Introduction to Oscillators

8/13/2019 Introduction to Oscillators

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1

OscillatorsOscillator: An oscillator is a circuit that generates a repetitive waveform of fixed amplitude and

frequency without any external input signal [1, p. 279].

Function of Oscillator: The function of oscillator is to generate alternating current or voltage

wave forms such as sinusoidal, square wave, triangular wave, sawtooth wave,etc.

Used of Oscillator: Oscillators are used in radio, TV, computers, CRT, Oscillocope, and

communications.

Types of Oscillator: There are two main types of electronic oscillator:

(i) Harmonic oscillator and

(ii) Relaxation oscillator

Depending on the used elements of an oscillator, the types of oscillator are:

(i)RCoscillator,(ii)LCoscillator and

(iii) Crystaloscillator.

Harmonic Oscillator: The harmonic oscillator produces a sinusoidal output. The basic form of aharmonic oscillator is an electronic amplifier with the output attached to a narrow-band electronic

filter, and the output of the filter attached to the input of the amplifier. When the power supply to

the amplifier is first switched on, the amplifier's output consists only of noise. The noise travels

around the loop, being filtered and re-amplified until it increasingly resembles the desired signal.

Relaxation Oscillator: The relaxation oscillator is often used to produce a non-sinusoidal output,such as a square wave, sawtooth wave, and triangular wave. The oscillator contains a nonlinearcomponent such as a transistor that periodically discharges the energy stored in a capacitor or

inductor, causing abrupt changes in the output waveform.

Sinusoidal Oscillator: If the output signal of an oscillator circuit varies sinusodally, the circuit is

referred to as a sinusoidal oscillator [3, p. 757].

Pulse or Square-Wave Oscillator: If the output voltages of an oscillator circuit rises quickly toone voltage level and later drops quickly to another voltage level, the circuit is

generally referred to as a pulse or square-wave oscillator.

Oscillator Principles: An oscillator is a type of feedback amplifier in which part of the output is

fed to the input via a feedback circuit. If the signal fed back is of proper magnitude

and phase, the circuits produces alternating currents or voltages. To visualize therequirements of an oscillator, consider the block diagram of Figure 7-17[1].


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Fig. 7-17Block diagram of oscillator.

This diagram looks identical to that of the feedback amplifiers. However, here the input voltage is

zero (vin=0). Also, the feedback is positive because most oscillators use positivefeedback. Finally, the closed-loop gain of the amplifier is denoted by Av.

In the block diagram of Fig. 7-17,infd

vvv ; dvo vAv ; of vv

Using these relationships, the following equation is obtained:v

v

in

o

A

A

v

v

1

However, vin=0 and vo0 implies that

1vA (7-20)

expressed in polar form,oo 360or01

vA (7-21)

Equation (7-21) gives the two requirements for oscillations:

(1)The magnitude of the loop gainAvmust be at least 1, and(2)The total phase shift of the loop gainAv must be equal to 0oor 360o.

For instance, as indicated in Figure 7-17, if the amplifier causes a phase shift of 180o, the

feedback circuit must provide an additional phase shift of 180o, so that the total phase shift

around the loop is 360o.

The waveforms shown in Fig. 7-17 are sinusoidal and are used to illustrate the circuit

action. The type of waveform generated by an oscillator depends on the components in the circuit

and hence may be sinusoidal, square, or triangular. In addition, the frequency of oscillation isdetermined by the components in the feedback circuit.

The Barkhausen Criterion:

(1)Oscillations will not be sustained if, at the oscillator frequency, the magnitude of theproduct of the transfer gain of the amplifier (Av) and the magnitude of the feedback

factor () of the feedback network (the magnitude of the loop gain) are less than unity.(2)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). Stated more simply, the frequency of a sinusoidal

oscillator is determined by the condition that the loop-gain phase shift is zero.


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The condition of unity loop gain Av=1 is called the Barkhausen criterion. This

condition implies, of course, both that Av=1 and that the phase ofAvis zero.

What happens to the output voltage?If AB is less than 1, ABvin is less than vin and the output signal will die out. However AB is

greater than 1,ABvinis greater than vinand the output signal build up.

