0200 Harmonic Oscillator

8/3/2019 0200 Harmonic Oscillator

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Introductory Electronics Notes 200-1 Copyright M H Miller: 2000The University of Michigan-Dearborn revised

Har monic Oscillators

ObjectiveThere is a metaphysical quality to a sinusoidal oscillation. Sinusoidal oscillation is thenatural part of the first-order response to a small perturbation of virtually all physicalsystems. Moreover a non-pathological signal variation can be synthesized as asuperposition of sinusoids of appropriate amplitude and frequency. This latter property inparticular enables linear circuit performance to be characterized generally in terms of itsresponses to a set of sinusoidal inputs of varying frequency, rather than having to consider

separately each possible waveform. Indeed one reasonably effective way of generating asinusoidal oscillation is first to generate a convenient periodic waveform, and then filter thiswaveform to eliminate unwanted harmonics. A ubiquitous inexpensive commercialfunction generator first generates a square-wave output, integrates this to obtain atriangular-wave output, and then filters the triangular wave (diode shaping) to obtain a verylow distortion sinusoidal output.

The specific focus for this discussion is the generation of sinusoidal oscillations using(nearly) linear feedback circuits. The underlying principle for this process is a bootstrapprocess that may be described crudely as follows. The output of a unity-gain amplifier isby definition a copy of the input signal waveform that produces the output. If this output isfedback as a replacement for the input the amplifier drives itself! The initial input necessary

to start the process, i.e., to provide the initial amplifier output, is provided by inevitablethermal noise.

There is however one or two matters to consider more fully.

IntroductionPerhaps the most familiar harmonic oscillator is the idealized series LC circuit whose transient response isexamined in an elementary 'circuits' course. The natural response of this circuit (assuming the circuit isenergized, i.e., there is somehow an initial capacitor charge or inductor current) is a harmonic oscillation.In practice, or course, there is an inevitable loss associated with the circuit elements and any initial storedenergy is eventually dissipated; the waveform of the response of a lossy LC circuit is a damped sinusoid.On the other hand if an energy source is provided to replace losses an oscillation can be sustained. Byway of an interesting introduction, parenthetic to the main content of this note, we explore an example of a

negative resistance oscillator using a LC circuit for which losses are compensated.

Negative-Resistance OscillatorAn example of a negative resistance circuit is shown in the accompanyingfigure. Assuming an idealized opamp for simplicity, and because of thisneglecting the voltage difference across the amplifier input terminals, thevoltage drops across R1 and R2 are equal. With negligible amplifier inputcurrents the circuit input volt-ampere relation is as shown in the figure.

The description as a negative resistance is a reflection of the appearance ofthe volt-ampere relation. Note that in practice the idealized opampassumption has a limited range of validity, in particular the range ofapplicability of the negative resistance relationship is limited by amplifier saturation.

The resistance RS in the following series RLC circuit represents circuit losses, primarily the inductor wireresistance. This is compensated by adding an equal negative resistance. A PSpice netlist for arepresentative design accompanies the figure.


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*Negative Resistance OscillatorX741A 3 1 6 7 2 UA741V+ 6 0 DC 12V- 7 0 DC -12R1 1 2 1KR2 3 2 1K.PARAM RVAL = 1KR3 3 0 {RVAL}LS 4 1 10M

CS 5 4 1U IC=1RS 5 0 1K.LIB EVAL.LIB.STEP PARAM RVAL LIST 1K, 970, 700.TRAN .03m 3m 0 10u UIC.PROBE.END

The negative resistance oscillator is a circuit with just two poles; the poles are negative real for overdampedoperation, and complex with negative real parts for underdamped operation. For the special case of zeronet resistance the poles become a complex imaginary pair. Computed output data for oscillation, and forthe cases of over- and under-damping is plotted in the figure following.

The computed frequency of oscillation may be compared to the expected (calculated) value obtained from

2LC = 1. Note the initial amplifier saturation that limits the increase of energy stored in the circuit.

Although examination of a negative resistance harmonic oscillator offers insights into the nature ofoscillatory behavior there is another more productive point of view associated with the concept of feedbackwhich provides the principal content of what follows.

Har monic OscillatorsThere are three essential constituents of a practical harmonic oscillator:

a) An amplifier to sustain the oscillation, transferring energy (usually from a DC source) into theoscillation to replace circuit losses and oscillation energy supplied to a load;b) Circuit components that establish the frequency of the oscillation;c) A mechanism to define the amplitude of oscillation.

