01092018 Simple TransientResponse Measurement Determines PowerSupply BandwidthBob Sheehan Electronic Design Ascertaining the relationship between bandwidth and response in power supplies isn’t easy, but this Idea for Design presents a measurement approach that helps ease the process.
It’s normal to assume that there’s an easy way to relate the bandwidth of a powersupply control loop to its transient response – no good reference exists that defines this in simple terms. It seems like a straightforward problem, which should have a simple solution. The higher the bandwidth, the faster the loop responds, and with less voltage deviation. However, several limiting factors may get in the way of this simple relationship. The first one is the series resistance of the output capacitor. If that resistance is too high, then the load step creates a large voltage deviation before the control loop can respond. Equation 1 gives the peak voltage deviation:
Second, the inductor can cause slewrate limiting. This is related to the controlloop bandwidth by the voltage across the inductor, calculated with Equation 2:
Third, there’s a critical inductance limit beyond which the duty cycle will saturate. The peak transient voltage is then determined by the largesignal limiting of the inductor current into the output capacitor. This is related to the voltage across the inductor, output capacitor, and series resistance, as expressed by Equation 3:
Look at the design of power supply intended to avoid these issues, and use an electronic load to test the transient response. If your controlloop bandwidth is relatively high, the output voltage may follow the load current and isn’t limited by the control loop. In this case, you can use a MOSFET and load resistor on a small board for the load step, controlled by a function generator. A low duty cycle for the load ontime will minimize dissipation in the resistor.
It’s important to mount this as close to the powersupply output as possible in order to minimize wiring inductance; Figure 1 shows a typical setup. The small black wire connects to a surfacemount coaxial cable for the outputvoltage measurement.
Figure 2 shows the measured transient response, which is directly related to the bandwidth of the control loop of Figure 3. With no equivalent series resistance (ESR), or slewrate or dutycycle limiting, the initial response time is onefourth the effective controlloop period. This is the equivalent first quarter of a sinusoidal response at the unitygain frequency. The peak voltage will vary based on the topology and damping, but is easily predictable with a surprising degree of accuracy.
With no ESR, slew rate or dutycycle limiting, Equation 4 calculates t_{P} as:
For currentmode control, Equation 5 gives the singlepole approximation that results in the peak voltage deviation:
Equation 6 calculates the critically damped case for currentmode control (as shown in Fig. 2):
For voltagemode control, Equation 7 gives the peak voltage deviation:
It’s important to verify the performance over all operating conditions. Dutycycle limiting can cause a significant droop when operating the control loop outside its linear range (Fig. 4).
From this, we see that the relationship between bandwidth and transient response is simple and straightforward. By observing the transient response, you can quickly get a good estimate of the controlloop bandwidth. 

