Switching Power Supply Circuit Design Tips: Easily Estimate Load Transient Response

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 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 10.1 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. The peak is formed 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 can be 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 is essentially the same at high frequencies where the gain is low. The following points need to be noted: 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,

The peak closed-loop impedance will therefore depend on the output capacitor impedance at the crossover frequency.

Figure 10.1 The closed-loop output impedance peak Zout appears 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 caveat is related to the load step rise time. If the load step is slow (low), the response will be determined by the closed-loop output impedance in the low frequency region, which is related to the rise time. If the load step 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, i.e., the current in the inductor may not respond as quickly as the control loop expects, because the inductance and applied voltage will limit the current slew rate.

The figure above is an example of how to use the above relationship. The problem is to select an output capacitor based on a 50mV output change within the allowable range of a 10amp change in a 200kHz switching power supply. The peak output impedance allowed 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, 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 30kHz. The output filter capacitor then needs a reactance of less than 5 milliohms, or a capacitance greater than 1000uF at 30kHz. Figure 10.2 shows a load transient simulation of this problem for 5 milliohm ESR, 1000uF capacitance, and 30kHz voltage mode control conditions. For a 10amp load change to verify that this method is effective, the output voltage changes by about 52mV.