Introduction

A Band-Pass Filter has a single pass-band (ω1 < ω < ω2) and two stop-bands (0 < ω < ω1 and ω > ω2). The frequencies ω1 and ω2 are the cutoff frequencies (the point at which the amplitude equals 1/ 2  = 0.707 times its maximum value). The drawing on the right shows the ideal and realizable responses.

2nd Order Active Band-Pass Filter (MFB/VCVS)
Configuration
MFB
VCVS
Gain
Res. Tol.

F0: Hz BW: Hz
FL: Hz FH: Hz
F0: Hz Q:

Cxxx
C1 > xxx
SVC?
Yes No

The area on the left of the calculator is for defining your input requirements. Initially, you would want to set the Stage Gain and select the Configuration, MFB or VCVS. Like with the Low-Pass and High-Pass filters, both configurations contain a minimal number of circuit elements, have low output resistance, and are convenient for cascading with other stages. While the MFB has better stability than VCVS, the VCVS configuration is capabile of relatively high gains, and are relatively easy to adjust the characteristics. The MFB configuration provides "inverting" gain, whereas the VCVS configuration provides "non-inverting" gain.

Then select the way you want to define the Frequency Characteristics. The characteristics can be set in several ways. Using the buttons on the far left of the input area, you can set Frequency (F0) and Q, and let the calculator fill in the Bandwidth (BW), Low Frequency (FL), and High Frequency (FH). You can also just define Low Frequency (FL), and High Frequency (FH) or define Frequency (F0) and Bandwidth (BW). The calculator will take care of the other parameters.

For the most part, the capacitor values are arbitrary. However, going to far afield can cause the resistor values to be extreme. So to help in your selection, a suggestion is provided. For C,the suggestion is a Standard Value Capacitor (SVC) that is near 10/F0 (uF). For MFB filters, the value for C1 should be chosen so that C1 > C(K-Q2)/Q2. For VCVS filters, C = C1, so the input area for C1 is disabled.

The figures on the left and right show the equations used for the MFB and VCVS calculations.

When you are using standard 1%, 2%, 5% or 10% resistors, the filter parameters will be slightly different than specified. The center frequency (F0) may be slightly higher or lower than intended. And/or the Gain might be slightly higher or lower than intended. For the MFB filter, these parameters (F0 and Gain) can be fine tuned using an adjustable resistor for R1 and R2, respectively. To do this, reduce the resistor's value by about 10% and then add a variable resistor in series with that resistor. The value of the variable resistor should be about 20% of the initial resistor value. This should provide you with about a +/- 10% adjustment to the resistor value.

As an example, suppose that a calculated value for R2 is 1,378Ω, and a 1.3KΩ resistor (5%) is selected. If you then replace that resistor with a 1.2K resistor (next lower standard 5% resistance) and a 200Ω variable resistor in series. This should provide you with ample range to adjust the filter's center frequency (F0). A similar approach could be used to adjust the Gain (using R1) to a more precise value, should the need arise.

2nd Order Bi-Quad Band-Pass Filter

The circuit in the calculator is known as a Bi-Quad Filter. It is also known as a Tow-Thomas Biquad. While it needs more components than the previous filters, it is more flexible and easier to adjust. It is made up of three Op-Amps that provide High-Pass (Inverting and Non-Inverting) and Low-Pass outputs. U1 is a Integrator, that doubles as a Summing amplifier. U2 is a Inverting amplifier with a Gain of 1. U3 is an Integrator that feeds back to U1.

Using the Bi-Quad Filter calculator is very much the same as the other calculators. The information above the drawing is intended for defining your input parameters. First, select how you want to define the frequency response (F0, Q, Bandwidth, etc..). Then set the Gain (K) and values for C1, C2, R4, R5, and R6. A suggestion is provided for C1, C2 and a selector for C1, C2, R4, R5, and R6. But since the Cs and Rs are mostly arbitrary, you can select almost any value you like. Just pick values that keep the other component values in a useful range.

F0: Hz BW: Hz CA/CBxxx
Gain (K)
FL: Hz FH: Hz Res. Tol.
R4, R5, R6
F0: Hz Q: Standard Value Capactance
Yes No

It's important to note that the drawing shows a High-Pass output (Inverting at U1 and Non-Inverting at U2) and a Low-Pass output (Inverting) at U3. The position of the Integrator (R6, C2, and U3) and the Inverting Amplifier (R4, R5, and U2) can be swapped. This would then provide a Non-Inverting and Inverting Low-Pass output.

If you find it necessary to adjust the values of Gain (K), Q, and F0 more precisely, adjustable resistors can be used for R1, R2, R3, respectively. To obtain good resolution in the adjustment, reduce the resistor’s value by about 10% and add a series variable resistor, whose value is about 20% of the total value required. The order of adjustment is: first adjust F0 using R3 (this will affect Q), then adjust the Gain (K) using R1 (this will not affect F0 and Q ), and finally to adjust Q using R2 (this will not affect F0 and the Gain (K)).

A different biquad filter is shown at the left. In this circuit, the first integrator is also used as the summer. One might think that in this case only two op-amps would be required, but unfortunately a signal sign change is necessary, so a unity-gain inverting amplifier is included. The inverting amplifier and the second integrator may be in either order. A drawback is that the highpass response is not available, only the bandpass and lowpass functions.