Where Does the Starting Voltage Come From?Every conductive wire or resistor contains free electrons. Because of ambient temperature, these

free electrons move randomly in different directions and generate a noise voltage

over 1000 GHz. So the conductive wire or resistor acts as a small ac voltage source

producing all frequencies.When the power is turned on, the only signals in the system are noise voltages generated by the

conductive wire or resistor. These nose voltages are amplified and appear at the

output terminals. The amplified nose, which contains all frequencies, drives the

resonant feedback circuit. According to the design of an oscillator, the loop gain is

greater 1 and the loop phase shift is equal to 0oor 360o at the resonant frequency.Above or bellow the resonant frequency the phase shift is different from 0

oor 360

o.

As a result oscillation will build up only at the resonant frequency of the feedbackcircuit.

Phase-Shift Oscillator: An oscillator circuit that follows the basic development of afeedback circuit is thephase-shift oscillator.

Figure 7-18(and Fig. 14-29) shows a phase shift oscillator, which consists of an op-

amp as the amplifying stage and three RC cascaded networks as the feedback circuit. Thefeedback circuit provides feedback voltage from the output back to the input of the amplifier. Theop-amp is used in the inverting mode; therefore, any signal that appears at the inverting terminal

is shifted 180oat the output.

An additional 180o phase shift required for oscillation is provided by the cascaded RC

networks. Thus the total phase shift around the loop is 360o(or 0

o). At some specific frequency

when the phase shift of the cascaded RC networks is exactly 180oand the gain of the amplifier is

sufficiently large, the circuit will oscillate at that frequency. This frequency is called thefrequency of oscillation,fo, and is given by

)()(2

1

132112133221322121 CCCCRRCCCCCCRRCCRRfo

RCRCfo

065.0

62

1

(7-22a)

At this frequency, the gainAvmust be at least 29. That is,

291

R

RA Fv

Or1

29RRF (7-22b)


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Thus the circuit will produce a sinusoidal waveform of frequency foif the gain is

29 and the total phase shift around the circuit is exactly 360o.

Fig. 7-18Phase-shift oscillator.

For a desired frequency of oscillation, choose a capacitor C, and then calculate the valueof R from equation (7-22a). A desired output, however, can be obtained with back-to-back zeners

connected at the output terminal.

7-13 WIEN BRIDGE OSCILLATOR [R. A. Gayakward]Because of it simplicity and stability, one of the most commonly used audio-frequency

oscillators is the Wien Bridge. Figure 7-19 shows the Wien Bridge Oscillator in which theWien Bridge circuit is connected between the amplifier input terminals and the output terminal.

The bridge has a seriesRCnetwork in one arm and a parallelRCnetwork in the adjoining

arm. In the remaining two arms of the bridge, resistorsR1andRFare connected (see Figure 7-19).

The phase angle criterion for oscillation is that the total phase shift around the circuit

must be 0o. This condition occurs only when the bridge is balanced, that is, at resonance. The

frequency of the oscillationfois exactly the resonant frequency of the balanced Wien bridge andis given by

RCRCfo

159.0

2

1

(7-23a)

assuming that the resistors are equal in value, and the capacitors are equal in value in the reactive

leg of the Wien bridge. At this frequency the gin required for sustained oscillation is given by


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31

BAv

That is, 11

2;31 RRR

RF

F (7-23b)

Figure 7-19Wien bridge oscillator

For the derivation of Equations (7-23a) and (7-23b), refer to Appendix C. The Wienbridge oscillator is designed using Equations (7-23a) and (7-23b), as illustrated in Example 7-13.

Example 7-13: Design the Wien bridge oscillator of Figure 7-19 so thatfo=965Hz.

Solution: Let C= 0.05 F. Therefore, from Equation (7-23a),

K3.3105965

159.0159.08Cf

Ro

Now letR1=12K. Then, from Equation (7-23b),

K24K122FR

7-14 Quadrature Oscillator [R. A. Gayakward]As its name implies, the quadrature oscillartor generates two signals (sine and cosine) that

are in quadrature, that is, out of phase by 90o. Although the actual location of the sine and cosine

is arbitrary, in the quadrature oscillator of Figure 7-20the output ofA1is labeled a sine and theoutput ofA2is a cosine.


