Band Pass Filters
A band pass filter passes a specific range of frequencies — the passband — while attenuating both lower and higher frequencies. It is the combination of a high pass filter (which passes everything above a lower cutoff) and a low pass filter (which passes everything below an upper cutoff). The region where both are passing is the passband, bounded by the two cutoff frequencies.
Band pass filters are everywhere in electronics. A graphic equalizer is a bank of band-pass filters, each controlling one slice of the audio spectrum. An AM or FM radio uses band-pass filters to pick one station out of many. A wireless chip has band-pass filters that pass only its own frequency band. A sensor interface uses one to keep the frequencies its signal occupies and reject both slow drift and high-frequency noise. Understanding band pass filters is central to understanding how equipment picks out the signals it wants.
Band pass filter frequency response. The passband is centered on f0 with the −3 dB points at f1 (lower) and f2 (upper). Bandwidth BW = f2 − f1. Both the low-frequency stopband and high-frequency stopband are attenuated.
View LargerHow a Band Pass Filter Works
The simplest way to understand a band pass filter is to combine what you already know about high pass and low pass filters. A high pass filter with lower cutoff frequency f1 allows all signals above f1 to pass. A low pass filter with upper cutoff frequency f2 allows all signals below f2 to pass. Cascade them (connect them in series) and only signals that are above f1 AND below f2 can pass both filters — the passband lies between f1 and f2.
This cascaded approach works well for audio frequencies where wide passbands are needed. However, where very narrow passbands are needed (a 6 kHz filter at 455 kHz in an AM radio, for example), the cascaded RC filter approach would require an enormous number of poles to achieve the required selectivity. The solution is to use a resonant LC circuit as the basis of the bandpass filter — a circuit that naturally passes a narrow range of frequencies centered on the resonant frequency.
Two Approaches: Cascaded HPF/LPF vs. Resonant Circuit
Cascaded RC High Pass + Low Pass
This approach suits audio and low-frequency applications where the passband is relatively wide compared to the center frequency (wide fractional bandwidth).
Requirements for a valid cascade approach:
- f2 must be significantly greater than f1 (typically f2 > 3 × f1 for minimal interaction between the two filters)
- The impedance of the second filter must not significantly load the first filter (or a buffer amplifier must separate them)
- The rolloff in each stopband is determined by the order of each filter section
Resonant LC Bandpass Circuit
This approach suits high-frequency applications where the passband is narrow compared to the center frequency (narrow fractional bandwidth). A parallel LC tank circuit presents high impedance only at its resonant frequency — by using this high impedance as the load in a circuit, you create a bandpass filter with center frequency f0 = 1/(2π√LC) and bandwidth BW = f0/Q.
The LC bandpass filter is the dominant approach for narrow-band filtering at radio frequencies because it achieves narrow passbands efficiently with just two components per resonator stage.
Center Frequency and Bandwidth
For any band pass filter, the key specifications are:
Center frequency: f0 = √(f1 × f2) [geometric mean — exact for symmetric response]
For narrow bandpass (BW << f0): f0 ≈ (f1 + f2) / 2 [arithmetic mean — good approximation]
Bandwidth: BW = f2 − f1
Q factor: Q = f0 / BW
where f1 = lower −3 dB frequency, f2 = upper −3 dB frequency
For a single-pole LC bandpass filter (one resonant circuit):
f0 = 1 / (2π√(LC))
BW = f0 / Q = R / (2πL) [for series resonant bandpass]
BW = 1 / (2πRC) [for parallel resonant bandpass with load R]
The Q and bandwidth relationship you learned in the Q Factor and Bandwidth lesson applies directly here: the higher the Q of the resonant circuit, the narrower the passband. This is why crystal resonators (Q = 50,000–100,000) are used for very narrow filters, while LC circuits (Q = 50–300) are used for wider filters.
An AM broadcast radio uses a 455 kHz intermediate frequency. You want an LC bandpass filter with 6 kHz bandwidth (3 kHz each side of center, suitable for AM broadcast reception). Find the required Q and inductor/capacitor values.
Step 1 — Required Q:
Q = f0 / BW = 455,000 / 6,000 = 75.8 ≈ 76
Step 2 — Choose C and find L for f0 = 455 kHz:
Choose C = 100 pF (a convenient value for 455 kHz IF circuits)
L = 1 / (4π² × f0² × C) = 1 / (39.478 × (455,000)² × 100×10⁻¹²)
= 1 / (39.478 × 207.025×10⁹ × 100×10⁻¹²) = 1 / (817,235) = 1.224 µH
Step 3 — Required inductor series resistance for Q = 76:
XL = 2π × 455,000 × 1.224×10⁻⁶ = 3.498 Ω
RS = XL / Q = 3.498 / 76 = 0.046 Ω
Problem: An inductor resistance of 46 mΩ is essentially zero — unrealistically ideal. In practice, a 1.2 µH air-core inductor might have RS = 0.1–0.3 Ω, giving Q = 3.5/0.2 = 17.5 at best. This illustrates why LC filters cannot achieve the Q = 76 needed for a 6 kHz filter at 455 kHz. Solution: Use a ceramic resonator (Q ≈ 2,000–5,000) or crystal filter (Q ≈ 50,000) for narrow filtering. LC circuits are used for wider filters at 455 kHz.
For a wideband preselector with Q = 20:
BW = f0 / Q = 455,000 / 20 = 22,750 Hz ≈ 22.8 kHz
This 22.8 kHz bandwidth passes several adjacent stations at once but provides useful rejection of far-off-frequency signals. A wide LC filter like this, followed by a ceramic filter with 6 kHz bandwidth, provides excellent overall selectivity.
