VRM Stability - Part II: ESR

The old analog designer's lament, "All my amplifiers oscillate, and my oscillators amplify," bears more than a grain of truth. Between an amplifier and an oscillator there lies only a very fine line of distinction. Both circuits provide power amplification, and both involve feedback, the only difference is the exact type of feedback provided.

A voltage regulator module (VRM) acts as a very specialized amplifier. The input to this amplifier is a reference voltage (sometimes internally generated). The output provides, quite precisely, a greatly amplified quantity of current at the same voltage (or a voltage related to) the reference.

The VRM, therefore, falls subject to all the same foibles of any other type of amplifier, including the propensity to oscillate if improperly loaded—the subject of this letter.

By the time you read this I'll be winging my way to Dallas, Texas for my final public seminar of 2007. During this year, I taught my 10,000th student! Thank you everyone who has attended my classes, and encouraged others to go. It has been my pleasure to work with you all.

On a completely un-related subject, I would like to thank all the people who wrote to me about RoHS. I feel your pain.

I recently met Joe Fjeldstad, an expert in electronic interconnections. He knows way more about RoHS than me. On a recently visit to my lab, we conducted an extensive interview in which he explains how and why RoHS doesn't work. EDN has posted the complete interview, along with my previous article, here: Rollback the Lead-Free Initiative

VRM Stability — Part II: ESR

The system on my bench emits a hideous, demonic-sounding low buzz on both output channels. The buzz is not related to the 60-Hz power cycle. It undulates in a sickening fashion, rising slowly after power-on to eventually stabilize at about 20 Hz.

I'm listening to a very broken 1972 Fender Rhodes electronic piano. It belongs to my good friend Chris "Breathe" Frue, here with me to help debug the circuit (see: VRM Stability—Part I: Feedback).

We are investigating the keyboard pre-amplifier and its relation to the power supply (VRM) mounted behind the main panel in the base unit (Figure 1). The "OUT" connections go to a pair of power amplifiers and speakers, through which we hear the horrible buzz

Breathe unplugs connector J2, disconnecting the pre-amplifier. The buzzing stops. To check the power amplifiers and speakers, I hook up a pre-recorded audio signal to jack J2. The booming voice of Kinky Friedman, humorist and song-writer, issues forth, "I run for the job of governor of the great state of Texas because I want to serve the people, and besides, why the H___ not? Look who's doing the job now… how hard can it be?" Kinky sounds as clear and lucent as ever. It's my favorite test tape.

Apparently, the power amps work beautifully. Breathe stands ready to conclude that the buzzing problem lies within the pre-amplifier, but I feel differently.

The buzz exists as a kind of self-oscillation. We search, therefore, for some amplifier experiencing unintentional feedback at 20 Hz. My instincts tell me that amplifiers often, not always, but often, oscillate near the upper limits of their bandwidth capability. The only part of the system with time constants that slow is the power system. I therefore suspect a power problem.

"Breathe," I suggest, "let's check out the power supply".

Power for the pre-amplifier section of a Fender Rhodes 80W suitcase piano comes from a simple +25VDC regulator located on a PCB behind the main panel (Fig. 1, red wire). I re-connect the pre-amplifier at J2 and power up again.

A quick check of the VRM output reveals terrible buzzing corruption of the +25VDC power supply. With the pre-amplifier connected, the +25VDC rail pulsates wildly from 3V to 12V, never stabilizing at its intended regulating limit.

"Does it stop doing that when I disconnect the pre-amplifier?" Breathe asks, as he pulls the plug at J2. The VRM output immediately stabilizes at 24.2 volts.

Perfect. What we have here is a case of "conditional stability". Unloaded, the VRM works fine. With the load applied, the VRM oscillates wildly. That tells me a lot about what is going on inside the regulator.

    [ED. NOTE—At this point you face a choice. Dr. Johnson is about the make some remarks of vital interest to those of you with good analog design backgrounds (or at least a circuit theory course under your belt). He hopes the comments will be helpful. If analog design isn't your bag, you may want to step out for a cup of tea while he says what he has to say, and then pop back at the end to see what this all has to do with Breathe's piano.]

