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5 Oscillators and phase locked loops

The generation of a stable sinusoidal signal is a crucial function in most RF systems.

A transmitter will amplify and suitably modulate such a signal in order to produce its

required output. In the case of a receiver system, such a signal is fed into the mixercircuits for the purposes of frequency conversion and demodulation. A circuit that gen-

erates a repetitive waveform is known as an oscillator. Such circuits usually consist of

an amplifier with positive feedback that causes any input, however small, to grow until

limited by the non-linearities of the circuit. The feedback will need to be frequency

selective in order to control the rate of waveform repetition. This frequency selec-

tion is often achieved using combinations of capacitors and inductors, but can also

be achieved with resistor and capacitor combinations. In the present chapter, how-

ever, we will concentrate on feedback circuits based on capacitor/inductor combina-

tions. We consider a variety of oscillator circuits that are suitable for RF purposes and

investigate the conditions under which oscillation occurs. In addition, we consider

the issue of oscillator noise since this can often pose a severe limitation upon system

performance.

A particularly important class of oscillator is that for which the frequency can be

controlled by a d.c. voltage. Such an oscillator is an important element in what is known

as a phase locked loop. In such a system, there is a feedback loop that compares the

oscillator output with a reference signal and generates a control voltage based upon

their phase difference. When the system settles down, the oscillator is locked onto thereference signal. Phase locked loops form a generic class of system that can been used

for purposes such as frequency control and demodulation. Here we consider the basic

principles of phase locked loops and investigate some of their applications.

5.1 Feedback

A general amplifier (H) with feedback (G) is illustrated in Figure 5.1. For this system,

the relationship between input and output voltages is given by

vo =H( j)

1 G( j)H( j)vi. (5.1)

108


11/43

109 5.2 The Colpitts oscillator

H

G

vovi

+

Figure 5.1 General feedback system.

Negative feedback (the positive sign in Equation 5.1) provides a means of tailoring

the amplifier frequency response and controlling both gain and linearity. When there is

positive feedback, however, the system can provide a means of generating oscillations.

For oscillation, we require a system that produces a signal without input (except for an

initial excitation). Consequently, for oscillations to occur at frequency 0,

|G( j0)||H( j0)| = 1 (5.2)

and

arg{G( j0)H( j0)} = 0. (5.3)

This is the Barkhausen criterion and it ensures that any small component at frequency

0 will grow until limited by the non-linearities of the system.

5.2 The Colpitts oscillator

In a Colpitts oscillator, the feedback occurs via a series inductance -network. An

example, based on a common-emitter BJT amplifier, is shown in the circuit of Figure 5.2.

By neglecting all but the current source of the BJT (r = ro = and C = C = 0),we obtain the simplified small signal model of Figure 5.3. Current balance at the BJT

collector will imply

sC2v + gmv +

1

R+ sC1

vo = 0 (5.4)

and, at the base,

sC2v =vo v

s L. (5.5)

From Equation 5.5 we obtain vo

=v (1

+s2C2L) and then, eliminating vo from Equa-

tion 5.4,

sC2 + gm +

1

R+ sC1

(1+ s2C2L) = 0. (5.6)


12/43


CBL

R

L

C1

C2

VCC

RFC

bias

Figure 5.2 Colpitts oscillator.

cb

mC2 v R

C1

v vog

Figure 5.3 Simple model of Colpitts oscillator.

Noting that s = j, and separating the real and imaginary parts of Equation 5.6, weobtain

=

C1 + C2LC1C2

andC2

C1= gmR, (5.7)

where is the frequency of oscillation, C2/C1 is the feedback ratio and gmR is the volt-age gain of the amplifier. A practical design will normally set the transistor gain slightly

higher than the feedback ratio (C2/C1 < gmR) to take account of component variations.

In this case, the oscillations will grow until the non-linearities in the device cause suffi-

cient loss of gain for the Barkhausen criterion to be satisfied. This last point is important

as it means that the steady state operation of an oscillator is essentially non-linear. Ifv

is large, the transistor will make excursions into regions where it is switched off (we usu-

ally set the bias so that this is the case). Consequently, the transistor collector current It

will consist of periodic pulses with the peaks occurring where v is maximum. The bias

current Ibias will be the average of the transistor current pulses (Ibias = (1/T)T

0It dt,

where T = 2/) and, by Fourier techniques, the fundamental component of the cur-rent Ifund is given by Ifund = (2/T)

T0

It cos(t) dt. The major contribution to Ifund will

arise around the peak ofv and, as a consequence, Ifund 2Ibias. Ifv = Vosc cos(t)


13/43


DG

S

R1

RSR2

C2L

CBL

RD

VDD

CBL

CBP

C1

Figure 5.4 FET Colpitts with bias circuits.

C2 C1

L

Figure 5.5 General Colpitts oscillator.

(feedback only occurs at the fundamental frequency) there will be an effective large sig-

nal transconductance ofGm = 2Ibias/Vosc (see Lee). This is a general relationship, butfor a BJT it reduces to Gm = (2VT/Vosc)gm and for an FET to Gm = [(VGS Vt)/Vosc]gm, where VGS is the d.c. component of the gatesource voltage. In designing

an oscillator, we will need to ensure that the Barkhaussen criterion can be satisfied

somewhere between the extremes of the large and small signal transconductances.

An FET version of the Colpitts oscillator is shown in Figure 5.4. If we replace the FET

amplifier by a generic unit, we obtain the generic Colpitts oscillator shown in Figure 5.5.


14/43


C3

C1

L C

C2 L1 L2

Figure 5.6 Clapp and Hartley feedback circuits.

VDD

bias

Figure 5.7 FET differential oscillator.

(Note that the feedback requires the amplifier to have a phase shift of 180, whichis the case for common-source and common-emitter amplifiers.) The Colpitts circuit

employs a series inductance -network feedback, but there are alternative feedback

circuits that give rise to the Clapp and Hartley oscillators (see Figure 5.6). Whilst

oscillators based on a single-ended amplifier input are common, it is also possible to

base an oscillator on a differential amplifier. Figure 5.7 shows a design that is suitable

for CMOS implementation. Positive feedback is achieved by using both differential

input and output, the output of one side feeding the input of the other.

In practical applications, the oscillator will need to act as a source of RF signals.

Consequently, this will mean an additional load on the circuit. The signal is normally

taken from the output of the amplifier on which the oscillator is based (the collector in


15/43


Figure 5.2 and the drain in Figure 5.4). Unfortunately, the additional load can have an

adverse affect upon oscillator performance and so it is advisable to make this loading

as light as possible (a high impedance load). To this end, the additional load is normally

connected through a high input impedance buffer amplifier (emitter or source follower

circuits are often used).

Example Design a JFET Colpitts oscillator for operation at 30 MHz. Assume a supply

voltage of 6 V and FET parameters Vt = 3 V, K = 103 A/V2, CGS = 4pF, CGD =1.6pF, CDS = 0.1 pF and rd = 30 k.

