Active High Pass Filter

A vibration sensor's output contains a useful AC signal riding on top of a slow-drifting DC level. The active high-pass filter strips out that DC and low-frequency drift while amplifying the fast signal of interest.

Core Concept

A passive RC high-pass filter gives -20 dB/decade roll-off but has high output impedance and no gain. The active high-pass filter solves this by placing the frequency-determining RC network at the input of an inverting or non-inverting op-amp stage. The result is gain in the pass-band and low output impedance.

In the inverting configuration, capacitor C1 in series with resistor R1 forms the input network. DC is blocked by C1. Above fc = 1/(2πR1C1), C1's reactance is small and the stage behaves like a standard inverting amplifier with gain -Rf/R1. The negative sign means the output is 180 degrees out of phase with the input in the pass-band.

In the non-inverting or Sallen-Key configuration, two RC stages feed the non-inverting input. Feedback sets Q and enables a second-order response. The TL071 is preferred here because its FET input stage draws negligible current through R1, which would otherwise shift the DC operating point.

Key Equations

Cut-off frequency: fc = 1 / (2πR1C1) where R1 is the series resistor and C1 is the series capacitor.

Pass-band gain (inverting): A = -Rf / R1. Magnitude is |A| = Rf/R1. Phase is 180 degrees inverted in pass-band.

Transfer function (1st order inverting): H(s) = -Rf/R1 × (sR1C1) / (1 + sR1C1)

At f = fc: gain magnitude is |A| / sqrt(2) = 0.707 × pass-band gain, corresponding to -3 dB point.

Second-order Sallen-Key cut-off (equal R, equal C): fc = 1 / (2πRC) with roll-off of -40 dB/decade below fc.

Given:
  R1 = 3.3 kΩ = 3,300 Ω
  C1 = 47 nF = 47 × 10^-9 F
  Rf = 33 kΩ
  Vin = 1 V at f = 5 kHz

Why this formula:
  First-order inverting HPF: fc from R1, C1. Gain from Rf/R1.

Formula:
  fc = 1 / (2π × R1 × C1)
  |H| = (Rf/R1) × (f/fc) / sqrt(1 + (f/fc)^2)

Substitution for fc:
  fc = 1 / (2π × 3300 × 47×10^-9)
     = 1 / (2π × 1.551×10^-4)
     = 1 / 9.746×10^-4
     = 1026 Hz ≈ 1.03 kHz

Gain at 5 kHz:
  f/fc = 5000/1026 = 4.873
  |H| = (33k/3.3k) × 4.873 / sqrt(1 + 4.873^2)
      = 10 × 4.873 / sqrt(1 + 23.75)
      = 48.73 / sqrt(24.75)
      = 48.73 / 4.975
      = 9.796

Final Answer:
  Vout = 9.796 × 1 V = 9.80 V
  (Close to full gain of 10 since 5 kHz >> fc = 1.03 kHz)

Exam Tip: In GATE, the most tested point is that gain and cut-off frequency are set independently. fc = 1/(2πR1C1) and pass-band gain = Rf/R1. Changing Rf shifts gain but not fc. Changing C1 shifts fc but not the pass-band gain. Also remember the phase: inverting HPF has 180-degree phase inversion in the pass-band, not 0 degrees. Non-inverting HPF has 0-degree phase in the pass-band.

Key Properties

• Active HPF blocks DC and low-frequency signals while amplifying signals above fc in the pass-band.
• Cut-off frequency fc = 1/(2πR1C1) is independent of the gain-setting feedback resistor Rf.
• Inverting configuration gives pass-band gain = -Rf/R1, with 180-degree phase inversion.
• Non-inverting configuration gives pass-band gain = 1 + Rf/R1, with 0-degree phase in pass-band.
• TL071 op-amp is preferred due to its 3 MHz GBW and very low input bias current, which prevents DC offset from resistors in the bias path.
• Second-order Sallen-Key HPF achieves -40 dB/decade roll-off below fc using one op-amp.
• At fc, output is 3 dB below the pass-band gain level. This is the standard definition of cut-off for active filters too.

Quick Revision

• Active HPF uses an op-amp to provide gain and low output impedance, unlike passive RC.
• fc = 1/(2πR1C1). Gain does not affect fc.
• Inverting HPF: gain = -Rf/R1, output inverted by 180 degrees in pass-band.
• Non-inverting HPF: gain = 1 + Rf/R1, output in phase in pass-band.
• At fc: gain is 3 dB below maximum (0.707 × pass-band gain).
• First-order: -20 dB/decade roll-off. Second-order Sallen-Key: -40 dB/decade.
• Op-amp GBW must be well above A × fc to avoid gain droop at high frequency.
• Exam trap: Students think changing Rf shifts fc. It does not. fc depends only on R1 and C1. Rf sets only the pass-band gain.