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Figure 7-19Quadrature Oscillator.A1andA2dual op-amp: 1458/353.

This oscillator requires a dual op-amp and threeRCcombinations. The first op-ampA1isoperating in the noninverting mode and appears as a noninverting integrator. The second op-amp

A2is working as a pure integrator.

Furthermore, A2 is followed by a voltage divider consisting of R3 and C3. The divider

network forms a feedback circuit, whereasA1andA2form the amplifier stage.The total phase shift of 360

oaround thee loop required for oscillation is obtained in the

following way. The op-amp A2 is a pure integrator and inverter. Hence it contributes -270o (or

90o) of phase shift. The remaining -90

o(or 270

o) of phase shift needed are obtained at the voltage

dividerR3C3and the op-amp A1. The total phase shift of 360o, however, is obtained at only one

frequencyfo, called thefrequency of oscillation. This frequency is given by

RCRCfo

159.0

2

1

(7-24a)

whereR1C1=R2C2=R3C3=RC. At this frequency,

414.11

BAv (7-24b)

which is the second condition for oscillation.

Thus, to design a quadrature oscillator for a desired frequency fo, choose a value of C;then, from Equation (7-24a), calculate the value of R. To simplify design calculations, chooseC1=C2=C3 and R1= R2= R3. In addition, R1 may be a potentiometer in order to eliminate any

possible distortion in the output wave forms.

Example: Design the quadrature oscillator of Figure 7-20 so thatfo=159 Hz.

Solution: Let C=0.01 F. Then, from equation (7-24a),


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K10010159

159.0159.08Cf

Ro

Thus C1=C2=C3=0.01 F andR1=R2=R3=100 K. However, R1 may be a 200 K

potentiometer, which can be adjusted for undistorted output wave forms.

23.3 Twin-T Oscillator [3]

FigureTwin-T oscillator

An oscillator configuration that uses a Twin-T filter in the feedback path is shown in the

following Figure 23.12. The Twin-T filter consists of two Tee-shaped networks connected in

parallel. These Twin-T filters are also known as band-reject filters or notch filters. In the Twin-Tfilter shown in Figure 1.38, the elements connected to ground have values of 2C and R/2,

respectively. The twin-T circuits acts as a lead-lag circuit with a changing phase angle. There is a

frequencyfoat which the voltage gain drops to 0. The equation of the frequency is

RCfo

2

1

The positive feedback to the noninverting input is through a voltage divider. The negative

feedback is through the Twin-T filter.

To ensure that the oscillation frequency is close to the notch frequency fo, the voltagedivider should haveR2much larger thanR1.

The Twin-T oscillator is not popular circuity because it works well only at one frequency.

14-18 A General Form of Oscillator Circuit[J. Millman, C. C. Halkias]Many oscillator circuits fall into the general form shown in Figure 14-32a.


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Figure 14-32(a) The basic configuration for many resonant-circuit oscillators. (b) The linearequivalent circuit using an operational amplifier.

The active device may be a bipolar transistor, an operational amplifier or an FET. In the

analysis that follows we assume an active device with infinite input resistancesuch as an FET, oran operational amplifier.

Figure 14-32b shows the linear equivalent circuit of Figure 14-32a, using the amplifier

with negative gain Av and output resistance Ro. Clearly the topology of Fig. 14-32 is that ofvoltage-series feedback.

The Loop gain: The value ofABwill be obtained by considering the circuit of Fig. 14-32a to

be a feedback amplifier with output taken from terminals 2 and 3 and terminals 1 and 3. The loadimpedance ZL consists of Z2 in parallel with series combination of Z1 and Z3 that means

ZL=Z2(Z1+Z3)/(Z1+Z2+Z3). The gain without feedback is A=-AvZL/(ZL+Ro). The feedback factor isB=-Z1/(Z1+Z3). The loop gain is found to be

)()( 312321

21

ZZZZZZR

ZZAAB

o

v

(14-65)

Reactive Elements Z1, Z2, and Z3: If the impedances are pure reactances (either inductiveor capacitive), then Z1=jX1, Z2=jX2, and Z3=jX3. For an inductor X=L, and for a capacitor X=-

1/C. Then

)()( 312321

21

XXXXXXjR

XXAABo

v

(14-66)

For the loop gain to be real (zero phase shift)

0321 XXX (14-67)

and


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31

1

312

21

)( XX

XA

XXX

XXAAB vv

(14-68)

From Eq. (14-67) we see that the circuit will oscillate at the resonant frequency of the seriescombination ofX1,X2, andX3.

