Measuring the Feedback Loop Characteristics of DC-DC Converters Using a Network Analyzer

Basic working principle of DC-DC converter

First, let’s quickly review the basic operation of a DC-DC converter. As an example, we will use a simple, non-isolated, single-phase buck converter in voltage-mode control.

The schematic and timing diagram in Figure 1 show the basic working principle of the DC-DC buck converter. The MOSFET switch converts the DC input voltage Vin into a pulse voltage, and the on/off state of the switch is controlled by the feedback loop. This pulse voltage is then converted into a DC output voltage Vout through the charging and discharging process of the LC filter in the output stage of the circuit.

When the switch is turned on, the current Ion flows through the inductor L, transferring power to the output capacitor Cout and the load, and Vout rises.

When Vout reaches a certain voltage level, the switch will be turned off, and the electricity charged to L by current Ion will generate current Ioff and transfer the electricity to the load. At the same time, the electricity charged to Cout will also be transferred to the load, and Vout will drop. When Vout drops to a certain level, the switch will be turned on again, so that Vout will rise again. The output voltage level is determined by the pulse duty cycle.

The longer the time Ton is, the higher the output voltage of the circuit is. The shorter the time Ton is, the lower the output voltage of the circuit is. When a current higher than a certain level continues to flow through the inductor L, the average output voltage can be calculated according to the following formula: Vout = Ton/(Ton + Toff) x Vin. Repeating this on/off operation while monitoring the output voltage and adjusting the pulse duty cycle can obtain a stable output DC voltage regardless of how the load changes.

Figure 1. Basic operating principle of a DC-DC converter.

Figure 2 is an example of a detailed schematic of a DC-DC converter. The output voltage shared by R1 and R2 is fed back to the error amplifier, which compares the feedback voltage with a stable reference voltage Vref and produces an output voltage proportional to the difference between the two voltages. The pulse width modulator (PWM) provides a pulse with a duty cycle determined by the output voltage of the error amplifier, which can turn the MOSFET switch on or off.

Figure 2. Example of a DC-DC converter schematic

When the feedback voltage is lower than Vref, the feedback system will extend the period Ton to increase the output voltage. When the feedback voltage is higher than Vref, the feedback system will shorten the period Ton to reduce the output voltage. In this way, a stable DC output voltage can be obtained.

Components such as C1, C2, C3, R3, R4 as well as R1 and R2 can adjust the gain and phase delay of the error amplifier, thereby improving the stability of the feedback loop (feedback compensation).

Measuring Feedback Loop Characteristics of DC-DC Converters

This section describes how to use the E5061B-3L5 LF-RF network analyzer to measure feedback loop characteristics. Before we begin the measurement method, let’s take a quick look at the basics of feedback loop control.

Loop Gain

As shown in Figure 3, the DC-DC converter can be considered as a negative feedback control system with the input signal Vref and the output signal Vout. |G| is called the open-loop gain, |Vout/Vref| = |G/(1 + GH)| is called the closed-loop gain, and |GH| is called the loop gain. It should be noted here that the loop transfer function is GH x (-1) = -GH because it includes the inverse of the error amplifier. The transfer function G corresponds to the total transfer function from the error amplifier to the output LC filter, while the transfer function H corresponds to the resistor divider circuit including R1 and R2. Resistors R1 and R2 also determine the gain and phase delay of the error amplifier together with R3, C1, C2, C3, and R4.

This negative feedback loop control system can adjust the variable output voltage Vout to be close to Vref/H. The larger the loop gain |GH|, the stronger the voltage regulation ability. As the frequency of voltage changes increases, the loop gain will decrease; when the loop gain is less than 1, the regulation no longer works.

The frequency at which the loop gain |GH| equals 1 (or 0 dB) is called the crossover frequency, and this frequency is the bandwidth of the loop (Figure 4). The higher the crossover frequency, the faster the feedback loop can regulate voltage changes and the faster it can respond to load changes.

Figure 3. Negative feedback loop control system

Phase Margin and Gain Margin

When the feedback control loop operates at high frequencies, phase delay usually occurs. Now, let's look at the phase delay of the loop transfer function –GH. In the low frequency range close to DC, the error amplifier only experiences a 180° phase delay. As the frequency rises, the error amplifier phase delay becomes larger, and additional delays occur elsewhere in the loop.

As shown in Figure 5, a large phase delay occurs around the resonant frequency fc = 1/(2 * π * √ (L*C)) of the output LC filter. Especially in applications where capacitors with relatively low ESR values ​​are commonly used to reduce the ripple of the output voltage, the phase response of the LC filter will be close to that of an ideal LC filter due to the extremely low ESR, so the phase delay near the resonant frequency will become very large, close to 180°. If the total phase delay of the feedback loop is close to 360°, the feedback loop will exhibit positive feedback instead of negative feedback. Moreover, if the loop gain |GH| is still greater than 1, an unstable control loop will cause oscillation due to changes in the components used in the loop circuit and changes in other conditions (such as temperature).

