Post on 16-Apr-2022
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
Chapter III - EEL 7300 3
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
Chapter III - EEL 7300 4
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)
Chapter III - EEL 7300 5
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
Chapter III - EEL 7300 6
3.1 Introduction
E. Vittoz, Low-Power Crystal and MEMS Oscillators – The Experience
of Watch Developments, Springer, 2010.
Quartz resonator
Chapter III - EEL 7300 7
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
Chapter III - EEL 7300 8
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=
Chapter III - EEL 7300 9
( ) & ( ) 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−
3.2 The Barkhausen criterion for oscillation
xs Amplifier A
Frequency-
selective
network
xo
xf
xi( )( ) ( )
f jφ ω
s
xA jω β jω e
x=
Chapter III - EEL 7300 10
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
3.2 The Barkhausen criterion for oscillation
( )( )
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
Chapter III - EEL 7300 11
The magnitude of the oscillations does not grow indefinitely, but is limited by some
amplitude-limiting mechanism (opamp saturation, nonlinear voltage gain, etc)
3.2 The Barkhausen criterion for oscillation
( )( )
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)
Chapter III - EEL 7300 12
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
Chapter III - EEL 7300 13
( )j ( )H j
180
120
90
[o]
2
1
3RC = RC
-20 dB/decade
3.3 The phase shift oscillator
( )( )
( ) ( )
( )
2
2;
1
180 arctan
o
i
o
V jH j
V j RC
j RC
= =+
= −
http://www.allaboutcircuits.com/worksheets/opamp10.html
Chapter III - EEL 7300 14
( )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
3.3 The phase shift oscillator
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
Chapter III - EEL 7300 15
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?
Chapter III - EEL 7300 16
( )j ( )H j
0
?
90
[o]
1
?
?oscRC = RC
-20 dB/decade
3.3 The phase shift oscillator
Chapter III - EEL 7300 17
3.3 The phase shift oscillator
Chapter III - EEL 7300 18
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
Chapter III - EEL 7300 19
+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
Chapter III - EEL 7300 20
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
Chapter III - EEL 7300 21
3.5 Quadrature oscillators
Quadrature oscillator
Chapter III - EEL 7300 22
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
Chapter III - EEL 7300 23
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
Chapter III - EEL 7300 24
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
Chapter III - EEL 7300 25
Appendix 3.2 – Cross-coupled oscillator
MOSFET model
Chapter III - EEL 7300 26
Enhanced swing ring
oscillator (ESRO)
VDD=3.7 mVVDD=4.7 mV
Chapter III - EEL 7300 27
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
Chapter III - EEL 7300 28
+−
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
Chapter III - EEL 7300 29