If the oscillator circuit used were truly linear there would be no amplitude limiting mechanism other thanthat associated with the average energy stored in the system. In a linear system signal amplitudes can bescaled (i.e., multiplied by a constant) without affecting frequency, and will be whatever the availableenergy will sustain. In a practical oscillator however there is a continual infusion of energy into the


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oscillation by the amplifier, and signal amplitude grows until circuit nonlinearities constrain furthergrowth. Often a controlled nonlinearity is introduced deliberately for this purpose, rather than dependingon the unpredictable effect of inherent nonlinearities to limit oscillation amplitude

A productive method of studying linear oscillators is to view them as feedback amplifiers for which thesignal fedback from the output provides the total input necessary to produce the output. (This introducesthe question of how the oscillation, which drives itself, starts in the first place. Although the question isfundamental the answer is simple; inevitable thermal motion of electrons, i.e., electrical noise, provides animplicit startup signal current. The more pertinent question to consider is how the circuit must nurture this

signal appropriately to sustain oscillation.

The figure below is reproduced from earlier notes on feedback. Si is an input signal applied to an

amplifier of gain GS whose output is So. An output sample fSo is fed back and subtracted from the inputto produce the amplifier input signal Si - fSo.

For other purposes we previously assumed that the circuit designer made fGS >>1, i.e., designed for

degenerative feedback. Suppose however that a design makes 1+fGS -> 0, implying the singular gain

So/Si -> . Whereas for degenerative feedback the signal sample is subtracted from the input the physicabasis for this regenerative feedback is the addition of the output signal sample to the input, i.e., toreinforce the input signal. The increased input signal strength further increases the output signal, and so

causes still a larger reinforcing feedback signal. The condition 1+fGS -> 0 corresponds to a theoreticalfeedback signal strength sufficient by itself to produce the output level necessary to support the feedbacksignal; the operation becomes self-sustaining. However this is not a continuing sustainable condition; the

amplifier inevitably is driven into a saturated state for which the gain is essentially zero, and further signalgrowth stops. The Barkhausen Condition 1+fGS -> 0 therefore is interpreted as the condition for theonset of oscillation.

This is not really a 'bootstrap' affair; as a practical matter the oscillation starts because of signals generatedby random electron movement (i.e., currents) associated with thermal excitation. The circuit feedback,appropriately designed, continuously reinforces a particular frequency component to form a self-sustainingoutput. The oscillation energy is obtained by conversion from an energy source that is an integral part ofthe amplifier.

As noted before circuit nonlinearity, whether inherent or specifically introduced, eventually must limit thesignal amplitude. Nevertheless a linear analysis is meaningfully applicable until the signal amplitude islarge enough for nonlinearity to be significant, and so may be applied usefully to determining theconditions for the onset of oscillation. And, provided the nonlinear limiting is not too severe, a generalcontinuity in nature suggests the results of the linear analysis can be 'close' in some useful sense to theactual circuit performance.

Barkhausen ConditionsThe unity loop gain condition for the onset of oscillations (i.e., output = input) involves two distinctrequirements: the amplitude of the net loop gain must be 1, and the phase of the loop gain must be 0 (or amultiple of 360). These two independent requirements together form the Barkhausen conditions for the


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onset of oscillations. For a particular linear circuit to support an oscillation the roots of the circuitdeterminant must, by definition, have conjugate complex poles on the imaginary axis. Hence to make anoscillator we must start with a circuit whose determinant involves two poles, and specify circuit parameterso that these poles are placed properly on the imaginary axis. Unfortunately a circuit with just two poles isnot sufficient. The root locus for a two-pole system including loss simply does not cross the imaginaryaxis whatever the circuit element values used. At least one more singularity, either a pole or a zero, mustbe present as a minimal requirement. Oscillators meeting the minimal condition are studied first. Then weexamine illustrations of more involved sets of singularities which provide certain added benefits.

A linear system may be scaled in frequency without changing the relative amplitudes of circuit voltages orcurrents. Frequency is involved only as a factor in a product with either a circuit inductance orcapacitance, and only the product affects the voltage and current amplitudes. Hence the condition of unityloop gain magnitude can be maintained while frequency is scaled arbitrarily; simply scale inductance andcapacitance by the inverse of the factor the frequency is scaled. It follows then that it is the phase of theloop gain, and only the phase , which can determine the frequency of oscillation; the oscillation frequencymust be such that there is no net phase shift around the loop. Unity loop gain magnitude is necessary toinitiate the oscillation and to replace energy losses, but this requirement is not involved in the determinationof the oscillation frequency.

