Input Filter Calculator

Worked example: the same inputs, three designs

This example uses the same 12 V → 5 V @ 3 A, 500 kHz converter as this calculator's default. It's sized to the same 40 dB attenuation target, the same 6 dB separation requirement, and the same series-RL damping topology. From that one set of inputs, Design mode returns three designs, not two. Those are Middlebrook's own sizing, GMPM's own sizing, and a third design that keeps Middlebrook's Lf and Cf exactly, spending GMPM's freedom on the damping network instead of on R0.

Middlebrook's search converges on Lf = 3.9 µH, Cf = 2.2 µF, Rd = 13.3 Ω, and Ld = 39 µH. That design passes its own criterion at +6.882 dB of separation, which is 0.882 dB to spare over the 6 dB gate. Its peak |T_m| of 0.453 sits inside the 0.501-radius circle a 6 dB gain margin sets.

GMPM's search, given the identical inputs, converges on Lf = 15 µH, Cf = 0.68 µF, Rd = 49.9 Ω, and Ld = 150 µH. That's under a third of the capacitance and close to four times the inductance. It passes its own criterion with +6.965 dB of separation.

Its locus swings much wider, peaking at |T_m| ≈ 1.964, about 3.9× outside that same 0.501-radius circle. But its phase there is only +5.49°, nowhere near the ±60° wedge around 180° that GMPM actually forbids. Checked against Middlebrook's own criterion instead, GMPM's design fails outright, at −5.863 dB of separation against the same 6 dB gate. These aren't two designs that happen to both work. They're the two criteria giving genuinely different answers to the same sizing problem.

The third design spends that same freedom differently. It holds Middlebrook's exact Lf = 3.9 µH and Cf = 2.2 µF, leaving the corner frequency and R0 untouched. It simply deletes the damping network, no Rd and no Ld, for 2 components instead of 4. That design passes GMPM at +17.04 dB of separation and fails Middlebrook at +1.00 dB against the same 6 dB gate. It isn't a smaller version of the GMPM R0 design above. It's Middlebrook's own two reactive components, with the parts that damped the resonance removed.

GMPM's freedom is genuinely two-dimensional. These are two different ways to spend it, not two points on one slider. For a fixed corner frequency f_c, the attenuation target pins the product Lf·Cf, while stability bounds the ratio R0 = √(Lf/Cf). That's two equations and two unknowns. Push R0 up at a pinned product, and Lf rises while Cf falls together. So there's no design that keeps Middlebrook's inductor while also banking the capacitor saving, which is why GMPM's own R0 design above moves both.

The damping network is a separate freedom, independent of R0. Middlebrook's disc has to bring the whole resonance peak under one radius. GMPM's wedge only has to keep the locus clear of one direction of phase. So a design that needs damping to satisfy Middlebrook can sometimes satisfy GMPM with a lighter network, or none at all, without moving Lf or Cf at all. This calculator explores each of those freedoms from Middlebrook's own baseline separately. The R0 design never touches the damping topology, and the min-damping design never touches Lf or Cf. That's instead of searching the full, two-parameter space of R0-and-damping combinations at once, so neither design claims to be the single best use of GMPM's freedom. Each is simply a genuine, verified one.

Why does GMPM's R0 search push up at all, instead of stopping at Middlebrook's own value? Total energy stored in Lf and Cf at the DC operating point, E = ½·Lf·I_dc² + ½·Cf·V_in², is minimised at R0 = V_in²/P_in. That's exactly the constant-power load's own |Zin|, the same -9.6 Ω this page opened with for this operating point.

Stability requires the filter's peak output impedance to sit a gain margin below that same |Zin|, so you'd expect the stability constraint to bind before that energy minimum in general. It did, directly, at four operating points checked on parallel RC, this calculator's default topology. Those were 12 V → 5 V @ 3 A, 12 V → 1.2 V @ 30 A, 48 V → 12 V @ 5 A, and 5 V → 3.3 V @ 2 A. In every case, the largest stable R0 sat 1.7× to 3.6× below that operating point's own energy optimum, with energy falling monotonically as R0 rose toward it.

On the worked example above, Middlebrook's design stores 161 µJ, and 158 µJ of that is in the capacitor alone. At Vin, the capacitor's ½CV² term dominates a modest inductor current's ½LI², so that's expected. GMPM's design, by contrast, stores only 61 µJ. The largest-R0 design this calculator finds is also the one with the least stored energy.