A VRM is a type of feedback control system (Figure 2). It is designed to deliver a constant voltage into a very specific type of load. The VRM architecture may vary, but generally, a modern switching power supply in a high-speed digital product contains the following elements:

  1. A reference source (Vref). A good VRM includes a built-in temperature-controlled source. It may be laser-trimmed for accuracy. Sometimes the reference is supplied externally.
  2. A comparator. This tells the control circuit whether to raise or lower the delivered current, depending upon whether the sense voltage does or does not exceed the reference voltage. The output of the comparator equals the numerical difference between voltages Vref and Vsense. This output is also called the "error voltage".
  3. A control circuit—the "brains" of the regulator. Presumably, this is a high-gain amplifier equipped with a mechanism for activating the switches in an alternating fashion with a controlled duty cycle. In the context of this article, the "control gain" equals the incremental amount of duty cycle adjustment made for each unit of error voltage.
  4. Two or more switching transistors, each with on-state resistance R[on]. The figure shows a step-down architecture with two switches configured as a "chopper" circuit. The switches operate alternately. Since one switch or the other remains on constantly, the output impedance of the chopper remains fairly consistent at all times. Figure 2 represents the effective output impedance of the chopper as a single resistance R[on]. The style of VRM architecture shown here is called a "buck regulator". The switching arrangement is the most highly variable part of any VRM architecture. In some circuit diagrams the chopping arrangement looks really complicated, but the concept remains the same: it just modulates the average flow of current to the load.
  5. An inductor. This is the first stage of an L-C filter designed to attenuate the ripple generated by the "chopping" action of the switches.
  6. A capacitor. The regulator datasheet usually stipulates a minimum value of capacitance, and a maximum value of the crucial ESR parameter. Figure 2 separately depicts the capacitive (C1) and resistive (R1) parts of the capacitor model, even though in reality the two are inseparable parts of the same component. At the frequencies of interest to this article, up to about 100 KHz, you may safely ignore the ESL, as it makes only slight differences in the observed waveforms while adding little to your understanding of low-frequency VRM behavior. The simulations shown here omit ESL.
  7. A sense network (R2/R3). Some regulators leave this part of the design in the hands of the user. By implementing various divider ratios, the user may cause voltage Vreg to exceed Vref by any fixed ratio, thus providing some kind of adjustable-voltage feature. In a simple fixed-voltage VRM, the sense network may be embedded inside in the control IC.
  8. A feedback path, connecting the sense network to the Vsense input at the comparator. In direct DC-coupled implementations, this may be just a wire (as shown). In "floating" configurations the feedback path may involve a transformer-coupled or optically-coupled signaling link.

For our first simulation, let's try something a little unusual. I want you to completely disable the control circuit. Just erase it (Figure 3). Set the switches to operate by themselves, with no supervision or control, at a fixed duty cycle of 50 With that accomplished, the signal output from the choppers pounds up and down with a terrible rippling action. The inductor and capacitor filter out the rippling part of the waveform, leaving only the average DC voltage. In the absence of load current, the average DC voltage of a chopped signal exactly equals the chopper duty cycle times the voltage, Vraw, being switched.

If, for example, the input at Vraw equals 2.5VDC, then a fifty percent duty cycle makes the output at Vreg equal 1.25VDC (unloaded). There! You've made a down-converter. The feedback control circuit has nothing to do with the basic operation of the system, it exists only to fine-tune the dynamic response of the regulator—a problem we will examine in a moment.

For now, let's simulate the ripple waveform with a fixed 50% duty cycle. Figure 4 assumes the following values: R[on]=0.03 ohm, L1=1 uH, C1=1200 uF, and ESR1=0.01 ohm. Let the switching frequency equal 150 KHz.