The design will be that shown in Figure 5.4. Since the circuit employs a JFET, we

can use the option of self bias and remove resistor R1. We will take R2 = 100k sincethis is well below the d.c. input resistance of a typical JFET and hence will suitably

ground the gate. The small signal model of a JFET is essentially the same as that for aBJT, except that r can be neglected. Consequently,

=

C1 + C2LC1C2

andC2

C1= gmRD (5.8)

for oscillations to occur. To reduce the effect of transistor capacitance, C1 and C2

should have values much greater than the FET output capacitance (0.1 pF) and input

capacitance (4 pF), respectively. We choose bias conditions such that the d.c. component

of gatesource voltage VGS is

2V. Such close proximity to Vt will ensurethat operation

is pushed well into the non-linear regime. From the saturation region characteristic

equations, this will require a drain current of 1 mA and hence a source resistor RS of

2 k. The value of transconductance is calculated from gm = 2K(VGS Vt) and, forthe above bias conditions, will have the value 2 103 S. Resistor RD is given thevalue 1 k to satisfy the usual design rule that the quiescent VDS be approximately

VDD/2. For oscillations to start, we will require that C2/C1 < gmRD which implies

that C2/C1 < 2. To guard against component variations, it is advisable to make this

constraint well satisfied and so we take C1 = C2. Furthermore, to satisfy our originalconstraints on C1 and C2, we choose C1 = C2 = 50 pF. For 30 MHz oscillations, theconditions in Equation 5.8 will imply a value of 1.125H for inductor L . This value,

however, will need to be slightly reduced due to the effect of the parasitic reactances

within the transistor. As the oscillations grow, the large signal transconductance Gm will

become appropriate and we will have C2/C1 = GmRD when steady state is achieved.Since Gm = [(VGS Vt)/Vosc]gm at this point, the amplitude of oscillations at the gatewill be around 2 V. This value, however, will tend to be an overestimate since the

drain current will deviate from the ideal pulse behaviour that is assumed by the theory

(Figure 5.8 illustrates the drain current behaviour for this oscillator).

Another variant of the Colpitts oscillator, shown in Figure 5.9, is useful when the

amplifier has less than unity voltage gain. (Note that the phase shift of the feedback

is zero, consistent with the phase shifts of source and emitter follower amplifiers.)


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Time

Draincurren

t

Figure 5.8 The drain current for a Colpitts oscillator.

L

C2

C1

Figure 5.9 Alternative Colpitts oscillator.

Consider an oscillator based on the n-channel JFET source follower of Figure 5.10.

Capacitors C1 and C2 of the feedback circuit need to be chosen so that the total loop

gain is greater than 1 (choosing C1 = C2 is usually sufficient). As the amplitude ofoscillation rises, the positive swings will eventually be clipped by the action of the

diode (at a level of about 0.7 V for a silicon diode). This results in a d.c. voltage that

pushes the transistor gatesource voltage towards Vt. As a consequence, there will be

a reduction in gain that continues until equilibrium is reached (i.e., the Barkhausen

criterion is satisfied).

We have noted that C1 and C2 need to be chosen to give adequate loop gain, but

other considerations can affect their choice. Firstly, their values should not be so small


17/43


output

inputCBL

RFCRG

RD

VDD

CBP

Figure 5.10 Source follower amplifier.

gmvL C2 v

vC1R

i2

i1

i3

Figure 5.11 Simplified Colpitts oscillator model.

that the internal capacitance of the transistor becomes a significant factor and hence

introduces a susceptibility to device variations. Secondly, they should not be so large

as to prevent oscillation. The idealised model of Figure 5.11 (it ignores the diode and

internal parasitics of the FET) helps to explain the last point (note that R represents

the resistance of the inductor). The currents in the circuit are related to the voltages

through

i1 = vjC2 (5.9)i2 =

v + vR +jL , (5.10)

i3 = vjC1. (5.11)


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We must have i3 = i1 + gmv , from whichv ( jC2 + gm) = vjC1 (5.12)

and i1

+i2

=0, from which

vjC2 +v + v

R +jL = 0. (5.13)

If we eliminate v between Equations (5.12) and (5.13), we obtain

jC2 + gm +jC1[ jC2(R +jL) + 1] = 0 (5.14)and, on taking imaginary and real parts,

C2 3C1C2L + C1 = 0 (5.15)

and

gm 2C1C2R = 0. (5.16)Equation 5.15 implies that the oscillations will occur at frequency =

(C1 + C2)/C1C2L and, from Equation 5.16, it is clear that too large a value forC1C2will prevent the oscillator from finding equilibrium. We need to have C1C2 gm/2Rfor oscillation to occur.

Figure 5.12 shows an alternative source follower Colpitts oscillator in which a source

resistor is used to produce gain compression. As the level of oscillation rises, the d.c.component of the source current will also rise and force VGS towards Vt. As a conse-

quence, the gain will reduce until a point is reached where the Barkhausen criterion is

satisfied.

RF output

L

C2

VDD

C1

Figure 5.12 An FET Colpitts oscillator that allows output to be taken from the drain.


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117 5.3 Stability and phase noise in oscillators

Note that, since the drain resistor has no bypass, an RF voltage will develop at the

drain and output can be taken from this point with very little effect upon the quality

of oscillation. Because of its pulse nature, the drain current will be rich in harmonics

and, if the drain resistor is replaced by a suitably tuned circuit, it is possible to extract

power at a harmonic frequency.

5.3 Stability and phase noise in oscillators

Consider the positive feedback system of Figure 5.13. The amplifier (assumed to be

ideal) has voltage gain A and the feedback is provided by a series combination of

inductance, capacitance and resistance. The relationship between source and output

voltages (assuming A 1) will be given byvo =

viA

1 RiARi+R+j

L 1

C

. (5.17)We can rearrange this into the form

vo =viA

1 11+jQ

o o

A RiRi+R

, (5.18)

where o = 1/LC is the resonant frequency of the feedback loop andQ = oL

Ri + R(5.19)

is its quality factor. What is clear from Equation 5.18 is that a disturbance in the phase

transfer characteristics of the amplifier (i.e., in the characteristics of A) will require an

adjustment in oscillation frequency if the Barkhausen criterion is to remain satisfied.

Ri

A

R L C

vivo

Figure 5.13 Oscillator with series LCR feedback.


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Amplitud

e

0

D

Figure 5.14 Phase noise.

It will be noted, however, that the requisite adjustment in frequency will reduce as the

value of Q increases. Consequently, high Q circuits are clearly the key to frequency

stable oscillators.

Close to the oscillation frequency o,

vo viAo

2jQ( o), (5.20)

where A=

(Ri+

R)/Ri for the Barkhausen criterion to be satisfied. It is clear from

this that the quality of oscillations will be strongly affected by disturbances containing

frequencies close to resonance. In particular, noise in the oscillator amplifier can cause

a spread of output frequencies around that which is desired (see Figure 5.14). The

amplifier noise can be expressed as a voltage source vni at the input of the amplifier

with mean square value satisfying v2ni = kT R i F for a bandwidth of 1 Hz ( F is theamplifier noise factor). The corresponding output noise voltage vno will satisfy

v2no =A22okT R i F

4Q2(o)2

(5.21)

which indicates that the output noise density will fall away as ( o)2. Very close toresonance, however, the above behaviour will be moderated by gain compression and

far from resonance there will be a floor that is set by the amplifier noise itself. Due to the

effect of gain compression, the main contribution to noise will arise from fluctuations

in phase and, as a consequence, oscillator noise is often referred to as phase noise. It is

clear from the above considerations that phase noise can be reduced by using high Q

components in the oscillator circuit.

