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Measurement of Switching Power Supplies Based on FRA
Abstract: From a small-signal perspective, this paper evaluates product stability by examining the impact of injecting a perturbation signal into the feedback loop on the entire product, and then uses a loop analyzer to perform quantitative analysis of the loop. The results facilitate future improvements and enhance product reliability.
Keywords: feedback loop, bandwidth, phase margin, gain margin, stability
Title: A Measurement of Switching Power Supply Based on FRA
ABSTRACT: This article describes the impact on the whole product caused by injecting a perturbation signal into the feedback loop, in order to evaluate stability. Then, by using a frequency response analyzer, an accurate result is obtained, and the result is used for improvement as well as to enhance product reliability.
Keywords: feedback loop, bandwidth, phase margin, gain margin, stability
As technologies such as electronics, automatic control, aerospace, communications, and medical devices continue to develop in depth and breadth, the power supplies that feed them are inevitably required to have higher stability—that is, not only good line regulation and load regulation but also fast dynamic load response. These factors are all related to the control loop, which generally operates in a negative feedback state, referred to as voltage negative feedback. If no feedback control loop is used in the converter (i.e., the H section in Figure 1 below), its transfer function is generally as follows, where G is the product of factors such as the input filter, power conversion, and rectifier filter sections (since they are in cascade form, the total product factor G is used in this paper). It can be seen that the output changes linearly with the input; however, due to the nonlinearity of the rectifier and filter network in the time domain, this change is actually only approximately linear, so when the input voltage changes, it cannot effectively maintain voltage regulation. If the feedback loop is poorly designed, the loop cannot make timely and appropriate adjustments to transient load changes, and the output voltage will momentarily be too high or too low, possibly even causing oscillation of the power supply system and damaging the downstream stage. Therefore, the ability to measure the loop is very important. So how does the loop section affect the entire circuit? Referring to MORNSUN's wide-input or AC-DC series products, the figure below shows the loop gain model of the operational amplifier in the feedback loop control section, and its transfer function is
where G: open-loop gain, H: feedback factor, GH: loop gain (as can be derived from Figure 1)
Figure 1. Feed back
I. Criteria for Loop Gain Stability:
From the transfer function , since the open-loop gain G of the amplifier is a function of frequency and decreases as frequency increases, and is also related to the phase of the amplifier, when GH = -1, the value of the transfer function is ∞, meaning the gain is infinite. It can be considered that any arbitrarily small input disturbance can cause an infinite output. If such an infinite output signal is fed back to the power conversion stage, it will inevitably cause oscillation of the final output, and the entire system will no longer maintain voltage regulation. Therefore, the stability of the system can be judged by analyzing the gain and phase of GH.
Also, since oscillation occurs when GH = -1, the phase shift ∠GH is 180° (because negative feedback itself has a 180° phase shift), and the loop gain |GH| = 1 (0 dB).
Therefore, for the operational amplifier to be stable, the following conditions must be met: 1. The phase condition is that its phase shift must be less than 180°, i.e., there must be a margin of more than 45 degrees; 2. The gain condition must also be met, i.e., there must be a margin of more than 12 dB; 3. The crossover frequency should close at 20 dB/Dec. Relevant explanations are given below.
II. Fundamentals of the Bode Plot:
As can be seen from the above, the stability of the system can be judged by the frequency characteristics of the loop gain GH, and the frequency characteristics of the loop gain |GH| and the loop phase difference ∠GH can be represented by a Bode plot (see Figure 2). The stability of the system can be measured by the phase margin, gain margin, and crossover frequency in the Bode plot. The phase margin refers to the difference between the actual phase delay and 360 deg on the frequency-phase curve when the loop gain is unity gain, expressed in degrees (deg), as shown in Figure 2.
The gain margin refers to the amount by which the gain is below unity gain on the frequency-gain curve when the total phase delay is 360 deg, expressed in decibels (dB), as shown in Figure 2.
The crossover frequency, also referred to in some literature as bandwidth, refers to the frequency value corresponding to zero gain on the frequency-gain curve, as shown in Figure 2.