The min-damping design sits at the opposite corner of that same trade. Because it keeps Middlebrook's own Lf and Cf, its stored energy is exactly Middlebrook's 161 µJ, unchanged. The two designs save genuinely different things. One saves joules by moving L and C. The other saves parts by removing Rd and Ld.

That's not the same as saying max-R0 is the cheapest design. Stored energy is a physical quantity, not a cost. It favours max-R0 for a reason that has nothing to do with price. A big capacitor sitting at the full input rail stores more than a modest inductor carrying the DC input current, so cutting the capacitor down always looks good on this particular ledger.

In BOM terms, GMPM's R0 design above trades about 1.5 µF of ceramic for two inductors, Lf and Ld, that nearly quadruple. Read the stored-energy figure as physics that explains why GMPM's R0 search lands where it does, not as a recommendation for which design costs less to build.

That preference for a larger characteristic impedance is not specific to this one operating point. The capacitor's ½CV² term dominates the inductor's ½LI² at any typical buck input. So a lower stored-energy figure on your own results table means the same trade, ceramic capacitance down for a couple of inductors that are often several times larger, and not a cheaper bill of materials.

GMPM doesn't always change the answer. When it doesn't, that's usually a fact about your design, not a limitation of the tool. If the impedance ratio's phase never swings close to 180° anywhere its magnitude is large enough to matter, GMPM's wedge is never entered. There's no GMPM-specific R0 to find, so Design mode reports GMPM's column as Middlebrook's design, unchanged, rather than inventing a difference that isn't there. Two identical columns from this path is a correct read of your inputs, not a sign the tool ran out of anything to check.

Parallel RC is this calculator's default damping topology, not the series-RL used in the worked example above. On parallel RC, the R0 freedom doesn't go unused occasionally. It disappears entirely, once you round to what you can actually buy. At the same four operating points above, GMPM's ideal, unsnapped R0 came out under two parts in 10,000 above Middlebrook's own at three of the four points, and identical at the fourth. It's never larger by a margin that survives snapping to buyable E-series values, so the standard, buildable networks agree at all four.

Parallel RC's disc and wedge bind at effectively the same point on these designs, so there's no larger-R0, less-capacitance buildable design for GMPM to find there. That's a real result about parallel RC on these designs, not a gap in the search. The damping-network freedom is independent of that, and it still applies. This calculator's own default inputs still produce a min-damping design that deletes the damping network entirely, even though the R0 design finds nothing to change.

Occasionally the two sizing columns match for a different reason. The wedge was entered, but GMPM's own R0/fc search failed to converge on a design of its own. Middlebrook's design is provably GMPM-satisfying either way, so Design mode still shows it in GMPM's column. Here, though, that's a search limitation, not proof there's no better GMPM-specific design to find. The tool reports which case applies alongside the design.

One caveat holds regardless of which criterion wins, or which freedom a design spends. A GMPM pass that depends on a large |T_m| peak staying clear of the wedge, rather than on keeping |T_m| small in the first place, is only as good as that phase margin.

GMPM's R0 design above is exactly this case. Its peak |T_m| of 1.964 clears Middlebrook's disc by nearly 4×, and at that peak, near the filter's own resonance, phase sits at +5.49°, nowhere near the ±60° wedge. That point costs no margin at all. GMPM's actual +6.965 dB margin binds elsewhere, near 7.4 kHz, where the converter's own output-filter resonance drags phase to 169.7°, close to dead-on −1, while |T_m| there is smaller, at 0.448.

Shift the real Zin the converter presents, with load, temperature, or component tolerance, and phase, not just magnitude, is what can close that margin, whether or not the design carries a damping network at all.

The caveat bites hardest on the min-damping design, exactly because it came out undamped. With no engineered resistor at all, its own resonance peak, |T_m| of about 0.891 at around 48.5 kHz, is shaped purely by parasitic dcr, esr, and rs. It already sits well outside Middlebrook's 0.501-radius circle, which is exactly why that design fails Middlebrook at only +1.00 dB.

It clears GMPM only because that peak's phase, about +5.47°, sits nowhere near the wedge. Nothing in this design holds that phase in place, since there's no damping resistor left to do it. So a min-damping design that clears GMPM this way is only as safe as an unengineered phase angle, not a designed margin. Scrutinise any GMPM pass that leans on phase staying clear of 180°, whether the design carries a full damping network, a light one, or none at all.