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### ANALYSIS OF RC OSCILLATORS USING CASCADE CONNECTION OF LOW PASS AND HIGH PASS FILTERS.

•     Oscillators produce the desired output waveform when the voltage feedback is in phase with the in1 wave.
•     The phase shift between the voltage feedback and the input should be 0 degrees or 180 degrees.
•     If a transistor is used as the basic amplifying  element it 9introduces a phase shift of 1800 degrees.
•      A further phase shift of 1800 degrees should be introduced by the feedback network, so that the output of the feed back network is in phase with the input signal An RC network may be used as the feedback network Wein bridge oscillator and RC phase shift oscillator are the RC oscillators used.
HIGH PASS COMBINATION:

•     The basic high pass filter is shown in the figure Since the reactance of the capacitor decreases with increasing frequency the higher frequency components of the input signal appear at the output with less attenuation than the low frequency components .
•     At very high frequencies the capacitor acts almost as a short circuit and virtually all the input appears at the output.
•     At 0 frequency the capacitor has infinite reactance and hence behaves as an open circuit.
•     Any constant input voltage (D.C) is blocked and cannot reach the output.
•     Therefore C is called the blocking capacitor.
•      This basic  configuration is the most common coupling circuit to obtain d.c isolation between input and output.

SINUSOIDAL INPUT:

•     When a sinusoidal input Vin is applied the output signal Vo increases with the increasing frequency.
•     Even in the case of a transmission network where no amplification is involved and in which the output is always smaller than the input, the ratio Vo/Vin is called the amplification or gain A of the circuit.
•     For the given circuit the gain A and the angle θ by which the output leads the input are given by

LOW PASS COMBINATION:

•     The circuit shows a low pass filter. it passes the low frequencies but attenuates the high frequencies because of the reactance of the capacitor C decreases with increase in frequency.
•     At very high frequencies the capacitor acts as a short circuit and the output falls to 0.
•     This basic low pass filter represents the situation that exists in a basic signal source.
•      The terminals of the source are 0— 0’.
•      Looking back at the source the source may be replaced by a Thevinins equivalent.
•     The voltage Vi is the open circuit voltage and R is the output impedance of the source assumed purely resistive.
•      The capacitance C represents all the capacitance which appears in shunt across 0 -- 0’.
•     This capacitance may arise from the wire used to couple the terminals 0 — 0’ to a load or may arise as a result of the capacitive component of the admittance presented by the load or from stray capacitance across terminals at the signal source itself.
•     The network shown in the figure is similar to that of the high pass filter except that the output is now taken across C instead of R.
•     Hence the mathematical solution for the low pass circuit can be obtained in a similar way as for a high pass circuit.

SINUSOIDAL INPUT:

•     If the input voltage Vi is sinusoidal the magnitude ‘of the steady state gain A and the angle θ by which the output leads the input are given by