general, the interaction between an incoming signal and an outgoing signal in an electrical network is described using a transfer function. Common types of transfer functions are shown in the table below.

 Quantity Transformation T-parameters Voltage-to-voltage or current-to-current S-parameters Power to power ABCD parameters (Voltage + current) to (Voltage + current) Admittance matrix (Y-parameters) Voltage to current Impedance matrix (Z-parameters) Current to voltage H-parameters Mixes T-parameters, Z-parameters, and Y-parameters

To better see the correspondence between a transfer function and Bode plot, we can look at some specific parameter sets in the table above.

### T-parameters

Most designers will be familiar with T-parameters. For a 2-port, monodirectional network with no reflection, this is the typical transfer function that one calculates with a frequency sweep in a SPICE simulation. The T-parameters are also used to calculate the impulse response functions between ports, which are used to verify causality in signal integrity models. They are very useful in understanding signal propagation in terms of the real waveforms one would measure with an oscilloscope or VNA.

### S-parameters

As an example from the above table, S-parameters are a type of transfer function, and they transform power (incoming) into power (outgoing). In fact, S-parameters are normally plotted in a Bode plot, but no one calls a plot of the S-parameters a Bode plot. S-parameters are interesting in that they really define 2N transfer functions for an N-port network (insertion losses, return losses, and crosstalk, both for incoming and outgoing waves).

### Z-Parameters

The impedance parameters (Z-parameters) transform a current into a voltage. In other words, these network parameters tell you the voltage measured at the output port of a network for some current flowing into the input port of the same network. This is important for power integrity engineers, as a complex PDN must be modeled as an N-port network. When a transient current is drawn into the network at 1 port, it will create some voltage fluctuation at all N ports in the network. PDN impedance plots are really part of a Bode plot for the network’s Z-parameters.

### ABCD Parameters

Many signal integrity engineers focus on S-parameters, but the important underlying network parameters are the ABCD parameters. These parameters form the bridge in conversions between all other sets of network parameters, so they are extremely important. In fact, programs like Matlab will use ABCD parameters to create conversions between network parameters. For most PCB designers, they do not need to work with parameter sets to get a transfer function; the best SPICE simulator applications will do this for you during a frequency sweep.

No matter which type of parameters you want to use in your design process, you’ll need the best set of front-end design features from Cadence to calculate a transfer function from a Bode plot for your circuits. The PSpice Simulator application lets you create detailed circuits, run standard simulations, and optimize circuit behavior with parameter sweeps. Once you’re ready to create a PCB for your circuits, simply capture your schematics in a blank layout and use Cadence’s board layout utilities to finish your design.