Power Electronics

# Easy Calculation Yields Load Transient Response

Many electronic circuits can rapidly shift between a sleep state and full load operation, placing a heavy requirement on the power converter to stabilize its output voltage. An excessive voltage variation could cause the electronics to malfunction, reset, latch up or fail. It is typically the function of the power converter's output capacitance to provide output voltage holdup during any heavy load steps. The amount of capacitance placed on the converter's output often is miscalculated, increasing overall cost if too much capacitance is used and compromising system performance if too little is used. In addition, the effect of the output capacitance is frequently overlooked, and too much effort is spent trying to increase the converter's bandwidth to improve the load transient response.

Quite simply, the power supply's voltage variation to a load transient is its peak output impedance (in the frequency domain) times the change in load current. Fig. 1 presents a simplified model of the open-loop power supply power stage and the resulting output impedance of a typical filter. The buck power switching elements have been replaced by a voltage source with zero output impedance. The output impedance is determined by driving the output with a 1-A ac source and measuring the output voltage. Consequently, the output impedance at low frequency is set by the output inductor's equivalent series resistance (ESR) and increases with the inductor impedance. The impedance rises until the output inductor and capacitor resonate. At this frequency, the magnitude of the impedance is determined by the Q of the circuit and can be many times the impedance of the filter elements. Once past resonance, the output impedance drops with increasing frequency until the capacitor's ESR is reached. Finally, at higher frequencies, there is a parasitic or equivalent series inductance (ESL) in the capacitor that causes the impedance to increase. From Fig. 1, the output-voltage variation to a rapid load step can be quickly estimated, neglecting the effect of any feedback network. If the load step has a rate of rise much greater than one divided by the filter resonant frequency, the filter characteristic impedance sets the maximum impedance and the voltage variation in the open circuit case is just Zout × Istep, where:

The step-load repetition rate may play an important role and needs to be identified. If the pulse-load repetition rate is equal to the resonance frequency of the output filter, then the Q of the filter will make the output impedance much greater than Zout defined in Equation 1 and a rather large voltage variation can result. If the repetition rate is at a rapid rate and is above the bandwidth of the power supply, the ESL sets the maximum output impedance. However in a properly bypassed system, where distributed capacitance usually supplies pulse currents to the load, this is usually not the case.

### Current-mode Control

Fig. 2 presents a simplified current-mode buck power stage model and corresponding output impedance, assuming that the voltage loop is open. Note that in current-mode control, the control loop makes the output inductor behave as a constant current source feeding the output filter capacitor, negating its effect in the ac model. At low frequencies, the current-mode output impedance is much higher than with voltage-mode control; however, at higher frequencies, the impedances are identical.

Closing a feedback loop around the output voltage helps lower the output impedance. From classical feedback theory, the closed-loop output impedance is equal to the open- loop output impedance divided by one plus the loop gain. Fig. 3 shows this graphically. The output impedance of a voltage-mode power supply is presented and has a shape similar to Fig. 1. The closed-loop gain of the power supply is shown as an integrator-type response and, in general, has a -1 slope. The closed-loop output impedance also is graphed. As expected, at low frequencies, substantial reductions in output impedance are seen. However, near the crossover frequency, the closed-loop impedance approaches the open-loop impedance and at much higher frequencies they are identical.

An interesting effect can be seen in the closed-loop output impedance at the loop gain crossover frequency (where loop gain crosses 0 dB or where loop gain equals 1). Again, the closed-loop output impedance is set by the open-loop impedance divided by the loop gain of 1 at an angle related to the phase margin or simply:

While not really apparent in Fig. 3 due to the logarithmic scale, there is a 1.15 scale factor in peaking in the closed-loop-output impedance near the loop-gain crossover frequency due to a 55-degree phase margin. Fig. 4 presents this graphically. Peaking at the crossover frequency can be as little as 0.7 with 90 degree of phase margin or approaching infinity with 0-degree phase margin (but in reality, the system goes unstable with 0-degree phase margin).

