Methods for estimating power supply load transient response

A simple method for estimating the transient response of a power supply by knowing the control bandwidth and the output filter capacitor characteristics is presented. This method takes advantage of the fact that the closed-loop output impedance of any circuit is the open-loop output impedance divided by 1 plus the loop gain, or simply stated:

Figure 1 illustrates this relationship graphically, with both impedances in dB-Ω or 20*log [Z]. In the low frequency region of the open loop curve, the output impedance is determined by the output inductor impedance and the inductance. Peaking occurs when the output capacitor and inductor resonate. The high frequency impedance is determined by the capacitor output filter characteristics, the equivalent series resistance (ESR), and the equivalent series inductance (ESL). The closed loop output impedance is calculated by dividing the open loop impedance by 1 and adding the loop gain.

Since the graph is logarithmic, i.e. simple subtraction, the impedance is greatly reduced at low frequencies where the gain is high, and the closed-loop and open-loop impedances are essentially the same at high frequencies where the gain is less. It is important to note that 1) the peak loop impedance occurs near the power supply crossover frequency, or where the loop gain is 1 (or 0 dB), and 2) most of the time, the power supply control bandwidth will be above the filter resonance, so the peak closed-loop impedance will be determined by the output capacitor impedance at the crossover frequency.

Figure 1: The closed-loop output impedance peak Zout occurs at the control loop crossover frequency

Once the peak output impedance is known, the transient response can be easily estimated by multiplying the load step by the peak closed-loop impedance. A few caveats are in order; the actual peak value may be higher due to peaking caused by low phase margin. However, for a quick estimate, this effect is negligible [1].

The second consideration is related to the load ramp. If the load ramp is slow (low dI/dt), the response is determined by the closed-loop output impedance in the low frequency region related to the rise time. If the load ramp is very fast, the output impedance will be determined by the output filter ESL. If this is the case, more high-frequency bypassing may be required. Finally, for very high-performance systems, the power stage of the supply may limit the response time, that is, the current in the inductor may not respond as quickly as the control loop expects, because the inductance and applied voltage limit the current slew rate.
Here is an example of how the above relationship can be used. The problem is to pick an output capacitor based on a 50mV output change within the allowable range of a 10 amp change in a 200kHz switching supply. The peak output impedance allowed is: Zout = 50 mV / 10 amps or 5 milliohms. This is the maximum allowable output capacitor ESR. The next step is to establish the required capacitance. Fortunately, ESR and capacitance are both orthogonal and can be treated separately. An aggressive power supply control loop bandwidth can be 1/6 of the switching frequency or 30 kHz. The output filter capacitor then needs a reactance of less than 5 milliohms, or a capacitance greater than 1000uF at 30 kHz. Figure 2 shows a load transient simulation of this problem with 5 milliohm ESR, 1000uF capacitance, and 30 kHz voltage mode control. For a 10amp load change to verify that this approach is effective, the output voltage changes by approximately 52mV.

Figure 2: Simulation verification of estimated load transient performance