# Generating and Understanding Bode Plots in Filter and Amplifier Design

Picture the scene: a 10 year old child just received their first Fender electric guitar with matching amplifier for their birthday and proceeds to crank away. I had plenty of fun experimenting with the settings on the amplifier and listening to the sound change, all while annoying my parents in the process.

I don’t play guitar so much anymore. Instead, I get to learn more about amplifiers using Bode plots. This simple mathematical tool is vital for understanding the behavior of filters, amplifiers, control systems, oscillators, and other electronic devices. Understanding Bode plots helps you optimize your design for specific applications.

## What is a Bode Plot?

A Bode plot is a specific type of plot that is used to understand and summarize how an electronic circuit (specifically, a linear time-invariant circuit) affects AC signals with different frequencies. When examining the behavior of a filter or amplifier, you’ll need to calculate how signals with different frequencies are attenuated or amplified as they are sent through the device.

When designing an electronic filter or amplifier, the output amplitude and phase of the output signal will depend on the frequency of the input signal. Take, for example, an RLC circuit with all elements in series. This circuit acts as a bandpass filter that strongly attenuates signals with frequencies that are far from resonance. In the presence of weak damping, the output signal reaches a maximum amplitude at resonance. The output signal will also acquire a phase difference with respect to the input signal, and this phase difference will be a function of frequency.

This behavior can be summarized using an important function called a transfer function. This function tells you how the device modifies signals with different frequencies. The transfer function is defined as follows:

Note that this lumps the amplitude and phase into a very convenient function of frequency. Taking the magnitude of the transfer function tells you how the magnitude of the input signal changes, and the phase of the transfer function tells you the phase difference between the input and output signals:

A Bode plot shows the same behavior as the transfer function, but it is converted to decibels using a logarithm. The figure below overlays a transfer function and a Bode plot for a 2nd order low pass filter as an example. As you look into the high frequency tail of the transfer function, it becomes difficult to determine the effectiveness of the filter from looking at the transfer function. From the Bode plot, you can see that the attenuation increases almost linearly on a logarithmic scale as the frequency of the input signal increases.

*Transfer function and Bode plot for a 2nd order low pass filter*

## Understanding Bode Plots

Bode plots are very useful for understanding how a filter or amplifier affects an AC signal at a specific frequency. A filter will have a transfer function whose magnitude is less than or equal to 1 for all frequencies. In contrast, an amplifier will have a transfer function that increases above 1 at particular frequencies. Once you have the Bode plot for a circuit, you can easily convert it to its transfer function, and vice versa.

Using a Bode plot allows you to extract this behavior and quote it as a decibel value without working in the time domain. When working with harmonic signals, you can certainly look at the behavior of your filter and how it affects a harmonic signal in the time domain, but this is more difficult in the frequency domain. Working in the frequency domain allows you to quickly generate a Bode plot and immediately understand how your circuit affects a signal with any frequency.

Once you know the transfer function for your circuit, you can use it to determine the output waveform for a signal in the time domain using a Fourier transform. You can take an arbitrary input waveform and convert it to the frequency domain using a Fourier transform. Multiplying the input spectrum by the transfer function gives you the spectrum of the signal that is output from the circuit. You can then apply an inverse Fourier transform and convert the output spectrum back to a waveform in the time-domain.

## Building a Bode Plot With a SPICE Package

If you want to build a Bode plot for a circuit early in the design process, you’ll need to use a SPICE package that simulates your circuit using a frequency sweep. This tool allows you to calculate the amplitude and phase of your signal throughout a range of frequencies. You won’t have to manually read the amplitude and phase from multiple signals in the time domain.

*Conducting proper circuit analysis with a bode plot should be easier to read than some wacky analog waves*

Working with a powerful SPICE package simplifies frequency domain analysis of complex circuits, allowing you to tune and optimize the behavior of your analog system. OrCAD PSpice Simulator from Cadence allows you to perform frequency sweeps, transient analysis, and many other tasks for analyzing analog circuits for any application.

This unique package is adapted to complex PCB designs, interfaces directly with your design data, and helps you with understanding Bode plots for your circuit.

If you’re looking to learn more about the solutions Cadence has for you, talk to us and our team of experts.