The phase noise performance of an oscillator is normally described in terms of the

relative phase noise density, defined (in terms of dB/Hz) by

L = 10 logv2no

v2sig

, (5.22)


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119 5.3 Stability and phase noise in oscillators

IF

Frequency

Amp

litude

RF1

RF2LO

Figure 5.15 Illustration of reciprocal mixing.

where vsig is the noise free signal level. Phase noise imposes an important limitation

of oscillator performance and oscillator specification will normally include values of

relative phase noise at various frequency offsets from the intended frequency. In partic-

ular, for a receiver that down converts the input RF signal to an intermediate frequency

(IF), the phase noise performance of the receiverlocal oscillator(LO) can be importantbecause of the possibility of reciprocal mixing. Although the desired signal will mix

with the intended LO frequency to produce the IF frequency, there is the possibility

that energy from the local oscillator frequency skirts could mix with strong out-of-band

signals to also produce the IF frequency. This is known as reciprocal mixing and will

lead to interference that could be unacceptable in some applications. The concept is

illustrated in Figure 5.15 where RF1 represents the desired signal and RF2 the strong

out of band signal.

It is clear that we require high Q resonant circuits for good oscillator performance.

Extremely high Q resonators can be constructed out of quartz crystals and these are

used extensively in RF circuits. Such resonators are electromechanical in nature and use

the piezoelectric effect to translate high quality mechanical vibrations into electrical

oscillations. (Other devices with high Q arecoaxial and ceramic resonators.) Figure5.16

shows a typical circuit model for a quartz crystal (valid near the fundamental frequency

of resonance). The inductance is extremely large (hundreds of henries) and the shunt

capacitance C2 is typically tens of picofarads (the series capacitance C1 is very much

less). From the circuit model, we obtain the following expression for the impedance of

the crystal

Z= (1 2LC1) +jRC1

j(C1 + C2)

1 2LC1C2C1+C2

2RC1C2

. (5.23)


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crystal C2 C1

L

R

series

resonance

parallel

resonance

Z

Figure 5.16 Crystal resonator model.

C2

C1 R

Figure 5.17 Crystal Colpitts oscillator.

It is clear that the device will exhibit both series and parallel resonance (note that

R can be neglected due to the very high Q). The frequencies of these resonances will,

however, be very close. Figure 5.17 shows a typical example of a crystal controlled

Colpitts oscillator (bias components and d.c. supply not shown).

5.4 Voltage controlled oscillators

An oscillator with a voltage controlled frequency is often required in applications such

as phase locked loops. A varicap (variable capacitance) diode (sometimes known as a

varactor) can be used to achieve this. Diodes can be manufactured such that the reverse

bias junction capacitance changes quite dramatically with voltage

C(Vbias) =C0

1 VbiasVdiff

12

, (5.24)


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121 5.5 Negative resistance approach to oscillators

CBP

VDDCBP

output

tuning

bias

RFC

RFC

Figure 5.18 A Colpitts VCO.

cathode anode

n nn

Figure 5.19 Gunn diode.

where C0 is the zero bias capacitance and Vdiff has a value of about 0.6 V for a silicon

diode. Figure 5.18 shows a Colpitts voltage controlled oscillator (VCO) that is based

on such devices (note the addition of a source resistor to provide gain compression).

A variable frequency crystal oscillator (VXO) can be constructed by replacing the

inductance with a quartz crystal, but the achievable frequency variation is often very

small.

5.5 Negative resistance approach to oscillators

When a tuned circuit is excited by a pulse, it will ring at the resonant frequency. The

circuit resistance will, however, cause a rapid damping of these oscillations. This can be

overcome by introducing a device that has negative resistance in order to cancel out the

circuit resistance. An example of such a device is the Gunn diode shown in Figure 5.19.

A Gunn diode exhibits higher energy states for which the current carriers have lower

mobility and this will cause negative resistance under suitable bias conditions (see

Figure 5.20). The negative resistance can be used to cancel out the damping resistanceof a tuned circuit and hence create an oscillator (see Figure 5.20 and Collin).

We can also generate negative resistance using an FET (or a BJT) and this can provide

an alternative way of analysing oscillators. The circuit shown in Figure 5.21 is capable


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Gunndiode C R

L

Vtime

low mobility states

high mobility statesI

V

Figure 5.20 Oscillator based on the Gunn diode.

m

Zi vi

C1

RL

VDD

C1 v g

C2RL

ii

RFCC2

ii

vi

v

Figure 5.21 Negative resistance circuit based on an FET.

of generating negative resistance (gate bias not shown). This can be analysed through

the model that is also shown in the Figure 5.21 and from which

vi =ii

jC1+

ii + gmii

jC1

1

jC2. (5.25)

The input impedance Zi will be given by

Zi =vi

ii= 1

jC1+

1 + gmjC1

1

jC2(5.26)

which has a negative real part

Ri =gm

C1C22. (5.27)


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123 5.7 Analysis of a phase locked loop

phase

comparatorLFP VCO

VD VC VV

VR

reference

Figure 5.22 General phase locked loop.

By connecting the above impedance in parallel with an inductor, a Colpitts oscillator

is formed. The magnitude of Ri will need to be larger than the intrinsic resistance of

the inductor in order for oscillation to occur and this will lead to the same condition as

was derived in Section 5.2.

5.6 Phase locked loops

A phase locked loop (PLL) is a feedback system in which the feedback is based on

phase difference alone. These systems have a large variety of applications including

frequency control and demodulation. Figure 5.22 shows a typical PLL architecture.

The PLL compares the output of a voltage controlled oscillator (VCO) with a reference

signal and produces a control voltage that is proportional to the phase difference between

them. This voltage then adjusts the VCO such that it moves closer to the reference signal

in terms of phase. When the system settles down, the VCO is basically locked onto

the reference signal. The low-pass filter (LPF) helps remove unwanted high frequency

components that are present at the phase comparator output and the amplifier ensures

an adequate level of control voltage. In essence, a PLL produces a less noisy version

of the reference signal, but slightly out of phase (the phase difference can be reduced

by increasing the amplifier gain). The low-pass filter characteristics will be dictated by

the application and, in the case of demodulation, will need to exhibit a bandwidth that

is at least that of the baseband signal.

5.7 Analysis of a phase locked loop

If the reference signal vR has the form VR cos[0t+ R(t)] and the VCO signal vV hasthe form VV cos[0t+ V(t)], the output of the phase detectorvD will be

vD(t)

=kD[R(t)

V(t)], (5.28)

where kD depends on the nature of the phase detector. We assume that the phase detector

is linear and that 0 is the free running frequency of the VCO (the frequency for which

vC = 0). After passage through thefilter (transfer function H)andtheamplifier(gain A),


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ACTIVE FILTERS AND OSCILLATORS0 Chapter 5

1,Figure 5.46. Parasitic oscillation example.

capacitance of the transistor and the metercapacitance resonated with the meter in-ductance in a classic Hartley oscillator cir-cuit, with feedback provided by collector-emitter capacitance. Adding a small baseresistor suppressed the oscillation by re-ducing the high-frequency common-basegain. This is one trick that often helps.

