Free Buck Converter Input Filter & Middlebrook Tool

The Middlebrook Criterion & Input Filter Oscillation Physics

Switch-mode power supplies (buck, boost, flyback) with tight closed-loop output regulation act as constant power loads. If the input voltage drops, the controller increases duty cycle and draws more current: $$P_{in} = V_{in} \cdot I_{in} = \text{const} \implies \frac{dI_{in}}{dV_{in}} = -\frac{P_{in}}{V_{in}^2} < 0$$ This creates a negative dynamic input resistance ($R_{in} = -V_{in}^2 \cdot \eta / P_{out}$).

Why Negative Resistance Provokes Instability

If an undamped input LC filter has an output impedance peak ($|Z_{out}|$) at its resonant frequency that approaches or exceeds $|R_{in}|$, the net resistance of the combined system becomes negative ($R_{net} < 0$), triggering spontaneous, uncontrolled undamped oscillations that crash converter regulation or cause over-voltage component breakdown.

The Middlebrook Stability Rule: $$|Z_{out,filter}(f)| \ll |Z_{in,converter}(f)| \quad (\ge 6\text{ dB to } 10\text{ dB margin})$$

Sizing the Parallel RC Damping Network ($R_d, C_d$)

To suppress the high Q resonance of low-ESR ceramic caps without wasting DC power:

• Characteristic Impedance: $Z_0 = \sqrt{L_{in} / C_{in}}$
• Damping Resistor: $R_d \approx 1.0 \times Z_0$ to $1.2 \times Z_0$
• Damping Capacitor: $C_d \ge 4 \times C_{in}$ (provides AC coupling for $R_d$ while blocking DC)