Welcome to the topic Considerations for Measuring Loop Gain in Power Supplies. In the introduction, we will talk about what loop gain is and why we measure loop gain. Then we will discuss how to measure loop gain. And in the third part, we will cover test setups and provide test examples. Finally, we will summarize our discussion in this topic.

Introduction. Loop gain overview. Most power supplies use output feedback to achieve good regulation. On the left is a simple feedback system. It has a full working stage and a feedback path. Loop gain is the product of all gains around feedback loop. A power supply is a much more complicated system.

On the right is a voltage remote controlled buck converter. It comprises of power stage, an output feedback resister divider, an error amplifier, and a path with modulation comparator, which converts the continuous output of the error amplifier into pulsating [? gate-driving ?] signals. The looping of this buck converter is the product of all gains of all stages.

Why do we measure loop gain? The first reason is that loop gain is a good indicator of stability. Here is an example. On the left is the measure of the loop gain of a buck converter. This buck converter has a loop bandwidth of 22 kilohertz and a phase margin of 15 degrees. On the right is the corresponding load transient response. After load [? steps ?] this output [? ringing, ?] the ringing frequency matches the loop bandwidth. As we can see that, this system is on the brink of oscillation, which is indicated by the low phase module.

The second reason to measure loop gain is that the loop gain measurement result can guide us to [? tune ?] compensation for better load transient response. On the left is the measure of the loop gain of a buck converter with a [INAUDIBLE] remote control. On the right is the corresponding load transient response. The undershoot and overshoot of the load transient response is in reverse proportion to the loop gain bandwidth. Given a stable system, higher loop bandwidth improves transient response.

Loop gain measurement method. A transfer function gain by definition is the ratio between the input signal and the output response. Theoretically speaking, we can break open the power supply feedback loop, apply the DC bias, and inject AC disturbance to measure loop gain. However, it is not practical to measure loop gain in such an open loop setup.

The error amplifier has high DC gain. Even a tiny amount of error applied to point A will result in a tremendous amount of error at output point B. It is difficult to maintain the correct DC operating point. It is also easy to saturate the circuit by injecting too much AC disturbance.

To make our discussion easier, let us simplify our power supply. Given any feedback system, we can break open the feedback loop. Looking backward and applying Thevenin's theory, the circuit is equivalent to a voltage controlled voltage source in series with output impedance Z2. Looking forward, the circuit is equivalent to a network with impedance Z1. The loop gain of this system is the product of the gain of the voltage controlled voltage source in the ratio of the impedance dividers Z1 and Z2.

If we can break the loop at a point where the impedance looking backward is zero, the loop gain equals to the gain of the voltage controlled voltage source. Instead of breaking the loop, we keep the loop closed, and AC voltage is injected into the feedback loop. Since the loop is still closed, the response at point A and point B are closed loop response from the voltage injection, VINJ. The open loop gain can be derived from the ratio between the response of point B and point A.

This loop gain measurement method is called the voltage injection method. While loop gain can also be measured with a current injection, the voltage injection method is easy to implement and the most popular.

Now we understand the theory of loop gain measurement, let's talk about setting up the equipment, preparing the power supply, and measuring the loop gain accurately. To make a loop gain measurement, we need two pieces of equipment. One is the frequency analyzer, the other is the source injection isolator. To set up the power supply for loop gain measurement, we first identify the correct voltage injection point, and then we hook up the equipment with the power supply.

This is a frequency analyzer. It provides an AC voltage source for voltage injection. It has two receivers to measure the response at point A and point B simultaneously. It calculates the loop gain by the ratio of the measurement from the two receivers. Please be noted that the output resistance of the voltage source is 50 ohm. This means the voltage injected between point A and point B is not the same as the voltage source provided by the analyzer. We will discuss more later.

The frequency analyzer receivers measures signals most accurately when the signal is within the measurable range. For example, the measurable range of AP300, a frequency analyzer, is from 5 microvolts to 1.77 volts. We know that the response at point A and point B are related to both the voltage injection and the loop gain. Here, I use a buck converter to illustrate the relationship. I derive the transfer function gains from VINJ to point A and point B.