5.19 Quartz-crystal oscillators

RC oscillators can easily attain stabilitiesapproaching O.l%, with initial predictabil-ity of 5% to 10%. That's good enough formany applications, such as the multiplexeddisplay in a pocket calculator, in which amultidigit numerical display is driven bylighting one digit after another in rapidsuccession (a lkHz rate is typical). Onlyone digit is lit at any time, but your eyesees the whole display. In such an appli-cation the precise rate is quite irrelevant- you just want something in the ballpark.

As stable sources of frequency, LCoscil-lators can do a bit better, with stabilitiesofO.OlO/oover reasonable periods of time.That's good enough for oscillators in radio-frequency receivers and television sets.

For real stability there's no substitute

for a crystal oscillator. This uses a pieceof quartz (same chemical as glass, silicondioxide) that is cut and polished to vibrateat a certain frequency. Quartz is piezo-electric (a strain generates a voltage, andvice versa), so acoustic waves in the crys-tal can be driven by an applied electricfield and in turn can generate a voltageat the surface of the crystal. By platingsome contacts on the surface, you windup with an honest circuit element that canbe modeled by an RLC circuit, pretunedto some frequency. In fact, its equiva-lent circuit contains two capacitors, giving

a pair of closely spaced (within 1%) se-ries and parallel resonant frequencies (Fig.5.47). The effect is to produce a rapidlychanging reactance with frequency (Fig.5.48). The quartz crystal's high & (typ-ically around 10,000) and good stabilitymake it a natural for oscillator control,as well as for high-performance filters (seeSection 13.12). As withLCoscillators, the

crystal's equivalent circuit provides posi-tive feedback and gain at the resonant fre-quency, leading to sustained oscillations.

I

Figure 5.47

Figure 5.49 shows some crystal oscilla-tor circuits. In A the classic Pierce oscilla-tor is shown, using the versatile FET (seeChapter 3). The Colpitts oscillator, witha crystal instead of an LC, is shown in B.An npn bipolar transistor with the crystal


27/43

OSCILLATORS

5.19 Quartz-crystal oscillators 301

as feedback element is used in C. The re-maining circuits generate logic-level out-puts using digital logic functions (D and

El.

capacitive I

B

Figure 5.48

The last diagram uses the convenientMC1206011206 1series of crystal oscillatorcircuits from Motorola. These chips are in-

tended for crystals in the range lOOkHz to20MHz and are designed to give excellentfrequency stability by carefully limiting theamplitude of oscillation via internal ampli-tude discrimination and limiting circuitry.They provide sine-wave and square-waveoutputs (both "TTL" and "ECL" logic lev-els).

An even more convenient alternative,if you're willing to accept a square wave

output only, and if utmost stability isn'tneeded, is the use of complete crystal os-cillator modules, usually provided as DIPIC-sized metal packages. They come inlots of standard frequencies (e.g., 1, 2, 4,5, 6 , 8, 10, 16, and 20MHz), as well as

weird frequencies commonly used in mi-croprocessor systems (e.g., 14.3 1818MHz,used for video boards). These "crystalclock modules" typically provide accura-cies (over temperature, power supply volt-

age, and time) of only 0.01% (lOOppm),but you get it cheap ($2 to $ 9 , and youdon't have to wire up any circuitry. Fur-thermore, they are guaranteed to oscillate,which isn't by any means assured whenyou wire your own oscillator: Crystal os-cillator circuits depend on electrical prop-erties of the crystal (such as series versusparallel mode, effective series resistance,

and mount capacitance) that aren't alwayswell specified. All too often you may findthat your home-built crystal oscillator os-cillates, but at a frequency unrelated tothat stamped on the crystal! Our own ex-perience with discrete crystal oscillator cir-cuits has been, well, checkered.

Quartz crystals are available from aboutlOkHz to about lOMHz, with overtone-mode crystals going to about 250MHz.Although crystals have to be ordered fora given frequency, most of the commonlyused frequencies are available off the shelf.Frequencies such as 1OOkHz, 1 OMHz,2.0MHz, 4.0MHz7 S.OMHz, and 1O.OMHzare always easy to get. A 3.579545MHzcrystal (available for less than a dollar) isused in TV color-burst oscillators. Digitalwristwatches use 32.768kHz (divide by 215to get lHz), and other powers of 2 arealso common. A crystal oscillator canbe adjusted slightly by varying a seriesor parallel capacitor, as shown in Figure5.49D. Given the low cost of crystals(typically about 2 to 5 dollars), it isworth considering a crystal oscillator inany application where you would have tostrain the capabilities of RC relaxation

oscillators.If you need a stable frequency with a

very small amount of electrical tunability,you can use a varactor to "pull" thefrequency of a quartz-crystal oscillator.The resulting circuit is called a "VCXO"


28/43

ACTIVE FILTERS AND OSCILLATORS2 Chapter 5

2.5rnH1OOOpF

output

-A. Pierce oscillator 0. Colpitts oscillator C

CMOS inverter

Figure 5.49. Various crystal oscillators.

v output

(voltage-controlled crystal oscillator), and

combines the good-to-excellent stability ofcrystal oscillators with the tunability ofLCoscillators. The best approach is proba-bly to buy a commercial VCXO, rather

than attempt to design your own. Typ-ically they produce maximum deviationsoff Oppm to f OOppm from center fre-quency, though wide-deviation units (up tof 000ppm) are also available.

Without great care you can obtain fre-quency stabilities of a few parts per mil-

1OMR1

oscillator) with somewhat better perfor-

mance. Both TCXOs and uncompensatedoscillators are available as complete mod-ules from many manufacturers, e.g., Bliley,CTS Knights, Motorola, Reeves Hoffman,Statek, and Vectron. They come in varioussizes, ranging down to DIP packages andTO-5 standard transistor cans. TCXOsdeliver stabilities of lppm over the range0C to 50C (inexpensive) down to 0.lppmover the same range (expensive).

- + 5 v

lion over normal temperature ranges withcrystal oscillators. By using temperature- Temperature-stabilized oscillatorscompensation schemes yo; can- make a For the utmost in stability, you may need aTCXO (temperature-compensated crystal crystal oscillator in a constant-temperature

T r i r 15

11 161OOk

1 ) 10; 4 )32,768Hz -7

sine1OPF g outputs 9 1- -- MC12060 (100kHz-2MHz) --L-- MC12061 (2MHz-20MHzlD E

+ slne42 -0-sine 3 -d}e ut10 TL out


29/43

SELF-EXPLANATORY CIRCUITS5.20 Circuit ideas 303

ven. A crystal with a zero temperatureoefficient at some elevated temperature80C to 90C) is used, with the thermo-at set to maintain that temperature. Suchscillators are available as small modulesor inclusion into an instrument or asomplete frequency standards ready forack mounting. The 10811 from Hewlett-ackard is typical of high-performance

modular oscillators, delivering 1OMHzwith stabilities of a few parts in 10'' over

eriods of seconds to hours.When thermal instabilities have been

educed to this level, the dominant ef-

ects become crystal"aging"(the frequencyends to decrease continuously with time),ower-supply variations, and environmen-al influences such as shock and vibrationthe latter are the most serious problemsn quartz wristwatch design). To give andea of the aging problem, the oscillator

mentioned previously has a specified agingate at delivery of 5 parts in 10'' per day,

maximum. Aging effects are due in parto the gradual relief of strains, and theyend to settle down after a few months,articularly in a well-manufactured crystal.

