Library / Electrical / ELE-41
Electrical
Gain of a first-order RC high-pass filter
Ratio of output amplitude to input amplitude for an unloaded RC high-pass filter, taken across the resistor.
- Ref
- ELE-41
- Rev
- v1
- Field
- Electrical
- Result
- |HHP|
The amplitude gain of a first-order RC high-pass filter is |H| = (f/fc)/√(1 + (f/fc)²). It tends to 0 at low frequency, is 0.707 (−3 dB) at fc and tends to 1 well above it. The phase leads the input, from +90° at low frequency to +45° at fc. It blocks DC and slow drifts and passes fast changes, which is why it is used as a coupling stage.
Ratio of output amplitude to input amplitude for an unloaded RC high-pass filter, taken across the resistor. It is near zero at low frequency, 0.707 at the cutoff and approaches 1 above it.
Calculator
- Gain in decibels, 20 log₁₀|H|−0.9691 dB
- Phase of the output relative to the input26.57 °
- Low-pass gain squared plus high-pass gain squared (always 1)1
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 2,000 Hz, 2 times the cutoff frequency, a high-pass filter passes 89.4 % of the amplitude (−0.969 dB), and the output leads the input by 26.6°.
- For the same R and C the high-pass and low-pass gains obey |HLP|² + |HHP|² = 1: where one passes most of the signal, the other blocks most of it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| f Signal frequency | 1 to 100,000 Hz | 1e-3 | 1 | 0.999 | +10% | +1.78 % |
| fc Cutoff frequency | 1 to 100,000 Hz | 0.02 | 1 | 0.98 | +10% | −2.04 % |
Sweep of Signal frequency
| Signal frequency (Hz) | High-pass gain |
|---|---|
| 1 | 1e-3 |
| 4.217 | 0.004217 |
| 17.78 | 0.01778 |
| 74.99 | 0.07478 |
| 316.2 | 0.3015 |
| 1,334 | 0.8 |
| 5,623 | 0.9846 |
| 23,710 | 0.9991 |
| 100,000 | 1 |
Sweep one variable
Variables and how each one contributes
High-pass gain
What the equation gives you. Everything else feeds it.
Signal frequency
The signal frequency, in Hz. Raises the gain: higher frequencies pass more freely. Raising it by 200 Hz (to 2,200 Hz) moves the result by +1.78 %, from 0.8944 to 0.9104.
Cutoff frequency
The filter’s cutoff frequency, 1/(2πRC), in Hz. Lowers the gain: a higher cutoff blocks more of the low-frequency signal. Raising it by 100 Hz (to 1,100 Hz) moves the result by −2.04 %, from 0.8944 to 0.8762.
Worked example with the values above
- Frequency ratio: f/fc = 2,000 / 1,000 = 2.
- Under the root: 1 + 2² = 5, and its root is 2.236.
- |H| = 2 / 2.236 = 0.8944.
Assumptions and limits
- A first-order RC high-pass filter, driven from a low-impedance source into a high-impedance load.
- Steady-state sinusoidal signals.
Common mistakes
- Using a high-pass filter to remove a DC offset from a signal that must be measured as DC.
- Confusing which of R and C sets the corner: both do, as 1/(2πRC).
Statistics and uncertainty
Below the corner the gain falls in proportion to frequency, so a 10 % error in fc gives about a 10 % error in the gain of a signal well below the corner. DC offsets and slow drift are removed completely, which suits AC coupling but not measurements of steady values.
Where it is used
- AC coupling between amplifier stages and in sensor conditioning.
- Removing drift and low-frequency interference from a signal.
Maths you need
- Cutoff frequency: where the gain is 0.707 of the passband value
- Decibel: 20 log₁₀ of an amplitude ratio