# How to design Active Filters- Low Pass Filter & High Pass Filter

There are two broad classes of electronic filter, passive filters and active filters. A passive filter is designed using only passive electronic components(resistors, capacitors and inductors). An active filter is designed using operational amplifiers(op-amps), capacitors and resistors. That is in active filters no inductors are needed. Such electronics filters are used for electrical noise suppression filters, as EMI suppression filters, in communication systems to stop or allow signals of certain frequency ranges and there are many more applications.

The active filters are preferred because they have advantages over passive filters. Some of the advantages of active filters are:

- bulky inductors are not required and filters can be implemented using only resistors and capacitors which saves cost and reduces losses

- active filters provides gain and hence higher SNR(Signal to Noise Ratio)

- active filters provides high input impedance and low output impedance

- Q-factor is comparatively higher in active filters than passive filters

Here we illustrate 1st, 2nd and 3rd order active filters using operational amplifier. Filter design procedure, online active filter calculator and design examples are illustrated.

### Active Low Pass Filter

The frequency response graph(Gain or Attenuation vs Frequency) of a Low Pass Filter is shown below.

### First Order Active Low Pass filter Design

An active first order Low Pass Filter(LPF) circuit diagram is shown below.

The design procedure for active LPF design is as follows,

1.Choose desired cutoff frequency

$f_{c}= \frac{1}{2 \pi R C }$

2. Select value of capacitor C (usually between 0.001uF and0.1uF)

3. Calculate value of R,

$R = \frac{1}{2 \pi f_{c} C}$

4. Select R1 and R2 to set the gain in the passband,

$A_{p}=1+\frac{R_{1}}{R_{2}}$

We can use the following online 1st order LPF calculator

#### 1st order active LPF Calculator

Inputs:

Theoretical Results:

Example
Let us consider design of first order LPF with cutoff frequency of 15.9KHz and gain of 11. Let use C1=0.001uF and R3=10KOhm then we get using the online calculator R1=10KOhm and R2=100KOhm.

### Second Order Active Low Pass Filter Design

The circuit diagram of second order LPF using Sallen-Key filter topology is shown below,

The second order active LPF circuit is obtained by adding another RC network(R2 and C2 in above circuit diagram) to the first order active LPF circuit. With the addition of another RC network the rolloff is increased to -40dB/decade which means higher sharp roll-off at the cutoff frequency.

The design procedure for 2nd order active LPF design is as follows,

1.Choose desired cutoff frequency

$f_{c}= \frac{1}{2 \pi \sqrt(R_{A} C_{A} R_{B} C_{B})}$

2. For simplicity, let RA=RB=R and CA=CB=C. Then select value of capacitor C (usually between 0.001uF and0.1uF)

3. Calculate value of R(=RA=RB),

$R = \frac{1}{2 \pi f_{c} C}$

4. Choose gain in the passband (Ap) and calculate R2 by selecting R2 and using the following

$A_{p}=1+\frac{R_{1}}{R_{2}}$

Note: To obtain Butterworth filter response, the gain Ap has to be equal to 1.586.  With this constraint we obtain the condition for R1 and R2 as R1=0.586*R2.

We can use the following online 2st order LPF calculator

#### 2nd order active LPF Calculator

Inputs:

Theoretical Results:
Example:
Let the cutoff frequency be 1KHz and the passband gain 1.586. Let C=CA=CB=0.005uF then we get R=RA=RB=31.8KOhm.Let R2=33KOhm then R1=19.3KOhm.

### Third Order Active Low Pass Filter Design

The circuit diagram of third order low pass filter is shown below,

The third order active LPF circuit is obtained by cascading a first order LPF(RA2 and CA2 with op-amp in above circuit diagram) with the second order LPF circuit(RA1,CA1,RB1,CB1 with op-amp in the figure above). The rolloff of 3rd order active filter is -60dB/decade which means higher sharp roll-off at the cutoff frequency comparatively to first or second order filter, hence less transition band between the cutoff frequency and stop band frequency.

The design procedure for 3rd order active LPF design is as follows,

1.Assume for simplicity, RA1=RB1=RA2=R and CA1=CB1=CA2=C.

Choose desired cutoff frequency. The equation for cutoff frequency is given by which is same for both the 2nd order and 1st order filter.

$f_{c}= \frac{1}{2 \pi R C}$

2. Select a value for capacitor C(=CA1=CB1=CA2)  which is usually between 0.001uF and0.1uF.

3. Calculate value of R(=RA1=RB1=RA2),

$R = \frac{1}{2 \pi f_{c} C}$

4. Assume R2=R4, R1=R3 for simplicity. Choose gain in the passband(Ap). Then select R2 to obtain the value of R1 using the passband gain equation,

$A_{p}=1+\frac{R_{1}}{R_{2}}$

Solving for R1,

$R_{1}= R_{2}(1+A_{p})$

Note: To obtain Butterworth filter response for second order filter, the gain Ap of the first stage has to be equal to 2. With this constraint we obtain the condition for R1 and R2 as R1=R2. For the second stage the gain has also to be 1. Therefore R3=R4.