### Current- vs. Voltage-mode Control

Fig. 5 presents a comparison between current-mode control and voltage-mode control on the power supply's closed loop output impedance. In both cases, the loops cross 0 dB at a frequency approximately equal to the loop gain shown in Fig. 3. However, the current-mode loop has a better phase margin (90 degrees versus 45 degrees). Notice that at lower frequencies, the output impedance using voltage-mode control is lower. This is due to the lower impedance of the output inductor in voltage mode versus the current source equivalent of current-mode control. As the frequency nears the overall loop-gain crossover or bandwidth of 30 kHz, the voltage-mode loop exhibits higher output impedance due to the peaking caused by the lower phase margin. Finally, at high frequency, the two output impedances converge as the improvement made by closing the loop wanes.

Fig. 6 shows the transient response of these two power supplies to a 5-A load step. There are two distinct differences between the two. First, the current-mode supply has a smaller voltage deviation than the voltage-mode supply. This is due to the voltage-mode converter's lower phase margin and corresponding higher output impedance near its crossover frequency. This impedance sets the initial transient voltage spike during a load step; hence, the voltage spike is a little larger than seen with current-mode control. The second difference is the voltage-mode supply recovers much quicker than the current-mode supply. This is due to the lower output impedance of the voltage mode at the lower frequencies, which allows an output-voltage recovery time that is faster when compared to a similar design using current-mode control.

### Output Filter Design

Some simplifications can be made designing the output filter. If one neglects the peaking caused by less than 60 degrees of phase, the closed-loop output impedance is approximately equal to the open-loop output impedance at the crossover frequency. The transient response to a load step can be calculated by multiplying the load step by the filter output impedance at the crossover frequency. In both voltage mode and current mode, this impedance usually is set by the output capacitor. Given this fact, Fig. 7 provides a graphical method of establishing loop bandwidth and output capacitance requirements. In this figure, the capacitance is assumed to have no ESR. First the required closed-loop output impedance is calculated from the equation below for a given transient load. For example, if a 50-mV voltage transient is desired for a 1-A pulse load, then the required closed-loop output impedance is 50 mΩ.

Next, select the loop-gain crossover frequency for the converter. If the loop crossover is not known, then a typical estimate would be to use one-fifth of the switching frequency. Find the intersection between the required closed-loop impedance and the loop-gain crossover frequency to find the output capacitance needed to meet the voltage transient requirement. This also can be worked from the opposite direction by starting with the loop-gain crossover frequency, finding the intersection with the desired output capacitor value (a known output capacitor) and then working back to find the capacitor impedance. This allows for the transient voltage to be calculated for any given step current and output capacitor used.

In reality, many capacitors have an ESR that cannot be neglected and that needs to be accounted for in the output filter design. In fact, many times, ESL needs to be accounted for as well. But looking at the effects of the ESR and output capacitance only, the output voltage variation is equal to the load-current step multiplied by the output impedance of the filter at crossover. For large value capacitors (polymer, tantalum, aluminum electrolytic) and reasonably high crossover frequencies, the combined output impedance is usually dominated by the ESR of the output capacitor. This is because the impedance due to the capacitance itself at the crossover frequency is typically much lower than the ESR. To design the output filter, just determine the impedance requirement, choose a capacitor with that ESR, and guarantee that the crossover frequency is above the zero formed by the output capacitor and its ESR. Note that this crossover frequency can be relatively low at the expense of response time.

To summarize the important points, a power supply's response to a load step is approximately equal to its output impedance at its crossover frequency multiplied by the magnitude of the load step. The type of control strategy can impact this impedance. Typically, a current-mode power supply will have smaller output impedance and a smaller transient response; however, it will be slower to recover from a load step. Finally, it is a simple calculation to determine output filter capacitance or ESR from a transient load-step requirement and an estimation of the loop-crossover frequency. 