This power supply design tip introduces a simple method to estimate the transient response of a power supply by understanding the control bandwidth and output filter capacitor characteristics. This method makes full use of the fact that the closed-loop output impedance of all circuits is the open-loop output impedance divided by 1 plus the loop gain, or simply expressed as:

Figure 10.1 illustrates the above relationship graphically. Both impedances are in dB-Ω or 20 * log [Z]. In the low frequency region on the open loop curve, the output impedance depends on the output inductor impedance and inductance. When the output capacitance and inductance resonate, a peak is formed. High-frequency impedance depends on the characteristics of the capacitor output filter, the equivalent series resistance (ESR), and the equivalent series inductance (ESL). Divide the open-loop impedance by 1 plus the loop gain to calculate the closed-loop output impedance.

Because the graph is expressed in logarithm, that is, simple subtraction, the impedance will be greatly reduced in the low frequency region where the gain is high; the closed-loop and open-loop impedances are basically the same in the high frequency region where the gain is small. The following points need to be explained here: 1) the peak loop impedance appears near the crossover frequency of the power supply, or where the loop gain is equal to 1 (or 0dB); and 2) most of the time, the power supply control bandwidth will be Higher than the filter resonance,

Therefore the peak closed loop impedance will depend on the output capacitance impedance at the crossover frequency.

Figure 10.1 The closed-loop output impedance peak Zout appears at the control loop crossover frequency

Once you know the peak output impedance, you can easily estimate the transient response by multiplying the magnitude of the load variation with the peak closed-loop impedance. There are a few things to note, because the low phase margin will cause peaking, so the actual peak value may be higher. However, in terms of quick estimates, this effect is negligible [1]. The second thing to note is related to the increase in the magnitude of the load change. If the load change amplitude changes slowly and low), the response depends on the closed-loop output impedance in the low frequency region related to the rise time; if the load change amplitude changes extremely quickly, the output impedance will depend on the output filter ESL. If it does, more high-frequency bypassing may be required. Finally, for very high-performance systems, the power stage of the power 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 inductor and the applied voltage limit the current Conversion rate.

The figure above is an example of how to use the above relationship. The problem is to choose an output capacitor based on the 50mV output variation within the allowable range of the 10amp change amplitude of the 200kHz switching power supply. The allowed peak output impedance is: Zout = 50mV / 10amps or 5 milliohms. This is the maximum allowable output capacitor ESR. The next step is to establish the required capacitance. Fortunately, both ESR and capacitors are orthogonal and can be processed separately. A high (Aggressive) power control loop bandwidth can be 1/6 or 30kHz of the switching frequency. So at 30kHz, the output filter capacitor needs a reactance of less than 5 milliohms, or a capacitance higher than 1000uF. Figure 10.2 shows a load transient simulation of this problem under 5 milliohm ESR, 1000uF capacitor, and 30kHz voltage mode control conditions. As far as the 10amp load fluctuation range for verifying the validity of this method is concerned, the output voltage change is approximately 52mV.