Using Eq. (14-67) in Eq. (14-68) yields

2

1

X

XAAB v (14-69)

SinceABmust be positive and at least unity in magnitude, then X1andX2must have the

same sign (Avis positive). In other words, they must be the same kind of reactance, either both

inductive or both capacitive. Then, from Eq. (14-67), X3=-(X1+X2) must be inductive if X1andX2are capacitive, or vice versa.

IfX1andX2are capacitors andX3is an inductor, the circuit is called a Colpits oscillator.IfX1andX2are inductors andX3is an capacitor, the circuit is called aHartley oscillator.

In this latter case, there may be mutual coupling betweenX1andX2(and the above equations willthen not apply).

Transistor versions of above types of LC oscillators are possible. As an example, atransistor Colpits Oscillator is indicated in Fig 14-33a. Qualitatively, this circuit operates in themanner described above. However, the detailed analysis of a transistor oscillator circuit is more

difficult, for two fundamental reasons.

First, the low impedance of the transistor shunts Z1in Fig. 14-32a, and hence complicatesthe expressions for the loop gain given above.

Second, if the oscillation frequency is beyond the audio range, the simple low-frequency

h-parameter model is no longer valid. Under these circumstances the more complicated high-

frequency hybrid-model of Fig. 11-5 must be used. A transistor Hartley oscillator is shown inFig. 14-33b.

Figure.(a) An Op-amp Colpits Oscillator. (b) An op-amp Hartley Oscillator.


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10

231

2

1

31

1

31

1 XXXX

X

XX

X

ZZ

ZB

BAX

XAAB v

v

2

1 1R

RAA Fv

1

1RR

BA F

From (14.67)

For Hartley: 0321 XXX ;

021

cLL jXjXjX ;

01

21

CjLjLj

;

01)(21

2 CLL ;

CLL )(1

21 ;

CLLf

o

)(2

1

21

;

CLf

eq

o

2

1 ;

21 LLL

eq

2

1

L

LB ;

1

2

1

1

L

L

R

R

BA F ; 1

1

2 RL

LRF

For Colpits: 0321 XXX ;

021

LCC jXjXjX ;

011

21

LjC

jC

j

; 021

21

Lj

CC

CC

;

0)(21

2

21 CLCCC ;21

212

CLC

CC ;

eq

LC

12 ;

21

21

CC

CCC

eq

;

eqLC

1 ;

eq

o

LCf

2

1

1

2

C

CB ;

2

1

1

1

C

C

R

R

BA F ; 1

2

1 RC

CRF


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11

Clapp Oscillator:The Clapp oscillator is a Colpitts oscillator

with an additional capacitor placed in series

with the inductor. The oscillation frequency inhertz (cycles per second) for the circuit of

Clapp oscillator, is

)111

(1

2

1

210 CCCLfo

A Clapp circuit is often preferred over a

Colpitts circuit for constructing a variablefrequency oscillator (VFO). In a Colpitts

VFO, the voltage divider contains the variable

capacitor (either C1 or C2). This causes the

feedback voltage to be variable as well,sometimes making the Colpitts circuit less

likely to achieve oscillation over a portion of

the desired frequency range. This problem isavoided in the Clapp circuit by using fixedcapacitors in the voltage divider and a variable

capacitor (C0) in series with the inductor.Figure.An Op-amp Clapp Oscillator.

Crystal Oscillators18-9 [Robert Boylestad, Louis Nashelsky]

A crystal oscillator is basically a tuned-circuit oscillator using piezoelectric crystal as aresonant tank circuit. The crystal (usually quartz) has a greater stability in holding constant at

whatever frequency the crystal is originally cut to operate. Crystal oscillators are used whenever

great stability is required, for example, in communication transmitters and receivers.

