Pi Filter

A single capacitor filter leaves a noticeable ripple on the DC output. The pi filter adds an inductor and a second capacitor to form a low-pass LC network that attenuates ripple far more effectively. High-fidelity audio amplifiers and CRT monitor power supplies used pi filters to achieve ripple below 1 mV on their DC rails.

Core Concept

The pi filter is named for its shape: two capacitors and one inductor arranged like the Greek letter pi. The input capacitor C1 shunts ripple current to ground before it reaches the inductor. The inductor L blocks the remaining AC ripple from passing through. The output capacitor C2 then shunts any residual ripple to ground.

Because it is a second-order LC low-pass filter, the pi filter attenuates ripple at 40 dB per decade above its cutoff frequency. A simple capacitor filter is only first-order (20 dB per decade). This steeper rolloff means far less ripple reaches the load. For a 100 Hz ripple frequency, the attenuation can be 100 times better than a capacitor alone.

The choke (inductor) in the pi filter must handle the full DC load current without saturating. Audio power supply chokes are wound on iron cores and rated at 5 to 10 H with DC current ratings of 200 mA to 1 A. For modern compact designs, the pi filter is often replaced by an LC filter or a voltage regulator IC like the LM317, but pi filters still appear in tube amplifier power supplies.

Key Equations

Cutoff frequency of LC section: f_c = 1 / (2*pi*sqrt(L*C2))

Ripple attenuation factor (at ripple frequency f_r): A = 1 / (1 - (f_r/f_c)^2) — becomes very large when f_r >> f_c.

For f_r >> f_c (high attenuation case): V_out_ripple ≈ V_in_ripple / (L * C2 * omega_r^2) where omega_r = 2*pi*f_r

DC output voltage: V_dc_out = V_dc_in - I_dc * R_choke — the choke's DC winding resistance R_choke causes a small voltage drop.

Given:
Bridge rectifier output (after C1): V_dc = 20 V
Ripple voltage after C1: Vr1 = 2 V peak-to-peak
Ripple frequency f_r = 100 Hz
Choke L = 10 H, DC resistance R_L = 50 Ω
C2 = 1000 µF = 1000e-6 F
Load current I_dc = 100 mA

Why this formula:
LC low-pass filter attenuates ripple by omega_r^2 * L * C in high-attenuation case.

Formula:
Attenuation factor = omega_r^2 * L * C2
omega_r = 2 * pi * 100 = 628.3 rad/s

Attenuation factor:
A = (628.3)^2 * 10 * 1000e-6
A = 394,817 * 0.01
A = 3948

Output ripple:
Vr2 = Vr1 / A = 2 / 3948 = 0.000507 V = 0.5 mV

DC output voltage:
V_dc_out = 20 - (0.1 * 50) = 20 - 5 = 15 V
(The choke's 50 Ω winding resistance drops 5 V at 100 mA)

Final Answer:
Output ripple = 0.5 mV (excellent), V_dc_out = 15 V

Exam Tip: The pi filter is frequently confused with the L-filter (inductor input) and the T-filter. In the pi filter, the first element seen from the rectifier is a capacitor (shunt), then an inductor (series), then a capacitor (shunt). The L-filter starts with the inductor. Also, the choke's DC winding resistance causes a load-dependent voltage drop that reduces V_dc_out. GATE sometimes asks to calculate V_dc_out including this drop. Do not forget it.

Key Properties

• Pi filter topology: C1 (shunt) - L (series) - C2 (shunt). Resembles the letter pi.
• Second-order low-pass filter: rolloff is 40 dB/decade above cutoff. A capacitor-only filter rolls off at only 20 dB/decade.
• Ripple attenuation A = omega_r^2 * L * C2 in the high-attenuation region (f_r >> f_c).
• The choke's winding resistance R_choke drops voltage at full load. A 50 Ω choke at 200 mA drops 10 V, significantly reducing V_dc_out.
• Choke must not saturate at peak current. Iron-core chokes for 100 mA typically have inductance of 5 to 20 H.
• At no load, V_dc_out rises to near Vm (no drop across choke). This must be within capacitor and load voltage ratings.
• Pi filters are preferred in valve (tube) amplifiers and high-voltage DC supplies where excellent ripple rejection is needed without a regulator IC.

Quick Revision

• Pi filter: C1 - L - C2. Named for its shape.
• Second-order filter: 40 dB/decade rolloff. Better than capacitor-only filter.
• Ripple attenuation A = omega_r^2 * L * C2.
• DC drop across choke = I_dc * R_choke. Reduces output voltage at load.
• Output ripple = Input ripple / A. Very small with large L and C2.
• No-load output voltage ≈ Vm (choke drop disappears at zero current).
• Used in audio and valve amplifier supplies for very low ripple without ICs.
• Exam trap: Forgetting the choke winding resistance voltage drop when calculating V_dc_out under load. The inductor is not ideal and its DC resistance reduces the output voltage.