It is instructive to analyze this circuit, which is straightforward since there is a virtual ground node at each amplifier. The feedback impedance of the first integrator is RB/(1 + jωRB·C), which can be used in the usual expressions for the gain of an inverting amplifier. Some algebra is necessary, but the appropriate biquad transfer function is obtained.

RF (R3) and C (C1 = C2) determine the center frequency F0. The value of R4, R5, and R6 is unimportant; 10k is a usual value. RB (R2) determines Q = RB/RF (R2/R3) and the bandpass gain ABP = RB/RG (R2/R1). The bandwith ΔF = f0/Q = 1/2·π·C·RB (R2) depends only on RB (R2), which can be said to set the bandwidth instead of Q, independent of frequency. RF (R3) can be varied to change the center frequency, perhaps using ganged rheostats. However, to maintain a high Q the resistances must track very accurately.

For a test circuit, I used RF = 16k and C = 0.01µF as for the previous circuit. R was 10k, and RB = RG = 33k. These values gave fo = 995 Hz, ?f = 483 Hz, Q = 2.06 and ABP = 1.

For a test circuit, I used R3 = 16k and C1 = C2 = 0.01µF as for the previous circuit. R4, R5, and R6 was 10k, and R2 = R1 = 33k. These values gave F0 = 995 Hz, ΔF = 483 Hz, Q = 2.06 and ABP = 1.

Tow-Thomas Biquad Example

W0 = 1 / SQRT(R2·R4·C1·C2)

Q = SQRT( (R22·C1) / (R3·R4·C2) )

The bandwidth is approximated by B=W0/Q (W0 = 2·π·FC), and Q is sometimes expressed as a damping constant zeta = 1/2Q. If a noninverting low-pass filter is required, the output can be taken at the output of the second operational amplifier, after the order of the second integrator and the inverter has been switched. If a noninverting bandpass filter is required, the order of the second integrator and the inverter can be switched, and the output taken at the output of the inverter's operational amplifier.

If it is necessary to adjust the values of F0, Q, and Gain very precisely, adjustable resistors can be used for R3, R2, and R1 respectively. To obtain good resolution in the adjustment it is sensible to reduce the resistor’s value by about 10% and to add a series variable-resistor whose value is about 20% of the total value required. The order of adjustment is: first adjust f0 using R 5 (which will affect Q, then adjust the gain using R 1 (which will not affect f0 and Q ), and finally to adjust Q using R 11 (which will not affect f0 and the gain).

A practical way to adjust f0 is to make use of the fact that at f = f0 there is a 90° phase shift between the output and the input voltages. The gain can be adjusted at a frequency much lower than f0 (e.g f0 / 100). The value of Q can be adjusted by setting the input frequency to f0 and using the fact that at f = f0, Vo / Vi = H0LPQ.

Power (V+/V-)

Its common practice to unclutter a schematic by showing the design (analog or digital) and power connections, on a separate sheets. On small designs, power connections are often show at the bottom of a drawing away from the main design. Further, in many cases, especially where a package may contain multiple devices, only one of the devices may show power connections. But the power connections are equally as important as the main design. The same approach is used here, with the calculators shown above having references to +V, -V, and Ground, but not containing any description of the references.

The specific connections that you use, depends on the your power supply arrangement and the device you have selected. Most garden variety Op-Amps are designed to work with Dual Power Supplies. Dual Power Supplies are simply two equal voltage power supplies, with the negative (-V) of one supply connected to the positive (+V) of the other. This effectively forms a three wire connection between the power supplies and the Op-Amp circuit. These power supplies can be +9V/-9V, +15V/-15V, +18V/-18V, or anything in between. The intent is to provide the output of the Op-Amp with an equal voltage swing above and below zero.

The diagram on the left shows how you might connect a dual (+12V/-12V) power supply. The 0.1 uF capacitors are used to decouple the Op-Amp from any noise spikes that might be on the power lines. These capacitors are connected to the power and ground pins of the Op-Amp and must be located as close to the Op-Amp as possible.

But suppose you only had a Single Voltage Power Supply. This can certainly be used, but there will be some small changes to the filter drawings and possibly some limitations. The diagram on the right shows how it can be done. First a reference voltage (V-Ref) is created with a voltage divider using R1, R2, and C3. Because there is very little current required by the Op-Amp, the resistors can be almost any value. But a good choice is shown. The capacitor C3 is used as a filter to stabilize V-Ref, and not reflect changes in the Op-Amps current draw.

The MFB filters show the "+" input of the Op-Amp connected to ground. For use with the Single Voltage Power Supply, this need to be disconnected and connected to V-Ref. This biases the Op-Amp at +V/2. But this also means that the filter input and output connections will no longer be referenced to zero voltage (Ground). To fix this issue, the input and output of the filter can be coupled with series capacitors. The input capacitor and output capacitor will serve as DC isolators for the filter. Along with the input/output impedances, the capacitors will form a low frequency High-Pass filter. With the components specified (1 uF), the the low frequency response will start rolling off at around 16 Hz. The exact value depends on the frequency you are using. If you are working a low frequencies, you might want to increase the series capacitors.