Filter design and circuit are mainly divided into four categories

1 Second-order Voltage-Controlled Low-Pass Filter

The second-order voltage-controlled low-pass filter circuit is shown in the figure. R1, C1, R2, and C2 form two first-order low-pass filters, respectively. However, C1 is connected to the output terminal to introduce positive voltage feedback, which helps create a voltage-controlled filter.

(1) Transfer Function

(2) Frequency Characteristics

It can be seen that the low-pass filter is characterized by the damping coefficient ζ, which is determined by the ratio of resistors R1, R2, C1, and C2. The natural frequency ω₀ is related to the specific values of these components, meaning that ω₀ and ζ are independently adjustable and do not interfere with each other.

(3) Parameter Selection

To simplify parameter matching, it's common to choose C1 = C2 = C because the nominal capacitance values are small. The natural frequency ω₀ and damping coefficient ζ can then be adjusted by selecting different R1 and R2 values.

2 Unit Gain Second-Order Voltage-Controlled Low-Pass Filter

When the passband amplification factor Aup = 1 (unit gain), the circuit becomes as shown in the figure, where RF = R1 + R2.

(1) Basic Relationship

(4) Parameter Selection

If the natural frequency ω₀ and damping coefficient ζ are known, the design steps are as follows:

3 Second-Order Low-Pass Filter

(1) Transfer Function

(2) Frequency Characteristics

4 Infinite Gain Multiplicative Feedback Low-Pass Filter

(1) Transfer Function

(2) Frequency Characteristics

High Pass Filter

The general form of the second-order high-pass filter transfer function is shown below.

1 Second-Order High-Pass Voltage-Controlled Filter

(1) Transfer Function

2- Pass Voltage-Controlled High-Pass Filter with 2 Gains Aup

When the passband amplification factor Aup = 1, the op amp is connected as a voltage follower, as shown.

(1) Transfer Function

(2) Frequency Characteristics

3 Infinite Gain Multi-Feedback High-Pass Filter

This circuit can operate on dual power supplies or single power supplies, making it widely used.

(1) Transfer Function

(2) Design Steps

First, based on the cutoff frequency, initially identify a value for C3. C1 is determined according to the passband gain Aup (Note: capacitors C1 and C3 take certain nominal values).

Second, when C1 and C3 are fixed, and given the natural frequency ω₀ and damping coefficient, the resistance values of R1 and R2 can be determined using the following analytical expressions.

3 Second-Order High-Pass Filter

(1) Transfer Function

(2) Frequency Characteristics

Bandpass Filter

A bandpass filter (BPF) allows signals within a specific frequency range to pass through while attenuating signals outside that range. Its amplitude-frequency characteristics are shown in the figure.

1 Second Order Bandpass Filter

The simple second-order bandpass filter circuit is shown in the figure, where R1 and C1 form a low-pass filter, and C2 and R3 form a high-pass filter.

(1) Transfer Function

(2) Frequency Characteristics

(3) Design Steps

The following is a concrete example explaining the design process of this type of bandpass filter circuit.

This circuit requires few components, operates on dual power supplies, or can work on a single supply (with a bias potential of 1/2Vcc). It is widely used in single-supply systems. However, the quality factor Q cannot be too high, and this configuration is typically used for bandpass filters with large bandwidths (i.e., small Q values).

2 High Q Value Second-Order Bandpass Filter

The high-Q second-order bandpass filter circuit is shown in the figure. It can operate with dual power supplies or single-supply operation and is easy to use in single-supply systems.

Since the Q value can be made larger, it is particularly suitable for point frequency filtering.

(1) Transfer Function

(2) Frequency Characteristics

(3) Design Steps

The following is a concrete example explaining the design process of this type of bandpass filter circuit.

3 High Q BPF Composed of Dual Op Amps

A high-Q BPF (DABPF) circuit composed of dual op amps is shown. Since the circuit uses few components, a very high Q value can be achieved when the passband gain Aup is fixed at 2, making it a commonly used BPF circuit.

(1) Transfer Function

(2) Frequency Characteristics

Compared to the standard second-order BPF transfer function, the following parameters can be obtained:

4 Second-Order Voltage-Controlled Bandpass Filter

The second-order voltage-controlled bandpass filter is shown in the figure, where R1 and C1 form a low-pass filter, R2 and C2 form a high-pass filter, and positive voltage feedback through R3 creates a voltage-controlled bandpass filter.

(1) Transfer Function

In order to stabilize the system, the coefficient of the first term in the denominator must be > 0, i.e., 3 − Aup > 0, meaning Aup < 3.

(2) Amplitude-Frequency Characteristics

(3) Design Steps

In high-pass and bandpass filters, the op amp output does not need to be zero in static conditions, so single-supply operation can be selected. In low-pass and band-stop filter circuits, a DC-to-AC amplifier usually requires dual-supply operation.

Band Stop Filter

The band rejection filter (BEF) has exactly the opposite characteristics of the bandpass filter. Signals between f p1 and f p2 are blocked, and it is mainly used to suppress signals in a specific frequency band.

It can also be obtained by subtracting the BPF output signal u0 from the input signal ui, as shown in (b). The BEF filter obtained by subtraction has the same characteristics as the BPF filter, meaning the central angular frequency ω₀ and the quality factor Q are the same as the BPF filter. This makes it easy to design, but it requires an additional op amp for signal subtraction.

For example, a BEF circuit formed by a simple second-order inverting input BPF filter by subtraction is shown in the figure. According to the BPF filter characteristics, when ω = ω₀, the BPF filter output signal

When generating BEF by subtraction, a simple second-order BPF is generally used instead of a high-Q BPF, as the circuit requires more matching parameters and debugging is difficult.

The BEF utility circuit using summation operation mostly uses a double T-type network active filter, as shown in the figure.

(1) Transfer Function

(2) Frequency Characteristics

State Variable Filter

The state variable filter comes in two types: inverting input and non-inverting input.

1 Inverting Input State Variable Filter

(1) Basic Relationship

(2) Transfer Function

2 Non-Inverting Input State Variable Filter

(1) Basic Relationship

(2) Transfer Function

It can be seen that the outputs of the non-inverting input state variable filters, uo1, uo2, and uo3, are the outputs of the second-order high-pass, band-pass, and low-pass filters, respectively. Their natural frequencies ω₀ and Q are the same, only the passband amplification factors AupH, AupB, and AupL differ.

(3) Parameter Selection

Band-Stop Filter Formed by a 3-State Variable Filter

By summing the low-pass filter and the high-pass filtered output of the low-pass filter, a band-reject filter can be created. As long as the state of the tunable filter and the high-pass filtered output of the low-pass filter are fed into the summing amplifier (or noninverting and inverting), a band-stop filter can be obtained, as shown.

Since the passbands of the low-pass and high-pass filters in the state variable filter are the same, R8 = R9 is taken in the BEF filter.

These filters are integrated and come in many varieties. Some can also be programmed by controlling internal electronic switches and selecting different capacitors.