The control theory of switching power supplies can be quite abstract and intimidating. System instability is a common issue, but how do you fix it? Why do some systems fail during mass production? There are plenty of resources that discuss the theory, but they often involve complex mathematical concepts that can be hard to grasp. So, how can we really understand the loop stability of a switching power supply?
Cedar once shared a great metaphor with me, and I thought it was very insightful. Let me share it with you:
The feedback loop is like a wife giving her husband a massage. At first, the wife says, “I didn’t eat, so I don’t have the strength.†She starts working harder, adding more pressure, and the husband says, “You’re working too hard, you’ll kill me!†This cycle repeats, and it becomes an example of loop instability.
Today, I came across another analogy involving a man and a woman, which I found interesting. Here’s my take on it:
Let’s start with the concept of a pole. A simple example is an RC filter. It acts as an open circuit for DC (because the capacitor blocks DC) and a short circuit for high frequencies (because the capacitor conducts at higher frequencies). On a Bode plot, the amplitude remains flat before the pole and then decreases by -20 dB/dec after the pole. Think of this as a man — he starts off passionate, but over time, his energy fades. Whether in love or in life, men often lose their initial intensity.
On the other hand, a zero is like a woman. A classic example is the ESR zero of a capacitor. At DC, the capacitor has infinite resistance, but as frequency increases, its resistance drops. After the pole, the ESR resistor no longer decreases. In this analogy, a man is fire, and a woman is water. Women may not be as strong as men, but they tend to last longer. They also show deep emotional commitment. It's easy to see the difference between men and women through stories like Anna Karenina and her lover.
Now, moving on to two men. Unfortunately, I’ve been near San Francisco for a long time, and I don’t want to talk about "comrades" (good words, bad things). An LC filter creates a double pole. When two men are involved, conflicts arise — much like a high Q value. An LC circuit without parasitic resistance has an infinitely high Q, which amplifies signals at resonance. That’s why you can tune into different radio stations by adjusting a small capacitor. The conflict between the two men can cause instability. The level of conflict depends on damping — the resistance or how well the system is controlled. A high Q system has a rapid phase shift from 0° to -180°, making it prone to instability. Therefore, systems with high efficiency (low resistance) are harder to stabilize.
To compensate for instability, we usually add a zero, but this also introduces a high-frequency pole. For example, adding a capacitor to the feedback path creates a zero-pole pair. Understanding this is like talking about love: the zero comes first, just like a woman entering a relationship early on. Most men are only interested in accompanying a woman to a store at the beginning. Once they commit, they stay together — either out of truth, loyalty, or even a promise. If the zero occurs before the pole in life, it might end in tragedy. But if it happens after, it could become immortal love.
In a feedback system, a pole reduces the amplitude (which helps stability) but also reduces the phase margin (which is bad). A zero increases the amplitude (bad for stability) but improves the phase margin (good). So both have their pros and cons. However, a right-half-plane zero is the worst — it increases amplitude and reduces phase margin, doing two bad things. Such a woman is like a witch.
An example of this is in a boost converter. When the switch is on, the inductor stores energy; when the diode is on, it delivers that energy to the load. The current drawn by the load is approximately IL(1-D). As D increases, the output current actually decreases. That negative sign turns a woman into a witch.
We use negative feedback systems because if the output increases, we reduce it in the control loop. However, the loop itself introduces phase lag. If the phase shifts by 180°, and the gain is greater than 1, the system becomes unstable. The stability condition is that when the loop gain is 1 (0 dB), the phase lag must be less than 180° (ideally less than 135° for a safety margin).
In the book *Dust Settled*, the character who dies at 0 dB can be compared to a man with 90° phase lag (a single pole) or a half-man with 135° phase lag (half a pole).
Let’s look at an example: in a current-mode buck converter, the inductor current is limited, so the inductor loses its dominance. The main circuit is left with one pole from Rload and Cout (man 2) and the ESR zero of the output capacitor (woman 1). The control part also has a low-frequency pole (man 1). So we have two men and one woman. The key is where the ESR zero is located. Electrolytic capacitors have a very low zero frequency, so they may partially neutralize the pole, requiring little or no compensation. Ceramic capacitors, on the other hand, have a high zero frequency, so we often need to add a zero-pole pair to stabilize the system.
For a voltage-mode buck converter, L and Cout form a double pole (men 2 and 3), plus the control pole (man 1). Now we face three men. To avoid instability, we need to add one or two zero-pole pairs. Voltage-mode buck converters are inherently harder to stabilize — like a teenager swaying between emotions, needing to mature through multiple experiences.
Finally, let’s touch on how the poles and zeros in switching circuits are derived. This step is crucial, and I find it fascinating. When a switching power supply operates, it doesn't just exist in one state — sometimes it switches between multiple states. Each state has its own set of equations. How do we describe the overall behavior?
It’s actually quite intuitive. Imagine walking east for three steps, then north for one. Your path can be described as 75% east + 25% north. Similarly, in a switching circuit, the different states are weighted and averaged. Then, we apply small-signal perturbations. By combining these, we derive the poles and zeros of the system.
This is how we model the dynamic behavior of a switching power supply — blending different states, applying small variations, and observing how the system responds. It’s a powerful way to analyze and design stable power supplies.
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