Figure 4 shows the resulting ripple under five conditions. First, it shows the nominal case just described. Then, it shows the result of ±50% variations applied to the value of C1. These variations produce almost no change to the picture. Finally, in red and green, the figure shows the effect of ±50% variations applied to the ESR of C1.

Regardless the size of the capacitor, it makes almost no difference to ripple! That's typical. At the switching frequency of 150 KHz, compute the impedance of C1.

The value of ESR, which equals 10 milli-ohms, looms 10 times larger than the impedance of C1, so it exerts a greater influence on ripple.

ESR is not, of course, the only thing that affects ripple, but it is an important part. The VRM design process works like this.

  • Pick the switching frequency based on how fast your FETs can switch, and what efficiency you need.
  • Set the duty cycle to get the correct voltage down-conversion ratio.
  • The inductance controls the magnitude of ripple current flowing through the chopper. Adjust L1 so the ripple current falls just below the maximum current rating of the chopper switches. Ripple current also affects the amount of EMI radiating from the supply, and the size of current surges demanded from the raw DC source.
  • From the ripple current, pick a value of ESR small enough to control the output ripple voltage.
  • Set the minimum value of capacitance according to the equations for loop stability criteria.

This choice of C1 happens almost as an afterthought. The capacitance itself is not the most important part of the design.

  • POINT TO REMEMBER: ESR controls ripple. Do not exceed the MAXIMUM ESR specified on your VRM datasheet.

The presentation so far suggests that at low-load conditions your VRM circuit does not really need a fancy control circuit. The control circuit only helps in two situations:

  1. When the input voltage fluctuates, or
  2. When you load the circuit with something that pulls massive amounts of current from the output—let's focus on this effect next.

When you pull current out of a VRM, the output droops. With a fixed duty cycle chopper, the effective series resistance of the circuit controls the degree of droop. The effective series resistance depends mostly on the R[on] of the switching transistors, and the DC resistance of the inductor, and the effective source impedance of the Vraw supply. Of those three parameters, R[on] usually dominates the problem.

For example, Figure 5 shows what happens to our brain-dead 50-percent fixed duty-cycle controller when you try to pull 8 amps out of it. Here I assume that for each switch, R[on]=0.03 ohm. Since one switch or the other stays on constantly, the effective series resistance of the chopper output equals R[on] at all times.

With a nominal value of R[on]=0.03 ohms, the circuit produces a peak droop of almost 300 mV in response to an 8-Amp step load. That's way too much droop for a 1.25VDC power system. To reduce the droop, you could try using bigger switching transistors to lower the R[on], but even if you triple their size to 0.01 ohms, the droop still looks terrible. Reducing R[on], by itself, is not a winning strategy.

Introducing…. the control system!

The control system exists to improve the droop problem. The controller is a negative-feedback system. When it sees droop at the sense point, the controller boosts the duty-cycle at the chopper, hopefully just enough to bring the output back up to its nominal level.

Let's try various amount of control gain to see what happens. In all cases, I assume a control equation that sets the duty cycle like this:

DutyCycle = 1/2 + (Gain)*(Vref-Vreg)

The additive factor of 1/2 in the control equation establishes a baseline duty cycle of 50, as required by our down-conversion ratio,

Vreg/Vraw = (1.25/2.50) = 0.50

Changes from the normal operating point occur in response to errors in the output voltage multiplied times a gain factor. The nominal gain factor for this system is 4, meaning that if Vref droops 10mV below Vreg, the controller increases the duty cycle by 4 percent, making it 54 [NOTE: My technical model also inserts a processing delay of one switching cycle in the control loop to model the mean delay of the chopper circuit.]

Figure 6 illustrates the response for four different gain settings.

With the control loop operative, regulation definitely improves, at least in the long term (i.e., longer than 100 usec). In the short term, the control circuit is not as effective. This happens because the chopper circuit, working through the smoothing filter, requires a certain amount of time to respond to changes in the load. Even if you increase the gain tremendously, the circuit can't just instantaneously jerk up output. Figure 7 makes that point very clear.