The 'stability' of the frequency of oscillation is quite often a matter of special concern. 'Stability' hererefers to the extent to which a variation of the oscillator circuit elements from design values causes thefrequency of oscillation to change, perhaps because of an environmental temperature change or because of

component tolerances. One measure of this stability is rate of change of the phase of the loop-gain withfrequency. Since the oscillation frequency is determined entirely by the phase shift requirement the largerthis rate of phase change the smaller the frequency change required to compensate for a given circuitphase-shift variation.

'T uned Cir cuit ' OscillatorThe parallel tuned circuit illustrated to the right isoften used in one form or another to satisfy the phaseshift condition for a sinusoidal oscillator; Grepresents inevitable inductor (primarily) andcapacitor losses rather than an intended circuitelement. An expression for the admittance Y lookinginto the circuit is shown to the right of the tunedcircuit.

An illustrative parallel tuned circuit oscillator isdrawn to the right. The amplifier on the left providesan adjustable output voltage, of which a fraction1/(1+RY) is passed on to the voltage follower andreturned to drive the amplifier; Y is the admittance ofthe parallel combination of G, L, and C. Frequencyselectivity is provided by the frequency-dependentvoltage divider.

Assume for simplicity that the amplifier itself does not contribute any phase shift, i.e., it simply provides a

voltage gain A. Then the Barkhausen requirements are A/(1+RY) = 1. Note that Y contributes two polesand a zero to the expression, a minimal requirement for oscillation to be achievable. The Barkhausen

conditions are satisfied at (radian) frequencyo (defined above), and the (theoretical) gain requirement issimply A 1+RG.


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The rate of change of phase of the tuned circuit evaluated at its resonant frequency is:

Values for the tuned circuit Q vary from perhaps 10 or so when discrete components are used, to severalthousand for a crystal equivalent circuit. If, as is not uncommon at higher frequencies, the amplifierintroduces some phase shift then the resonant circuit would have to provide a compensating change inphase shift so as to maintain the net phase shift zero. The higher the Q the less the change in oscillationfrequency needed for such phase corrections.

Tuned Cir cuit O scillator ComputationA representative tuned-circuit oscillator circuit is drawn below. The Q of the tuned circuit as calculatedfrom the circuit element values is 17.3. The gain amplifier has a nominal voltage gain of 3.5; this isgreater than the theoretical threshold requirement that the gain be greater than 2 (the inverse of theattenuation of the frequency-sensitive voltage divider at resonance). Note the inclusion of an initial voltageacross the capacitor to provide an equivalent for a thermal energy starting perturbation.

*TunedCircuitOscillatorL1 1 0 10m

RL1 1 0 10K

C1 1 0 .03U IC= 0.1R1 3 1 10KX1 1 2 4 5 3 UA741V+ 4 0 DC 12V- 5 0 DC -12R20 2 0 1KR32 3 2 2.5K.TRAN 1U 3m UIC.LIB EVAL.LIB.PROBE.END

The PSpice transient computation is plotted below. Note the exponential growth of the signalcorresponding to the regenerative feedback used. The growth is limited eventually by saturation of theamplifier.


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An expanded scale view of the transient is plotted next. Compare the computed oscillation period with thetheoretically expected value. Note the greater purity of the waveform at node 1, where the frequencyselectivity discriminates against higher harmonics present in the saturated amplifier output. Note also theclipping levels and relate these to the amplifier saturation limits.

The next computation plotted is for a reduced amplifier gain of 1.5, i.e., less than the thresholdrequirement for oscillation. The amplifier is unable to compensate fully for the transfer loss across thevoltage divider, and the initial energy stored in the circuit is continuously dissipated. The consequence isthe damped oscillation shown.

Finally we plot the oscillation when the amplifier gain is increased to 101; the larger gain is chosen (moreor less arbitrarily) so as to cause the amplifier to contribute some phase shift into the transfer properties ofthe circuit. (Locate the approximate dominant pole for the 741 amplifier from the manufacturersspecifications. What does this imply about the phase shift introduced by the opamp, which is assumed to

be zero in the analysis?) Since the oscillation condition for the net phase shift around the loop remains thesame the tuned circuit must operate off resonance to provide a compensatory phase adjustment. As aconsequence the frequency of oscillation changes a small but noticeable amount.