Our specimen of the 108 11 oscillator agesbout 1 part in 1011 per day.

Atomic frequency standards are usedwhere the stability of ovenized-crystaltandards is insufficient. These use a mi-

rowave absorption line in a rubidium gasell, or atomic transitions in an atomic ce-ium beam, as the reference to which auartz crystal is stabilized. Accuracy andtability of a few parts in 1012can be ob-ained. Cesium-beam standards are the of-icial timekeepers in this country, with tim-ng transmissions from the National

Bureau of Standards and the Naval Ob-

ervatory. Atomic hydrogen masersave been suggested as the ultimate intable clocks, with claimed stabilities ap-roaching a few parts in loi4. Recentesearch in stable clocks has centered onechniques using"cooled ions" to achieve

even better stability. Many physicists be-lieve that ultimate stabilities of parts in1018may be possible.

SELF-EXPLANATORY CIRCUITS

5.20 Circuit ideas

Figure 5.51 presents a variety of circuitideas, mostly taken from manufacturers'data sheets and applications literature.

ADDITIONAL EXERCISES

1. Design a 6-pole high-pass Bessel filterwith cutoff frequency 1kHz.2. Design a 60Hz twin-T notch filter withop-amp input and output buffers.3. Design a sawtooth-wave oscillator, todeliver 1kHz, by replacing the charging re-sistor in the 555 oscillator circuit with atransistor current source. Be sure to pro-vide enough current-source compliance.What value should RB (Fig. 5.33) have?4. Make a triangle-wave oscillator witha 555. Use a pair of current sources I.(sourcing) and 210 (sinking). Use the 555'soutput to switch the 210 current sink onand off appropriately. The following figureshows one possibility.

0 - 555 - -U-Lr

outpu t

fl

A

2 + TFigure 5.50


30/43

C H A P T E R

11 Oscillators

Oscillators are autonomous dc-to-ac converters. They are used as the frequency-

determining elements of transmitters and receivers and as master clocks in

computers, frequency synthesizers, wristwatches, etc. Their function is to

divide time into regular intervals. The invention of mechanical oscillators

(clocks) made it possible to divide time into intervals much smaller than the

Earths rotation period and much more regular than a human pulse rate.

Electronic oscillators are analogs of mechanical clocks.

11.1 Negative feedback (relaxation) oscillators

The earliest clocks used a verge and foliot mechanism which resembled

a torsional pendulum but was not a pendulum at all. These clocks operated as

follows: torque derived from a weight or a wound spring was applied to a

pivoted mass. The mass accelerated according to Torque = Id2/dt2 (the angular

version of F= ma). When reached a threshold, 0, the mechanism reversed

the torque, causing the mass to accelerate in the opposite direction. When it

reached 0 the torque reversed again, and so on. The period was a function of

the moment of inertia of the mass, the magnitude of the torque, and the

threshold setting. These clocks employed negative feedback; when the con-

trolled variable had gone too far in either direction, the action was reversed.

Most home heating systems are negative feedback oscillators; the temperature

cycles between the turn on and turn off points of the thermostat. Negative

feedback electronic oscillators are called relaxation oscillators. Most of

these circuits operate by charging a capacitor until its voltage reaches an

upper threshold and then discharging it until the voltage reaches a lower

threshold voltage. In Figure 11.1(a), when the voltage on the capacitor builds

up to about 85 V, the neon bulb fires. The capacitor then discharges quickly

through the ionized gas (relaxes) until the voltage decays to about 40 V. Thebulb then extinguishes and the cycle begins anew.

120


31/43

The circuit of Figure 11.1(b) alternately charges the capacitor, C, until its

voltage reaches 2.5 V, and then discharges it until the voltage has fallen to

2.5 V. (V1(t) decays alternately toward +5 or5 volts. When it reaches zero

volts, the left-hand op-amp abruptly saturates in the opposite direction, kicking

V1(t) to the voltage it had been approaching. The voltage then begins to decay in

the opposite direction, and so forth.) Voltage-to-frequency converters are usu-

ally relaxation oscillators in which the control voltage determines the slope, and

hence the oscillation period, of a fixed-amplitude sawtooth wave. Relaxation

oscillators typically contain waveforms that are ramps or exponential decays. In

the verge and foliot clock, the angle(t) consists of a sequence of parabolic arcs.

Note that relaxation oscillators are nonlinear circuits which switch alternately

between a charge mode and a discharge mode. Positive feedback oscillators, the

main subject of this chapter, are nominally linear circuits. They generate sine

waves.

11.2 Positive feedback oscillators

Clock makers improved frequency stability dramatically by using a true pen-dulum, a moving mass with a restoring force supplied by a hair spring or

gravity.1 As first observed by Galileo, a pendulum has its own natural fre-

quency, independent of amplitude. It moves sinusoidally in simple harmonic

motion. A pendulum clock uses positive feedback to push the pendulum in the

direction of its motion, just as one pushes a swing to restore energy lost to

friction.

Figure 11.1. Relaxation

(negative feedback) oscillators.

V1(t)

5

+5

1/2 TL082

1/2 TL082

R

(b)

C

Vout

5V

5

0 f= 1/(2 R CIn2)

(a)

5V

5V2k

2k

++

90V

90V

NE-2

Neon bulb

V0

t

1 The Salisbury Cathedral clock, when installed around 1386, used a verge and foliot

mechanism. Some 300 years later, after Christian Huygens invented the pendulum clock

based on Galileos observations of pendulum behavior, the Salisbury clock was converted

into a positive feedback pendulum clock, its present form.

121 Oscillators


32/43

Electronic versions of the pendulum clock are usually based on resonators

such as parallel or series LC circuits or electromechanical resonators such asquartz crystals. They use positive feedback to maintain the oscillation. A

resonator with some initial energy (inductor current, capacitor charge, or

mechanical kinetic energy) will oscillate sinusoidally with an exponentially

decaying amplitude as shown in Figure 11.2. The decay is due to energy loss in

the load and in the internal loss of the finite-Q resonator. In Figure 11.2 the

resonator is a parallel LCR circuit.

To counteract the exponential decay, a circuit pumps current into the reso-

nator when its voltage is positive and/or pulls current out when its voltage is

negative. Figure 11.3 shows how a transistor and a dc supply can provide this

energy. In this example circuit, the transistor is shown in the emitter-follower

configuration simply because it is so easy to analyze; the emitter voltage tracks

the base voltage and the base draws negligible current. The single transistor

cannot supply negative current but we can set it up with a dc bias as a class-A

amplifier so that current values less than the bias current are equivalent to

negative current.