For point A, the worst case happens at low frequencies, where loop gain is high. For example, at 100 hertz, the magnitude of a loop gain is 60 dB, which is 1,000 times. In order to have the signal level at point A be greater than 5 microvolts, the voltage injection level should be greater than 5 millivolts.

For point B, The worst case happens at high frequencies, where loop gain is low. In this case, at 300 kilohertz, in order to have a signal level at point B be greater than 5 microvolts, the voltage injection level should be greater than 15 microvolts.

While voltage injection should be strong enough to make accurate loop gain measurement, too much voltage injection would saturate the power supply on the test. For [INAUDIBLE] power supply, too much voltage injection causes saturation to duty-cycle and error amplifier. It can even trigger over-current protection and over-voltage protection during measurement. For linear regulators, excessive voltage injection could saturate path device and its driver.

Here I'm going to use an LDO TPS7A8300 to demonstrate how voltage injection level affects loop gain measurement. I start with a voltage source with amplitude constant over the frequency range. I set the amplitude to 2 millivolts. The loop gain magnitude drops at low frequencies and phases shift strangely. This is unexpected for an LDO with a good output regulation and power supply ripple rejection.

I increase the source amplitude to 10 millivolts. The measurement at low frequencies is different, which indicates the measurement error is related to the voltage injection level. The frequency analyzer, AP300, provides voltage source programmability. I programmed the voltage source so that the amplitude is high at low frequencies. With the programed voltage source, the loop gain measurement result is normal.

With the same LDO, I'm going to show you how excessive voltage injection affects loop gain measurement. I set the voltage source level to a 100 millivolt constant over the frequency range. Around 6 kilohertz, the magnitude drops suddenly and there's a bump at phase measurement. I decrease the amplitude to 15 millivolts. The measure of the loop gain result changes, which indicates the error is related to the voltage injection level.

I used the programed voltage source for which decreases the amplitude as the frequency increases. The measurement result is smooth. Some frequency analyzers do not offer programmability. They could adjust voltage source amplitude automatically.

Here is the control manual of a frequency analyzer [INAUDIBLE]. We can use the signal control to adjust the source amplitude. We first set the minimum source level. It is usually from 2 millivolts to 10 millivolts. The analyzer will start measurement with this minimum source level.

I set the servo control on the smaller of the two receivers and set the threshold to 200 microvolts. When the signal level received is lower than 200 microvolts, the AC output will be increased by the step size until the received signal level is greater than 200 microvolts.

The other important setting is the intermediate frequency bandwidth, or the integration time. Narrowed intermediate frequency bandwidth, or longer integration time, means less random noise, but longer sweeping time. This graph shows the measurement results with a different intermediate frequency bandwidth. This is especially important for power supplies of which loop bandwidth is low, such as a power [INAUDIBLE] correction, or some isolated power converters.

The other important piece of equipment is the injection isolator. Different isolators covers different frequency range. Please use the injection isolator with the correct frequency range.

Now we set up the frequency analyzer and have the correct injection isolator, let's prepare the power supply for loop gain measurement. The static operating point affects the converter frequency response. We need to maintain the same static operating point during measurement.

A small resistor is inserted between point A and B to keep the feedback loop closed, regardless of the isolator and analyzer source output impedance. This resistor value is usually from 10 ohm to 100 ohm. By doing so, we keep the output the same as the converter's normal operation.

This slide shows a loop gain measurement result of a buck converter, with the test setup in slide number 21. Measure crossover frequency and phase margin match simulation well.

Let's locate the correct voltage injection point. This is the simplified model we used earlier. When impedance Z2 is not zero, it will affect loop gain measurement. Here is a [INAUDIBLE] the loop gain measurement. The correct voltage injection point should have Z to much smaller than Z1. Let's use a voltage remote control buck converter as an example, and examine each possible injection point.

