Ćuk, SEPIC, and Fourth-Order Converters

Because of where the coupling capacitor sits, and this is a genuine safety property rather than an incidental one.

The boost converter. Trace the path from source to load: \( V_s \rightarrow L \rightarrow \) diode \( \rightarrow \) output. Every element is a DC conductor. Short the output and that path carries whatever the source can deliver, limited only by the wiring. The switch is not in the path, so turning it off changes nothing. The diode fails first, usually within milliseconds.

The SEPIC. The same trace passes through \( C_1 \), which blocks DC by construction. A short at the output cannot pull steady current from the source, because a capacitor will not pass it. Stop switching and the fault current stops.

Why this matters commercially. Automotive, aerospace and medical standards frequently require that a single output fault must not draw unlimited current from the supply. A boost converter needs an added series disconnect device to comply; the SEPIC complies inherently. When a SEPIC is specified over a boost that would otherwise do the job, this is often the reason.

Two caveats, so the property is not oversold:

• The transient discharge of \( C_1 \) into the short is still substantial — the protection is against sustained current, not against the initial surge.
• The Ćuk shares the property (its \( C_1 \) is in the path too); the Zeta and the buck–boost do not, since their switch is in series and must be commanded off.

Almost certainly because the replacement is an electrolytic and the original was a film or ceramic part. Capacitance was never the specification that mattered.

What \( C_1 \) actually does. It carries the full converter current as a two-level square wave — \( I_{L2} \) while the switch is on, \( -I_{L1} \) while it is off. Its RMS current is

\[ I_{C1(rms)} = \sqrt{D\,I_{L2}^2 + (1-D)I_{L1}^2} \]

which for a typical design is close to the converter's full load current.

What that does to an electrolytic. A 100 μF aluminium electrolytic has perhaps 200 mΩ of ESR at 100 kHz. At 2.6 A RMS:

\[ P = I^2R = (2.6)^2(0.2) = 1.35\ \text{W} \]

dissipated inside a sealed can with poor thermal conductivity. The core temperature rises far above ambient, the electrolyte dries out, the ESR rises further, and the heating accelerates. It is a thermal runaway with a lifetime measured in weeks.

What the original part was doing. A film capacitor of one-tenth the capacitance might have 5 mΩ of ESR, dissipating 34 mW — forty times less, in a package that sheds heat far better.

The rule to carry away: in Ćuk and SEPIC converters, select \( C_1 \) on RMS ripple current rating first, then check that the capacitance is sufficient for the ripple voltage. Reversing that order is one of the most common ways to build a converter that works on the bench and fails in the field.

Because it combines the phase penalty of a pole with the gain increase of a zero, and there is no way to cancel it.

Compare the three:

• A left-half-plane zero at \( \omega_z \) adds +20 dB/decade of gain and adds up to +90° of phase. Helpful — it is what compensator design uses to buy phase margin.
• A pole adds −20 dB/decade and subtracts up to 90°. Harmful in phase, but at least the gain falls, so the loop naturally rolls off.
• A right-half-plane zero adds +20 dB/decade and subtracts up to 90°. The worst of both: the gain rises just where the phase is collapsing.

Why it cannot be cancelled. A left-half-plane zero can be nulled by a compensator pole at the same frequency. To do the same for an RHP zero you would need a compensator pole in the right half-plane — an unstable filter. Even if it were implementable, any tiny mismatch would leave an unstable mode. So it is not a matter of designing more carefully; it is impossible.

What is done instead. Roll the loop gain off below the zero so the loop simply never operates there. The rule of thumb \( f_{bw} \le f_{RHPZ}/5 \) leaves enough margin for the zero's phase contribution to be negligible at crossover.

And the physical reading. The RHP zero is not a modelling artefact — it is the crouch before the jump. Asking the loop to respond faster than the zero is asking it to correct the output before the inductor has had time to build up the current that will correct it. No controller can do that, because the information does not exist yet.

On part count, board area and cost — and for small auxiliary rails those often win outright.

The comparison, concretely. A buck–boost needs one inductor, one switch, one diode and two capacitors. A SEPIC needs two inductors (or one coupled part), one switch, one diode, and three capacitors including a coupling capacitor with a demanding ripple rating.

