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`INTEGRATED ELECTRONICS:
`ANALOG AND DIGITAL CIRCUITS
`AND SYSTEMS
`
`Jacob Millman, Ph.D.
`
`Professor of Electrical Engineering
`
`Columbia University
`
`Christos C. Halkias, Ph.D.
`
`Professor of Electrical Engineering
`
`Columbia University
`
`New York St. Louis San Francisco Dusseldorf Johannesburg Kuala Lumpur
`London Mexico Montreal New Delhi Panama Rio de Janeiro Singapore
`Sydney Toronto
`
`McGRA W-HILL BOOK COMPANY
`
`1
`
`Micro Motion 1032
`
`

`

`INTEGRATED ELECTRONICS: ANALOG AND
`DIGITAL CIRCUITS AND SYSTEMS
`
`Copyright @ 1972 by McGraw-Hili, Inc. All rights
`reserved. Printed in the United States of America.
`No part of this publication may be reproduced,
`stored in a retrieval system, or transmitted, in any
`form or by any means, electronic, mechanical, photo(cid:173)
`copying, recording, or otherwise, without the prior
`written permission of the publisher.
`
`Library of Congress Catalog Card Number 79-172657
`
`07-042315-6
`
`67890 MAMM 7987654
`
`This book was set in Modern by The Maple Press.
`Company, and printed and bound by The Maple
`Press Company. The designer was Richard Paul
`Kluga; the drawings Were dane by John Card €Os,
`J & R Technical Services, Inc. The editors Were
`Michael Elia, Charles R. Wade, and Madelaine
`Eichberg. Sally Ellyson supervised production.
`
`2
`
`

`

`1·14
`
`Sec. 14·15
`
`STABILITY AND OSCILLATORS I 483
`
`and the low-frequency gain with feedback .Avl = -95.68, or 39.61 dB. The
`complete response is shown in Fig. 14-27.
`c. From the plot of part b we see that the voltage gain peaks at f. = 8 MHz.
`If we place the zero of the (3 network at f., we find
`
`1
`Cf=-~""'4pF
`27rRtf.
`
`The frequency response of AVf with RI = 5 K and Cf = 4 pF is plotted in Fig.
`14-27, from which we see that there is no peaking.
`
`14-15
`
`SINUSOIDAL OSCILLATORS
`
`Many different circuit configurations deliver an essentially sinusoidal output
`waveform even 'without input-signal excitation. The basic principles govern(cid:173)
`ing all these oscillators are investigated.
`In addition to determining the con(cid:173)
`ditions required for oscillation to take place, the frequency and amplitude
`stability are also studied.
`Figure 14-28 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 x" as a consequence of the signal Xi applied directly to the ampli(cid:173)
`(3xo = A!3Xi,
`fier input terminal. The output of the feedback network is Xf
`and the output of the mixing circuit (which is now simply an inverter) is
`
`x~ = -XI = -A!3Xi
`
`From Fig. 14-28 the loop gain is
`
`X'
`Loop gain = .J. =
`Xi
`
`Xi
`
`= -!3A
`
`(14-59)
`
`Suppose it should happen that matters are adjusted in suc.h a way that the
`signal x~ 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, it would appear that, if the external source were removed and if
`terminal 2 were connected to terminal 1, the amplifier would continue to
`
`Fig. 14-28 An amplifler with
`transfer gain A. and feed(cid:173)
`back network !3 not yet con(cid:173)
`nected to form a dosed loop.
`(Compare with Fig. 13-8.)
`
`Mbdngor
`inverting
`network
`
`vith
`
`cal(cid:173)
`oles
`
`fhe
`
`~ck
`
`3
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`

`

`484 /
`
`INTEGRATED ELECTRONICS
`
`Sec. 14-15-
`
`provide the same output signal Xo as before. Note, of course, that the state(cid:173)
`ment x; = Xi means that the instantaneous values of x; and Xi are exactly
`equal at all times. Note also that, since in the above discussion no restriction
`was made on the waveform, it need not be sinusoidal. The amplifier need
`not be linear, and the waveshape need not preserve its form as it is transmitted
`through the amplifier, provided only that the signal x; has the waveform and
`frequency of the input signal Xi. The condition x; = Xi is equivalent to
`- A,6 = 1, 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 condition Xi = x; is equivalent to the condition that
`the amplitude, phase, and frequency of Xi and x; be 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 proceeds 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 2'1I} 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
`It might be noted parenthetically that it is
`identical with that stated above.
`not inconceivable that the above condition might be satisfied for more than a
`single frequency.
`In such a contingency there is the possibility of simultane(cid:173)
`ous 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 x; must be identical. 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 feed(cid:173)
`back factor of the feedback network (the magnitude of the loop gain) are less than
`unity.
`The condition of unity loop gain - A,6 = 1 is called the Barkhausen
`criterion. This condition implies, of course, both that IA,61 = 1 and that the
`phase of - A,6 is zero. The above principles are consistent with the feedback
`formula Aj = AI(1 + ,6A). For if -,6A = 1, then Aj ~ 00, which may be
`interpreted to mean that there exists an output voltage even in the absence of
`an externally applied signal voltage.
`
`Sec.
`
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`4
`
`

`

`Sec. 14-16
`
`STABILITY AND OSCiLlATORS / 485
`
`Practical Considerations Referring to Fig. 14-8, it appears that if
`I,sAI at the oscillator frequency is precisely unity, then, with the feedback
`signal connected to the input terminals, the removal of the external generator
`If I,sA I is less than unity, the removal of the external
`.. ill make no difference.
`generator will result in a cessation of oscillations. But now suppose that I,sA I
`is greater than unity. Then, for example, a I-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 1 V. This larger voltage
`will then reappear as a still larger voltage, and so on. It seems, then, that
`if I,sAI 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 opera(cid:173)
`tion in the active devices associated ,vith the amplifier. Such a nonlinearity
`becomes more marked as the amplitude of oscillation increases. This onset
`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 I,sAI = 1 does not give a range of acceptable values of
`I,sAI, but rather a single and precise value. Now suppose that 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 I,sAI will become either 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 accordingly necessary, in the adjustment of a practical
`oscillator, always to arrange to have I,sAI somewhat larger (say 5 percent)
`than unity in order to ensure that, with incidental variations in transistor
`and circuit parameters, I,sA I shall not fall below unity. While the first two
`principles stated above must be satisfied on purely theoretical grounds, we may
`add a third general principle dictated by practical considerations, i.e.:
`In ellery practical oscillator the loop gain is slightly larger than unity, and
`the amplitude of the oscillations is limited by the onset of nonlinearity.
`The treatment of oscillators, taking into account the nonlinearity, is very
`difficult on account of the innate perverseness of nonlinearities generally.
`In many cases the extension into the range of nonlinear operation is small, and
`we simply neglect these nonlinearities altogether.
`
`14-16
`
`THE PHASE-SHIFT OSCILLATOR9
`
`We select the so-called phase-shift oscillator (Fig. 14-29) as a first example
`because it exemplifies very simply the principles set forth above. Here an
`FET amplifier of conventional design is followed by three cascaded arrange(cid:173)
`ments of a capacitor C and a resistor R, the output of the last RC combination
`
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