Active Band Stop Filter

A power line communication receiver must block the 50 Hz fundamental and its harmonics while passing the data signal above 1 kHz. The active band-stop filter does exactly that, carving out a deep notch at a precise frequency.

Core Concept

A band-stop filter (also called a notch filter or band-reject filter) passes all frequencies except a specific band centered on the notch frequency fn. Outside the stop-band, gain is approximately unity. Inside the stop-band, attenuation can exceed 40 dB for a well-designed active version.

The Twin-T passive network is the heart of most active notch filters. It consists of two T-networks, one made of resistors and one made of capacitors, connected in parallel. At the notch frequency, the two paths cancel exactly, giving theoretically infinite attenuation. Passive Twin-T alone has poor selectivity (low Q). Adding an op-amp voltage follower or feedback sharpens the notch significantly.

Active notch filters using TL071 can achieve Q values from 1 to over 50 depending on the feedback fraction. A 50 Hz notch filter with Q = 10 has a stop-band from 47.5 Hz to 52.5 Hz, rejecting mains hum while passing everything else from DC to 47.5 Hz and above 52.5 Hz.

Key Equations

Twin-T notch frequency: fn = 1 / (2πRC) where R is each equal series resistor and C is each equal shunt capacitor. The center shunt has R/2 and the center series has 2C.

Transfer function of passive Twin-T: H(jω) = (1 - (ω/ωn)^2) / (1 - (ω/ωn)^2 + j4(ω/ωn)) where ωn = 2πfn.

Quality factor with positive feedback fraction k: Q = 1 / (4(1-k)) where 0 < k < 1. At k = 0 (no feedback), Q = 0.25. At k = 0.9, Q = 2.5.

Bandwidth of stop-band: BW = fn / Q. Narrower BW means more selective rejection.

Given:
  Target notch frequency fn = 50 Hz
  C = 100 nF = 100 × 10^-9 F
  Feedback fraction k = 0 (simple buffer, passive Twin-T)

Why this formula:
  fn = 1/(2πRC) for Twin-T network

Formula:
  R = 1 / (2π × fn × C)

Substitution:
  R = 1 / (2π × 50 × 100×10^-9)
    = 1 / (2π × 5×10^-6)
    = 1 / (3.1416 × 10^-5)

Calculation:
  R = 1 / 3.1416×10^-5
    = 31,831 Ω ≈ 31.8 kΩ

Component values for 50 Hz Twin-T:
  Series R = 31.8 kΩ (two of these)
  Shunt R/2 = 15.9 kΩ (one)
  Series C = 100 nF (two of these)
  Shunt 2C = 200 nF (one)

Final Answer:
  R = 31.8 kΩ, C = 100 nF gives fn = 50 Hz
  Q = 0.25 (passive only), BW = 200 Hz at k = 0

Exam Tip: GATE questions on notch filters test component values for the Twin-T network. Remember the exact component ratios: two equal R, two equal C, one R/2, and one 2C. The formula fn = 1/(2πRC) uses the single R and single C, not the halved or doubled values. Getting the component scaling wrong is the most common mistake. Also remember: Q of passive Twin-T alone is only 0.25. Active feedback is what raises Q above 1.

Key Properties

• Band-stop filter passes all frequencies except a narrow band around fn where attenuation is maximum.
• Twin-T network achieves theoretically infinite attenuation at fn when component ratios are exact.
• Passive Twin-T Q = 0.25, giving a wide stop-band. Active feedback with fraction k raises Q to 1/(4(1-k)).
• Stop-band bandwidth BW = fn/Q. At Q = 5 and fn = 50 Hz, the stop-band is only 10 Hz wide.
• Outside the stop-band, an ideal notch filter has unity gain and zero phase shift. Real op-amps introduce small deviations.
• TL071 is used as the buffer and feedback amplifier. Its high input impedance does not load the Twin-T network.
• Component tolerances critically affect notch depth. 1% resistors and 2% capacitors are minimum for a notch deeper than 40 dB.

Quick Revision

• Band-stop (notch) filter rejects a narrow frequency band and passes all others.
• Twin-T notch frequency: fn = 1/(2πRC).
• Twin-T component set: two R, two C, one R/2, one 2C.
• Passive Twin-T Q = 0.25. Active feedback raises Q = 1/(4(1-k)).
• BW = fn/Q. Higher Q = narrower stop-band = more selective notch.
• Common application: 50 Hz mains hum rejection, 60 Hz rejection in US systems.
• Component matching is critical. 1% tolerance or better required for deep notch.
• Exam trap: Students use the halved or doubled component values in the fn formula. Always use fn = 1/(2πRC) where R and C are the basic (non-halved, non-doubled) element values from the network specification.