## First Order Low Pass Butterworth Filter:

The first order low pass butterworth filter is realised by R-C circuit used alongwith an op-amp, used in the noninverting configuration. The circuit diagram is shown in Fig. 2.74. This also called one pole low pass butterworth filter.

The resistances R_{f} and R_{1} decide the gain of the filter in the pass band.

### Analysis of the Filter Circuit

The impedance of the capacitor C is – j Xc where Xc is the capacitive reactance given by

By the potential divider rule, the voltage at the noninverting input terminal A which is the voltage across capacitor C is given by,

As the op-amp is in the noninverting configuration,

is the transfer flinction of the filter and can be expressed in the polar V in –

form as,

The phase angle Φ is in degrees. The equation (7) describes the behaviour of the low pass filter.

Thus, for the range of frequencies, 0 < f < f_{H}, the gain is almost constant equal to f_{H} which is high cut off frequency. At f = f_{H}, gain reduces to 0.707 AF i.e. 3 dB down from AF. And as the frequency increases than f_{H}, the gain decreases at a rate of 20dB/decade. The rate 20 dB/decade means decrease of 20 dB in gain per 10 times change in frequency. The same rate can be expressed as 6 dB/octave i.e. decrease of 6 dB per two times change in the frequency. The frequency f_{H} is called cut off frequency, break frequency, — 3dB frequency or corner frequency. The frequency response is shown in the Fig. 2.75.

The rate of decrease in gain is 20 dB/decade i.e. the decrease can be indicated by a negative slope in the frequency response, as — 20 dB/decade.

### Design Steps

The design steps for the first order low pass Butterworth filter are

1) Choose the cut off frequency, f_{H}.

2) Choose the capacitance C usually betwen 0.001 and 1 μF. Generally, it is selected as 1 μF or less than that. For better performance, mylar or tantalum capacitors are selected.

3) Now, for the RC circuit,

Hence, as f_{H} and C are known, calculate the value of R.

4) The resistances R_{f} and R_{1} can be selected depending on the required gain in the pass band.

### Frequency Scaling

Once the filter is designed, sometimes, it is necessary to change the value of cut-off frequency f_{H}. The method used to change the original cut-off frequency f_{H} to a new cut-off frequency f_{H1} is called as frequency scaling.

To achieve such a frequency scaling, the standard value capacitor C is selected first. The required cut-off frequency can be achieved by calculating corresponding value of resistance R. But to achieve frequency scaling a potentiometer is used as shown in Fig. 2.75. Thus, the resistance R is generally a potentiometer with which required cut-off frequency f_{H} can be adjusted and changed later on if required.