2nd Order Active Filters, Low-Pass/High-Pass

This page contains two calculators. The first calculator can be used to aide in the design of 2nd Order Low-Pass and High-Pass Active Filters. It uses the equations that are listed below the calculator. This maked it flexible when specifying the requirements.

The second calculator, while similar to the first calculator, is based on Tables and uses a Scale Factor to calculate the components. While the same filter types can be designed, the Gain is limited to the available table entries, which is usually Gains of 2 and 10 for 2nd order filters.

The drawings on the right show the Ideal and Realizable filter responses for 2nd Order Low-Pass and High-Pass Active Filters.

The frequency ω_C is the cutoff frequency. This is the point at which the amplitude equals 1/√ 2  = 0.707 times its maximum value.

The Low-Pass filter, left side, has a single passband, 0 < ω < ω_C, and a single stopband, ω > ω_C. So signals below ω_C are passed without attenuation and signals above ω_C are attenuated.

The High-Pass filter, right side, has a single passband, ω > ω_C, and a single stopband, 0 < ω < ω_C. So signals above ω_C are passed without attenuation and signals below ω_C are attenuated.

The flat response in the passband is a feature of a Butterworth response. A Chebishev response would have Ripple in the passband. The calculators provide selectable passband ripple of 0.1 dB, 0.5 dB, and 1.0 dB.

There are two configurations, Multiple Feed Back (MFB) and Voltage Controlled Voltage Source (VCVS). Each configuration allows for Butterworth and Chebishev response characteristics.

Both MFB and VCVS 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 is relatively easy to adjust the characteristics. The MFB configuration provides "inverting" gain, whereas the VCVS configuration provides "non-inverting" gain.

2nd Order Low-Pass/High-Pass Filter Calculater - Equation Method

A Low-Pass Filter allow signals, below a specified Cutoff Frequency (F_C), to pass with minimum attenuation. And then blocks signals above the Cutoff Frequency. A High-Pass Filter allow signals, above a specified Cutoff Frequency (F_C), to pass with minimum attenuation. And then blocks signals below the Cutoff Frequency.

The area on the left of the calculator is for defining your input requirements. Initially, you would want to select the Filter Type, the Configuration, and the Characteristics. Then enter your required Gain and Cutoff Frequency.

For the capacitors, the program offers suggestions. Underneath the suggestions you can enter the capacitor values you would like to use. The suggestion for C or C_A/C_B can vary widely. However, the suggestion for C1 should not be exceeded. Exceeding the suggested value will cause a "divide by zero" error, which will generate bogus values for the resistors.

At the bottom of the entry area there are selectors for defining the Resistor Tolerance (Res. Tol.) and whether the program will use Standard Value Capacitors (SVC). You can choose 1%, 2%, 5% or 10% tolerance resistors. You can even choose exact values, if you want to see exactley what the equations come up with. If you leave the selector for SVC set to "YES", your capacitor selections will be changed upon entry, to a standard value.

2nd Order Low-Pass/High-Pass Filter Calculater - Equations

2nd Order Filter Coefficient Values
	Butt.	Cheb. 0.1db	Cheb. 0.5db	Cheb. 1.0db
a	1.41421	2.37209	1.42562	1.09773
b	1.00000	3.31329	1.51620	1.10251

The drawing on the left shows the equations used for calculating component values for MFB and VCVS 2nd Order Low-Pass filters. The drawing on the right shows the equations used for calculating component values for MFB and VCVS 2nd Order High-Pass filters.

The coefficients "a" and "b" are taken from a table, similar to the one that is below the High-Pass filter equations. The table is based on the type of filter response you require. For a Butterworth response, the "b" coefficient is equal to "1.0000" and can be dropped from the equations. But if your looking for a Chebishev response, the "b" coefficient is required to calculate the component values. The "db" entry in the table indicates the amound of ripple in the filter's passband. Note that this table is only for 2nd Order filters. Higher order filters will have a expanded coefficient table.

2nd Order Low-Pass/High-Pass Filter Calculater - Table Method

Another method for determining filter components is the Table method. It is intended to be a quick method, as long as your needs are simple. While you could certainly make a set of tables to handle any Gain figure, published tables for 2nd Order filters are usually only available for Gains of 2 and 10. Initially a Scale Factor (SF) is calculated using SF = 500/(PI×F₀×C) where F₀ is the filter cut-off frequency and C is the initial capacitance, in uF

On the left side of the calculator is a selection panel. Start by selecting the Filter Type (Low-Pass or High-Pass), Configuration (MFB or VCVS), and Response Characteristics (Butterworth or Chebishev). Then, enter the Cutoff Frequency (F₀), select the Gain (2 or 10), and enter the initial Capacitance (C). At the bottom of the selection panel, the Resistor Tolerance (Exact, 1%, 2%, 5%, or 10%) and whether to use a Standard Value Capacitor (SVC) can be selected. The Exact setting will show the exact calculated values for the componenets.

On the right side of the calculator is the schematic and a table. The schematic will match the Filter Type (Low-Pass or High-Pass) and Configuration (MFB or VCVS) selected, and the component values will be listed. The default resistor tolerance is 5% but that can be changed at any time. The table below the schematic will show the two entries for each component. The first entry is the calculation used for that component. The second entry is the result of the calculation, adjusted according to the resistor tolerance selected.

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.

For example, the MFB Low-Pass filter show the "+" input of the Op-Amp, and the input capacitor "C" connected to ground. For use with a Single Voltage Power Supply, both of those connections need to be disconnected from ground 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.