Band Pass Filter: What is it? (Circuit, Design & Transfer Function)

Contents

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Key learnings:

Band Pass Filter Definition: A band pass filter is a device that permits frequencies within a specific range while attenuating those outside of it.
Circuit Components: The circuit of a band pass filter typically involves both high pass and low pass filters to determine its frequency limits.
Transfer Function: The transfer function of a band pass filter helps in understanding how different frequencies are treated, emphasizing its capability to isolate a specific range.
Filter Types: Active and passive filters differ primarily in the inclusion of an amplifier component in active filters, which affects their handling of signals.
Practical Application: Band pass filters are essential in many technologies, such as audio systems and communication devices, to filter out unwanted frequencies.

What is a Band Pass Filter?

A band pass filter, also called a BPF, passes a selected range of frequencies while attenuating frequencies below and above that range. Real filters have finite attenuation, insertion loss and transition regions rather than perfect pass and stop bands.

A low pass filter passes frequencies below its cutoff and attenuates higher frequencies. A high pass filter passes frequencies above its cutoff and attenuates lower frequencies.

The cascade of a high pass stage and a low pass stage creates a Band Pass Filter. This requires the high pass cutoff to be below the low pass cutoff. The two stages can interact unless a buffer or suitable impedance separation limits loading.

A band pass filter has a lower cutoff and an upper cutoff, usually defined at specified attenuation levels such as the -3 dB points. The high pass stage sets the lower limit, while the low pass stage sets the upper limit. Their difference is the bandwidth.

Band Pass Filter Circuit

The circuit below cascades a passive RC high pass stage and a passive RC low pass stage. Its actual response depends on the source impedance, interaction between the stages and the connected load.

The first section is a passive RC high pass stage. The protected cutoff equation below is exact for an isolated first-order stage with a low source impedance and a sufficiently high load impedance.

$F_c_l_o_w = \frac{1}{2 \pi R_1 C_1}$

The second section is a passive RC low pass stage. Its protected cutoff equation also assumes negligible loading. In the unbuffered cascade, calculate the complete transfer function or verify the response by simulation and measurement.

$F_c_h_i_g_h = \frac{1}{2 \pi R_2 C_2}$

Bandwidth is the difference between the specified upper and lower cutoff frequencies. The geometric mean of those cutoffs gives the centre frequency for a conventional second-order band pass response.

$Bandwidth = F_c_h_i_g_h - F_c_l_o_w$

Band Pass Filter Types

Band pass filters can use active RC, passive RC, RLC, state-variable, switched-capacitor or digital structures. The sections below cover the circuits shown on this page.

Active Band Pass Filter

An active band pass filter uses an amplifier with resistors and capacitors. It can buffer stages, provide gain or realise a second-order response in one topology, but its usable range is limited by amplifier bandwidth, noise, slew rate and signal range.

The illustrated circuit places a high pass network before an op-amp stage and a low pass network after it. Other active band pass topologies combine filtering and gain around one or more op amps.

Circuit Diagram of Active Band Pass Filter

Passive Band Pass Filter

A passive filter uses components such as resistors, capacitors and inductors. It requires no powered amplifier and cannot provide power gain. Source and load impedances form part of its response.

The illustrated passive RC circuit combines a high pass section with a low pass section. It is only equivalent to two independent first-order stages when loading between them is negligible.

Circuit Diagram of Passive Band Pass Filter

The first section provides the high pass behaviour and the second provides the low pass behaviour.

RLC Band Pass Filter

An RLC band pass filter uses resistance, inductance and capacitance to create a resonant response. The total resistance includes source, load and component losses, not only a labelled resistor.

A series-resonant branch has minimum impedance at resonance and can pass energy to a load. A parallel-resonant branch has maximum impedance at resonance and can be arranged to pass the resonant band. The exact transfer function depends on where the output is measured.

The protected series-RLC bandwidth equation below applies when RL represents the total series resistance and bandwidth is measured in radians per second.

Bandwidth for series RLC filter

$\Delta \omega = \frac{R_L}{L}$

The two half-power frequencies are not each equal to the undamped resonant frequency. Their separation defines bandwidth, while the ideal LC resonant angular frequency is 1/√LC. The protected parallel-RLC formula below applies only when RL is the effective parallel resistance; source, load and loss resistances can change the result.