Figure 2 Bode plot• Gain margin• Phase margin
The role of phase margin is to ensure that the system remains stable under certain conditions (including component tolerances, input voltage variations, load variations, temperature rise, etc.). At nominal input, rated load, and room temperature, a margin of 45 degrees is required; if the input voltage, load, and temperature variation ranges are very large, the phase margin should not be less than 30 degrees.
For gain margin, in order not to approach the instability point, it is generally considered that 12 dB or more is necessary.
The crossover frequency (bandwidth) can reflect the speed of the control loop response. It is generally believed that the wider the bandwidth, the better its ability to suppress dynamic load changes, the smaller the overshoot and undershoot, and the faster the recovery time, thus making the system more stable. However, due to the influence of the right-half-plane zero, as well as comprehensive factors such as the finite bandwidth of raw materials and operational amplifiers, the bandwidth of the power supply cannot be increased without limit. It is generally taken as 1/20 to 1/6 of the switching frequency.
III. Loop Testing
For the measurement of loop gain and phase, a frequency response analyzer (FRA) or gain-phase analyzer can generally be used. These instruments preprocess the sampled analog signals, then perform A/D conversion, and then use DFT (Discrete Fourier Transform) computation to obtain the gain and phase, finally presenting them as a curve (Bode plot).
This paper uses the frequency response analyzer (PSM1700) from N4L, UK, for analysis, mainly following the wiring diagram below. The key points are the location and size of the injection resistor. To reduce measurement error, a resistor of 50–100 Ω is generally selected for the experiment; the magnitude of the perturbation signal can be read through an oscilloscope during testing, or set using the Amplitude compression function of the FRA. However, the perturbation amplitude must not exceed 5% of the output voltage; otherwise, the measured results will be inaccurate.
Figure 3 Location of the injection resistor and wiring for FRA measurement
IV. Example of Switching Power Supply Stability Testing:
After understanding the factors affecting product stability, the judgment criteria, and how to test the product, the results of the tested product can be analyzed. Below, an experiment is conducted on a product from MORNSUN's wide-input series. The actual wiring is shown in Figure 3, and the test results are shown in Figure 4.
Figure 4 Magnitude and phase of loop gain for nominal input voltage at full rated power and 25℃
In this example, it can be seen from the above figure that the gain margin is -39.3 dB, the phase margin is 87.8 deg, and the bandwidth is 315.5 Hz, and the gain above 0 dB rolls off at -20 dB/decade. According to the above criteria, the system is preliminarily judged to be stable, but a phase margin of 87.8 deg means the dynamic response is underdamped. In fact, this can be achieved by making the error amplifier compensation network roll off at a slightly lower frequency. Under the premise of ensuring the phase margin, the gain below 0 dB rolls off at -40 dB/decade, which is more conducive to suppressing high-frequency interference and achieving optimal response;
The frequency corresponding to the 0 dB crossover point is 315.5 Hz. From a theoretical perspective, for a switching frequency of 300 kHz, the measured bandwidth may cause the control loop response to be somewhat slow, which is not conducive to the transient stability of the system. A higher bandwidth allows a smaller output filter capacitor to produce a smaller ripple voltage, which can also reduce raw material usage and module power supply size, thereby making the module power supply more energy-efficient. Considering the very conservative gain and phase margins, some minor modifications can of course be made to the compensation section of the error amplifier to reasonably increase the bandwidth, enabling the system to achieve faster response and improved stability.
V. Conclusion:
This paper reflects the stability of switching power supplies from the perspective of injecting a perturbation signal into the feedback loop, presents the stability criteria, the understanding of Bode plots, the use of the FRA including perturbation signal injection and loop characteristic measurement, and a brief analysis of experimental results, and finally proposes areas requiring modification. Of course, system stability can also be qualitatively analyzed in the time domain, but the frequency response analyzer enables quantitative determination in the frequency domain, thereby allowing more intuitive judgment of system performance.
References:
[1] "Op Amp Stability" Tim Green, Texas Instruments.
[2] "Measuring the Closed-Loop Feedback Response of Switching Power Supplies" Ken Salz, Clarke-Hess Communication Research Corp.
[3] "Frequency Response Analyzer" instruction, NF Corporation.
[4] "Small-signal analysis and control design of isolated power supplies with optocoupler feedback" Yuri Panov and Milan Jovanovich. Power Electronics Laboratory, Delta Corporation.