All that remains to complete this oscillator circuit is to provide the transistor

drive, i.e., the base-to-emitter voltage. We want to increase the transistors

conduction when the output voltage (emitter voltage) increases, and we see

that the emitter voltage has the correct polarity to be the drive signal. Since anemitter followers voltage gain is slightly less than unity, the base needs a drive

V(t)V

t

Figure 11.2. Damped oscillation

in a parallel LCRcircuit.

Base-to-emitter

drive voltage

RL

VdcFigure 11.3. Transistor and dc

supply replace energy lost to

damping.

122 Radio-frequency electronics: Circuits and applications


33/43

signal with slightly more amplitude than the sine wave on the emitter.

Figure 11.4 shows three methods to provide this drive signal. The Armstrong

oscillator adds a small secondary winding to the inductor. The voltage induced

in the secondary adds to the emitter voltage. The Hartley oscillator accom-

plishes the same thing by connecting the emitter to a tap slightly below the top

of the inductor. This is just an autotransformer version of the Armstrong

oscillator if the magnetic flux links all the turns of the inductor. But the top

and bottom portions of the inductor do not really have to be magnetically

coupled at all; most of the current in the inductor(s) is from energy stored in

the high-Q resonant circuit. This current is common to the two inductors so they

essentially form a voltage divider. (Note, though, that the ratio of voltages on the

top and bottom portions of the inductor ranges from the turns ratio, when theyare fully coupled, to the square of the turns ratio, when they have no coupling.)

If we consider the totally decoupled Hartley oscillatorno mutual inductance

and then replace the inductors by capacitors of equal (but opposite) reactance

and replace the capacitor by an inductor, we get the Colpitts oscillator. Note that

each oscillator in Figure 11.4 is an amplifier with a positive feedback loop. No

power supply or biasing circuitry is shown in these figures; they simply indicate

the ac signal paths.

Using the Hartley circuit as an example, Figure 11.5 shows a practical circuit.

It includes the standard biasing arrangement to set the transistors operating

point. (A resistor voltage divider determines the base voltage and an emitter

resistor then determines the emitter current, since Vbe will be very close to

0.7 V.) A blocking capacitor allows the base to be dc biased with respect to the

emitter. A bypass capacitor puts the bottom of the resonant circuit at RF ground.

In practice, one usually finds oscillators in grounded emitter circuits, as

shown in Figure 11.6. The amplitude of the base drive signal must be much

smaller than the sine wave on the resonant circuit. Moreover, the polarity of the

base drive signal must be inverted with respect to the sine wave on the collector.

You can inspect these circuits to see that they do satisfy these conditions. But,

on closer inspection, you can note the circuits are identical to the circuits ofFigure 11.4, except that the ground point has been moved from the collector to

(a) Armstrong (b) Hartley (c) Colpitts

RL RLC2

C1

RL

Figure 11.4. Feedback loop

details define (a) Armstrong;

(b) Hartley; (c) Colpitts

oscillators.

RL

Vdc

Figure 11.5. Hartley oscillator

circuit including bias circuitry.

123 Oscillators


34/43

the emitter. In these oscillators, an amplifier is enclosed in a positive feedback

loop. But, because there is no input signal to have a terminal in common with

the output signal, oscillators, unlike amplifiers, do not have common-emitter,

common-collector, and common-base versions.

The Colpitts oscillator, needing no tap or secondary winding on the inductor,

is the most commonly used circuit. Sometimes the transistors parasitic

collector-to-emitter capacitance is, by itself, the top capacitor, C1, so this

capacitor may appear to be missing in a circuit diagram. A practical design

example for the Colpitts circuit of Figure 11.6(c) is presented later in this

chapter.

11.2.1 Unintentional oscillators

In RF work it is common for a casually designed amplifier to break into

oscillation. One way this happens is shown in Figure 11.7. The circuit is a

basic common-emitter amplifier with parallel resonant circuits on the input andoutput (as bandpass filters and/or to cancel the input and output capacitances of

the transistor). When the transistors parasitic base-to-collector capacitance is

included, the circuit has the topology of the decoupled Hartley oscillator. If the

feedback is sufficient, it will oscillate. The frequency will be somewhat lower

than that of the input and output circuits so that they look inductive as shown in

the center figure. This circuit known as a TPTG oscillator, form Tuned-Plate

Tuned-Grid, in the days of the vacuum tube.

Figure 11.6. Grounded-emitter

oscillator circuits.

(a) Armstrong

C2

C1

(c) Colpitts(b) Hartley

Figure 11.7. Tuned amplifier as

an oscillator

Cbc Cbc

Cbc

Hartley oscillatorTuned amplifier



35/43

With luck, the loop gain of any amplifier will be less than unity at any

frequency for which the total loop phase shift is 360 and an amplifier will be

stable. If not, it can be neutralized to avoid oscillation. Two methods of

neutralization are shown in Figure 11.8.

In Figure 11.8(a), a secondary winding is added to provide an out-of-phase

voltage which is capacitor-coupled to the base to cancel the in-phase voltage

coupled through Cbc. In Figure 11.8(b), an inductor from collector to base

resonates Cbc to effectively remove it (a dc blocking capacitor would be placed

in series with this inductor). In grounded-base transistor amplifiers and

grounded-grid vacuum tube amplifiers the input circuit is shielded from the

output circuit. These are stable without neutralization (but provide less powergain than their common-emitter and common-cathode-counterparts).

11.2.2 Series resonant oscillators

The oscillators discussed above were all derived from the parallel resonant

circuit shown in Figure 11.2. We could just as well have started with a series

LCR circuit. Like the open parallel circuit, a shorted series LCR circuit executes

an exponentially damped oscillation unless we can replenish the dissipated

energy. In this case we need to put a voltage source in the loop which will be

positive when the current is positive and negative when the current is negative,

as shown in Figure 11.9.

CbcFigure 11.8. Amplifier

neutralization.

Figure 11.9. Series-mode

oscillator operation.

V(t)

RL

RL

125 Oscillators


36/43

While a bare transistor with base-to-emitter voltage drive makes a good

current source for a parallel-mode oscillator, a low-impedance voltage source

is needed for a series-mode oscillator. In the series-mode oscillator shown in

Figure 11.10, an op-amp with feedback is such a voltage source.

Since no phase inversion is provided by the tank circuit, the amplifier is

connected to be noninverting. An emitter-follower has a low output impedance

and can be used in a series-mode oscillator (see Problem 11.4). When the series

LCcircuit is replaced by a multisection RC network, the resulting oscillator is

commonly known as a phase-shift oscillator (even though every feedback

oscillator oscillates at the frequency at which the overall loop phase shift is360). An RC phase-shift oscillator circuit is shown in Figure 11.11. Op-amp

voltages followers make the circuit easy to analyze.

For the three cascaded RCunits, the transfer function is given by V2(t)/V1(t) =

1/(RC+1)3. The inverting amplifier at the left provides a voltage gain of16/2

= 8, so V1(t)/V2(t) = 8. Combining these two equations yields a cubic

equation with three roots: RC= 3j,ffiffiffi

3p

, and ffiffiffi3p . The first root correspondsto an exponential decay of any initial charges on the capacitors while the two

imaginary roots indicate that the circuit will produce a steady sine-wave oscil-

lation whose frequency is given by RC ffiffiffi3p

. In practice, the 16k resistor

would be increased to perhaps twice that value to ensure oscillation. Note thatFigure 11.11. An RCphase-shiftoscillator.

