# Power Supply Control Loop Transfer Function Construction

Power supplies that provide active regulation of power delivery have two very important quantities: the output impedance and loop gain. The former is important for ensuring efficient power delivery to a load, while the latter determines the stability of the power supply as a function of frequency. The two qualities are actually related to each other due to the fact that regulated power supplies are closed loop systems. These are related via a transfer function, just as is the case in an amplifier circuit.

Why worry about a loop transfer function? If you’re designing a VRM or qualifying a VRM for a digital system, RF system, or mixed signal system, you may need to evaluate the loop gain and phase of your proposed power supply design. Here is how to relate direct measurements of the output impedance to the loop gain in order to evaluate stability.

## Loop Gain Starts With a Transfer Function

A common way to evaluate a power supply circuit or module for an application is to measure its loop gain and output impedance. Ideally, the output impedance should be zero, and the loop gain should be flat with no change in phase. In reality, the loop gain will limit the bandwidth of the power supply’s ability to provide power at higher frequencies, which might eliminate that circuit from being useful in a high-speed or high-frequency system.

The loop gain, output impedance, and loop transfer function of a power supply are all related with a very simple formula:

F(s) is the transfer function of the control loop, which is equal to the loop gain. In this formula, the CL superscript refers to closed-loop, and the OL superscript refers to open-loop.

The two impedances can be easily measured with and without the control loop completed in a regulator circuit. The standard technique is to inject a small signal into the control loop via an isolation transformer. A signal generator injects the signal, and the oscilloscope is used to compare measurements at each node.

Points A and B mark where signal injection can occur.

Next, the signal generator frequency is swept up to some very high value, usually high MHz frequencies. This frequency sweep limit should be near the bandwidth limit for many systems where a regulator may start to become unstable. The measured impedance values are plugged into the above formula at each frequency, and this gives the complete function for the loop gain.

Care needs to be taken when measuring the loop gain in this way. The placement of cables, resistive connections on measurement points, and bandwidth of probes all contribute to any errors seen in this type of measurement. The measurement itself modifies the loop gain one would want to measure, especially at high frequencies near the gain crossover.

## Now You Have the Loop Gain, What’s Next

After the loop gain is determined for the system under test, the loop gain can be qualified by checking the phase margin. The phase margin is found by reading off the value of the phase in the loop gain plot at the 0 dB crossover point. The phase margin is equal to:

Phase margin = Loop gain phase + 180°

The reason we care about phase margin has to do with the stability of the regulator circuit or module being qualified. If phase margin is > 0, then the regulator will be stable. Conservative designs generally demand a higher phase margin of at least 45° in order to be considered stable and acceptable for use in all situations. Note that phase margin can be affected by the presence of compensation networks, i.e., such as anything attached to the COMP pin in a regulator.

The COMP pin on a regulator is used to compensate the control loop with a capacitor or an RC circuit. (Source: LM2577ADJ datasheet)

At high frequencies, it is always the case that PCB layout or package layout parasitics affect the ability of the power regulator to maintain stable power and avoid oscillation. If there is not enough phase margin in the loop transfer function up to the regulator’s bandwidth, then the regulator package and/or PCB layout may need to be redesigned. Circuit components and compensation networks might also require changes if there is not enough phase margin.

Once you have the loop gain determined for your power regulator circuit, you can evaluate your power systems using the complete set of system analysis tools from Cadence. Only Cadence offers a comprehensive set of circuit, IC, and PCB design tools for any application and any level of complexity. Cadence PCB design products also integrate with a multiphysics field solver for thermal analysis, including verification of thermally sensitive chip and package designs.

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Solving electromagnetic, electronics, thermal, and electromechanical simulation challenges to ensure your system works under wide-ranging operating conditions

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