Wien bridge oscillator-:

Normally, frequency determining network provides 180^{0} phase shift and the amplitude determining network provides the another 180^{0} phase shift, totaling a net 360 degree phase shift; so that the positive feedback takes place as in RC phase shift oscillator.

But,alternatively oscillation could also take place when the amplitude as well as frequency determining network both introduces zero phase shifts. This is the principle of wien bridge oscillator.

The above figure shows a schematic diagram of wien-bridge oscillator.

The above figure shows the frequency determining network for the above wien bridge oscillator circuit.

Now, let us proceed to derive the required relation for wien-bridge oscillator circuit as follows-:

For series:

Z_{1}=R+jX

For parallel;

Z_{2=}RjX/(R+jX)

Then we have, the relation of V_{f} as-:

V_{f}=V_{out}* Z_{2}/total sum

= V_{out*} {RjX/R+jX)}/{(R+jX)+ RjX/(R+jX))

Keep solving this; then you will reach to a point where you will get,

V_{f}= V_{out}*RjX/((R+jX)²+RjX)

= V_{out}*-RX/(R²-X²+3RjX)

For the circuit to oscillate, we have a condition is-:

R²-X²=0

That is R=X

So you can get that

R=1/(2*pi*f*C);

Which is the required expression in case of wien bridge oscillator.

Now, let us discuss about the 4:1 Multiplexer

S_{1} |
S_{0} |
Y |

0 | 0 | I_{0} |

0 | 1 | I_{1} |

1 | 0 | I_{2} |

1 | 1 | I_{3} |

The above figure shows the required final expression for 4:1 Multiplexer.

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