Chapter III SINUSOIDAL OSCILLATORS

CONTENTS:

1. Introduction

2. The Barkhausen criterion for oscillation

3. The phase shift oscillator

4. The Colpitts oscillator

5. Quadrature oscillators

6. Wien bridge oscillators

7. Cross-coupled oscillators

Chapter III

SINUSOIDAL OSCILLATORS

Chapter III - EEL 7300 1

3.1 Introduction

Types of oscillators:

• Sinusoidal oscillators produce (nearly) sinusoidal outputs

• Relaxation oscillators operate by alternately charging and

discharging an energy storage element (capacitor)

Oscillator: signal-generating circuit which produces its

own periodic signal.

Applications:

▪ Clock generation for timing

▪ Frequency synthesis

▪ Voltage/current/temperature/radiation-controlled oscillators

▪ Ring oscillators for IC technology characterization

▪ Carrier generation for FM/AM transmission Chapter III - EEL 7300 2

3.1 Introduction"A picture of the water clock

created by Su Sung.” 1094

Water clock of the

Ancient Egyptian

“Clepsydra”

The Science of Timekeeping, Application Note 1289, Hewlett Packard


3.1 Introduction

https://www.youtube.com/watch?v=7otHVM-ZCEM&t=50s

Water clock (clepsydra) – São Paulo– Shopping Iguatemi

also in Porto Alegre– Shopping Iguatemi

https://iguatemi.com.br/saopaulo/contato

Dear students

Please, send a message to

asking the administrator of the Shopping Iguatemi website to

post a video of the water clock.

https://www.youtube.com/watch?v=rjiHY7pqPHk


https://iguatemi.com.br/saopaulo/contato

https://www.youtube.com/watch?v=rjiHY7pqPHk

3.1 Introduction

E. Vittoz, Low-Power Crystal and MEMS Oscillators – The Experience of

Watch Developments, Springer, 2010.

Time duration for modern science:

(femto) fs (10-15 s) to

13.799±0.021)×109 years (~440 x 1015 s)

Accuracy of atomic clock ~ 10-14 =10 parts/quadrillion

Achievable accuracy of purely

electronic circuits ~ 10-3

Error of ~1.5 minute/day

Wristwatches need an

accuracy of ~ 10-6 (1 ppm)

Error of ~30 seconds/year

Wristwatch consumption ~ 0.1 - 1 W

wikipedia

Lithium Coin Cell - CR2016: 3 V 90 mAh

Question: How many years would last the CR2016 for 1 W-power

wristwatch? 31 years!!!!!! (Optimistic assumption that the total energy

of the cell can be used)


3.1 Introduction

A. Sheikholeslami, “A capacitor Analogy – Part 3,” IEEE Solid-State Circuits Mag., pp. 7-8, 51, Winter 2017.

Challenge: Assume R=0, V1(t=0)=1 V (left capacitor), V2(t=0)=0 V (right

capacitor). Draw the waveforms of I(t), V1(t), V2(t), and VL(t) and of the energy

in each component for C= 1 F and L = 1 H.

Liquid Oscillator & Electrical Oscillator

1


3.1 Introduction

E. Vittoz, Low-Power Crystal and MEMS Oscillators – The Experience

of Watch Developments, Springer, 2010.

Quartz resonator


3.1 Introduction

E. Vittoz, Low-Power Crystal and MEMS Oscillators – The Experience of

Watch Developments, Springer, 2010.wikipedia

Crystal oscillators

The Pierce oscillator

1. Very accurate frequency

2. Low loss

3. But the crystal cannot be integrated

Implementation of a CMOS oscillator


VB

0

3.2 The Barkhausen criterion for oscillation

A feedback

amplifier

xs Amplifier A

Frequency-

selective

network

xo

xf

( )( )

1 ( ) ( )

( ) ( ) ( ) : loop gain

of

s

x A sA s

x A s s

L s A s s

= =−

=

o i

i s f

f o

x Ax

x x x

x x

=

= +

=

xi

( )( ) ( ) ( ) ( ) jφ ωA jω β jω A jω β jω e=


( ) & ( ) A s s are, in most cases, stable functions (poles in the complex left half-plane).

, however, can give rise to poles in the complex right half-plane! 1 ( ) ( )A s s−


xs Amplifier A

Frequency-

selective

network

xo

xf

xi( )( ) ( )

f jφ ω

s

xA jω β jω e

x=


Loop gain

The Barkhausen criterion for oscillation is:

( )

( )

2 ;

1

osc

osc

φ ω ω πN N

Aβ jω a

= =

=

xs(t) 1 a<1

txf(t)

a=1

a>1

t

t

Response to a sine wave: In-phase signals

add constructively. Let us use superposition.