Characteristics of a Quartz CrystalA quartz crystal (one of a number of crystal types) exhibits the property that If a

piezoelectrical crystal, usually quartz, has electrodes plated on opposite faces when mechanicalstress is applied across the faces of the crystal, a difference of potential develops across opposite

faces of the crystal. This property of a crystal is called the piezoelectric effect. Similarly, a

voltage applied across one set of faces of the crystal causes mechanical distortion in the crystalshape.

When alternating voltage is applied to a crystal, mechanical vibrations are set up-these

vibrations having a natural resonant frequency dependent on the crystal. Although the crystal has

electromechanical resonance, we can represent the crystal action by an equivalent resonant circuit

as shown in Fig. 14-35.


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(a)

(b)(c)

Fig. 14-35A piezoelectric crystal. (a) symbol; (b) electrical model; (c) the reactance function(ifR=0)

The inductor L and capacitor C represent electrical equivalents of a crystal mass and

compliance while R is an electrical equivalent of the crystal structures internal friction. Theshunt capacitance CMrepresents the capacitance due to the mechanical mounting of the crystal.

Because the losses of crystal, represented

byR, are small, the equivalent crystal Q(qualityfactor) is high. Values of Qup to almost 10

6can

be achieved by using crystals.

The crystal as represented by the

equivalent electrical circuit of Fig. 14-35 canhave two resonant frequencies. One resonantcondition occurs when the reactances of series

RLC leg are equal (and opposite). For this

condition theseries-resonantimpedance is verylow (equal toR).

Figure 18.32Crystal impedance versusfrequency.

The other resonant condition occurs at a high frequency when the reactance of the series-

resonant leg equals the reactance of capacitor CM. This is a parallel resonance or anti-resonance

condition of the crystal. At this frequency the crystal offers a very high impedance to the external

circuit. The impedance versus frequency of the crystal is shown in Fig. 18-32.In order to use he crystal properly it must be connected in a circuit so that its low

impedance in the series operating mode or high impedance in the anti-resonant operating mode isselected.

Series-Resonant CircuitsTo excite a crystal for operation in the series-resonant mode it may be connected as a

series element in a feedback path. At the series-resonant frequency to the crystal its impedance is


13/19

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smallest and the amount of (positive) feedback is largest. A typical transistor circuit is shown in

Fig. 18.33. Resistors R1, R2, and RE provide a voltage-divider stabilized dc bias circuit.Capacitor CE provides ac bypass of the emitter resistor and the RFC coil provides for dc biaswhile decoupling any ac signal on the power lines from affecting the output signal. The voltage

feedback from collector to base is a maximum when the crystal impedance is minimum (in series-

resonant). The coupling capacitor CChas negligible impedance at the circuit operating frequencybut blocks any dc between collector and base.

Figure 18.33Crystal-controlled oscillator using crystal in series-feedback path.

The resulting circuit frequency of oscillation is set, then, by the series-resonant frequencyof the crystal. Changing in supply voltage, transistor device parameters, and so on, have no effect

on the circuit operating frequency which is held stabilized by the crystal. The circuit frequency

stability is set by the crystal frequency stability, which is good.

Parallel-Resonant CircuitsSince the parallel-resonant impedance of a crystal is a maximum value, it is connected in

shunt. At the parallel-resonant operating frequency a crystal appears as an inductive reactance of

largest value. Figure 18-34 shows a crystal connected as the inductor element in a modified

Colpits circuit. The basic dc bias circuit should be evident. Maximum voltage is developed acrossthe crystal at its parallel-resonant frequency. The voltage is coupled to the emitter by a capacitorvoltage divider-capacitors C1and C2.

Figure 18.34Crystal-controlled oscillator operating in parallel-resonant mode.


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A Miller crystal-controlled oscillator is shown in Fig. 18.35. A tuned LC circuit in thedrain section is adjusted near the crystal parallel-resonant frequency. The maximum gate-sourcesignal occurs at the crystal anti-resonant frequency controlling the circuit operating frequency.

Figure 18-35Miller crystal-controlled oscillator.

