 Interested?

R&S®Essentials | Digital oscilloscope and probe fundamentals

# Understanding Bode plots

## What are Bode plots?

Bode plots were originally devised by Dr. Henrik Wayne Bode while he was working for Bell Labs in the 1930s. They are most used to analyze the stability of control systems, for example when designing and analyzing power supply feedback loops. The advantage of using Bode plot is that they provide a straightforward and common way of describing the frequency response of a linear time invariant system.

## How to read Bode plots?

Bode plots show the frequency response, that is, the changes in magnitude and phase as a function of frequency.

This is done on two semi-log scale plots. The top plot is typically magnitude or “gain” in dB. The bottom plot is phase, most commonly in degrees.

### Phase and gain margins

The information in a Bode plot can be used to quantify the stability of a feedback system by using the phase and gain margins.

Phase margin is measured at the frequency where gain equals 0 dB. This is commonly referred to as the “crossover frequency”. Phase margin is a measure of the distance from the measured phase to a phase shift of -180°. In other words, how many degrees the phase must be decreased in order to reach -180°.

Gain margin, on the other hand, is measured at the frequency where the phase shift equals -180°. Gain margin indicates the distance, in dB, from the measured gain to a gain of 0 dB. These values, 0 dB and -180° are important because system instability occurs if these two values meet.

Gain and phase margins represent the distance from the points at which instability could occur. The greater the distance or margin the better, because higher gain and phase margins mean greater stability. A loop with a gain margin of zero or even less would only be conditionally stable and could easily become unstable if gain changed. A typical goal for phase margin is to have at least 45 degrees, and even higher values might be desirable in more critical applications.

In addition to safety considerations, performance is also affected by values that can be determined from Bode plots. For example, a higher 0 dB crossover frequency usually means a faster response to load changes. And lower gain at higher frequencies means better noise immunity or lower output ripple.

### Stable and unstable closed loop systems

The measured phase at 0 dB is -135°, so the phase margin is 45°. The gain at -180° degrees is -9 dB, so the gain margin is 9 dB. Since phase margin is positive, this system is stable.

The measured gain is +13 dB when phase is -180°, so the gain margin is minus 13 dB. At a gain of 0 dB, the measured phase is minus 215°, so the phase margin is minus 35° at the gain crossover point. This system is unstable.

## Bode plot vs. load transient test and step response tests

There are other common ways of quantifying or measuring the stability of power supplies, such as load transient or step response tests. Although this method is well-understood and widely used, it can be difficult to build a circuit to generate a fast load step, especially if there is inductance between the power supply unit and the load step generator.

• Step response only shows large scale behavior, whereas Bode plots can also show behavior on a smaller scale.
• Bode plots can also easily be made at different load levels or operating points. This is important because loop stability often depends on the operating point. A power supply might appear to be stable, but approaches instability under different load conditions.

## Measuring closed loop stability with bode plots

To better describe the application of Bode Plots, closed loop stability of a DC/DC power supply is measured by determining the closed loop response. This can be tested using the voltage injection method. This method adds a very small resistor – usually on the order of 10 ohms – into the feedback loop. A point should be chosen such that the impedance looking in the direction of the feedback loop is much larger than the impedance looking back. A small disturbance signal is then injected across the resistor. This is normally done using a so-called injection transformer to avoid influencing the loop. The response is then measured and Bode plots are generated.

### Instruments for measuring closed loop response

Two different categories of instruments can be used when measuring closed loop response. The first of these is a vector network analyzer or VNA. A VNA usually has a very high dynamic range, which allows it to make very precise impedance measurements. One drawback to using a VNA, other than cost and complexity, is that VNA’s are best suited for the characterization of 50 ohm components. Oscilloscopes, on the other hand, are already commonly used in the development of power supplies and allow direct characterization of noise and output ripple. Scopes can now also make stability measurements such as gain and phase margin, power supply rejection ratio, and step response.

### Test configuration: How to measure control loop response with an oscilloscope

To measure the loop response of the DC-DC power supply, a disturbance signal must be injected into the loop. Thus, a point should be chosen where the impedance looking in the direction of the loop is much larger than the impedance looking backwards. A small resistor is placed at the injection point and the disturbance voltage is applied in parallel to the injection resistor using a wideband injection transformer. The disturbance signal is created by the internal generator of the oscilloscope. Two channels of the oscilloscope are connected to either side of the injection point. Based on the measured values, the oscilloscope generates and displays the Bode plots.

When measuring closed loop response, it’s important to use the right probes. The peak to peak amplitudes at the measuring points can be very low at some test frequencies. For this reason, 1x passive probes are recommended over the more common 10x probes. If the signal is increased to noise ratio, this also improves the dynamic range of the frequency response measurements. It is also important to use a ground spring or a very short ground lead in order to reduce switching noise pickup and inductive ground loops.

Summary

Bode plots are useful in analyzing magnitude and phase changes introduced by a linear time invariant system (LTI system) e.g. the control loop response of a power supply.

A Bode plot makes it easy to determine the phase and gain margins: Phase and gain margins are important for determining system stability (the more margin, the better)

Testing closed loop response with an oscilloscope:

• Inject a disturbance voltage into the loop
• Measure the voltage across the resistor
• Generate and display Bode plots on the oscilloscope 