Oscillators

Electronic Engineering

© University of Wales Newport 2009 This work is licensed under a Creative Commons Attribution 2.0 License.

The following presentation is a part of the level 5 module -- Electronic Engineering. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1 st year undergraduate programme.

The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.

Contents Oscillator Circuits Wien Bridge Oscillator R. C. Phase Shift Oscillator. Tuned Collector L C Oscillator. Generalised Oscillator Circuit Colpitts Oscillator Hartley Oscillator Crystal Oscillator Crystal Modelling Credits

In addition to the resource below, there are supporting documents which should be used in combination with this resource. Please see:Clayton G, 2000, Operational Amplifiers 4th Ed, Newnes James M, 2004, Higher Electronics, Newnes

Oscillators

Oscillator CircuitsAn amplifier will become unstable if it has positive

feedback applied to itSee below

VoutA

B

A is the gain of the amplifier and B is the proportion of the output fed back to the input.

Oscillators

A will amplify a signal on the input and then a proportion of the amplified signal will be fed back. If this is large enough to replace the original signal the system keeps generating an output. There are a few rules that determine the nature of the output.

These are stated in the Barkhausen Criterion.It states:

For Sinusoidal Oscillation to occur at a single frequency we must have:

1. A loop gain of unity (A x B = 1)2. A loop phase shift of zero (A + B = 0 or 360)3. Condition 1 or 2 to be true only at a single

frequency.

We will examine a range of sinusoidal oscillators.Oscillators

Wien Bridge Oscillator

This uses a feedback network of the following form:Note the resistors have the same value as do the capacitors.

R

R

C

C

ZS

ZP

Vout

Vin

As the input frequency is varied the output will have a different gain and phase relationship with the input. At one frequency the input and output will be in phase.

Cj

CRj

CjRZS

11

11

1

CRj

R

CjR

CjR

ZP

Cj

CRj

CRj

RCRj

R

VinZZ

ZVinVout

SP

P

1

1

1

11

11

1

1

CRj

CRj

CRjCRj

CRj

Cj

CRj

CRj

RCRj

R

Vin

VoutGain

22 311 )()( CRCRj

CRj

CRjCRj

CRjGain

Oscillators

CRjCR

CRj

CRCRj

CRjGain

3131 22

)()(

222

23

222

2

31

3

31

31

)())((

)()(

)())((

))((

CRCR

CRCRjCRj

CRCR

CRjCRCRjGain

222

2

222

2

31

1

31

3

)())((

))((

)())((

)(

CRCR

CRCRj

CRCR

CRGain

222

2222

31

13

)())((

)))((())((

CRCR

CRCRCRGain

2

21

3

1

)(

))((

CR

CRCRTanPhase

Oscillators

Example R = 10k and C = 10nF

Frequency Gain Phase

900 0.3093889 2.34789

1000 0.3173753 2.864766

1100 0.3232355 3.289809

1200 0.3274285 3.527987

1300 0.3303008 3.459755

1400 0.3321152 2.938197

1500 0.3330734 1.785935

1600 0.3333313 -0.20661

1700 0.3330114 -3.27647

1800 0.3322112 -7.6651

1900 0.331009 -13.5567

2000 0.3294684 -20.9608

Wien Bridge

0.305

0.31

0.315

0.32

0.325

0.33

0.335

900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Frequency

-25

-20

-15

-10

-5

0

5

Gain curve

Phase curve

1.59kHz

0.333

Oscillators

From the graph:

Zero degrees phase shift occurs at 1.59kHz

The gain at this point equals approximately 0.333

Can we determine this from the equations?

For this to equal 0 then:

2

21

3

1

)(

))((

CR

CRCRTanPhase

01 2 )( CR12 )( CR 1CR

CR

1

kHznFkCR

f 591110102

1

2

1.

Gain at this frequency:

But –

Which leaves us with:

To use this to produce an oscillator we need an amplifier with the following characteristics:

1. Gain = 3 (to give loop gain of unity)2. Phase shift of 0 (to give a loop phase shift of

zero)

222

2222

31

13

)())((

)))((())((

CRCR

CRCRCRGain

01 2 )( CR

3

1

9

3

3

3

3

32

2

2

22

)()(

)())((

CR

CR

CR

CRGain

Oscillators

R2

R1

R

RC

C

Vout

Feedback

Amplifier

32

11

R

RGain 221 RR

If the gain is too large the sine wave will clip at the supply rails and if it is too small it will not oscillate at all.

