# Bandpass Filter Bode Plot and Analysis

November 12, 2020

### Key Takeaways

• Every filter, linear amplifier, impedance matching network, and other reactive LTI circuits will have a transfer function.

• Bode plots are one way to visualize the magnitude and phase of a transfer function for one of these circuits.

• A bandpass filter for some circuits will start to look like a low-pass filter or a high-pass filter, and this behavior can be seen in a Bode plot. Keep your circuits as clean as the air in your house with a bandpass filter.

Filters are critical circuits for any engineer to understand, and they have simple mathematical representations that help designers visualize their functionality. As part of filter design, simulation, and evaluation, a Bode plot is a basic tool for visualizing a filter’s output for a harmonic input. In particular, for linear time-invariant (LTI) systems, a Bode plot shows the transfer function for a circuit, which is a basic part of simulation of causal systems in PCBs and integrated circuits.

One fundamental filter that can be constructed from simple passive circuit elements is a bandpass filter. The graph for a bandpass filter Bode plot can transition to low-pass behavior if the system’s resistance is large enough, and this is one aspect of the filter that can be seen visually. Here is how to interpret and use the Bode plot for a bandpass filter, as well as an example for a simple circuit.

## Building a Bandpass Filter Bode Plot

A Bode plot is simply a logarithmic plot of the transfer function for a circuit. This includes a bandpass filter Bode plot, which can be used to view the resonant or non-resonant behavior of a system. There are a few important points that can be determined from a filter’s Bode plot:

• Gain and attenuation: A linear circuit with gain, such as an op-amp operating in the linear regime or a bandpass filter near resonance, will have an output that is larger than the input (positive dB value on a logarithmic scale), and vice versa for attenuation. This can be easily seen in a Bode plot.

• Resonance, bandwidth, and rolloff: These features are seen by looking at the magnitude of the transfer function, as shown in a Bode plot. Resonance only occurs within a particular bandwidth, which can be used to calculate a Q-factor for the circuit. In addition, the system’s response will have some rolloff above and/or below the bandwidth limit (normally taken as the -3 dB frequencies).

• Phase shifting or reversal: This is viewed in the phase section of a Bode plot, which will show how the phase of the output is related to the phase of the input. This is quite important in transmission lines that are series-matched to the driver; once the phase delay is extracted from the output, a Bode plot for the line’s transfer function shows where resonances occur at different frequencies in the presence of impedance mismatches.

The transfer function for a circuit can be calculated by hand using Kirchoff’s laws and Ohm’s law, or it can be determined from a SPICE simulation. Note that transfer functions are only defined for LTI systems, although there is plenty of research literature on nonlinear time-dependent representations for transfer functions. For a simple linear circuit like a bandpass filter, it’s easy to calculate a Bode plot, as shown in the following example.

## Example for a Series RLC Circuit

Perhaps the simplest bandpass circuit you can construct is a series RLC resonant circuit, where a resistor, capacitor, and inductor are placed in series. This circuit provides bandpass behavior with gain inside a narrow bandwidth. The maximum gain and phase of the signal depends on the value of the resistor in this network. Finally, the output from this circuit, which is normally taken across the capacitor, can be connected to a load component. The load impedance then determines the exact transfer function in the system, which can be visualized in a Bode plot.

The circuit below shows an example of a series RLC circuit with the output connected to a 20 Ohm load resistor. The load is arranged in parallel with the capacitor, i.e., the filter’s output is taken across the capacitor. The series resistor R will determine the level of damping in the circuit and the circuit’s transfer function, as well as the total power delivered to the load. Simple series RLC bandpass filter circuit with a 20 Ohm load

The bandpass filter Bode plot below shows when the resonance occurs for various values of the series resistor R. We can see strong resonance at R = 0.2 Ohms, which is to be expected as the damping in the circuit is proportional to R. At larger values of R, we see low-pass filter behavior; this is because the capacitor will have the highest impedance at low frequencies, so all the input voltage will be dropped across the capacitor and load resistor. We can also see that phase reversal occurs beyond the resonant frequency, i.e., the output and input are nearly completely out-of-phase. Bandpass filter Bode plot for a series RLC circuit with output taken across the capacitor

Although the output is taken across the capacitor in this example, it could also be taken across the inductor. In this case, we’d still have a resonance at the circuit’s natural frequency. In addition, we’d have high-pass behavior above the resonant frequency, and the same type of phase reversal seen above at low frequencies. At high frequencies, the output voltage and the input voltage are completely in-phase. This is to be expected from Ohm’s law, i.e., all the voltage would be dropped across the inductor because it has the largest impedance. Bandpass filter Bode plot for a series RLC circuit with output taken across the inductor

From these two plots, we learn something quite important about series RLC circuits and the phase reversal seen in their Bode plots: The zero output voltage corresponds to complete phase reversal on the output voltage compared to the input. In other words, the input and output voltage interfere destructively and cancel each other out. This is why, in each configuration, no power will be dissipated across the load resistor within a certain frequency range.

After you use the front-end design features from Cadence to build your circuits, the electrical behavior shown here can be viewed in the frequency domain and time domain using PSpice Simulator. You can easily create a bandpass filter Bode plot and many other simulations with a powerful SPICE simulator. You’ll also have access to a range of circuit analysis and optimization features.

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