+

+ ++RRR

V2(t)V1(t)2k

f=0.27/(RC)

16k

CCC

R1

R2

RL

+

AV=1+R2/R1=just over 1

Figure 11.10. An op-amp series-

mode oscillator.



37/43

this circuit is a positive-feedback sine-wave oscillator even though it does not

contain a resonator. When the 16k resistor value is increased, the loop gain for

the original frequency becomes greater than unity, but for the new gain, there

will be a nearby complex frequency, j, for which the loop gain is unity. The

time dependence therefore becomes ej(j)t

= ejt

et

, showing that the oscil-lation amplitude grows as e

t. This circuit illustrates how any linear circuit with

feedback will produce sine-wave oscillations if there is a (complex) frequency

for which the overall loop gain is unity and the overall phase shift is 360. (Of

course must be positive, or the oscillation dies out exponentially.)

11.2.3 Negative-resistance oscillators

In the circuits described above, a transistor provides current to an RLCcircuit

when the voltage on this circuit is positive, i.e., the transistor behaves as a

negative resistance. But the transistor is a three-terminal device and the third

terminal is provided with a drive signal derived from theLCR tank. Figure 11.12

shows how two transistors can be used to make a two-terminal negative

resistance that is simply paralleled with the LCR tank to make a linear sine

wave oscillator that has no feedback loop.

The two transistors form an emitter-coupled differential amplifier in which

the resistor to Vee acts as a constant current source, supplying a bias current, I0.

The input to the amplifier is the base voltage of the right-hand transistor. The

output is the collector current of the left-hand transistor. The ratio of input to

output is

4VT/I0, where VT is the thermal voltage, 26 mV. This ratio is just thenegative resistance, since the input and output are tied together. This negative-

resistance oscillator uses a parallel-resonant circuit, but a series-resonant ver-

sion is certainly possible as well.

Any circuit element or device that has a negative slope on at least some

portion of its IV curve can, in principle, be used as a negative resistance.

Tunnel diodes can be used to build oscillators up into the microwave frequency

range. At microwave frequencies, single-transistor negative-resistance oscilla-

tors are common. A plasma discharge exhibits negative resistance and provided

a pre-vacuum tube method to generate coherent sine waves. High-efficiency

R

Vee

RL

RL

R

(a) (b)

Vcc

VccFigure 11.12. A negative-

resistance oscillator.

127 Oscillators


38/43

Poulsen arc transmitters, circa World War I, provided low-frequency RF power

exceeding 100 kW.

11.3 Oscillator dynamics

These resonant oscillators are basically linear amplifiers with positive feedback.

At turn-on they can get started by virtue of their own noise if they run class A.

The tiny amount of noise power at the oscillation frequency will grow expo-

nentially into the full-power sine wave. Once running, the signal level is

ultimately limited by some nonlinearity. This could be a small-signal non-

linearity in the transistor characteristics. Otherwise, the finite voltage of the

dc power provides a severe large-signal nonlinearity, and the operation will shift

toward class-C conditions. The fact that amplitude cannot increase indefinitely

shows that some nonlinearity is operative in every real oscillator. Any non-

linearity causes the transistors low- frequency 1/ noise to mix with the RF

signal, producing more noise close to the carrier than would exist for linear

operation. An obvious way to mitigate large-signal nonlinearity is to detect the

oscillators output power and use the detector voltage in a negative feedback

arrangement to control the gain. This can maintain an amplitude considerably

lower than the power supply voltage. Alternatively, if the oscillator uses a

device (transistor or op-amp circuit) with a soft saturation characteristic, the

amplitude will reach a limit while the operation is still nearly linear. For

example, the amplifier in the oscillator of Figure 11.10 might have a smallcubic term, i.e., VOUT = AVIN BVIN

3, where B/A is very small (see Problem

11.5).

11.4 Frequency stability

Long-term (seconds to years) frequency fluctuations are due to component

aging and changes in ambient temperature and are called drift. Short-term

fluctuations, known as oscillator noise, are caused by the noise produced in

the active device, the finite loaded Q of the resonant circuit, and nonlinearity in

the operating cycle. The higher the Q, the faster the loop phase-shift changes

with frequency. Any disturbances (transistor fluctuations, power supply varia-

tions changing the transistors parasitic capacitances, etc.) that tend to change

the phase shift will cause the frequency to move slightly to reestablish the

overall 360 shift. The higher the resonatorQ, the smaller the frequency shift.

Note that this is the loaded Q, so the most stable oscillators, besides having the

highestQ resonators, are loaded as lightly as possible. In LCoscillators, losses

in the inductor almost always determine the resonator Q. A shorted piece of

transmission line is sometimes used as a high-Q inductor. Chapter 24 treatsoscillator noise in detail.



39/43

11.5 Colpitts oscillator theory

Let us look in some detail at the operation of the Colpitts oscillator. Figure 11.13

shows the Colpitts oscillator of Figure 11.6(c) redrawn as a small-signal

equivalent circuit (compare the figures). The still-to-be-biased transistor is

represented as a voltage-controlled current source. The resistor rbe represents

the small-signal base-to-emitter resistance of the transistor.

The parallel combination ofL and the load resistor,R, is denoted asZ, i.e.,Z=

jLR/(jL+R) = jLS +RS, where LS and RS are the component values for the

equivalent series network. Likewise, it is convenient to denote rbe1 as g. The

voltage Vbe, a phasor, is produced by the current I (a phasor) from the current

source. This is a linear circuit, so Vbe can be written as Vbe = I ZT, where ZT is a

function of. We will calculate this transfer impedance using standard circuit

analysis. Since the current I is proportional to Vbe, we can write an equationexpressing that, in going around the loop, the voltage Vbe exactly reproduces

itself :

gmVbeZT Vbe or 1ZT

gm: (11:1)

This equation will let us find the component values needed for the circuit to

oscillate at the desired frequency, i.e., the values that will make the loop gain

equal to unity and the phase shift equal 360.

We can arbitrarily select L, choosing an inductor whose Q is high at the

desired frequency. Equation (11.1), really two equations (real and imaginary

parts), will then provide values forC1 and C2. To derive an expression forZT,

we will assume thatVbe = 1 and work backward to find the corresponding value

ofI. With this assumption, inspection ofFigure 11.13 shows that the currentI1 is

given by

I1 jC2 g: (11:2)Now the voltage Vc is just the 1 volt assumed for Vbe plus I1Z, the voltage

developed across Z:

Vc 1 jC2 gZ: (11:3)

Vc

l1

C2C1

Z

R

L

I=gmVbe

Vbe

rbe =1/g

Figure 11.13. Colpitts oscillator

small-signal equivalent circuit.