Wien Bridge Oscillator

Oscillators

R. C. Phase Shift Oscillator.As we are aware, an R C network will alter the amount of

signal it passes as the frequency varies. It also introduces different amounts of phase shift.

R

C

VIN VOUT

The maximum amount of phase shift available from a single R C combination is 90.If we have three such combinations with the same resistors and capacitors we will have up to 270 of phase shift.

Oscillators

VOUTR

C

R

C

R

C

VIN

V1 V2

To derive a relationship between input and output use Nodal Analysis.

@ VOUT R

VCjVV OUT

OUT )( 2

)()(CRj

CRjV

CRjVV OUTOUT

1112

@ V2 CjVVR

VCjVV OUT )()( 2

221

OUTVVCRj

VV 2

21 2

but )(CRj

CRjVV OUT

12

1

12

121 CRj

CRj

CRj

CRjVV OUT

Oscillators

@V1 CjVVR

VCjVVIN )()( 21

11

211 2 VVCRj

VVIN

but

1

12

121 CRj

CRj

CRj

CRjVV OUT

)(CRj

CRjVV OUT

12

CRj

CRj

CRj

CRj

CRj

CRj

CRjCRj

CRj

CRj

CRjVV OUTIN

11

12

12

112

1223

2

2344123 CRj

CRj

CRj

CRj

CRj

CRjVV OUTIN

2

2344123 CRj

CRj

CRj

CRj

CRj

CRjVV OUTIN

3

3232 223441

CRj

CRjCRjCRjCRjCRjCRjVV OUTIN

3

32651

CRj

CRjCRjCRjVV OUTIN

32

3

651 CRjCRjCRj

CRj

V

VGain

IN

OUT

As the frequency is varied the gain magnitude and phase will vary.

Oscillators

32

3

651 CRjCRjCRj

CRj

V

VGain

IN

OUT

32

3

561 CRCRjCR

CRjGain

2322

323

561

561

CRCRCR

CRCRjCRCRjGain

2322

6453

561

56

CRCRCR

CRCRCRjCRjGain

2322

35

2322

46

561

6

561

5

CRCRCR

CRCRj

CRCRCR

CRCRGain

2322

235246

561

65

CRCRCR

CRCRCRCRGain

Oscillators

46

351

5

6

CRCR

CRCRTanPhase

C= 10 nF and R = 10k

Range 100 Hz to 1600 Hz

Frequency Gain Phase

100 0.00024187 -107.824

200 0.00180268 -124.679

300 0.00547793 -139.942

400 0.01144164 -153.412

500 0.01946552 -165.165

600 0.02916192 -175.403

700 0.04013019 -184.346

800 0.05201965 -192.201

900 0.06454692 -199.142

1000 0.07749321 -205.315

1100 0.09069431 -210.839

1200 0.10402935 -215.813

1300 0.11741055 -220.316

1400 0.13077482 -224.417

1500 0.14407717 -228.17

1600 0.15728573 -231.621

RC Phase Shift

0

0.05

0.1

0.15

0.2

0.25

0 200 400 600 800 1000 1200 1400 1600

Frequency

-240

-220

-200

-180

-160

-140

-120

-100

~650Hz

Gain curvePhase

curve

Gain = ~0.035

356 CRCR

46

351

5

6

CRCR

CRCRTanPhase

16 2 CR

From the graph:-180 degrees phase shift occurs at 650HzThe gain at this point equals approximately 0.035Can we determine this from the equations?

For this to equal 0 (-180°)

222

6

1

RC

CR6

1 Hz

nFkRCf 7649

101026

1

26

1.

Oscillators

Now the gain – use: 6

1CR

2322

235246

561

65

CRCRCR

CRCRCRCRGain

22

22

661

65

66

1

66

1636

6365

2161

Gain 0345029

1

2162921629

6629

21629

22

2

.

To use this to produce an oscillator we need an amplifier with the following characteristics:

1. Gain = 29 (to give loop gain of unity)2. Phase shift of 180 (to give a loop phase shift of zero

(360)) Oscillators

R RR

C C C

Rin

Rf

Vout

AmplifierFeedback

R. C. Phase shift Oscillator

The values of Rin and Rf must be selected so that the gain of the amplifier equals twenty nine.

