2 Theory and Background

The loop gain $\mathrm{T}\left(\mathrm{S}\right)$ can be derived from the open-loop impedance ${\mathrm{Z}}_{\mathrm{OL}}\left(\mathrm{S}\right)$and the closed-loop impedance ${\mathrm{Z}}_{CL}\left(\mathrm{S}\right)$. The open-loop impedance is determined by the power stage and is composed of passive components that form the topology, such as the inductor, input and output capacitors, FET resistances, and board parasitics. It is measured when all conversion switches are off. The closed-loop impedance is the output impedance measured when the control loop is actively regulating in steady state. Figure 2-1 visualizes the closed-loop system for a converter.

Figure 2-1 Converter Loop Gain

The loop gain and the output impedances are related by the following equation (refer to [8, Ch. 9.2.1]):

This equation reveals a fundamental design principle: closed-loop impedance is inversely proportional to loop gain. At frequencies below the crossover frequency where loop gain is high, the closed-loop impedance is significantly reduced by the feedback control. At higher frequencies where loop gain drops below 0 dB, the open-loop impedance becomes dominant and closed-loop impedance approaches ${\mathrm{Z}}_{\mathrm{OL}}\left(\mathrm{S}\right)$. A critical design objective for any power supply is maintaining low output impedance across the frequency range to enable good load-transient response. Understanding the relationship between $\mathrm{T}\left(\mathrm{S}\right)$, ${\mathrm{Z}}_{\mathrm{OL}}\left(\mathrm{S}\right)$, and ${\mathrm{Z}}_{\mathrm{CL}}\left(\mathrm{S}\right)$ is essential to achieve this goal.

Figure 2-2 LM5126EVM Output Impedance: Open Loop vs. Closed Loop

By rearranging Equation 2 to solve for T(S), the loop gain can be derived as:

This relationship is the foundation of the output-impedance reconstruction method. By measuring both the open- and closed-loop output impedances, the loop gain can be directly computed without requiring access to internal feedback nodes or breaking the control loop.

Calculating the loop-gain requires separating the real and imaginary components of the output impedances to compute the loop-gain magnitude and phase. Using the following formulas in a spreadsheet tool the measured data can be easily processed.