**First Order Low Pass Butterworth Filter:**

The first order low pass butterworth filter is realized by R-C circuit used along with 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 for First Order Low Pass Butterworth Filter Circuit:**

The impedance of the capacitor C is – j X_{C} where X_{C} 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,

i.e.

where

and

and

The V_{o}/V_{in }is the transfer function of the filter and can be expressed in the polar form as,

where

and

The phase angle Φ is in degrees.

The equation (7) describes the behavior of the low pass filter.

1. At very low frequencies, f < f_{H}

2. f = f_{H},

3. At f > f_{H}

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 A_{F} i.e. 3 dB down from A_{F}. 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 between 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.

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.