WAVE-FORM GENERATORS1.The Basic Priciple of Sinusoidal Oscillators Many different circuit configurations deliver an essentially sinusoidal output waveform even without input-signal excitation. The basic principles governing all these oscillators are investigated. In addition to determining the conditions required for oscillation to take place, the frequency and amplitude stability are also studied. Fig.1-1 show an amplifier, a feedback network, and an input mixing circuit not yet connected to form a closed loop. The amplifier provides an output signal as a consequence of the signal applied directly to the amplifier input terminal. The output of the feedback network is and the output lf the mixing circuit (which is now simply an inverter) is Form Fig.1-1 the loop gain is Loop gain=Fig.1-1 An amplifier with transfer gain A and feedback network F not yet connected to form a closed loop.Suppose it should happen that matters are adjusted in such a way that the signalis identically equal to the externally applied input signal. Since the amplifier has no means of distinguishing the source of the input signal applied to it, it would appear that, if the external source were removed and if terminal 2 were connected to terminal 1, the amplifier would continue to provide the same output signal as before. Note, of course, that the statement =means that the instantaneous values of andare exactly equal at all times. The condition=is equivalent to, or the loop gain must equal unity. The Barkhausen Criterion We assume in this discussion of oscillators that the entire circuit operates linearly and that the amplifier or feedback network or both contain reactive elements. Under such circumstances, the only periodic waveform which will preserve, its form is the sinusoid. For a sinusoidal waveform the conditionis equivalent to the condition that the amplitude, phase, and frequency ofandbe identical. Since the phase shift introduced in a signal in being transmitted through a reactive network is invariably a function of the frequency, we have the following important principle:The frequency at which a sinusoidal oscillator will operate is the frequency for which the total shift introduced, as a signal proceed from the input terminals, through the amplifier and feedback network, and back again to the input, is precisely zero(or, of course, an integral multiple of 2). Stated more simply, the frequency of a sinusoidal oscillator is determined by the condition that the loop-gain phase shift is zero.Although other principles may be formulated which may serve equally to determine the frequency, these other principles may always be shown to be identical with that stated above. It might be noted parenthetically that it is not inconceivable that the above condition might be satisfied for more than a single frequency. In such a contingency there is the possibility of simultaneous oscillations at several frequencies or an oscillation at a single one of the allowed frequencies.The condition given above determines the frequency, provided that the circuit will oscillate ta all. Another condition which must clearly be met is that the magnitude of and must be identical. This condition is then embodied in the follwing principle:Oscillations will not be sustained if, at the oscillator frequency, the magnitude of the product of the transfer gain of the amplifier and the magnitude of the feedback factor of the feedback network (the magnitude of the loop gain) are less than unity.The condition of unity loop gainis called the Barkhausen criterion. This condition implies, of course, both that and that the phase of A F is zero. The above principles are consistent with the feedback formula . For if , then, which may be interpreted to mean that there exists an output voltage even in the absence of an externally applied signal voltage.Practical Considerations Referring to Fig.1-2, it appears that if at the oscillator frequency is precisely unity, then, with the feedback signal connected to the input terminals, the removal of the external generator will make no difference. If is less than unity, the removal of the external generator will result in a cessation of oscillations. But now suppose that is greater than unity. Then, for example, a 1-V signal appearing initially at the input terminals will, after a trip around the loop and back to the input terminals, appear there with an amplitude larger than 1V. This larger voltage will then reappear as a still larger voltage, and so on. It seems, then, that if is larger than unity, the amplitude of the oscillations will continue to increase without limit. But of course, such an increase in the amplitude can continue only as long as it is not limited by the onset of nonlinearity of operation in the active devices associated with the amplifier. Such a nonlinearity becomes more marked as the amplitude of oscillation increases. This onset of nonlinearit