The response to cos(t) is acos(t);

The response to acos(t) is a2cos(t);

The response to a2cos(t)is a3cos(t)…

The superposition of all these signals is:

( )2 3 coscos 1 ... 1;

1

1

f

f

ωtx ωt a a a a

a

x a

= + + + + = −

if

diverges if


( )( )

1 ( ) ( )

of

s

x A sA s

x A s s= =

− xs Amplifier A

Frequency-

selective

network

xo

xf

xi

Let us consider the frequency 2 for which (2)=2N, N

( )

( )

2

2

1

1

πN

πN

Aβ jω

Aβ jω

→

→

stable Af (j2) finite

oscillator Af (j2) =

(unstable)

Loop gain

The Barkhausen criterion for oscillation

( )2 1;πNAβ jω N


The magnitude of the oscillations does not grow indefinitely, but is limited by some

amplitude-limiting mechanism (opamp saturation, nonlinear voltage gain, etc)


( )( )

1 ( ) ( )= =

−

of

s

x A sA s

x A s s

http://wikieducator.org/Sinusoidal_Oscillator

What about the start

up of the oscillator?

▪Thermal noise

▪Switching noise

▪Stored energy (L, C)


http://wikieducator.org/images/7/7c/Concept_RCPhaseshift1.JPG

http://wikieducator.org/images/6/62/Concept_Wienbridge.JPG

3.3 The phase shift oscillator

What is this for?

Vi Vo

( )( )

( ) ( )

( )

2

2;

1

180 arctan

o

i

o

V jH j

V j RC

j RC

=+

= −

http://www.allaboutcircuits.com/worksheets/opamp10.html


( )j ( )H j

180

120

90

[o]

2

1

3RC = RC

-20 dB/decade


( )( )

( ) ( )

( )

2

2;

1

180 arctan

o

i

o

V jH j

V j RC

j RC

= =+

= −

http://www.allaboutcircuits.com/worksheets/opamp10.html


( )3 j ( )

3

H j

540

360

270

[o]

8

1

3RC = RC

-60 dB/decadeSingle

stage

( )( )

( ) ( )

3

3

2

2;

1

3 3 180 arctano

H jRC

j RC

= +

= −

3 stages

( )3

H j

3=RC

Loop gain

at

3

1exp( 2 / 3) 1= =j

Errors in R, C, opamp can result in gain <1

at the frequency for which =2. Solution:

increase slightly the voltage gain


Gain>1

Phase=2/3

Gain1

Phase=2/3

Gain 1

Phase=2/3

Vi Vo

3=RC

Loop gain

at

1 The amplitude of

oscillation tends to

increase without limit,

but the opamp output

voltage is limited Loop gain>1 Loop gain=1


3.3 The phase shift oscillatorProblem: Determine the

oscillation frequency in terms of R

and C. What is the minimum value

of RF/RG for oscillation? What are

the relative magnitudes of the

signal at the opamps outputs?


( )j ( )H j

0

?

90

[o]

1

?

?oscRC = RC

-20 dB/decade





LIMITER

Problem: Determine the oscillation frequency

in terms of R and C. What is the minimum

value of RF/R for oscillation?

Phase-shift oscillator with a limiter

for amplitude stabilization

Analysis of a limiter example. Determine the voltage transfer characteristic of

the limiter shown below. Use the ideal model of the diode


+V

─V

Rf=39 k,

R1=10 k,

R2=R5=15 k,

R3=R4=10 k,

V= 12V

( )3 2

2 3 2 3

A O

R RV V v

R R R R= + +

+ +

If vI=0, vO=0, D1 and D2 OFF. Thus,

( )10 15

12 0 4.8 V15 10 15 10

AV = + + =+ +

On the edge of conduction of D1 we have VD1=VA=0, and VD1=0.