To avoid such problems, feedback compensation is required to stabilize the loop. Feedback compensation components (such as R3, R4, C1, C2, and C3 in Figure 3) are added to adjust the gain and phase of the error amplifier near the resonant frequency of the LC filter. As shown in Figure 4, at the crossover frequency where the loop gain |GH| = 1, the difference between the phase angle of -GH and -360° (that is, the difference between the phase angle of GH and -180°) is called the phase margin. The phase margin is an important parameter that indicates loop stability. The larger the phase margin, the more stable the feedback loop.

In real applications, the feedback loop must have a large enough phase margin to ensure that the system can operate stably under any load conditions.

However, if the crossover frequency is lowered due to excessive feedback compensation, the response speed of the feedback system to load changes will also be reduced. Therefore, when designing the feedback compensation circuit, it is necessary to achieve an optimal balance between system stability and response speed to meet the requirements of the target application. In order to optimize these parameters, it is very important to use a low-frequency network analyzer to verify the true characteristics of the feedback loop.

Similar to the definition of phase margin, at the frequency where the phase angle is 0°, the difference between the gain of -GH and 0 dB is called gain margin. Gain margin is also an important parameter for measuring loop stability.

Figure 4. Gain-phase characteristics of GH

Figure 5. Gain-phase characteristics of the output stage LC filter.

6. Gain-phase characteristics of the error amplifier and feedback compensation circuit

How to measure loop gain using a network analyzer

Low-frequency network analyzers can measure the feedback loop in operation by injecting a source signal into the feedback loop through an additional injection circuit. The network analyzer measures the ratio of the AC voltage across the injection circuit (receiver ports R and T with high-impedance inputs). When applying the stimulus signal, the signal is injected into a place where the input impedance (Zin) is high and the output impedance (Zout) is low.

When it comes to testing DC-DC converters, a floating excitation circuit consisting of a transformer and a resistor is usually used to apply the test signal before the voltage divider circuit on the feedback circuit path, as shown in Figure 7. By applying the excitation signal at a point that satisfies Zin >> Zout and letting the resistor R satisfy the condition of Zin >> R >> Zout, we can obtain the characteristics of the loop transfer function –GH through the measurement results of the T/R ratio, and this measurement method will not interfere with the original characteristics of the feedback loop.

The injected signal level should not be too high to avoid the feedback loop entering the nonlinear region. Probing should be done with a probe with high input impedance so as not to affect the operation of the feedback loop.

In terms of the measurement frequency range, it is usually measured from a low frequency of 10 Hz or 100 Hz. But generally speaking, the most important frequency range for measuring the loop characteristics of a DC-DC converter is mainly between a few kHz and several hundred kHz. The resonant frequency of the LC filter and the crossover frequency of the loop are both within this range. Therefore, it is not necessary to be so strict in the measurement of the low frequency range.

Note that the measurement method discussed here is based on linear voltage mode control loops only. It is not applicable to current mode control loops and nonlinear control loops.

Figure 7. Loop gain measurement method for negative feedback control system.

Loop Gain Measurement Configuration Example

Figure 8 shows a configuration example for measuring loop gain using the gain-phase test port of the E5061B-3L5 LF-RF network analyzer. The gain-phase test port provides a switchable direct receiver input with a 5 Hz to 30 MHz frequency range and 1 MΩ/50 Ω impedance.

Transformer T1 and resistor R5 are used to form a floating signal application circuit. The resistance of R5 should be much smaller than Zin (usually several kΩ or tens of kΩ). In addition, if the resistance of R5 is too small, the injected test signal will be excessively attenuated. Generally, 20 Ω to 100 Ω is widely used, but low resistance such as 5 Ω can improve the bandwidth of the transformer, depending on the transformer used. When measuring, set the R and T ports of the receiver to 1 MΩ input mode (input impedance Zin = 1 MΩ // 30 pF). Use coaxial test cables to connect the R and T ports to the device under test. For this loop gain measurement configuration, it is recommended to use coaxial test leads instead of 10:1 passive probes, because in this configuration, both the signal source port and the receiver port are floating with respect to the ground of the device under test, and 10:1 passive probes will cause measurement errors related to stray coupling. (Note: Ports R and T are semi-floating with respect to their chassis ground, with a floating impedance of approximately 30 Ω, as detailed in Figure 22.) In this case, it is not a problem if the probing capacitance of the coaxial test cable is relatively large, because the frequency range required for such measurements is usually less than 1 MHz, and we can obtain a sufficiently high probe input impedance even with coaxial test cables. If you use a 10:1 passive probe in this measurement configuration that includes injection of a floating signal source, it is recommended to connect the outer shield of the LF OUT port (chassis ground of the analyzer) to the ground of the device under test using a short lead, as shown in the dashed portion of Figure 9.