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Pha se-Shift OscillatorThe tuned circuit oscillator makes use of the minimum number (three) of singularities necessary to form anoscillator, in particular two poles and a zero are involved. Since at least two poles, i.e., a conjugatecomplex pair, are necessary in any case a three-pole system is the only other combination of just three

singularities that can be used to form an oscillator. The three-pole 'phase shift' oscillator uses three RCfilter sections to obtain adequate phase shift necessary to meet the Barkhausen requirement. An illustrativeoscillator configuration is drawn below.

The inverting amplifier provides gain (and a virtualground at node 5 to isolate the output) to compensate forsignal attenuation across the ladder filter, and 180 phaseshift to compensate for an anticipated 180 phase shiftacross the RC filter. Connecting the amplifier output todrive the phase-shift filter closes the feedback loop.Provided adequate gain is introduced the circuit shouldoscillate at a frequency for which the phase shift acrossthe ladder is 180.

To determine the Barkhausen oscillation conditioncalculate the voltage transfer ratio vo/vi across the phase-shift network; separate real and imaginary parts of theexpression to obtain the gain and frequency conditionsindicated by the arrows.

A netlist for representative circuit element values follows:*Phase Shift OscillatorC12 1 2 47N IC=5C23 2 3 47NC34 3 4 47NR20 2 0 1KR30 3 0 1KR45 4 5 1KR51 5 1 33KX741 0 5 7 8 1 UA741

V+ 7 0 DC 12V- 8 0 DC -12

.LIB EVAL.LIB

.TRAN .1u 15m UIC

.OP

.PROBE

.END


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Rapid signal growth into steady-state oscillation is shown below. The initial energization of the circuit, thecomputational equivalent of noise energization, is again provided here by an initial capacitor charge.

The oscillation amplitude at node 4 is shown on an expanded scale in the following figure to illustrate theamplitude-limiting effect caused by amplifier saturation. When comparing amplitudes keep in mind that

v(4) is shown magnified by a factor of 33, about equal to the amplifier gain. As the amplifier just enterssaturation, i.e., when the output amplitude is about 12 volts, the gain decreases. Hence the rate ofchange of v(1) changes, and as a consequence a small capacitative current spike is fed into node 1. Thusalthough v(1) is largely clamped by amplifier saturation a reaction to that current spike shows up in v(4).

Wien Bridge OscillatorAlthough a minimum of three transfer-function singularities is required for a feedback oscillator more thanthe minimum can and often is used. A principal reason for this is that the additional singularities affect theroot locus of the singularities of the circuit determinant, providing means for controlling the sensitivity of

the circuit phase-shift with frequency, and so the stability of the circuit oscillation.

A number of frequency-selective networks used in oscillators take the form of a bridge network. Theparticular RC network drawn below is called a Wien Bridge, and the voltage transfer ratio Vout/Vininvolves four singularities, two zeros and two poles; the figure also shows the loop gain expression to theright.


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With the proper setting of the resistance (potentiometer) divider ratio k the parameter Q* -> (theoretically), with the consequence of moving the zeros of the transfer expression to the imaginary axis

at jo. The root locus of the poles of the expression converges on these zeros, adding an inexpensivestability to the oscillation frequency by use of precision resistors and capacitors. This stabilization is

indicated by the same stability measure used before. Thus at resonance ( = o)

and theoretically approaches infinity for the special case Q* ->

. As a consequence a variation of phase-shift because of, say, environmental factors or active component tolerances is compensated by the WienBridge with a comparatively small frequency change.

Wien Bridge Oscillator Computa tionA representative Wien Bridge oscillator circuit design is drawn below, accompanied by the correspondingPSpice netlist. In practice (as is done in the illustration) one usually makes R1=R2 and C1=C2. Then

oRC= 1, and the value of k for the optimum stability(i.e., Q*->) is 1/3 independent of the component values.Note that an initial capacitor voltage is used to provide asmall initial circuit energization to start the oscillationcomputation. In addition a diode limiting circuit is used to

control the oscillation amplitude as described below.The nonlinear mechanism limiting oscillation amplitude inthe test oscillator circuit is saturation of the amplifier whenthe signal amplitude is large enough. This produces adistorted waveform as the amplifier output. Lessdistortion may be obtained by adding a controllednonlinearity, 'soft' limiting the signal amplitude to asmaller value. The nonlinearity here is provided by theparalleled diodes; when the oscillation amplitude becomeslarge enough the diodes conduct at the peaks of the cyclemodifying the feedback (gain) of the amplifier. The 10 kpotentiometer can be adjusted so that the oscillation gain requirement will not be met at the very peaks of

the cycle. The oscillation stops briefly at the waveform peaks, limiting signal growth but with limitedwaveform distortion. The signal amplitude then decays a small amount to a point where the diodes arecutoff and the oscillation is restarted.


