Step-down converters have basic blocks that contribute to the closed loop system. Figure 2-1 shows a closed loop system for a generic step-down converter.

Figure 2-1 Closed Loop System for a Generic Step-down Converter

$Z_o^{ol}\left(s\right)$ is the open loop output impedance of the power stage loop gain and $T\left(s\right)$ is defined as the product of all gains around the loop. When the loop is open, the power stage is running at a constant operating point meaning the controller signal stays constant.

When the loop is closed, converter is running in normal operation. The closed loop output impedance is $Z_o^{cl}\left(s\right)$:

From this equation we can deduct that the higher the loop gain is, the more the output impedance is reduced by the feedback loop. While designing a power supply, reducing the output impedance is possible by adding output capacitance to lower the open loop impedance. Especially for high-current operation, making sure that the output impedance is low across the frequencies is important to enable good load transients. Though, this method comes to a size and bill of material cost, it might also degrade performance for some designs where a maximum output capacitance should not be exceeded.

From the previous formula we can derive the loop gain:

Figure 3-1 shows the output impedance measurement test setup. For more details regarding the impedance measurement method and different steps for a good calibration, refer to the How to Measure the Power Delivery Network Impedance of a DC-DC Converter.

Figure 3-2 shows the output impedance comparison when the loop is open $Z_o^{ol}\left(s\right)$ and when loop is closed $Z_o^{cl}\left(s\right)$ with a 500mΩ resistive load connected to GND at the output. The respective Phases are plotted in Figure 3-3.

Figure 3-2 Output Impedance Measurements Magnitude for TPS62816

Figure 3-3 Output Impedance Measurements Magnitude for TPS62816

For the open loop impedance, at low frequencies the impedance curve is influenced by the constant output resistance of 500mΩ. Above 10 kHz, the measured impedance tracks the impedance of the output capacitor bank, flattening out with its ESR value at 500 kHz. The far right part of the curve is dominated by the inductive behavior of the capacitance at very high frequencies.

For the closed loop impedance, at low frequencies a high DC loop gain value can be observed which results in a low output impedance. At higher frequencies, the measured impedance curve shows a similar behavior as in an open loop condition.

A particular characteristics of loop gain stability is that for the frequencies of interests the closed loop impedance is higher than the open loop. This behavior is due to control loop stability and can be observed between 100 kHz and 500 kHz. This specific frequency range will help us determine by calculation the phase margin as well as the crossover frequency of the TPS62816.

To facilitate further processing, a math function can be used to divide the open loop output impedance $Z_o^{ol}\left(s\right)$ by the closed loop output impedance $Z_o^{cl}\left(s\right)$:

Figure 3-4 Open Loop Impedance Divided by Closed Loop Impedance

The loop gain is a complex number with a real and an imaginary part due to the complex impedance. The information needed to calculate the loop gain are the magnitude Mag[T(s)] and the phase Phase[T(s)] respectively shown in Equation 4 and Equation 5.

To visualize data and apply formula to taken values, the use of an analytics solution may be required. All Network Analyzer do not provide flexible interface to analyze data.

Figure 4-1 shows a comparison of the reconstructed gain to measured voltage injection method.

The reconstructed gain using the Equation 4 from the impedance measurements shows a very good matching result until the lowest frequency. The red curve shows the real loop gain and the blue curve the calculated loop gain.

Figure 4-1 Loop Gain Magnitude with Output Impedance Reconstruction (blue) and with Voltage Injection Method (red)

Operating Conditions: $V_{in}^{}$ =3.3 V, $V_{out}^{}$ =1.0V, $I_{out}^{}$ =2 A

Crossover Frequency: 141.25 kHz (Red curve), 146.27 kHz (Blue curve)

Figure 4-2 shows a comparison of the reconstructed phase to measured voltage injection method.

The red curve of Figure 4-2 shows the measured voltage injection method phase and the blue curve shows the calculated phase, reconstructed using the Equation 5 from the impedance measurements.

Figure 4-2 Loop Gain Phase with Output Impedance Reconstruction (blue) and with Voltage Injection Method (red)

Operating Conditions: $V_{in}^{}$ =3.3 V, $V_{out}^{}$ =1.0V, $I_{out}^{}$ =2 A

Phase Margin: 69.28° C (Red curve), 61.17° C (Blue curve)

• Output Impedance Measurements and Loop Gains
• Output Impedance for Stability Analysis
• Deviation of Loop Gain from Output Impedances in DC-DC Buck Converter
• Texas Instruments, How to Measure the Control Loop of DCS-Control Devices application report
• Texas Instruments, A Topical Index of TI Low Power Buck DC/DC application report