1、WAVE-FORM GENERATORS 1 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 cond
2、itions required for oscillation to take place, the frequency and amplitude stability are also studied. Fig. 1-1 shows an amplifier, a feedback network, and an input mixing circuit not yet connected to form a closed loop. The amplifier provides an output signal X0 as a consequence of the signal Xi ap
3、plied directly to the amplifier input terminal. The output of the feedback network is Xf =FX0=AFXi, and the output of the mixing circuit (which is now simply an inverter) is Xf -Xf =-AFXi From Fig. 1-1 the loop gain is Loop gain=Xf/Xi=-Xf/Xi=-FA Suppose it should happen that matters are adjusted in
4、such a way that the signal Xf is identically equal to the externally applied input signal Xi. Since the amplifier has no means of distinguishing the source of the input signal applied to it at would appear that, if the external source were removed and if terminal 2 were connected to terminal 1, the
5、amplifier would continue to provide the same output signal Xo as before. Note, of course, that the statement Xf =Xi means that the instantaneous values of Xf and Xi are exactly equal at all times. The condition Xf =Xi is equivalent to AF=1, or the loop gain, must equal unity. Fig- 1-1 An amplifier w
6、ith transfer gain A and feedback network F not yet connected to form a closed loop. 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, t
7、he only periodic waveform which will preserve, its form is the sinusoid. For a sinusoidal waveform the condition Xi = Xf is equivalent to the condition that the amplitude, phase, and frequency of Xi and Xf be identical. Since the phase shift introduced in a signal in being transmitted through a reac
8、tive 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 proceeds from the input terminals, through the amplifier and feedback netwo
9、rk, 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 de
10、termine 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 sim
11、ultaneous 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 at all. Another condition which must clearly be met is that the magnitude of Xi and Xf must be iden
12、tical. This condition is then embodied in the following 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)
13、are less than unity. The condition of unity loop gain -AF = 1 is called the Barkhausen criterion. This condition implies, of course, both that |AF| =1 and that the phase of -A is zero. The above principles are consistent with the feedback formula Af=A/(1+FA). For if FA=1, then Af , which may be inte
14、rpreted 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 |FA| at the oscillator frequency is precisely unity t then, with the feedback signal connected to the input terminals,
15、the removal of the external generator will make no difference* If I FA I is less than unity, the removal of the external generator will result in a cessation of oscillations. But now suppose that |FA| is greater than unity. Then, for example, a 1-V signal appearing initially at the input terminals w
16、ill, 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 j then, that if |FA| is larger than unity, the amplitude of the oscillations will continue to increa
17、se 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 ons
18、et of nonlinearity to limit the amplitude of oscillation is an essential feature of the operation of all practical oscillators, as the following considerations will show: The condition |FA|=1 does not give a range of acceptable values of |FA| , but rather a single and precise value. Now suppose that
19、 initially it were even possible to satisfy this condition. Then, because circuit components and, more importantly, transistors change characteristics (drift) with age, temperature, voltage, etc., it is clear that if the entire oscillator is left to itself, in a very short time |FA| will become eith
20、er less or larger than unity. In the former case the oscillation simply stops, and in the latter case we are back to the point of requiring nonlinearity to limit the amplitude. An oscillator in which the loop gain is exactly unity is an abstraction completely unrealizable in practice. It is accordin
21、gly necessary, in the adjustment of a practical oscillator, always to arrange to have |FA| somewhat larger (say 5 percent) than unity in order to ensure that, with incidental variations in transistor and circuit parameters , |FA| shall not fall below unity. While the first two principles stated abov
22、e must be satisfied on purely theoretical grounds, we may add a third general principle dictated by practical considerations, i.e.: Fig. 1-2 Root locus of the three-pole transfer functions in the s plane. The poles without feedback (FA0 = 0) are s1, s2, and s3, whereas the poles after feedback is ad
23、ded are s1f, s2f, and s3f. In every practical oscillator the loop gain is slightly larger than unity, and the amplitude of the oscillations is limited by the onset of nonlinearity. 2 Op-amp Oscillators Op-amps can be used to generate sine wave, triangular-wave, and square wave signals. Well start by
24、 discussing the theory behind designing op-amp oscillators. Then well examine methods to stabilize oscillator circuits using thermistors, diodes, and small incandescent lamps. Finally, our discussion will round off with designing bi-stable op-amp switching circuits. 11.2.1 Sine-wave oscillator In Fi
25、g.2-1, an op-amp can be made to oscillate by feeding a portion of the output back to the input via a frequency-selective network and controlling the overall voltage gain. For optimum sine-wave generation, the frequency-selective network must feed back an overall phase shift of zero degrees while the
26、 gain network provides unity amplification at the desired oscillation frequency. The frequency network often has a negative gain, which must be compensated for by additional amplification in the gain network, so that the total gain is unity. If the overall gain is less than unity, the circuit will n
27、ot oscillate； if the overall gain is greater than unity, the output waveform will be distorted. Fig- 2-1 Stable sine-wave oscillation requires a zero phase shift between the input and output and an orerall gain of 1. As Fig. 2-2 shows, a Wien-bridge network is a practical way of implementing a sine-
28、wave oscillator. The frequency-selective Wien-bridge is coostructed from the R1-C1 and R2-C2 networks. Normally, the Wien bridge is symmetrical, so that C1=C2=C and R1 =R2=R. When that condition is met, the phase relationship between the output and input signals varies from-90to 90, and is precisely
29、 0 at a center frequency, f0, which can be calculated using this formula: f0=1/(2CR) Fig. 2-2 Basic wein-bridge sine-wave oscillator. The Wien network is connected between the op-amps output and the non-inverting input, so that the I circuit gives zero overall phase shift at f0, where the voltage ga
30、in is 0.33； therefore, the op-amp must be given a voltage gain of 3 via feedback network R3-R4, which gives an overall gain of unity. That satisfies the basic requirements for sine-wave oscillation. In practice, however, the ratio of R3 to R4 must be carefully adjusted to give the overall voltage ga
31、in of precisely unity, which is necessary for a low-distortion sine wave. Op-amps are sensitive to temperature variations, supply-voltage fluctuations, and other conditions that carse the op-amps output voltage to vary. Those voltage fluctuations across components R3-R4 will also use the voltage gai
32、n to vary. The feedback network can be modified to give automatic gain adjustment (to increase amplifier stability) by replacing the passive R3-R4 gain-determining network with a gain-stabilizing circuit. Figs. 2-3 through 2-7 show practical versions of Wien-bridge oscillators having automatic amplitude stabilization.
