SMPS Input Filter Designer: Middlebrook Stability and Damping

Converter and impedance screen

Use $P_{in}=P_{out}/\eta$ and $Z_{in}=-V_{in}^{2}/P_{in}$, with voltage measured at the converter terminals. MIT 6.622 Lecture 29, pp. 5–6, explains the constant-power approximation and impedance separation; Middlebrook’s extra element theorem supplies the interaction framework.

With the source voltage set to zero, solve the passive ladder for $Z_o$. A main capacitor contributes $Y_C=sC/(1+sC\,ESR)$. An RC damper adds $sC_d/[1+sC_d(R_d+ESR_d)]$ in shunt. An RL damper places $R_d+sL_d$ in parallel with the main winding impedance. Source resistance precedes the first section. Cascaded stages are coupled exactly by series/parallel impedance reduction.

Report $M=20\log_{10}(|Z_{in}|/\max|Z_o|)$ from DC to the declared constant-power frequency. A logarithmic search locates candidate resonances, then bounded optimization refines each sampled peak. The desired separation is a user-selected engineering allowance, not a standard requirement.

Damping selection

For an ideal isolated RC-damped stage, let $Z_c=\sqrt{L/C}$ and $n=C_d/C$:

This minimizes output-impedance peaking (MIT Lecture 29, p. 4; Erickson 1999). Subtract damping-capacitor ESR to suggest a discrete resistor. A 1 mΩ lower bound is flagged when ESR exhausts the resistance budget.

For the dual RL topology with $n=L_d/L$, equal-fixed-point analysis gives

These are starting values for each isolated stage. They do not globally optimize a cascaded filter or account for parasitics in the optimum itself; the subsequent response does include the entered parasitics.

Attenuation, loss and transient

The unloaded passive voltage transfer $H$ also equals the reverse source-current / injected-converter-current transfer by reciprocity. For an independently injected current harmonic within the declared constant-power band:

Above that band the incremental converter loading is omitted, so $G_I=H$ and $V_{rms}=|I_hZ_o|$. Current-harmonic attenuation is $-20\log_{10}|G_I|$. The passive insertion and in-band CPL-loaded curves are identified separately.

Use $V_{pk}=\sqrt{2}V_{rms}$ to test the 5% operating-point boundary. No passing stability or target verdict is issued outside that boundary. Emission-baseline verdicts are also withheld inside the regulation band; they need loaded measurements rather than an assumed unchanged emission source.

Resistor ripple loss is $|I_d|^2R_d$ using RMS phasors and the same loading factor. For the first RL leg, recover the source-side node after source resistance, $v_u=v_1R_s/(R_s+Z_{L1})$, then use $(v_u-v_1)/(R_d+sL_d)$. This retains the voltage across the inductor instead of grounding its upstream node. DC sharing includes winding resistance in the RL bypass.

The transient solves KCL and KVL with load conductance $-1/|Z_{in}|$. A normalized state matrix and matrix exponential give the sampled source-step response, equivalent to $H/(1-Z_o/|Z_{in}|)$. Stable cases propagate through the entire requested window. If a sample brackets a 5% crossing, bisection with matrix-exponential propagation locates the crossing and sets an explicit event flag. A separate flag reports a computation limit; elapsed duration alone never implies a crossing. Peak perturbation is a sampled value. Animation only interpolates the precomputed trace.

Poles come directly from a diagonally balanced real Schur decomposition, extracting its 1×1 and 2×2 blocks. The calculation checks the reconstruction residual, orthogonality, neglected lower-block entries and balancing spread. Shifted inverse iteration estimates left/right eigenvalue condition numbers and residuals; a conservative numerical allowance must leave the pole sign resolved. Failed checks or marginal signs produce an unresolved verdict.

• The converter alone is stable and operates around a fixed DC point with constant efficiency.
• Constant-power impedance is applicable only over the declared band for the magnitude screen. The transient and pole calculation extend that approximation to all frequencies.
• Components are linear and lumped. Main capacitor ESR and inductor DC resistance are constant with frequency. The damping inductor is ideal; damping-capacitor ESR is entered separately.
• Both stages use the same damping topology and storage ratio. Local converter capacitance belongs in the final shunt capacitor.
• Source impedance is a real resistance; no upstream control dynamics or cable inductance beyond the entered filter inductors is modeled.
• The harmonic input is an independent sinusoidal current source excluding incremental converter response. Converter loading is $-1/|Z_{in}|$ within the declared band and neglected above it. The 5% limit applies to peak terminal voltage, not RMS.

Use this as a small-signal design screen, then verify the converter’s measured input impedance and the assembled filter. Meeting a magnitude criterion within a limited band is not a proof of complete system stability. A failed screen is inconclusive by itself; an unstable ideal-CPL pole is a prediction of that stated model.

Enter a source step no larger than 1% of DC terminal voltage. Traces stop at an explicitly located 5% crossing, where linear predictions should no longer be extrapolated. A computation limit has its own status and never claims a boundary crossing. This is not a startup, hot-plug, inrush or protection simulation. Review damping-resistor pulse energy separately for those events.

ESL, inductor self-resonance, saturation, core loss, temperature dependence, common-mode paths, layout coupling, switching dynamics and detailed control impedance are excluded. High-frequency attenuation is therefore an estimate below component self-resonance. DC losses include source and winding resistance; the damping-resistor result excludes capacitor ESR and magnetic losses.

Harmonic voltage is checked using peak = √2 × RMS, with converter loading included inside the model band. Outside the 5% boundary, numerical loss and ripple values are linear extrapolations and the passing stability/target verdicts are withheld. Unresolved numerical pole signs likewise withhold target acceptance.

For conducted emissions, enter the applicable target and use the prescribed measurement setup. Baseline-reduction verdicts inside the converter model band are withheld pending loaded measurements. No CISPR limits or detector models are implemented. Check against the standard in force; a calculated attenuation does not certify compliance.