We can use the following online 3rd order LPF calculator.

#### 3rd order active LPF Calculator

Inputs:

Theoretical Results:
Example:
Let the cutoff frequency be 1KHz and the passband gain 1.586. Let C=C1=C2=0.01uF then we get R=RA1=RB1=RA2=15.9KOhm. Let R2=R4=10KOhm then R1=R3=10KOhm.

### Active High Pass Filter

The frequency response graph(Gain or Attenuation vs Frequency) of a High Pass Filter is shown below.

### First Order Active High Pass filter Design

An active first order High Pass Filter(HPF) circuit diagram is shown below.

The design procedure for active HPF design is as follows,

1.Choose desired cutoff frequency

$f_{c}= \frac{1}{2 \pi R C }$

2. Select value of capacitor C (usually between 0.001uF and0.1uF)

3. Calculate value of R,

$R = \frac{1}{2 \pi f_{c} C}$

4. Select R1 and R2 to set the gain in the passband,

$A_{p}=1+\frac{R_{1}}{R_{2}}$

We can use the following online 1st order LPF calculator.

#### 1st order active HPF Calculator

Inputs:

Theoretical Results:

Example
Let us consider design of first order HPF with cutoff frequency of 15.9KHz and gain of 11. Let use C1=0.001uF and R2=10KOhm then we get using the online calculator R=10KOhm and R1=100KOhm.

### Second Order Active High Pass Filter Design

The circuit diagram of second order HPF using Sallen-Key filter topology is shown below,

The second order active HPF circuit is obtained by adding another RC network(RA and CA in above circuit diagram) to the first order active LPF circuit(with RB and RC). With the addition of another RC network the rolloff is increased to -40dB/decade which means higher sharp roll-off at the cutoff frequency.

The design procedure for 2nd order active HPF design is as follows,

1.Choose desired cutoff frequency

$f_{c}= \frac{1}{2 \pi \sqrt(R_{A} C_{A} R_{B} C_{B})}$

2. For simplicity, let RA=RB=R and CA=CB=C. Then select value of capacitor C (usually between 0.001uF and0.1uF)

3. Calculate value of R(=RA=RB),

$R = \frac{1}{2 \pi f_{c} C}$

4. Choose gain in the passband (Ap) and calculate R2 by selecting R2 and using the following

$A_{p}=1+\frac{R_{1}}{R_{2}}$

Note: To obtain Butterworth filter response, the gain Ap has to be equal to 1.586.  With this constraint we obtain the condition for R1 and R2 as R1=0.586*R2.

We can use the following online 2st order LPF calculator.

#### 2nd order active HPF Calculator

Inputs:

Theoretical Results:
Example:
Let the cutoff frequency be 1KHz and the passband gain 1.586. Let C=CA=CB=0.005uF then we get R=RA=RB=31.8KOhm.Let R2=33KOhm then R1=19.3KOhm.

### Third Order Active High Pass Filter Design

The circuit diagram of third order high pass filter is shown below,

The third order active HPF circuit is obtained by cascading a first order HPF(RA2 and CA2 with op-amp in above circuit diagram) with the second order HPF circuit(RA1,CA1,RB1,CB1 with op-amp in the figure above). The rolloff of 3rd order active filter is -60dB/decade which means higher sharp roll-off at the cutoff frequency comparatively to first or second order filter, hence less transition band between the cutoff frequency and stop band frequency.

The design procedure for 3rd order active HPF design is as follows,

1.Assume for simplicity, RA1=RB1=RA2=R and CA1=CB1=CA2=C.

Choose desired cutoff frequency. The equation for cutoff frequency is given by which is same for both the 2nd order and 1st order filter.

$f_{c}= \frac{1}{2 \pi R C}$

2. Select a value for capacitor C(=CA1=CB1=CA2)  which is usually between 0.001uF and0.1uF.

3. Calculate value of R(=RA1=RB1=RA2),

$R = \frac{1}{2 \pi f_{c} C}$

4. Assume R2=R4, R1=R3 for simplicity. Choose gain in the passband(Ap). Then select R2 to obtain the value of R1 using the passband gain equation,

$A_{p}=1+\frac{R_{1}}{R_{2}}$

Solving for R1,

$R_{1}= R_{2}(1+A_{p})$

Note: To obtain Butterworth filter response for second order filter, the gain Ap of the first stage has to be equal to 2. With this constraint we obtain the condition for R1 and R2 as R1=R2. For the second stage the gain has also to be 1. Therefore R3=R4.

We can use the following online 3rd order HPF calculator.

#### 3rd order active HPF Calculator

Inputs:

Theoretical Results:
Example:
Let the cutoff frequency be 1KHz and the passband gain 1.586. Let C=C1=C2=0.01uF then we get R=RA1=RB1=RA2=15.9KOhm. Let R2=R4=10KOhm then R1=R3=10KOhm.

Below are links to more tutorials on Filters.