First, is the point between the phase [? null ?] and output inductor. The impedance looking backward is the [INAUDIBLE] of the [? MOSFET, ?] which is much smaller than the impedance of the inductor. The impedance looking forward, this point meets the impedance requirement. However, the phase [? null ?] voltage is discontinued and non-linear. It is not possible for signal measurement.

The second point is between the output and the top of the feedback resistor divider. The impedance looking backward is the impedance of the output capacitors in parallel to the output inductor. That impedance looking forward is the resistor divider. The divider resister is usually in K ohms, which is much greater than the impedance of the output capacitors in parallel to the output inductor. This injection point is the most widely adopted.

The third point is between the midpoint of the resistor divider and the feedback pin. The voltage remote control converter uses a type 3 compensation. The feedback pin is virtually shorted to the AC ground. The impedance looking forward can be considered zero, which does not meet the impedance requirement.

The fourth point is between the output of the error amplifier and the [? P to M ?] comparator. The output impedance after error amplifier is very low, and the input impedance of the comparator is very high. It meets the impedance requirement. However, this point is usually buried inside the controller. It is not accessible.

This is a 1 [? amp ?] LED driver. An LED driver behaves like a current source. The output impedance of the converter is very high. The output of the converter is not a good voltage injection point. As we examine the converter, we notice that the point between the current sensing resistor and the compensator meets the impedance requirement.

The impedance looking backward is the impedance of the current sensing resistor, which is only 0.7 ohm. That impedance looking forward is the impedance of R6, which is 10 kiloohm. Here is the measure of the loop gain.

We mentioned earlier the second reason for loop gain measurement is to [? guide ?] power supply designer to improve load transient response. The load transient response performance is determined by the closed loop output impedance. The closed loop output impedance is determined by both the output capacitors and the open loop gain.

For a converter with multiple feedback paths, for the loop gain to be useful as a guideline for load transient performance or improvement, all feedback paths should be included in the measurement. If we measure the loop gain with some feedback paths closed, the measurement result might not tell how transient response performs.

A shunt regulator, such as TL431, is widely used for isolated converter output regulation. Here is the schematic of a compensator for isolated converter. From the output, there are two feedback paths. One is through the resistor divider and the shunt regulator. The other path is through the optocoupler [INAUDIBLE] resistor.

I selected a point to include both the feedback paths. Here is the measure of the loop gain and the corresponding low transient response. The transient response timing matches the loop gain bandwidth. Here is another example. This is a buck converter with a DCAP control. To stabilize the converter, DCAP control injects ripple to the feedback pin during loop gain measurement.

With setup number one, the voltage injection point is between the output and the top of the resistor divider. The measure the loop gain bandwidth is only 14 kilohertz, which does not match the transient response. A closer exam shows that there is another output feedback path, which is through the ripple injection circuit and the capacitor, CFF. Setup number two selects an injection point to include both the feedback paths. They loop gain measurement matches the loop transient response and simulation.

In an earlier slide, we emphasized keeping the same DC operating point for loop gain measurement. We also need to maintain the same similar AC operating point. This is especially important for advanced control topologies, such as DCAP control. The falling slope of the ripple at a feedback pin determines the path with modulation gain for a DCAP control. It is important to keep the AC ripple at feedback pin same or similar to that during normal operation.

In this example, we insert a 20 ohm resistor, RINJ, for loop gain measurement. At frequencies higher than switching frequence, we can consider capacitor CP shorted. The pulsating voltage across the phase [? null ?] and output will be distributed among the resistors RP and RINJ, even though RINJ is only 1/100 of RP.

The ripple voltage [? across ?] RINJ is still 119 millivolts, given an input of 12 volts. This pulsating ripple is coupled to the feedback pin by CFF. The AC ripple at feedback pin is greatly distorted. The distorted ripple at feedback pin affects the loop gain measurement. The solution is to add a bypassing capacitor in parallel to RINJ.

In this example, when I add a [? 0.1 ?] [INAUDIBLE] capacitor in parallel to RINJ, the ripple [? across ?] RINJ is reduced to 1.35 millivolts. This will maintain a signal AC ripple at feedback pin during measurement.