Nasty, isn't it? Further increasing the control gain does nothing for short-term regulation. It helps with long-term regulation, but the system fights back with terrible undulations. Taken to an extreme, the system self-oscillates. All control systems do this.

For example, consider what happens when steering your car. At parking-lot speeds, you must crank the wheel vigorously to accomplish much change in direction on the ground. The steering gain is quite low. On the highway, steering gain increases in proportion to your speed. At 70 mph, small motions of the wheel can send you careening off the road in seconds. At extreme speeds, the steering gain becomes so high that the car loses stability. It becomes no longer controllable by an ordinary human (especially if he has been drinking).

Most manufacturers set the VRM control gain at a reasonable compromise between regulating ability and stability.

If changing the system gain affects stability, do you suppose anything else might have that same effect? The answer is yes, definitely, everything affects stability. The sense network, the control circuit, the switching duty cycle, the smoothing filter, and the load on that filter—change any of those parameters and the system becomes either more or less stable.

Just for fun, let's mess with the ESR of our filter capacitor. As I mentioned earlier, near the switching frequency the ESR dominates the behavior of the filter capacitor. If you raise the ESR, a larger signal squeaks through the filter. Effectively, the ESR controls the gain of the filter.

But, isn't increasing the gain of the smoothing filter pretty much, from a control theory perspective, the same as increasing the gain of the control amplifier?

What I am suggesting is simple: an excessively large value of ESR can induce instability. Of course, you wouldn't do that, because the VRM manufacture has already given you a maximum value of ESR for the purpose of controlling ripple. If you follow that guideline, then the circuit should remain stable.

How about making ESR too small? What would that do?

Reducing ESR has the effect of (eventually) changing the nature of the capacitive element. With C1=1200uF and ESR1=0.01 ohm, at the frequency 150 KHz the component presents a resistive (ESR-dominated) character. The phase angle of the component equals (roughly) zero degrees. If you reduce ESR to ridiculous extremes by using a specially-made microwave grade silver-plated low-ESR capacitor, the picture changes. When ESR vanishes the component presents a capacitive character. The phase angle changes to -90 degrees. That delays the output signal, and the additional delay leads to another kind of instability.

Figure 8 shows the result. Here I reduce the ESR in steps to a value of 0.004 ohms, at which point a long, ringing tail becomes visible in the step response. One more step beyond this point the circuit breaks into self-oscillation.

  • POINT TO REMEMBER: ESR affects stability. Do not violate the minimum ESR specified on your VRM datasheet.

Breathe stands close to my bench, watching intently as I de-solder two electrolytic capacitors from his power regulator. I suspect that due to the extreme age of his equipment, the capacitors have deteriorated. It often happens with old electrolytic capacitors that the ESR drifts up over time.

Right now the whole power system sits perched near the cusp of self-oscillation, so close that even a small change, like connecting the load of a preamplifier circuit, tips the system into a state of continual self-flagellation. If I am right, the increase in ESR of these two parts has somehow driven the system to this edge of insanity.

The first capacitor has a nominal value of 100 uF. I connect it across the output of a 50-ohm pulse generator and set the output for a 500Hz square wave, about 1V amplitude. Figure 9 shows the voltage measured across the capacitor (blue waveform).

In this test a good 100 uF capacitor should show only a gentle ramp undulating up and down. The bad capacitor from Breathe's equipment shows a reasonable ramp but also an abrupt jump at every change of state. The size of the jump varies in proportion to the ESR of the capacitor under test. From this waveform I compute the ESR of his capacitor is about 20 ohms. That's a terrible, huge value. His second capacitor checks out just as bad.

I expected his capacitors to have a lot of ESR, but not THAT much. Replacing both his bad parts with new ones fixes the power problem. Now the piano works great. No more "low-frequency buzz of death."

I don't know what Breathe learned going through this exercise, except who to go see when something breaks, but he sure seems happy right now banging away at the keys. He's playing my favorite Chuck Mangione song, "Feels So Good."

Best Regards,
Dr. Howard Johnson