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1、WAVE-FORM GENERATORS1 The Basic Priciple of Sinusoidal OscillatorsMany 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 condit
2、ions 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 appli
3、ed 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) isXf-Xf =-AFXiFrom Fig. 1-1 the loop gain is Loop gain=Xf/Xi=-Xf/Xi=-FA Suppose it should happen that matters are adjusted in such a
4、 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 amplif
5、ier 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 with tra
6、nsfer 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, the only
7、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 reactive net
8、work 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 network, and b
9、ack 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 th
10、e 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
11、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 identical. This
12、 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) are less tha
13、n 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 interpreted to me
14、an 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, the removal of
15、 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 will, after a t
16、rip 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 increase without lim
17、it, 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 nonlinea
18、rity 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 initially it
19、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 either less or lar
20、ger 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 accordingly necessary,
21、 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 above must be sati
22、sfied 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 added are s1f, s
23、2f, 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 OscillatorsOp-amps can be used to generate sine wave, triangular-wave, and square wave signals. Well start by discussing the
24、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 oscillatorIn Fig.2-1, an op-amp
25、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 gain network prov
26、ides 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 not oscillate; if t
27、he 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-wave oscillator. Th
28、e 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 0 at a center freq
29、uency, 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 gain is 0.33; therefor
30、e, 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 gain of precisely unit
31、y, 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 gain to vary. The feedba
32、ck 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. F
33、ig. 2-3 Thermistor-stabilized 1kHz Wein-bridge oscillator.Fig. 2-4 Lamp-stabilized Wien-bridge oscillator.Fig. 2-5 Diode-regulated Wien-bndge oscillator.Fig. 2-6 Zener-regulated Wien-bridge oscillator.Fig. 2-7 Three decade 15 Hz15 kHz Wien-bridge oscillator. 2. 2 Thermistor stabilizationFig. 2-3 sho
34、ws a 1-kHz fixed-frequency oscillator. The output amplitude is stabilized by a Negative Temperature Coefficient (NTC) thermistor Rt which, together with R3 forms a gain-determining feedback network. The thermistor is heated by the mean power output of the op-amp The desired feedback thermistor resis
35、tance value is triple that of R3, so the feed-back gain is X3. When the feedback gain is multiplied by the frequency networks gain of 0.33, the overall gain becomes unity. If the oscillator output amplitude starts to rise, RT heats up and reduces its resistance, thereby automatically reducing the ga
36、in of the circuit, which stabilizes the amplitude of the output signal.An alternative method of thermistor stabilization is shown in Fig. 2-4, In that case, a low-current lamp is used as a Positive Temperature Coefficient (PTC) thermistor, and is placed in the lower part of the gain-determining feed
37、back network. If the output amplitude increases, the lamp heats up thereby increasing its resistance, reducing the feedback gain, and providing automatic amplitude stabilization. That circuit also shows how the Wien network can be modified by using a twin-ganged potentiometer to make a variable-freq
38、uency oscillator over the range 150 Hz to1.5 kHz. The sine-wave output amplitude can be made variable using R5.A slightly annoying feature of thermistor-stabilized circuits is that, in variable-frequency applications, the output amplitude of the sine wave tends to jitter or bounce as the frequency c
39、ontrol potentiometer is swept up and down its range. 2.3 Diode stabilizationThe jitter problem of variable-frequency circuits can be minimized by using the circuits of Figs. 2-5 or 2-6 which rely on the onset of diode or Zener conduction for automatic gain control. In essence, R3 is for a circuit ga
40、in slightly greater than unity when the output is close to zero, causing the circuit to oscillate; as each half-cycle nears the desired peak value, one of the diodes starts to conduct, which reduces the circuit gain, automatically stabilizing the peak amplitude of the output signal. That limiting” t
41、echnique typically results in the generation of 1% to 2% distortion on the sine-wave output. The maximum peak-to-peak output of each circuit is roughly double the breakdown voltage of its diode regulator element.In Fig- 2-5, the diodes start to conduct at 500 mV, so the circuit gives an output of ab
42、out 1-volt peak-to-peak. In Fig, 2-6, the Zener diodes D1 and D2 are connected back-to-back, and may have values as high as 5 to 6 volts, giving a p-p (peak-to-peak) output of about 12 volts. Each circuit is set up by adjusting R3 for the maximum value (minimum distortion) at which oscillation can b
43、e maintained across the frequency band.The frequency range of Wein-bridge oscillators can be altered by changing the C1 and C2 values; increasing C1 and C2 by a decade reduces the output frequency by a decade. Fig. 2-7 shows the circuit of a variable-frequency Wien oscillator that covers the range 1
44、5 Hz to 15 kHz in three switched-decade ranges. The circuit uses Zener-diode amplitude regulation, and its output is adjustable by both switched and fully-variable attenuators. Notice that the maximum useful operating frequency is restricted by the slew-rate limitations of the op-amp. The limit is a
45、bout 25 kHz using a LM741 op-amp, or about 70 kHz using a CA3140. 2. 4 Tvuin-T oscillatorsAnother way of designing a sine-wave oscillator is to wire a twin-T network between the output and input of an inverting op-amp, as shown in Fig, 2-8. The twin-T network comprises R1-R2-R3-R4 and C1-C2-C3. In a
46、 balanced circuit, those components are in the ratios R1=R2=2(R3R4), and C1=C2=C3/2. When the network is perfectly balanced, it acts as a notch filter that gives zero output at a center frequency (f0), a finite output at all other frequencies, and the phase of the output is 180 inverted. When the ne
47、twork is slightly unbalanced by adjusting R4, the network will give a minimal output at f0.Fig. 2-8 1kHz twin-T oscillator.By critically adjusting R4 to slightly unbalance the network, the twin-T gives a 180 inverted phase shift with a small-signal f0. Because the inverting op-amp also causes a 180
48、input-to-output phase shift, there is zero overall phase inversion as seen at the inverting op-amp input, and the circuit oscillates at a center frequency of 1 kHz, In practice R4 is adjusted so that oscillation is barely sustained, and under that condition the sine wave has less than 1% distortion.
49、Fig. 2-9 shows an alternative method of amplitude control, which results in slightly less distortion. Here, DY provides a feedback signal via potentiometer R5. That diode reduces the circuit gain when its forward voltage exceeds 500 mV. To set up the circuit, first set R5 for maximum resistance to ground, then adjust R4, so that oscillation is just sustained. Under those conditions, the o