Ćuk, SEPIC and Fourth-Order Converters

The duty ratio is unchanged from the buck–boost, because the volt-second balance on \(L_1\) is the same:

Now, crucially, \(L_1\) carries 2.5 A and \(L_2\) carries 2 A — the terminal currents themselves. Neither inductor carries the 4.5 A the buck–boost's single inductor had to.

The input inductor. With the switch on, the full input voltage appears across it:

The output inductor sees the same voltage, which is not obvious. With the switch on, \(L_2\) sees \(V_{C1}-\left|V_o\right|\):

Both inductors see \(V_{in}\) during the on-time and \(-\left|V_o\right|\) during the off-time. That identical excitation is what makes the coupled-inductor trick of Problem 3 possible.

The transfer capacitor, which is discharged by \(I_{L2}\) during the on-time and recharged by \(I_{L1}\) during the off-time:

The two intervals give the same swing, as charge balance requires. Note that \(C_1\) is not a filter — a 5% ripple on it is entirely acceptable, because it is the energy-transfer element and its ripple never reaches the output.

And now the payoff — the output capacitor. Because \(L_2\) feeds the output directly, the capacitor sees only a triangular ripple, as in a buck:

Five microfarads, against 55.6 for the buck–boost at the same specification — eleven times smaller. And its RMS current is \(\Delta I_2/\sqrt{12} = 0.12\) A rather than the buck–boost's 2.2 A, so ESR is no longer a constraint at all.

The full comparison:

Identical switch stresses, both terminals now smooth, and the output capacitor eleven times smaller. What has happened is that the difficult capacitor has moved inside the converter, where a 5% ripple is harmless, instead of sitting on the output where it is not.

The duty ratio and inductor currents are unchanged:

But the transfer capacitor sits at a different voltage. In a SEPIC the capacitor is in series between the two inductors on the input side of the diode, and volt-second balance on \(L_2\) gives:

Against 27 V for the Ćuk. A lower-voltage capacitor is cheaper and smaller, which is a genuine if minor advantage — and it means the SEPIC's capacitor voltage rating is set by the input alone, so a wide-input design does not need a 27 V part.

Device stresses are the same as the Ćuk's:

Both converters place the sum across the devices, because both must block the input and the output simultaneously. There is no escaping that in any topology derived from the buck–boost.

Now the difference that matters. In a SEPIC the diode feeds the output capacitor directly — there is no inductor between them — so the output current is pulsed, exactly as in a boost:

Against 5 µF for the Ćuk — twenty-two times larger, and with a substantial ESR requirement on top. That single row is the price of the non-inverted output.

The four options compared:

SEPIC and Zeta are mirror images: each keeps one terminal smooth and pulses the other, and each gives a positive output. Only the Ćuk keeps both, and only the Ćuk inverts. That trade appears to be structural — no known fourth-order arrangement gives a non-inverted output with both terminals continuous.

Choosing between SEPIC and Zeta. The rule follows from which terminal matters more:

SEPIC dominates in practice, and the reason is regulatory rather than technical: conducted emissions limits are enforced at the supply terminals, so a continuous input current saves an input filter that would otherwise be mandatory. The output ripple is the designer's own problem and can be fixed with a capacitor.

The starting observation from Problem 1: both inductors see identical voltage waveforms — \(+V_{in}\) during the on-time and \(-\left|V_o\right|\) during the off-time. So if they are wound on the same core with the same turns orientation, they can be coupled:

Solve the pair simultaneously:

Two independent nulls are available. Setting \(M = L_2\) makes the input ripple vanish; setting \(M = L_1\) makes the output ripple vanish. Both cannot be satisfied unless \(L_1 = L_2\).

Convert to a coupling coefficient:

A perfectly achievable value — 0.894 is loose coupling by transformer standards and is set by deliberately spacing the windings or by the leakage of a gapped core. The design is tuned by adjusting that leakage.

Why the other null is unavailable here:

Impossible — a coupling coefficient cannot exceed unity, since that would mean more mutual flux than either winding produces. So the null is only available in the winding with the smaller inductance. Nulling the output instead simply requires \(L_2 > L_1\), which is a design choice made before the core is wound.

What this achieves and what it costs:

The first cost is the important one: the ripple does not disappear, it moves. With \(M = L_2\) the output winding takes all of it, so \(L_2\) and \(C_2\) must be sized for the increase. And because the null depends on a ratio of inductances, a 5% coupling error leaves roughly 5% of the original ripple — still a twentyfold improvement, which is why the technique works in production despite the sensitivity.

Where it is used. The same principle applies beyond the Ćuk:

Which includes the SEPIC (whose inductors also see matched waveforms) and the Zeta. In a SEPIC, coupling the two windings on one core is near-universal practice — it saves a component and halves the ripple in both windings even without deliberate tuning, because the effective inductance of each doubles.