Crystal Oscillator

An op-amp can be used in a crystal oscillator as shown in Fig. 18.36. The crystal isconnected in the series-resonant path and operates at the crystal series-resonant frequency. Thepresent circuit has a high gain so that an output square-wave signal results as shown in the figure.

A pair of Zener diodes is shown at the output to provide output amplitude at exactly the Zener

voltage (VZ).

Figure 18.36Crystal oscillator using op-amp.

14-20 [J. Millman, C. C. Halkias]If a piezoelectrical crystal, usually quartz, has electrodes plated on opposite faces and if a

potential is applied between these electrodes, forces will be exerted on the bound charges within


15/19

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the crystal. If this device is properly mounted, deformations take place within the crystal, and an

electromechanical system is formed which will vibrate when properly excited.

The resonant frequency and the Qdepend upon the crystal dimensions, how the surfaces

are oriented with respect to its axes, and how the device is mounted.Frequency ranging from a few kilohertz (kHz) to a few megahertz (MHz), and Qs in the

range from several thousand to several hundred thousand, are commercially available.These extraordinarily high values of Qand the fact that the characteristics of quartz areextremely stable with respect to time and temperature account for the exceptional frequency

stability of oscillators incorporating crystals.

The electrical equivalent circuit of a crystal is indicated in Fig. 14-35. The inductor L,capacitor C, and resistor R are the analogs of the mass, the compliance (the reciprocal of the

spring constant), and thee viscous-damping factor of the mechanical system. CM is mountingcapacitance.

Typical values for a 90-kHz crystal are L=137 H, C=0.235 pF, and R= 15 K,

corresponding to Q=5,500. The dimensions of such a crystal are 30 by 4 by 1.5 mm. Since CM

represents the electrostatic capacitance between electrodes with the crystal as a dielectric, itsmagnitude (3.5 pF) is very much larger than C.

If we neglect the resistance R, the impedance of the crystal is a reactance jX whose

dependence upon frequency is given by

22

22

p

s

MC

jjX

(14-75)

whereLC

s

12 is the series resonant frequency (the zero impedance frequency), and

)

11

(

12

M

pCCL is the parallel resonant frequency (the infinite impedance frequency).

Since CM>>C, then sp . For the crystal whose parameters are specified above, the

parallel frequency is only three-tenth of 1 percent higher than the series frequency. For

ps , the reactance is inductive, and outside this range it is capacitive, as indicated in

Fig. 14-35.

A variety of crystal-oscillator circuit is possible. If in the basic configuration of Fig. 14-

32a a crystal is used forZ1, a tunedLCcombination forZ2, and the capacitance Cdgbetween drain

and gate forZ3, the resulting circuit is as indicated in Fig. 14-36.From the theory given in the preceding section, the crystal reactance, as well as that of the

LCnetwork, must be inductive. For the loop gain to be greater than unity, we see from Eq. (14-

69) thatX1cannot be too small. Hence the circuit will oscillate at a frequency which lies between

s and p but close to the parallel-resonance value. Since ps, the oscillator frequency isessentially determined by the crystal, and not by the rest of the circuit.

References:


16/19

16

[1] Ramakant A. Gayakward, Op-Amps and Linear Integrated Circuits (Fourth Edition),

Pearson Education, Inc., 2000. pp. 279

[2] Jacob Millman, and Christos C. Halkias, Integrated Electronics: Analog and Digital Circuits

and Systems, Tata McGrew-Hill Publishing Conpamy Ltd., 1972. pp. 486[3] Albert Paul Malvino, Electronic Principles, Tata McGraw-Hill, 1999

[4] Robert Boylestad, and Louis Nashlsky, Electronic Devices and Circuit Theory, Prentice-Hall inc., 1994.

