# Special functions of LPF and HPF

Tutorial on electronic circuits
2020-11-20 00:07:26
# Special Functions of LPF and HPF

The low pass and high pass filter circuits are used as special circuits in many applications. The low pass filter (LPF) can function as an ** integrator **, while the high pass filter (HPF) can function as a ** differentiator **. These two mathematical functions are only possible with these circuits which reduce the efforts of an electronics engineer in many applications.

## Low pass filter as integrator

At low frequencies the capacitive reactance tends to become infinite and at high frequencies the reactance becomes zero. Therefore, at low frequencies the LPF has a finite output, and at high frequencies the output is zero, which is the same for an integrator circuit. We can therefore say that the low-pass filter works like an ** integrator **.

For the LPF to behave like an integrator

$$tau gg T $$

Where $ tau = RC $ the time constant of the circuit

Then the voltage variation at C is very small.

$$ V_ {i} = iR + frac {1} {C} int i: dt $$

$$ V_ {i} cong iR $$

$$ Since:: frac {1} {C} int i: dt ll iR $$

$$ i = frac {V_ {i}} {R} $$

$$ Since:: V_ {0} = frac {1} {C} int i dt = frac {1} {RC} int V_ {i} dt = frac {1} {tau} int V_ { i} dt $$

$$ Output propto int input $ $

Therefore, an LPF with a high time constant produces an output proportional to the integral of one input.

### Frequency response

The frequency response of a low pass filter, when it operates as an integrator is as shown below.

### Output waveform

If the integrator circuit receives a sine wave input, the output will be a cosine wave. If the input is a wavesquare, the output waveform changes shape and appears as in the figure below.

## High pass filter as differentiator

At low frequencies, the output of a differentiator is zero while at high frequencies, its output is of a certain finite value . This is the same as for a differentiator. Therefore, the high pass filter is said to behave like a differentiator.

If the time constant of the RC HPF is much lower than the time period of the input signal, then the circuit behaves like a differentiator. Then the voltage drop on R is very small compared to the drop on C.

$$ V_ {i} = frac {1} {C} int i: dt + iR $$

But $ iR = V_ {0} $ is small

$$ since V_ {i} = frac {1} {C} int i: dt $$

$$ i = frac {V_ {0}} {R} $$

$$ Since: V_ {i} = frac {1} {tau} int V_ {0}: dt $$

Where $ tau = RC $ the time constant of the circuit.

Differentiation on both sides,

$$ frac {dV_ {i}} {dt} = frac {V_0} {tau} $$

$$ V_ {0} = tau frac {dV_ {i}} {dt} $$

$$ Since: V_ {0} propto frac {dV_ {i}} {dt} $$

The output is proportional to the differential of the input signal.

### Frequency response

The frequency response of a practical high pass filter, when it works as a differentiator is as shown below.

### Output waveform

If the differentiation circuit receives a sine input, the output will be a cosine wave. If the input is a square wave, the shape of the output wave changes shape and appears as in the figure below -below.

These two circuits are mainly used in