# Interpreting the Phase in a Bode Plot

January 8, 2021

### Key Takeaways

• Reactive components in LTI systems will produce a phase shift between the current and voltage in the component.

• When a phase shift is introduced into a filter or amplifier, the magnitude of the phase shift and attenuation/gain can be determined over a range of frequencies.

• The phase is one of two pieces of information shown in a Bode plot, where the output voltage is shifted in time with respect to the input voltage. As reactive components, inductors and capacitors induce a phase shift in a filter or amplifier circuit, creating a phase shift that can be seen in a Bode plot.

Reactive circuits need careful analysis to determine how they interact with signals of different frequencies. Among the various tools used to understand reactive circuits, a primary tool is the Bode plot. This simple plot is just the transfer function of a circuit plotted on a logarithmic scale, but the two portions of this plot tell you important information about signal behavior.

In a filter or amplifier, the phase curve in the Bode plot indicates separation between the input and output signals, which becomes quite important when we consider the role of feedback in a circuit. However, an often unconsidered aspect of phase in a Bode plot is its effect on stability, which arises in negative feedback amplifiers. To better see what the phase in a Bode plot says about your circuits, let’s analyze these systems briefly and show how you can identify important quantities in Bode plot phase curves.

## Important Phase Points in a Bode Plot

A Bode plot is a simple way to show some important information in the transfer function for a linear time-invariant (LTI) system. In short, the frequency response for any LTI system can be summarized using a Bode plot. The information one finds in a Bode plot depends on a few factors:

• Type of circuit: Filters and amplifiers are similar, but slightly different effects can occur depending on the circuit topology and whether active components are used.

• Presence of feedback: If feedback is present in the circuit, the circuit may transition to instability. One can determine this from looking at the magnitude and phase of the Bode plot.

• Presence of poles or zeros: These frequencies appear as peaks and valleys in a Bode plot magnitude curve, respectively, and the phase curve may pass through certain points at these frequencies.

As filters and amplifiers follow similar ideas, it helps to look at these two types of circuits to see what information is contained in the Bode plot phase curve.

### High Pass and Low Pass Filters

The image below shows the Bode plot for a 1st-order low pass filter (top: magnitude, bottom: phase). The phase plot shows how the phase shift develops when the source frequency starts to enter the cutoff region. Here, when the phase shift in the Bode plot is 45 degrees, the magnitude curve passes through approximately -3 dB. The same characteristics would be seen for a high pass filter, although the magnitude curve would be reversed. Magnitude and phase (Bode plot) for a 1st-order low pass filter.

When looking at these filters, we normally focus on the magnitude curve as this will have standard roll-off values and attenuation at the 45 degree phase shift point. For the 1st-order low pass filter shown above, the magnitude curve has standard -20 dB/decade rolloff and -3 dB attenuation at the 45 degree phase shift. For an Nth order filter, where N filter stages are cascaded in series, these roll-off and attenuation values would be multiplied by N.

For a passive filter circuit, this basically summarizes the important points in a Bode plot. More complicated filters with multiple cutoff stages can also be analyzed in a similar way. In an amplifier circuit where feedback is present, there is other information contained in the phase curve of the Bode plot that needs to be considered to ensure amplifier stability.

### Operational Amplifiers

In operational amplifier circuits with negative feedback, we can determine at which frequencies the amplifier may be driven unstable. If the phase in the feedback loop shifts by 180 degrees, feedback in the system switches from negative to positive feedback, the feedback switches from negative to positive and the amplifier will begin to oscillate. Eventually, the output will saturate and the output signal will be highly clipped (this is what happens in a comparator circuit).

The transition to instability can be summarized by looking at the magnitude and phase in the Bode plot. When there is a region in the plot with positive gain and 180 degree phase shift, the output will become unstable as feedback has switched from negative to positive. An example for an op-amp circuit with negative feedback is shown below. The metrics “gain margin” and “phase margin” are often used to define when an amplifier circuit becomes unstable; these are indicated in the example below. Bode plot magnitude and phase for stable and unstable amplifier circuits.

## Bode Plot Simulations During Circuit Design

Whether you’re designing a filter, amplifier, matching network, or other circuit, the circuit design tools you use should help you stay productive. When you need to generate a Bode plot, your circuit design tools should include a SPICE simulator with a frequency sweep simulation. A SPICE simulator can calculate the voltage and current in each net in a circuit with sweeping through a range of frequencies. Results for each frequency in the time domain can then be shown in a standard Bode plot for further analysis.

A related simulation that immediately picks out critical points in a Bode plot is pole-zero analysis. This calculation will give the frequencies that produce resonances (called poles) and total attenuation (called zeros). The advantage of this technique is it also returns the decay constant for each of these frequencies, which describes the transient response in the time domain.

When you need to design circuits for LTI systems, the front-end design features from Cadence can help you build circuit models and run a range of simulations. You can use the modeling and simulation features in PSpice Simulator to calculate the phase in a Bode plot for your system, and to perform a range of other analyses. Once you’re ready to create a PCB from your system, simply capture your schematics in a blank layout and use Cadence’s board design utilities.

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

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