129 Oscillators


40/43

Finally, the currentIis just the sum ofI1plus VcjC1, the current going into C1:

I jC2 g 1 jC2 gZjC1: (11:4)Since we had assumed that Vbe = 1, we have ZT = 1/Ior

1

ZT jC2 g 1 jC2 gZjC1: (11:5)

Using this, the condition for oscillation, Equation (11.1) becomes

gm jC2 g 1 jC2 gZjC1 0: (11:6)The job now is to solve Equation (11.6) forC1 and C2. If we assume that is

real i.e., that the oscillation neither grows nor decays, we find from the imag-

inary part of this equation, that

C2 C1 gRSC1LSC1C2

2 (11:7)

and, from the real part, that

2C1C2RS g2LSC1 1 gm: (11:8)Solving Equations (11.7) and (11.8) simultaneously forC2 and C1 produces

C2 gmLS2RS

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi

4RS1 gRSgm ggm2LS2

s !(11:9)

and

C1 C22LSC2 1 gRS : (11:10)

Normally C2 will have a much larger value than C1 and % 1=ffiffiffiffiffiffiffiffi

LC1p

.

Moreover, the second term in the square root of Equation (11.9) is usually

much less than unity so C2gm LS/RS.

11.5.1 Colpitts oscillator design exampleLet us design a practical grounded-emitter Colpitts oscillator. Suppose this

oscillator is to supply 1 mW at 5 MHz and that it will be powered by a 6 V dc

supply. Assuming full swing, the peak output sine wave voltage will be 6 V. The

output power is given by 0.001 W = (6 V)2/(2RL) so the value of the load

resistor, RL, will be 18 k ohms. Assuming class-A operation, the bias current

in the transistor is made equal to the peak current in the load: I = Ipk = 6 V /

18 k = 0.33 mA. If we let the emitter biasing resistor be 1.5 k, the emitter bias

voltage will be 1500 0.33 mA = 0.5 V. Assuming the typical 0.7 V offset

between the base and emitter, the base voltage needs to be 1.2 V. A voltagedivider using a 40 k resistor and a 10 k resistor will produce 1.2 V from the 6 V



41/43

supply. These bias components are shown in the schematic diagram of

Figure 11.14.

A 0.05 F bypass capacitor pins the base to ac ground and another bypass

capacitor ensures that the dc input is held at a firm RF ground. Note that the

1.5 k emitter bias resistor provides an unwanted signal path to ground. This path

could be eliminated by putting an inductor in series with the bias resistor as an

RF choke, but this is not really necessary; the1.5k resistor is in parallel with C 2,

which will have such a low reactance that the resistor will divert almost no

current from it.

With the biasing out of the way, we now deal with the signal components. The

transconductance of the transistor is found by dividing the bias current, I0, by26 mV, the so-called thermal voltage,2 i.e., gm = 0.33 mA/ 26 mV = 0.013 mhos.

The small-signal base-to-emitter resistance, r, is given by r= Vthermal/I0. For a

typical small-signal transistor, such as a 2N3904, is about 100, so rin =

1000.026 V/0.33 mA = 8000 ohms.

Using Equation (11.9) and (11.10), the values ofC1 and C2 are 102 pF and

0.023 F, respectively. These are the values for which the oscillator theoretically

will maintain a constant amplitude. In practice, we increase the feedback by

decreasing the value ofC2 to ensure oscillation. This produces a waveform that

grows exponentially until it reaches a limit imposed by circuit nonlinearity. The

frequency becomes complex, i.e., becomes j and the time dependence

therefore becomes ej(j)t = ejtet. Suppose we want to be, say 105, which

will cause oscillation to grow by a factore every 10sec. (Fast growth would be

important if, for example, the oscillator is to be rapidly pulsed on and off.) How

do we find the value of C2 to produce the desired ? To avoid doing more

analysis, it is convenient to use a standard computer program such as Mathcad

to find the root(s) ofEquation (11.6) for trial values ofC2. In this example, if we

decrease C2 to 0.020 F, we obtain the desired .

+5V

18k

C20.020 F

C1102 pF

2N3904

1.5k

40k

10k0.05 F

0.05 F 10Hy

0.01 F

Figure 11.14. Colpitts oscillator:

5 MHz, 1mW.

2 The thermal voltage is given by Vthermal = 0.026V = kT/e, where k is Boltzmans constant,

Tis the absolute ambient temperature, and e is the charge of an electron.

131 Oscillators


42/43

Problems

Problem 11.1. Draw a schematic diagram (without component values) for a bipolar

transistor Colpitts oscillator with the collector at ground for both dc RF. Include the

biasing circuit. The oscillator is to run from a positive dc supply.

Problem 11.2. Design (without specifying component values) a single-transistor

series-mode oscillator based on the emitter follower circuit.

Problem 11.3. A simple computer simulation can illustrate how an oscillator builds up

to an amplitude determined by the nonlinearity of its active element. The program shown

below models the negative-resistance oscillator of Figure 11.12(a). The LC resonant

frequency is 1 Hz. This network is in parallel with a negative-resistance element whose

voltage vs. current relation is given by I= (1/Rn)*(VV3), to model the circuit of

Figure 11.12. The small-signal (negative) resistance is justRn. The termV3 makes the

resistance become less negative for large signals. The program integrates the second-

order differential equation for V(t) and plots the voltage versus time from an arbitrary

initial condition, V= 1 volt.

Run this or an equivalent program. Change the value of the load resistorR. Find the

minimum value ofR for sustained oscillation. Experiment with the values of R and Rn.

You will find that when the loaded Q of the RLCcircuit is high, the oscillation will be

sinusoidal even when the value of the negative resistor is only a fraction of R. When Q is

low (as it is for R = 1), a low value of Rn such as Rn = 0.2 will produce a distinctly

distorted waveform.

QBasicsimulationofnegative-resistanceoscillator of Figure11.12a.SCREEN2

R=1:L=1/6.2832:C=L theparallelRLCcircuit:1 ohms,1/2pihenries,1/2pifarads

RN=.9 run program alsowithRN=.2 tosee non-sinusoidalwaveform

E=.01 negativeresistance:I=(1/RN)*(V-EV^3)

V=1:U= 0 initial conditions, V isvoltage,U is dV/dt

DT = .005 stepsizeinseconds

FOR I = 1 TO 3000

T=T+DT incrementthetime

VNEW=V+U*DT

U=U+(DT/C)*((1/RN)*(U-3*E*V*V*U)-V/L-U/R)

V=VNEW

PSET(40*T,100+5*V)plotthepoint

NEXTI

Problem 11.4. In the oscillator shown below, the voltage gain of the amplifier decreases

with amplitude. The voltage transfer function is Vout= 2 Vin 0.5Vin3. This characteristic

will limit the amplitude of the oscillation.



43/43

R1

R2

VoutVout=2 Vin0.5Vin

3

Vin

Vin

1.5

1.5

1 1

Find the ratio R2/R1 in order that the peak value of the sine wave Vin will be one volt.

Hint: assume Vin = sin(t). The amplifier output is then 2 sin(t) 0.5sin3(t). The

second term resembles the sine wave but is more peaked. The LC filter will pass the

fundamental Fourier component of this second term. Find this term and add it to 2sin(t).

Then calculate the ratio R2/R1 so that the voltage divider output is sin(t).

133 Oscillators

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