Nonivasive loop gain approximation from output impedance measurements.
Introduction
It is always desirable to simplify the measurement process of power supplies, and characterize them without overly invasive testing. In this article, Dr. Ridley shows how this can work for loop gains with limited success, and shows the pitfalls of trying to measure loops noninvasively.
Power Supply Output Impedance Measurements
For decades now, engineers have followed Middlebrook’s techniques for breaking and measuring feedback loop gains while keeping the feedback intact for regulation. However, in some cases, chip designers no longer make loop feedback points available, or board layouts prevent proper direct measurement. Or,you may just have a black box power supply with no access to interior nodes of the circuit.
It is well known (again from Middlebrook) that you can get the loop gain indirectly by looking at the output impedance of the power supply. Figure 1 shows the conceptual idea behind measuring output impedance.
Fig. 1: Output impedance is measured by driving a current into the output terminals of a power supply
If you can measure both open- and closed-loop impedances, the loop gain can be calculated, even if it is not directly accessible. The quantities are related by the following equation:
Note that two measurements are always needed, and it is not possible to get any meaningful loop stability information from just a single impedance measurement.
In order to calculate the loop gain from the above equation, we need accurate measurements of both the closed and open-loop output impedances. As we will see later, this can present problems in the measurement techniques used.
These two measurements are shown in Figure 2 for an example switching power supply.
Fig. 2: Open-loop and closed-loop output impedance of a power supply.
Figure 3 shows how this output impedance measurement is implemented practically using a frequency response analyzer. No special injection fixtures are needed [2], and as will be shown later, it is important to have flexibility on where measurements are made if you want to get good results.
Fig. 3: Practical Test Setup for Injecting Current and Measuring Impedance.
Figure 4 shows multiple test points where the output impedance can be measured. As power levels go up, it becomes crucial where you inject current, and where the test signal are measured if you want to get reliable results.
Fig. 4: Different Test Points for the Output Voltage.
The isolated current source is injected into the test pins indicated by the blue arrows. For best results, it is important not to make the measurements at these same pins, and a four-wire Kelvin connection is preferred to extract proper results.
The three arrows, red, yellow and blue, show points on the output voltage where impedance measurements can be made. (These are all the same node of the circuit, the output voltage, but they are separated from each other by parasitic impedances which become very significant for high power supplies.) Once the impedances are measured, the loop gain of the circuit can be calculated from:
Fig. 5 shows the measured loop results for a switching converter.
Fig. 5: Direct Loop Gain and Phase Margin Measurement Compared with Noninvasive Measurement
There are two loop gains shown in Figure 5: the first is the loop calculated from impedance measurements, shown in gold. The second is a direct loop measurement made by breaking the loop in the conventional way. Note that the phase curve shows the phase margin for each of the loop gains.
The results look very good from these measurements. The measurement point for the output impedance was taken from the point on the board where the feedback loop samples the output voltage, and this gives the best results. However, the estimated loop gain is very sensitive to the exact measurement location, as is shown in Figure 6.
Fig. 6: Different Loop Gain and Phase Margin Results Depending on Impedance Measurement Point
There are very different loop results obtained when the test point for the output voltage is moved, even a seemingly insignificant amount. The red loop gain curve, which corresponds to the red test point in Figure 4, picks up the impedance of the test pin through which the injected current is driven. This is a relatively high impedance compared to the expected output impedance of the power supply, and the result is a very degraded estimate of the gain of the loop.
The blue loop gain and phase curves are even more interesting. Even though the gain is closer to the gold curve (and hence the true loop gain measurement), the starting phase is 180 degrees away from the other estimates of the loop phase. This is caused by a test point just before the output of the power supply and the closed-loop output impedance at that point is actually negative. (Engineers who use remote sensing may be well aware of this kind of result.)
Open Loop Measurement Issues
There is a complication in measuring the open-loop output impedance, especially if you are trying to look at a point of load supply where you don’t have control over the input voltage. The best measurement of open-loop output impedance is made when the input line is too low for regulation, and the error amplifier of the feedback loop is railed high. However, it may be impossible to generate this condition when testing a finished power supply. Instead, you may have to measure the impedance of the unpowered system. Unfortunately, that will usually disconnect the inductor from the load, resulting in a higher than expected low-frequency impedance.
If you have to test this way, it will result in a large error in calculated loop gain, as shown by the red curve of Figure 6. This observation will vary whether the converter uses current-mode or voltage mode control. If you are testing a black box supply, you will have no idea what the controller is doing, and hence no way to get the proper open-loop impedance. There are no guarantees of the accuracy of the loop gain in such a case.
Summary
Predicting loop gain from output impedance requires two measurements, one for the open-loop, and one for the closed-loop impedance. The predictions of loop gain that result are very sensitive to the measurement location of the output impedance. It is important to be able to move the test probe detecting the output voltage to the proper feedback point if you want to get reliable results that agree with the true loop measurement. Unfortunately, this proper test point is not always accessible.
Problems also arise in measuring the open-loop output impedance. Ideally this is done with the converter delivering output power, but without regulation active, either with low line input or by breaking the feedback path. If these approaches aren’t possible, then unpowered measurements must be made, resulting in large loop gain errors.
Noninvasive measurements can be useful as a last resort. However, they should not be relied on to guarantee the loop gain accurately, or to optimize control loops during power supply development.
References
Ridley Engineering Frequency Response Analyzer from AP Instruments, www.ridleyengineering.com/index.php/analyzer.html
Ridley Engineering Design Center, [63] Why You Only Need One Injection Isolator, www.ridleyengineering.com/index.php/design-center.html
Measuring high-performance loops, https://www.youtube.com/watch?v=CbjtGZtaUaQ
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