Rin

RfGain

29

RinRf 29Oscillators

Tuned Collector L C Oscillator. This is a development of the four-resistor biased

single stage transistor amplifier:

From equivalent circuit analysis it can be shown that the gain of the amplifier depends (for a limited range) on the value of the collector resistor. If this is replaced by a tuned L C parallel network then we will have a gain which is dependent on frequency.

CL

R

The impedance of this network can be determined in the following way:

CL XXRZ //)(

11

1

2

LCCRj

LjR

CjLjR

CjLjRZ

)(

)(

222

2

21

1

1 CRLC

CRjLCLjR

CRjLC

LjRZ

222

22322

1 CRLC

LCRCLjLjCRjLCRRZ

222

232

222 11 CRLC

CLCRLj

CRLC

RZ

222

232

222 11 CRLC

CLCRLj

CRLC

RZ

222

22322

1 CRLC

CLCRLRZ

R

CLCRLTanPhase

2321

Example

L = 10 mH, R = 10 , C = 10 nF.

Plot the value of Z over the range:

Oscillators

Frequency Impendance Phase12000 1747.19495 88.2391

12500 2049.5822 88.09622

13000 2453.70559 87.89268

13500 3022.7584 87.5925

14000 3885.71647 87.12191

14500 5352.88937 86.30287

15000 8405.51864 84.57067

15500 18571.4156 78.7095

16000 68655.401 -47.2146

16500 13729.3845 -82.6617

17000 7558.08851 -86.202

17500 5253.32655 -87.5099

18000 4049.04035 -88.1861

18500 3308.55693 -88.597

19000 2806.81681 -88.8716

19500 2444.08824 -89.0672

20000 2169.40337 -89.2129

LC Phase Shift

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

12000 13000 14000 15000 16000 17000 18000 19000 20000

Frequency

-90

-70

-50

-30

-10

10

30

50

70

90

15.9 kHzOscillators

Zero phase shift occurs at about 15.9 kHz and this approximately coincides with the peak in the gain (within the limits of our results)Can we determine this from the equations?

R

CLCRLTanPhase

2321 This must equal 0 so:

0232 CLCRL CLCRL 232

CLCRL 222 2

2

2

22 1

L

R

LCCL

CRL

kHzL

R

LCf

L

R

LC9115

1

2

112

2

2

2

.

Impedance at this frequency:

Oscillators

222222

22322

11 CRLC

R

CRLC

CLCRLRZ

Note: 62

210

2

2

10110111

L

Rand

LCL

R

LC

so LC

1

kCR

L

LCRC

RZ 10022

If we use this parallel network instead of the collector resistor we will have an amplifier whose gain is frequency dependent.

Oscillators

Amplifier Characteristic

12000 13000 14000 15000 16000 17000 18000 19000 20000

Frequency

Gain

GMAX

FR

The peak gain GMAX occurs at the resonant frequency FR.We now need to introduce feedback and this is done by converting the inductor of the parallel L C combination into a transformer and applying the feedback signal to the base. An extra capacitor is required to connect the other side of the transformer to ground for ac signals.

Amplifier

Feedback

Oscillators

When the dc bias is set up the new capacitor has no effect and the new inductor acts like a short circuit.When operating a fraction of the output is generated across the new inductor and this is effectively between the base and ground. The amount of feedback is determined by the turns ratio TR of the transformer.If we arrange for the turns ratio of the transformer to have a value given by:

MAXR GT

1

Then we will have a Loop Gain of unity only at the peak gain and therefore this part of the criterion is met only at a single frequency. The Loop Phase Shift can be kept at 0 for all frequencies. Oscillation frequency is given by:

LCFrequency

21

Oscillators

Generalised Oscillator Circuit

Z2 Z3

Z1

-A

The circuit shows an inverting amplifier with three complex components connected to it.

(These components could be inductors or capacitors)

Oscillators

This can be redrawn in the following way –Note Z1 = jX1 etc.

ROUT is the output resistance of the amplifier

jX1

jX2

jX3

ROUT

VinAVin -

+~

Oscillators

The load on the output ZLOAD of the amplifier is given by )//( 213 jXjXjXZLOAD

321

213 )(

jXjXjX

jXjXjXZLOAD

This allows us to determine the gain of the amplifier.