Where the buck–boost is the right answer:

• Low power. At 1 W the pulsed port currents are milliamps. The EMI they generate is trivially filtered, and the capacitors are tiny.
• An inverted rail is wanted anyway. Op-amp bias supplies, LCD panel drivers and many analogue front ends need a negative rail, so the "problem" of inversion is the requirement.
• Board area is the binding constraint. A wearable or a sensor node may simply have no room for a second inductor.
• Cost is the binding constraint. A high-volume consumer product will not pay for a coupled inductor and a film capacitor to save a few decibels of EMI it can filter more cheaply.

Where it is the wrong answer: above roughly 20–50 W, when the source is a battery or PV panel that dislikes pulsed current, when conducted-emissions limits are strict, or when the load shares a ground with the source.

The engineering lesson generalises. The fourth-order converters are strictly better on port behaviour and strictly worse on part count. "Better" is never absolute — it is relative to what the application is short of.

Coupling changes the small-signal dynamics as well as the ripple, and a loop tuned on the uncoupled converter may no longer be stable. Three specific mechanisms are worth checking.

1. A new resonance. With the windings coupled, the leakage inductance and the coupling capacitor \( C_1 \) form a series resonant circuit. Leakage is small, so the resonance is at a high frequency — potentially inside the loop's bandwidth — and it is lightly damped, since there is little resistance to damp it. A sharp peak in the plant response at that frequency can push the gain above unity where the phase has already passed −180°.

2. The effective inductance changed. Two coupled windings do not behave as two independent inductors. Depending on the winding polarity, the effective inductance seen by the ripple can be roughly \( L + M \) or \( L - M \) — which for tight coupling can differ from the uncoupled value by a factor of two or more. That moves both the resonant frequency and the RHP zero, and the compensator was placed for the old ones.

3. Polarity wired backwards. If the windings are connected in the wrong sense, the mutual term adds to the ripple instead of cancelling it. The symptom is ripple that got worse rather than better, along with the changed dynamics — and it is the first thing to check, because it is the easiest to get wrong.

How to diagnose it properly. Measure the loop gain with a network analyser and compare against the uncoupled measurement. A resonant peak that was not there before points to mechanism 1; a general frequency shift points to mechanism 2; worse ripple points to mechanism 3.

The fix is usually a small damping network — a resistor in series with a capacitor across \( C_1 \), sized to damp the leakage resonance without dissipating much at the switching frequency — plus re-tuning the compensator for the new plant.

Because in a buck converter, increasing the duty ratio sends more energy to the output immediately. There is no interval during which the load is deliberately disconnected.

Trace the buck's response to a step increase in \( D \). The switch is on for longer, so the source is connected to the inductor for longer, so more charge flows to the output within the same cycle. The output rises straight away and keeps rising. Correct action, correct direction, no delay.

Now trace the boost's. The switch is on for longer, so the diode — which is the only path to the output — is off for longer. Less charge reaches the output in that cycle, and the output dips. Only after the inductor current has built up over several cycles does the output recover and overshoot. Correct action, wrong direction first.

The structural distinction. In a buck, the inductor is in series with the load, so its current is the load current and it is never interrupted. In a boost, buck–boost, Ćuk, SEPIC or Zeta, the load is fed only during part of the cycle, and increasing \( D \) shortens exactly that part.

The consequences you can predict from this alone:

• A buck's loop bandwidth is limited by the switching frequency (typically \( f_s/5 \) to \( f_s/10 \)) — a design choice.
• An indirect converter's is limited by \( f_{RHPZ}/5 \), which may be a hundredth of \( f_s \) — a physical constraint.
• So a buck's transient response can be an order of magnitude faster for the same switching frequency.

Why this shapes real architectures. When a fast-responding rail is needed from a wide-ranging input, designers frequently use two stages — a slow boost or SEPIC to make an intermediate bus, then a fast buck to regulate. The buck does the responding; the indirect stage only has to hold a rough average. That is a direct consequence of everything above, and it is worth recognising the pattern when you meet it.