Wave Shaping
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- Limiter & Voltage Multiplier
- Clamper Circuits
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- Nonlinear Wave Shapping
- Special Functions of LPF and HPF
- Linear Wave Shapping
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- Electronic Circuits - Filters
- Full Wave Rectifiers
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Special Functions of LPF and HPF
Low-pass and high-pass filter circuits are used as special circuits in many apppcations. Low-pass filter (LPF) can work as an Integrator, whereas the high-pass filter (HPF) can work as a Differentiator. These two mathematical functions are possible only with these circuits which reduce the efforts of an electronics engineer in many apppcations.
Low Pass Filter as Integrator
At low frequencies, the capacitive reactance tends to become infinite and at high frequencies the reactance becomes zero. Hence at low frequencies, the LPF has finite output and at high frequencies the output is nil, which is same for an integrator circuit. Hence low pass filter can be said to be worked as an integrator.
For the LPF to behave as an integrator
$$ au gg T$$
Where $ au = RC$ the time constant of the circuit
Then the voltage variation in C is very small.
![Low Pass Filter as Integrator](/electronic_circuits/images/lpf_integrator.jpg)
$$V_{i}=iR+frac{1}{C} int i :dt$$
$$V_{i}cong iR$$
$$Since :: frac{1}{C} int i :dt ll iR$$
$$i=frac{V_{i}}{R}$$
$$ Since :: V_{0}=frac{1}{C}int i dt =frac{1}{RC}int V_{i}dt=frac{1}{ au }int V_{i} dt$$
$$Output propto int input$$
Hence a LPF with large time constant produces an output that is proportional to the integral of an input.
Frequency Response
The Frequency response of a practical low pass filter, when it works as an Integrator is as shown below.
![LPF Frequency Response Working as Integrator](/electronic_circuits/images/lpf_frequency_response_working_as_integrator.jpg)
Output Waveform
If the integrator circuit is given a sinewave input, the output will be a cosine wave. If the input is a square wave, the output wave form changes its shape and appears as in the figure below.
![Output Waveform Of LPF integrator](/electronic_circuits/images/output_waveform_of_lpf_integrator.jpg)
High Pass Filter as Differentiator
At low frequencies, the output of a differentiator is zero whereas at high frequencies, its output is of some finite value. This is same as for a differentiator. Hence the high pass filter is said to be behaved as a differentiator.
If time constant of the RC HPF is very much smaller than time period of the input signal, then circuit behaves as a differentiator. Then the voltage drop across R is very small when compared to the drop across C.
![HPF Differentiator](/electronic_circuits/images/hpf_differentiator.jpg)
$$V_{i}=frac{1}{C}int i :dt +iR$$
But $iR=V_{0}$ is small
$$since V_{i}=frac{1}{C}int i :dt$$
$$i=frac{V_{0}}{R}$$
$$Since : V_{i} =frac{1}{ au }int V_{0} :dt$$
Where $ au =RC$ the time constant of the circuit.
Differentiating on both sides,
$$frac{dV_{i}}{dt}=frac{V_0}{ au }$$
$$V_{0}= au frac{dV_{i}}{dt}$$
$$Since :V_{0}propto frac{dV_{i}}{dt}$$
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.
![HPF Frequency Response Differentiator](/electronic_circuits/images/hpf_frequency_response_differentiator.jpg)
Output Wave form
If the differentiator circuit is given a sinewave input, the output will be a cosine wave. If the input is a square wave, the output wave form changes its shape and appears as in the figure below.
![Output Wave form HPF Differentiator](/electronic_circuits/images/output_waveform_of_hpf_differentiator.jpg)
These two circuits are mostly used in various electronic apppcations. A differentiator circuit produces a constant output voltage when the input appped tends to change steadily. An integrator circuit produces a steadily changing output voltage when the input voltage appped is constant.
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