Power Electronics

# FRA Measures Closed-Loop Response

Injecting a signal from a frequency response analyzer determines the gain and phase margin of a switching power supply.

Switching power supplies rely on feedback control loops to ensure the supply maintains the required voltage and current with varying load conditions. While design of the feedback control loop influences stability and transient response, it's important to verify the theoretical loop transfer function to ensure supply stability and transient response. One way to verify this is by injecting a disturbance signal from a frequency response analyzer (FRA) that then yields the loop gain and phase margin.

A feedback control loop oscillates when there's a frequency at which the loop gain is unity and the total phase lag equals 360°. Stability is usually measured by two factors:

• Phase margin — the difference between actual phase lag and 360° when the loop gain is unity (usually expressed in degrees).
• Gain margin — the amount the gain has fallen below unity when the total phase lag is 360° (usually expressed in dB).

For most closed-loop feedback control systems, there are two basic rules:

• Phase margin is greater than 45° (less than 315°) when the loop gain is greater than 0dB.
• Gain margin is -20dB or lower when the loop phase delay reaches 360°.

If you meet the above conditions, the control loop will have near optimum response and the control loop will be unconditionally stable, neither under- nor over-damped. To reveal all likely conditions, you can usually perform a frequency response measurement well beyond the control-loop operational bandwidth.

The Bode Plot of Fig. 1, on page 29, shows control loop response curves for a single output switching power supply. We used measurements from a GP102 gain phase analyzer (see photo), then imported them into a spreadsheet.

Comparing these gain and phase margins to the target values of -20dB gain margin and 60° phase margin, room is available for change without approaching areas of instability. The crossover point is 160 Hz, where the transient response and regulation of the tested power supply would be over-damped and unacceptable. Ideally, a positive loop gain up to 1kHz to 2 kHz would be required. After modifications to the error amplifier compensation components, the control loop can be re-tested to ensure stability.

Generally, you can use an FRA or gain-phase analyzer to perform these measurements, using discrete fourier transform (DFT) techniques. The DFT is used to extract the signal of interest.

### Selecting the Injection Point

To make a measurement, an FRA injects a disturbance at a known frequency (error signal) into the control loop, shown in Fig. 2, on page 29. Two FRA measurement channels determine the time for the disturbance to go from the error amplifier input to the power supply output.

The injection point should be where there's a single path control-loop feedback signal fed from a low impedance source. The feedback path connecting to the power supply output or error amplifier output are good places to inject a disturbance signal.

In operation, the FRA signal generator introduces a small disturbance into the control loop at a particular frequency. Often, an isolation transformer connects the signal generator to the circuit under test, ensuring electrical isolation between the FRA signal generator and circuit under test.

Selection of a suitable injection point also requires careful selection of injection amplitude by connecting an oscilloscope across the power supply output. Set the FRA signal generator amplitude to zero and to a low frequency, usually at the lower part of the control-loop bandwidth. Slowly increase the amplitude of the FRA generator. A good starting point for the FRA signal generator amplitude adjustment is when the oscilloscope shows a small disturbance, around 5% of the nominal power supply output voltage. Repeat this process at the upper part of the control-loop bandwidth. Avoid a situation where the FRA generator is under- or over-driving the control loop; measurements made under these conditions are likely to be incorrect.

It's unlikely you can use the same FRA signal generator setting over the entire control loop bandwidth. Instead use amplitude compression to maintain a steady disturbance signal while sweeping the frequency and the loop-gain changes.

If the power supply under test produces a high-output voltage, then injection method 1 doesn't apply. You can use injection method 2 (Fig. 3, on page 29) if the power supply output voltage exceeds the FRA input range (greater than 500Vpk). It allows FRA connections to be made where there's a low control-loop voltage to ground (common mode voltages). For injection method 1, connect the FRA CH2 to the control-loop output and FRA CH1 to the control-loop input, as shown in Fig. 4.

To use injection method 2, connect the system, as shown in Fig. 5. Make a sweep from typically 10 Hz to 30 kHz and look for gain and phase measurement repeatability as an indicator that you applied the correct level of injection to the control loop. Assess control-loop gain and phase margins, referring to earlier gain phase guidelines. You can apply suitable compensation components to the error amplifier stage. Performing a new sweep will show the effect of the new compensation values. The loop gain should ideally roll off at -20dB per decade, particularly where the loop gain passes through unity.