Tuned Oscillator:

A variety of oscillator circuits can be built using LC tuned circuits. A general form of tuned oscillator circuit is shown in figure 1. It is assumed that the active device used in the oscillator has very high input resistance such as FET, or an operational amplifier.

                                      Figure 1                                                                Figure 2

Figure 2 shows linear equivalent circuit of the figure 1 using an amplifier with the open circuit gain -Av and the output resistance R_O. It is clear from topology of the circuit that it is voltage series feedback type circuit.

The loop gain of circuit -Aβ can be obtained by considering the circuit to be a feedback amplifier with the output taken from terminals 2 and 3 and with input terminals 1 and 3. The load impedance Z_L consists of the Z₂ in parallel with series combination of Z₁ and Z₃. The gain of the amplifier without the feedback will be given as follows

The feedback circuit gain can be given by

Therefore, the loop gain can be given by

If impedances are pure reactances (inductive or capacitive), then Z₁ = jX₁, Z₂= jX₂ and Z₃= jX₃. Then

For loop gain to be real (zero phase shift around loop), X₁ + X₂ + X₃ = 0

and     

Thus, the circuit will oscillate at the resonant frequency of series combination of X₁, X₂ and X₃. Since -A β should be positive and at least unity in magnitude, then X₁ and X₂  should have the same sign (Av  is positive).Or we can say that they must be the same kind of reactance, both inductive or both capacitive.

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