321

213

321

213

)(

)(

jXjXjXjXjXjX

R

jXjXjXjXjXjX

A

ZR

ZAGain

OUTLOADOUT

LOAD

)()(

)(

213321

213jXjXjXjXjXjXR

jXjXjXAGain

OUT

Oscillators

Using j x j = -1

)()(

)(

213321

213XXXXXXjR

XXXAGain

OUT

The feedback ratio for the network involves X1 and X2.

21

2jXjX

jXVoutVin

21

2

XX

X

Vout

VinFeedback

The Loop Gain LG for the circuit is equal to Gain x Feedback

21

2

213321

213)()(

)(

XX

X

XXXXXXjR

XXXALG

OUT

)()( 213321

23XXXXXXjR

XXALG

OUT

For this to be an oscillator the equation must have no j component and therefore:

0321 XXX

This means that for the above equation to be true the three components cannot be the same type (e.g. three capacitors) as this will produce a positive or negative result – there must be a mix e.g. two inductors and one capacitor.

The Loop Gain equation is therefore:

3

2

21

2

213

23)( X

XA

XX

XA

XXX

XXALG

Oscillators

As this value must be positive it means that X2 and X3 must be of the same sign – they must be either capacitors or inductors.

From this general design we can generate two oscillators:

1. X1 = Inductor X2 = Capacitor X3 = Capacitor

2. X1 = Capacitor X2 = Inductor X3 = Inductor

Oscillators

Colpitts Oscillator

032

1 C

j

C

jLj

C2 C3

L1

-A

Oscillation Frequency.0321 XXX

011

321

CjCjLj

32

32

321

2 11

CC

CC

CCL

321

322

CCL

CC

321

32CCL

CC

321

322

1

CCL

CCf

Oscillators

Amplifier Gain

13

2

X

XALG

3

2

2

3

2

31

1

C

C

Cj

Cj

X

XA

Example

100pF470pF

L1

-A

If we need a 25kHz oscillator,

What value of L1 do we require and what should the gain of the amplifier equal?

Oscillators

Hartley Oscillator

0321

LjLjC

j

L2 L3

C1

-A

Oscillation Frequency.

0321 XXX

01

321

LjLjCj

132

2 1)(

CLL

)(

1

321

2

LLC

)(

1

321 LLC

)(2

1

321 LLCf

Oscillators

Amplifier Gain

13

2

X

XALG

2

3

2

3

2

3L

L

Lj

Lj

X

XA

L2 L3

1nF

-10

Example

Determine the values of L2 and L3 if the oscillator is to operate at 10kHz

Oscillators

Crystal OscillatorA crystal oscillator is an electronic circuit that

uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency. This frequency is commonly used to keep track of time (as in quartz wristwatches), to provide a stable clock signal for digital integrated circuits, and to stabilize frequencies for radio transmitters.

Using an amplifier and feedback, it is an especially accurate form of an electronic oscillator. The crystal used therein is sometimes called a "timing crystal". On schematic diagrams a crystal is sometimes labelled with the abbreviation XTAL.

Oscillators

Crystal Modelling *A quartz crystal can be modelled as an electrical

network with a low impedance (series) and a high impedance (parallel) resonance point spaced

closely together.

C0

C1 L1 R1

* The above text is taken from http://en.wikipedia.org/wiki/Crystal_oscillator and is available under the Creative Commons Attribution-ShareAlike License.

Oscillators

Adding additional capacitance across a crystal will cause the parallel resonance to shift downward. This can be used to adjust the frequency that a crystal oscillator oscillates at. Crystal manufacturers normally cut and trim their crystals to have a specified resonant frequency with a known 'load' capacitance added to the crystal. For example, a 6pF 32kHz crystal has a parallel resonance frequency of 32,768 Hz when a 6.0pF capacitor is placed across the crystal. Without this capacitance, the resonance frequency is higher than 32,768.

* The above text is taken from http://en.wikipedia.org/wiki/Crystal_oscillator and is available under the Creative Commons Attribution-ShareAlike License.

Oscillators

Oscillators

This resource was created by the University of Wales Newport and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme.

© 2009 University of Wales Newport

This work is licensed under a Creative Commons Attribution 2.0 License.

The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England & Wales Licence.  All reproductions must comply with the terms of that licence.

The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher.

The name and logo of University of Wales Newport is a trade mark and all rights in it are reserved. The name and logo should not be reproduced without the express authorisation of the University.