( )10 15

12 0 8 V15 10 15 10

A O OV v v= + + = = −+ +

For vO< -8 V (vI> (10/39)8 V) ,

R3 is in parallel with Rf . The

voltage gain becomes

─ (Rf // R3 ) / R1.

iD

vDVON=0

iD

+ -vD

+8 V

─8 V

(1)

(2)

iD

vDvON

+ -

iD

vD

D1 D2 vo vA vB

OFF OFF -RF/R1 vI

ON OFF (1) -vON

OFF ON (2) vON

3

2 3

2 FI ON

2 3 1

R+V -

R +R

R Rv -v

R +R R

4

4 5

2 FI ON

2 3 1

R-V -

R +R

R Rv <v

R +R R

Problem: Determine the voltage transfer characteristic of the limiter shown

in Fig (a). Use the Von model of the diode


3.4 The Colpitts oscillator

Colpitts oscillator – simplified analysis (lossless inductor)

Gm+1/R+1/(RB +r)

R=1k

+−

L

C1

C2

I=0

Io=GmVx

Vx

Vo

Vy

1. Open the loop at node VX;

2. Calculate the transfer function Vy/Vx (s);

3. Make s=j;

4. Verify the requirements for oscillation, i.e.

|Vy/Vx(josc)|>1 (Barkhausen criteria)

The AC analysis of the open-loop gain yields

2 1 21 2

1 2

1 2

1;

; / /

osc m

m m B B B

B

C CL G R C C

C C

rG g R R R

R r

= +

= =+

http://www.davidbridgen.com/Colpitts.htm


3.5 Quadrature oscillators

Quadrature oscillator


Inverting

integrator

Noninverting integrator

Problem:a) What’s the condition for

oscillation of the circuit shown? b) What is the oscillation frequency?c) Assuming equal Rs and Cs,

determine the oscillation frequency in terms of R and C.

d) What’s the condition for the quadrature signals to have equal amplitudes?

3.6 Wien bridge oscillators

20.3 103(1 1%)

10G

+= = +

Problem: Assuming that VON= 0.6 V, show that the

amplitude of vo is approximately 5.8 V


Problem: Determine the oscillation

frequency in terms of R and C.

Appendix 3.1 –

The Wien-Bridge oscillator: analysis of the loop gain

1

2

3

R

R1 R2

C

R C

vf

Ao→ vo

1

2

3

vs

Ao→

R1

R2

R C

CR

vf

Z1(s)

Z2(s)

ideal


vs +

+

vf

vo

feedback

network

(s)

G

1RCs

R)s(Z;

sC

1RCs)s(Z

)s(Z)s(Z

)s(Z)s(

)s(G1

G

v

vA

21

21

2

s

of

+=

+=

+=

−==

2 2 2 2 2 2

2 2 2

2 2 2

sRC GsRCβ(s)= 1-Gβ(s)=1-

s R C +3sRC+1 s R C +3sRC+1

s R C +(3-G)sRC+11-Gβ(s)=

s R C +3sRC+1 → G=3 the circuit oscillates

at RC1o =

If G=3 (R2=2R1) the circuit oscillates at .

To ensure that oscillations will start, R2=2R1+ (roots of

1-G(s) should lie in the RHP).

Note that (s)=1/3 for s=j/RC.

In a practical design, include op amp non-idealities

RC1o =

2

1

RG=1+

R


Appendix 3.2 – Cross-coupled oscillator

MOSFET model


Enhanced swing ring

oscillator (ESRO)

VDD=3.7 mVVDD=4.7 mV


App. 3.3 – Enhanced-swing cross-coupled oscillator

A tough problem: Given the scheme of the Colpitts

oscillator, check whether the formulas below are correct1

2 2 1 2

1 2 1 2

1 2 1 2

1 11;

; ; / /

osc osc

m m m B B B

B

C CL L

C C C C

rG R C C G g R R R

R r

−

+ = =

+

= =+

R


+−

L

C1

C2

I=0

Io=GmVx

Vx

Vo

Vy

Gm+1/R+1/(RB +r)

R=1k

David W. Allan, Neil Ashby, Clifford C. Hodge, The Science of Timekeeping Application Note 1289; Hewlett Packard, Online Available at http://www.allanstime.com/Publications/DWA/Science_Timekeeping/TheScienceOfTimekeeping.pdf

A. B. Grebene, Bipolar and MOS Analog Integrated Circuit Design, Wiley, 2003.

A. S. Sedra and K. C. Smith, Microelectronic Circuits, any edition.

R. C. Jaeger and T. Blalock, Microelectronic Circuit Design, McGraw-Hill, New York, any edition.

R. Mancini (editor-in-chief), Op Amps for Everyone, Texas Instruments.

http://www.ieee-uffc.org/frequency-control/learning-vig-tut.asp


http://www.ieee-uffc.org/frequency-control/learning-vig-tut.asp

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Chapter III SINUSOIDAL OSCILLATORS

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