To show:RCRC

fo

065.0

62

1

; 29

1

R

RF for Phase-shift oscillator

First consider the feedback circuit consisting of RC combinations of the phase shiftoscillator. For simplicity we use the Laplace transform. Thus, the circuit is represented in the S

domain as shown in Figure C-8. Let us determine Vf(S)/Vo(S) for the circuit.Writing Kirchhofs current law (KCL) at node V1(S), we get

)()()(321

SISISI

SC

SVSV

R

SV

SC

SVSVo

/1

)()()(

/1

)()(2111

Solving for V1(S), we have

12

)]()([)( 2

1

RCS

RCSSVSVSV o (C-8)

Writing KCL at node V2(S),

)()()(543

SISISI


17/19

17

To show:RCRC

fo159.0

2

1

; 12RRF for Wien bridge oscillator

First consider the feedback circuit of the Wien bridge oscillator of Figure 7-19. The circuit is

transformed in the S domain and redrawn in Figure C-10.Using the voltage-divider rule:

)()(

)()()(

SZSZ

SVSZSV

SP

oPf

whereSC

RSC

SCRSZ

RSC

R

SCRSZ SP

11)(;

1

1)(

Therefore, substitutingZP(S) andZS(S) values, we get

RCSRCS

SVRCSSV of

2)1(

)()()(

Or

13)(

)(

222

RCSSCR

RCS

SV

SVB

o

f (C-15)

Figure C-10Feedback circuit of the Wien bridge oscillator of Figure 7-19 represented in the Sdomain.

Next, consider the op-amp part of the Wien bridge oscillator. The circuit is redrawn in Figure C-

11.

Figure C-11Op-amp part of the Wien bridge oscillator.

The voltage gain Av of the op-amp is

1

1)(

)(

R

R

SV

SVA F

f

ov (C-16)

Finally, the requirement for oscillation is:

1BAv


18/19

18

Therefore using Equations (C-15) and (C-16), we have

113

)1(222

1

RCSSCR

RCS

R

RF

Substituting S=j in this equation and then equating the real and imaginary parts, we get thefrequency of the oscillationfoand the gain required for oscillation, as follows:

13)()1( 2

1

RCjRCjRCR

RF

22

2 1

CR (real part)

or

RCRCfo

159.0

2

1

(7-23a)

and

RCRCRRF 3)1(

1

(imaginary part)

1

1

2;31 RRR

RF

F (7-23b)

To show:RCRC

fo159.0

2

1

; Av=1/B=1.414for Quadrature oscillator

iVR

sCoV )

/11( ; oV

R

sCo

V /1

1 ;

1/1

/1o

VsCR

sCVf

To show22

22

p

s

MC

jjX

for crystal oscillator

The impedance of series branch is: )(CLs

XXjRZ

Neglecting R, )(CLs

XXjZ

The resonant occurs at: )/1(;;0 CLXXXXCLCL

;

)/1(2 LC if we denote the resonant frequency for series branch by s

then LCs /1 .

The impedance of overall circuit is:

MCL

MCL

MCL

MCL

Ms

Ms

pXXX

XXXj

jXXXj

jXXXj

jXZ

jXZZ

)(

)(

)(


19/19

19

)(

11

)(

11

11

1)

1(

2

2

2

2

MM

M

MM

M

M

M

pCCLCC

LCj

CC

CCLCC

CC

LC

j

CCL

CCL

jZ

)}]/1()/1){(/1([

)/1(1

]/)([

)/1(12

2

2

2

CCL

LC

Cj

CCCCLCC

LCLCjZ

MMMMM

p

22

221

p

s

M

pC

jZ

; )}/1()/1){(/1(2 CCL

Mp

OR

The impedance of series branch is: jssC

sCRZs 1

][

1]1[1

22

22

p

s

MM

ss

sL

Rss

sL

R

ssCsC

sCRsC

ZZ

The following steps should be followed to discuss or explain about the operation an oscillator:Step 1: Draw the circuit diagram and indicate the amplifier part and feedback network part

Step 2: Discuss or explain, what type of amplifier is used, what is the expression of amplifier

gain, and how much phase shift is occurred in the amplifier circuit.

Step 3: Discuss or explain, what type of feedback network is used, how much phase shift isrequired to obtain from the feedback network.

Step 4: Discuss or explain, what is the expression of frequency to satisfy the condition that the

angle of loop gain is 0 or 360 degree.Step 5: Discuss or explain, what is the requirement of the parameters of amplifier gain expression

to satisfy the condition that the amplitude or magnitude of loop gain is greater than or equal 1.

Date post:	04-Jun-2018
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Introduction to Oscillators

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