Band Pass Filter Calculator
Band Pass Filter Center Frequency and Bandwidth Calculator
Enter the lower and upper −3 dB frequencies to calculate center frequency, bandwidth, and Q factor. Or enter f0 and Q to find the bandwidth and −3 dB points.
Mode 1: From f₁ and f₂
Mode 2: From f₀ and Q
LC Bandpass Filters in RF Circuits
The single LC tank circuit is the simplest high-frequency bandpass filter. It passes frequencies near f0 and attenuates frequencies away from f0 — exactly the bandpass response needed for tuned amplifier stages. Adding more coupled resonators increases the order of the filter and steepens the rolloff in the stopbands.
Single-Resonator Bandpass
A single LC tank in the output of an amplifier stage acts as a single-pole bandpass filter. Its response is Lorentzian-shaped — a gentle bell curve with 20 dB/decade rolloff on each side. This provides moderate selectivity and is the standard tuned-amplifier configuration in many older radios.
Coupled Resonator Bandpass
Connecting two or more LC resonators through coupling capacitors or inductors creates a higher-order bandpass filter with much steeper skirts. A pair of coupled resonators forms a 2-pole bandpass filter with 40 dB/decade rolloff outside the passband. This is the standard approach for ceramic and crystal IF filters — multiple resonators are coupled together in a single package.
Crystal Bandpass Filters
Crystal filters use several quartz crystals coupled together. A well-designed 8-crystal filter at a few megahertz can provide a passband of a few kilohertz with attenuation of 80 dB or more at 10 kHz from center — a shape factor far beyond what any LC filter can achieve at these frequencies. SAW (surface acoustic wave) and BAW filters use the same principle at the gigahertz frequencies of phones and Wi-Fi.
Shape Factor and Selectivity
The shape factor (SF) is a single number that captures how quickly a filter's response rolls off outside the passband. It is the ratio of the bandwidth at a deeper attenuation level to the bandwidth at −3 dB:
SF = BW60dB / BW3dB (60 dB shape factor, most common)
SF = BW6dB / BW60dB (alternative definition — reciprocal)
A perfect rectangular filter (ideal, physically impossible) has SF = 1.0
Real filters: LC = 5–20; ceramic = 3–5; crystal = 1.5–2.5
Lower shape factor means steeper skirts — the filter transitions more abruptly from passband to stopband. Shape factor matters whenever a wanted signal sits close to an unwanted one. A filter with SF = 1.5 at 2.4 kHz bandwidth reaches 60 dB attenuation by 3.6 kHz from center — enough to strongly reject a signal only 2 kHz away from the wanted one.
| Filter Technology | Typical Shape Factor (SF60/3) | Application |
|---|---|---|
| Single LC resonator | 10–20 | Preselector, wideband IF |
| 4-pole Butterworth LC | 5–8 | Moderate-selectivity bandpass |
| Ceramic filter (455 kHz) | 3–5 | AM radio IF filter |
| 4-crystal filter | 2–3 | Narrow communication filters |
| 8-crystal filter | 1.5–2.0 | High-selectivity communication filters |
| Narrow crystal filter (500 Hz BW) | 1.3–1.6 | Very narrow test-equipment and data filters |
Band Pass Filters in Practice
Graphic Equalizers and Spectrum Displays
A graphic equalizer splits audio into a set of bands — typically spaced an octave or a third of an octave apart — with a band-pass filter for each, so each slider can boost or cut its own slice of the spectrum. A bar-graph spectrum display uses the same bank of filters to measure the level in each band.
Radio and Wireless Receivers
An AM or FM broadcast radio, a key-fob receiver and a Wi-Fi chip all start with a band-pass filter that passes only their own band of frequencies and rejects strong signals elsewhere that could overload the circuits that follow. Narrower ceramic, crystal or SAW filters then select the individual channel.
Sensor and Ultrasonic Filtering
An ultrasonic distance sensor transmits a 40 kHz burst and listens for its echo. A band-pass filter centered on 40 kHz passes the echo while rejecting audible noise below and electrical interference above, greatly improving the sensor's range and reliability. Heart-rate monitors and vibration analyzers similarly filter around the frequencies their signals occupy.
Frequently Asked Questions
Why is the center frequency the geometric mean of f1 and f2 rather than the arithmetic mean?
On a logarithmic frequency scale (which is the natural scale for frequency ratios), the geometric mean sits exactly at the center between f1 and f2. The bandpass response of a resonant circuit is symmetric on a log scale — equal attenuation at frequencies that are equal ratios above and below center (e.g., 2× f0 and f0/2 have equal attenuation). For narrowband filters (BW much smaller than f0), the arithmetic mean and geometric mean are nearly equal, and either is acceptable. For wideband filters (more than an octave of passband), the geometric mean is the correct definition.
Can I build a useful bandpass filter with just one LC resonant circuit?
Yes, for wideband applications. A single LC resonator provides a single-pole (20 dB/decade per side) bandpass response adequate for wide filters. For narrowband applications, the rolloff of a single LC resonator is far too gradual — you need multiple coupled resonators or crystal filters. A single LC resonator with Q = 50 at 14 MHz has 280 kHz bandwidth — useful as a wide filter but far too wide to separate channels a few kilohertz apart.
Does a bandpass filter affect the phase of the signal it passes?
Yes. All real filters introduce phase shift, and bandpass filters are no exception. Near the center of the passband the phase shift is relatively small and constant. Near the edges of the passband the phase shift changes rapidly with frequency — this is called group delay variation. In audio and data signals, group delay variation across the passband can cause a smeared sound or data errors, which is one reason filter design is an engineering art. High-quality filters are optimized for minimum group delay variation as well as good shape factor.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.