Bandwidth for parallel RLC filter

$\Delta \omega = \frac{1}{ R_L C}$

Wide Band Pass Filter

Wide and narrow are relative design terms, not universal categories separated at Q = 10. For a resonant band pass filter, Q is the centre frequency divided by the -3 dB bandwidth, so a lower Q means a wider fractional bandwidth.

A wide band pass response can be formed from separate high pass and low pass sections when their cutoff frequencies are well separated. Those sections may be passive or active.

The illustrated active arrangement cascades a high pass stage with a low pass stage. Buffering reduces stage interaction, while the op-amp gain-bandwidth product and component tolerances still limit the realised response.

Circuit Diagram of Wide band Pass Filter

Separate sections let the lower and upper cutoffs be adjusted with less coupling than in a high-Q single-stage design.

Narrow Band Pass Filter

A narrow band pass filter has a small fractional bandwidth and therefore a high Q. Multiple-feedback is one possible single-op-amp topology, but narrow responses can also use RLC, state-variable, biquad, switched-capacitor or digital implementations.

The circuit shown is called a multiple-feedback filter because more than one frequency-dependent path feeds the op-amp’s summing node.

The illustrated multiple-feedback band pass circuit is inverting, not non-inverting. Its gain, centre frequency and Q interact. The op amp must have enough open-loop gain and bandwidth at the centre frequency.

Circuit Diagram of Narrow Band Pass Filter

The figure compares a wider, lower-Q response with a narrower, higher-Q response.

Frequency Response of Wide Band Pass and Narrow Band Pass Filter

Band Pass Filter Transfer Function

First Order Band Pass Filter Transfer Function

A single-pole transfer function cannot attenuate both low and high frequencies, so a continuous-time band pass response needs at least two poles. Filter order counts poles, not simply the number of energy-storage components visible in a circuit.

Second Order Band Pass Filter Transfer Function

The circuit below is an inverting active RC band pass stage. Its derivation gives a numerator time constant of R2C1. The protected legacy equations incorrectly define τ3 as R3C3 and then introduce ω3, even though the circuit has no R3 or C3. The high pass pole must be below the low pass pole for a useful pass band.

$Z_1 = R_1 + \frac{1}{j \omega C_1}$

$Z_2 = R_2 || \frac {1}{j \omega C_2}$

$Z_2 = \frac{R_2 \frac{1}{j \omega C_2}}{R_2 + \frac{1}{j \omega C_2}}$

(1) $\begin{equation*} \begin{align} H(j \omega) &= - \frac{Z_2}{Z_1} \\ &= - \frac{\frac{R_2}{j \omega C_2}}{(R_2 + \frac{1}{j \omega C_2})(R_1 + \frac{1}{j \omega C_1})} \\ &= -\frac{\frac{R_2}{j \omega C_2}}{(\frac{R_2 j \omega C_2 + 1}{J \omega C_2})(\frac{R_1 j \omega C_1 + 1}{j \omega C_1})} \\ &= - \frac{j R_2 \omega C_1}{(1+j \omega C_2 R_2)(1+j \omega C_1 R_1)} \\ H(j \omega) &= - \frac{j \omega \tau_3}{(1+j \omega \tau_2)(1+j \omega \tau_1)} \end{align} \end{equation*}$

Where,

$\tau_1=R_1 C_1 \quad \tau_2=R_2 C_2 \quad \tau_3 = R_3 C_3$

$\omega_1 = \frac{1}{\tau_1} \quad \omega_2 = \frac{1}{\tau_2} \quad \omega_3 = \frac{1}{\tau_3}$

For band pass filter, following condition must satisfy,

$\omega_1, \omega_2 > \omega_3$

Band Pass Filter Cutoff Frequency

The high pass stage sets the lower cutoff. The protected legacy label F_c-high is reversed: the stage passes frequencies above F_c-high, but the value called F_c-high in the equation below is the lower cutoff from R1 and C1.

$F_c_l_o_w = \frac{1}{2 \pi R_1 C_1}$

The low pass stage sets the upper cutoff. The protected legacy label F_c-low is reversed: the stage passes frequencies below F_c-low, but the value called F_c-low in the equation below is the upper cutoff from R2 and C2.

$F_c_h_i_g_h = \frac{1}{2 \pi R_2 C_2}$

The pass band lies above the lower cutoff and below the upper cutoff. The legacy order F_c-highand F_c-low shown here is reversed; bandwidth is the upper cutoff minus the lower cutoff.

$Bandwidth = F_c_h_i_g_h - F_c_l_o_w$

The high pass cutoff defines the lower pass-band edge and the low pass cutoff defines the upper edge. For interacting or high-Q stages, find the actual -3 dB edges from the complete response rather than treating the sections independently.

