Filters are circuits that change a signal by allowing some frequencies to pass while reducing others. Low-pass filters keep low frequencies and attenuate high frequencies, while high-pass filters do the opposite. They are essential in audio systems, sensors, communications, power electronics, and data acquisition.
Engineers often describe filter behavior with a Bode magnitude plot, where gain is shown in decibels versus frequency on a logarithmic axis.
A passive RC filter uses a resistor and capacitor to make frequency-dependent voltage division. The cutoff frequency marks the point where the output power is half the passband value and the gain is usually -3 dB. For a first-order RC filter, the rolloff changes at about 20 dB per decade beyond the cutoff.
Band-pass and band-stop behavior can be made by combining low-pass and high-pass sections so that only a selected range is passed or rejected.
Key Facts
- RC cutoff frequency: fc = 1/(2πRC)
- Magnitude in decibels: gain dB = 20 log10(Vout/Vin)
- At cutoff for a first-order RC filter: |Vout/Vin| = 1/sqrt(2) = 0.707 and gain = -3 dB
- First-order low-pass magnitude: |H(f)| = 1/sqrt(1 + (f/fc)^2)
- First-order high-pass magnitude: |H(f)| = (f/fc)/sqrt(1 + (f/fc)^2)
- A first-order filter rolls off at approximately 20 dB per decade, which is 6 dB per octave
Vocabulary
- Low-pass filter
- A filter that passes frequencies below a cutoff frequency and attenuates frequencies above it.
- High-pass filter
- A filter that passes frequencies above a cutoff frequency and attenuates frequencies below it.
- Cutoff frequency
- The frequency where a filter output falls to 0.707 of the passband voltage, corresponding to a gain of -3 dB.
- Bode magnitude plot
- A graph of gain in decibels versus frequency, usually using a logarithmic frequency axis.
- Rolloff
- The rate at which a filter attenuates signals outside its passband, often measured in dB per decade.
Common Mistakes to Avoid
- Using f = 1/RC for cutoff frequency is wrong because the correct RC cutoff is fc = 1/(2πRC), and the factor 2π matters when converting angular frequency to hertz.
- Calling the cutoff frequency the point where the output becomes zero is wrong because a first-order filter still has 70.7 percent of the input voltage at cutoff.
- Reading equal spacing on a Bode frequency axis as equal hertz steps is wrong because the x-axis is logarithmic, so each equal interval usually represents a constant frequency ratio.
- Assuming a low-pass and high-pass filter with the same cutoff makes a band-pass filter automatically is incomplete because their connection and loading can change the actual response.
Practice Questions
- 1 A passive RC low-pass filter has R = 2.2 kΩ and C = 0.10 μF. Calculate the cutoff frequency in hertz.
- 2 A first-order high-pass filter has fc = 500 Hz. Estimate the gain in dB at 50 Hz using the 20 dB per decade rolloff approximation.
- 3 A sensor signal contains useful slow temperature changes plus unwanted high-frequency electrical noise. Explain whether a low-pass or high-pass filter is more appropriate and why.