In this slide, we will talk about the challenges in measuring loop gain for a power factor correction converter. First, the control bandwidth is low. It is usually below 10 hertz. We need to select an injection isolator that works in low frequency range, and set the frequency analyzer to intermediate frequency bandwidth, or integration time, accordingly.

Secondly, the output voltage is as high as 400 volts. The frequency analyzer should be able to support such a high common voltage. Finally, the actual input is alternating. We apply DC input for loop gain measurement.

Here are the loop gain measure before the power factor correction. SEPIC converter, with different DC inputs. If we do not have a frequency analyzer that can support high common voltage, an oscilloscope can be used. Please refer to the reference number five for more details.

[? After ?] we set up the equipment and power supply properly, now we can connect them together for loop gain measurement. To connect the equipment to the power supply, wires are soldered to the two ends of the injection point, and cables are used. You might be surprised at how these wires and cables affect the loop gain measurement results. I use a TPS53355, a 30 amp buck converter with a DCAP control, as an example.

I soldered a pair of 6-inch long wires to the two ends of resistor RINJ. Here is the measurement result. The gain margin is less than 10 dB. The gain increases at high frequencies, which is similar to the effect of an output capacitor, ESL. However, the phase drops dramatically, which indicates something else.

I shortened the connection wires to 0.5 inch and measured the loop gain again. The result is much different. The gain margin is much higher, and the phase is not drooping.

The connection wires are the culprit for the error in slide number 31. The connection wire is not ideal. When we apply the AC voltage source for measurement, there is a voltage drop across the wires when the receivers and the injection isolators share the same pair of wires. The receivers include the voltage drop across the wires into measurement. Thus, the impedance of the wire affects the loop gain measurement results.

When the wire impedance is so high, the measurement would be dominated by the impedance of the wires. It is recommend to use connection wires as short as possible. For occasions that short connection wires are not possible, use separate wires for measurement and voltage injection. That way the receiver will not include the voltage drop across the connection wire into measurement.

To verify this concept, I measured the loop gain with two pairs of long connection wires. I compared the result to that with one pair of short wires. The two results are similar. The two photos show the setup with one pair of connection wire and two pairs of connection wires, respectively.

When we talk about voltage, we are talking about voltage difference. We always need a reference point all other nodes refer to. This is an actual loop gain measurement setup. That B and C cables are used to connect the receivers to the connection wires.

The black clip is connected to the reference point. For a single-ended system, use a controller signal ground for reference. For a converter with a fully differential remote sense, please use a remote negative sense for reference.

Are the black clips always connected to the ground? I am going to show you a unique case. Here is an application circuit of the LM4041-N, For a shunt regulator. When I use the system ground as the reference point and inject voltage between the output and the top of the resistor divider, the measurement result is strange. I could not tell the stability directly.

As I examined the block diagram of the LM4041-N, I noticed that the internal reference voltage is from the output to the feedback pin. When system one is used as the reference point, the output feedback has two parts. One is through the resistor divider. The other is through the internal reference voltage. The injection points from the output at the top of the resistor divider would not include the feedback paths through the internal reference voltage.

Next, I used the output as the reference point. By doing so, the power supply has only one feedback path, which is from the ground through the bottom of the resistor divider. I use that point for loop gain measurement. The result is straightforward, and helps me to stabilize the power supply.

The stability of a power supply could change over the operating range. Here are the loop gain measurement results of an active [INAUDIBLE] forward converter over the low range. As the converter enters, its continuous current mode, the loop gain changes dramatically. Compensation network should be designed so that the system is stable over all conditions. It is necessary to measure loop gain over the whole operating range.

Here is the summary of this topic. To measure loop gain accurately, we first identify the correct injection point, which should have impedance looking backward much smaller than impedance looking forward. The injection point should include all output feedback paths. We also need to use the correct reference point.

Set up the frequency analyzer with the correct voltage source amplitude, and select the right injection isolator. During measurement, maintain the same DC and AC static operating point. Make sure the receiver do not include voltage drop along connection wires. Check stability over all conditions.