Band Pass Filter Bode Plot or Frequency Response

A bode plot shows magnitude and phase against logarithmic frequency. The pass band is the frequency interval between its defined lower and upper edges, not a region of zero loss.

For a basic second-order band pass section with one low-frequency zero, magnitude rises asymptotically at 20 dB per decade below FL. Higher-order designs can have steeper skirts.

When FL and F_H are well separated, the response can have a nearly flat middle region. Above the upper edge it falls asymptotically at 20 dB per decade for a basic second-order section. A narrow high-Q response instead has a peak near its centre frequency.

The basic band pass transfer function is second order because its denominator has two poles. Its phase changes across frequency, so it cannot be described as a constant 180˚ difference.

For the standard non-inverting second-order form, phase approaches +90˚ far below the centre frequency and crosses 0˚ at the centre. These are asymptotic limits, not one constant phase for all frequencies below the centre.

Far above the centre frequency, the same non-inverting form approaches -90˚. An inverting topology, including the protected transfer function above, adds a 180˚ inversion.

Ideal Band Pass Filter

An ideal band pass filter has zero response below F_L, a constant nonzero response from F_L to the upper edge and zero response above F_H. Its response changes discontinuously at each edge, so a dB-per-decade slope is not defined there.

No finite-order causal physical circuit can realise that brick-wall response exactly. Practical filters trade transition width, pass-band ripple, stop-band attenuation, phase response and implementation limits.

Band Pass Filter Equation

The protected piecewise equation models an ideal filter with unit gain inside the pass band and zero gain outside it. A real filter has loss and finite rejection, so its transfer function is continuous rather than this binary model.

For the idealised magnitude response:

(2) $\begin{equation*} P(\omega)=\begin{cases} 1, & \text{ $\omega_{clow}$ $<$ $\omega$ $<$ $\omega_{chigh}$}.\\ 0, & \text{ for \ all \ other \ frequencies }. \end{cases} \end{equation*}$

Band Pass Filter Applications

Band pass filters select a wanted frequency range and reduce out-of-band energy in applications such as these:

Audio crossovers and equalisers route or adjust a selected band. The realised response must account for source, amplifier and loudspeaker impedance.
Optical band pass filters select wavelength ranges in LASER and lidar systems. They use optical materials rather than the electrical RC and RLC circuits discussed above.
Communication receivers use band pass filters to select a channel or intermediate-frequency band and reject out-of-band interferers.
Audio signal processing uses band pass responses for tone shaping, analysis and noise reduction.
A receiver filter can improve in-band signal-to-noise ratio when it removes out-of-band noise, but it cannot remove noise already inside the wanted band.

Band Pass Filter Design Example

This example targets nominal lower and upper cutoffs of 80Hz and 800Hz using two first-order RC sections.

F1 = 80 Hz
F2 = 800 Hz

The isolated-stage equations provide starting values for R1, C1, R2 and C2. In the unbuffered passive circuit shown, stage and load impedances shift the combined response, so the complete circuit still needs analysis or measurement.

$f_1 = \frac{1}{2 \pi R_1 C_1}$

$R_1 C_1 = \frac{1}{2 \pi f_1}$

Similarly,

$R_2 C_2 = \frac{1}{2 \pi f_2}$

We have to assume the value of resistance or capacitance. Here, we will assume the value of C1 and C2. For simple calculation, we will assume the same value for C1 and C2 and that is 10^-6 F. And calculate the value of resistance according to this value of C1, C2, and F1, F2.

$C_1 = C_2 = 10^{-6} F$

Therefore,

$R_1 = \frac{1}{2 \pi f_1 C_1}$

$R_1 = \frac{1}{ 2 \pi \times 80 \times 10^{-6} }$

$R_1 \approx 500 \Omega$

The protected R1 result above is incorrect. With 80Hz and 1µF, R1 is approximately 1.99kΩ. Apply the upper-cutoff equation in the same way for R2.

$R_2 = \frac{1}{2 \pi f_2 C_2}$

$R_2 = \frac{1}{ 2 \pi \times 800 \times 10^{-6} }$

$R_2 \approx 5000 \Omega$

The protected R2 result above is also incorrect. With 800Hz and 1µF, R2 is approximately 199Ω, not 5000Ω. Use standard component values, include tolerances and loading, then verify the actual -3 dB edges.