Where the resonance comes from. The converter now has four reactive elements, so its transfer function is fourth order — two complex pole pairs rather than one. The lower pair comes from \(L_1\) ringing against \(C_1\):

Awkwardly placed — well above the loop's likely crossover but well below the switching frequency, so it lies inside the band where the compensator has to work.

And it is barely damped. The characteristic impedance is:

A \(Q\) of eighty means a 38 dB peak in the transfer function at 9.6 kHz, with the phase swinging through 180° in a fraction of a decade. No compensator can be designed around that; the resonance must be damped in the power stage.

Why the parasitics do not save it. Unlike the output \(LC\) of a buck, this loop is not loaded by the output resistance:

And every design improvement makes it worse. A lower-DCR inductor, a lower-ESR capacitor, a better switch — all reduce \(r\) and raise \(Q\). This is a resonance that gets sharper as the converter gets better.

The damping network. The standard solution is a series \(R_d\)–\(C_d\) branch in parallel with \(C_1\):

\(C_d\) must be several times \(C_1\) so that it looks like a short circuit at the resonant frequency — putting \(R_d\) across the loop where it can dissipate — while blocking DC so that \(R_d\) does not carry the 12 V transfer capacitor voltage continuously. With these values \(Q\) falls to roughly unity.

What the damper costs:

Small in practice, because at the switching frequency \(C_1\) itself is a much lower impedance than the damper branch, so most of the 2.24 A of Problem 5 flows in \(C_1\) rather than through \(R_d\). The damper only acts near 9.6 kHz, where there is little energy in normal operation — and a great deal during a transient, which is exactly when it is wanted.

Alternatives, and why the RC damper usually wins:

The third row is worth noting because it is how many designs get away without a damper: an electrolytic transfer capacitor has enough ESR to damp the loop by itself. Replacing it with a ceramic — an apparently obvious improvement — removes the damping and makes the converter ring.

The capacitor's current in each interval. During the on-time it supplies \(I_{L2}\); during the off-time it is charged by \(I_{L1}\):

Which is the condition that sets the duty ratio in the first place — a useful independent route to \(D\) that does not use volt-second balance at all.

The RMS value:

And there is a neat closed form, since \(D = I_{in}/(I_{in}+I_o)\):

The geometric mean of the two terminal currents — an elegant result and a useful one, because it can be evaluated before any component is chosen.

What that demands of the part. Compare the candidates at 4.1 µF, 27 V, 2.24 A RMS at 200 kHz:

The first row is why Ćuk converters have a reputation for unreliability among people who used an electrolytic there. And the last row is the trap of Problem 4: the best capacitor electrically is the one that leaves the input loop with a \(Q\) of eighty.

Why this scales badly. The capacitor's RMS current is \(\sqrt{I_{in}I_o}\), so it rises linearly with power at fixed voltages:

Two hundred amps of ripple current at 200 kHz through a capacitor is not a component that exists in any convenient form. Semiconductors scale far better — paralleling MOSFETs is routine — so the capacitor becomes the binding constraint long before the switches do.

Which sets the practical ceiling:

Above a kilowatt the answer is a transformer — which provides the same input-to-output isolation of the current path that \(C_1\) was providing, and does it with magnetic coupling instead of a component carrying the full throughput as AC.

Why nothing simpler will do. The output sits inside the input range:

A buck cannot reach 12 V from 9; a boost cannot reach it from 16. And an automotive supply is worse than these numbers suggest — cold-crank dips to 6 V and load-dump transients to 40 V are routine. The converter must cross the output voltage, which leaves a SEPIC, a Zeta, a Ćuk or a four-switch buck–boost.

The duty ratio range and inductor currents:

Note that the worst cases fall at opposite ends: the currents are worst at low input, the voltages at high input. Every component has to be checked at the correct extreme, and they are not the same extreme.

The inductors, at 9 V where the currents peak:

Both under 60 µH at low currents, so a single coupled component is practical — and standard for a SEPIC, as Problem 3 noted. A 47 µH coupled pair would suit, with the coupling itself halving both ripples.

The devices, at 16 V where the voltage peaks:

But automotive load dump forces a much higher rating:

The two capacitors:

A 1% output ripple is generous for LEDs, whose brightness follows current and whose eye response is slow — so the pulsed output that made the SEPIC expensive in Problem 2 costs almost nothing here. That is why LED drivers are the SEPIC's natural home.

Why SEPIC rather than the alternatives:

The continuous input current is decisive in a vehicle, where CISPR 25 conducted emissions limits are enforced on the supply leads and an input filter costs more than the converter. At 12 W the efficiency penalty against a four-switch design is a fraction of a watt.