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*Wien Bridge Oscillator

CA1 1 0 0.03U IC = 10URA1 1 0 4.7KCA2 2 1 0.03URA2 2 3 4.7KRB1 3 4 75KRB2 4 0 25K

V+ 6 0 DC 12V- 7 0 DC -12X741 1 4 6 7 3 UA741

* For the transfer function computation* reassign the opamp output from node 3 to* node 8, and activate R8 and VAC below.* Deactivate R45 and diode amplitude limiter.* R8 8 0 4.7K* VAC 3 0 AC 1* .AC DEC 50 100 10K

.TRAN 10u 8m 0 10u UIC

* Include the limiter for the transient computationR454 5 10KD1 5 3 D1N4004D2 3 5 D1N4004

.LIB EVAL.LIB

.OP

.PROBE

.END

A Bode plot of the transfer function of the bridge (amplitude limiter removed) is shown below. Netlistchanges for the computation are noted in the netlist shown. Note the phase shift changes roughly 16 perhertz around the calculated oscillation frequency (1 KHz).


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Quadrature OscillatorAnalysis of the preceding oscillators has been performed in the frequency domain, i.e., in terms of transferfunction poles and zeros. Another straightforward way of obtaining a sinusoidal oscillation is byassembling a circuit whose operation in the time domain mimics a second-order differential equation withconstant coefficients; the natural solution of such an equation is sinusoidal. Actually it is more practical tointegrate the second-order differential equation twice to obtain a more convenient second-order integralequation; assembling electronic integrators generally is easier than assembling differentiators. The circuitdiagram below illustrates one such an oscillator configuration.

The first stage to the left is an inverting Miller integrator (with a nonlinear diode amplitude-limitingnetwork added across the feedback capacitor). The other stage is a non-inverting integrator. It can beshown (show it!) that the current through the capacitance in the second stage is as indicated in the figure,and the integrating property of the amplifier follows directly from this.)

The output of the second stage is fedback to provide the input of the first, leading to a circuit with theintegral equation operating description desired. Note that because the output of one stage is the integral ofthe output of the other stage both sine and cosine signals are available; hence the name 'quadrature'oscillator.

Illustration of this type of oscillator is provided in a (guided) problem at the end of the notes.


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PROBLEMS

(1) The circuit diagram for a tuned-circuitoscillator circuit used as an illustration before isreproduced to the right. Diode limiting (1N4148) isadded. Recompute the oscillation, relating the signalamplitude to the effect of the diodes. Recall that afiltered waveform is available at node 1.

Shift the diodes to shunt the RLC elements andrepeat the problem.

(2) The dual of the phase shift oscillatorillustrated before interchanges the R and C elementsof the phase shift network (see figure ). Determinethe Barkhausen oscillation conditions. (Note: Theanalysis is revised only in that where before R wasused now 1/sC is used, and vice versa. Hence theresult of the new analysis may be obtained simply bymaking the indicated substitutions directly in theoriginal equations. Using the same element values asbefore compute the oscillator performance andcompare with theoretical expectations.

(3) The Colpitts circuit (drawn below) is a well-known three-pole oscillator configuration. A netlistfor a representative design also is shown. Assume an idealized opamp and C1 = C2); verify that thetransfer function is

Note that because node 1 is a virtual ground R41 effectively shunts C2. (To derive the transfer expression

consider assuming the voltage drop across R41 is 1 volt, and use a ladder analysis.)

The last term in the parentheses on the right in thedenominator ordinarily can be neglected (withcapacitance values in microfarads, resistances inkilohms, and inductance in millihenries the order of

magnitude of the term will be roughly 10-3 1+ (R/R41). Comparethis performance to the theoretical expectation.


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(4) The Hartley oscillator circuit is the dual of theColpitts oscillator; the roles of the inductor and thecapacitors are interchanged. Assign the same circuitelement values as for the Colpitts oscillator.Determine the (approximate) frequency of oscillation.Note: actually it is only necessary to replace sL by1/sC and vice versa in the Colpitts expression, i.e.,reinterpret the reactance expressions. Compute theoscillator response and compare with the theoretical

expectations.

(5) Assemble a quadrature oscillator using asnominal values C=0.03 f and R=10 k. (Omit thediode limiting for simplicity.) Compute the circuitwaveforms, and compare the calculated andcomputed frequencies of oscillation.

0200 Harmonic Oscillator

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