Transcript Oscillators - City University of Hong Kong

Lecture 3 Oscillator
• Introduction of Oscillator
• Linear Oscillator
– Wien Bridge Oscillator
– RC Phase-Shift Oscillator
– LC Oscillator
• Stability
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Oscillators
Oscillation: an effect that repeatedly and
regularly fluctuates about the mean value
Oscillator: circuit that produces oscillation
Characteristics: wave-shape, frequency,
amplitude, distortion, stability
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Application of Oscillators
• Oscillators are used to generate signals, e.g.
– Used as a local oscillator to transform the RF
signals to IF signals in a receiver;
– Used to generate RF carrier in a transmitter
– Used to generate clocks in digital systems;
– Used as sweep circuits in TV sets and CRO.
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Linear Oscillators
1.
2.
3.
4.
Wien Bridge Oscillators
RC Phase-Shift Oscillators
LC Oscillators
Stability
Ref:06103104HKN
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Integrant of Linear Oscillators
Vs
+
V

Amplifier (A)
Vo
+
Positive
Feedback
Vf
Frequency-Selective
Feedback Network ()
For sinusoidal input is connected
“Linear” because the output is approximately sinusoidal
A linear oscillator contains:
- a frequency selection feedback network
- an amplifier to maintain the loop gain at unity
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Basic Linear Oscillator
Vs
+

V
Vo
A(f)
+
Vf
Vo  AV  A(Vs  V f )
V
A
 o 
Vs 1  A
SelectiveNetwork
(f)
and
V f  Vo
If Vs = 0, the only way that Vo can be nonzero
is that loop gain A=1 which implies that
| A | 1
A  0
Ref:06103104HKN
(Barkhausen Criterion)
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Wien Bridge Oscillator
Let X C1 
1
1
X

and C 2
C1
C2
Frequency Selection Network
Z1
Z1  R1  jX C1
R1
Z2
C1
1
1
 jR2 X C 2
1 
Z2   


R

jX
R2  jX C 2
C2 
 2
Vi
C2
R2
Vo
Therefore, the feedback factor,
Vo
( jR2 X C 2 / R2  jX C 2 )
Z2
 

Vi Z1  Z 2 ( R1  jX C1 )  ( jR2 X C 2 / R2  jX C 2 )
 jR2 X C 2

( R1  jX C1 )(R2  jX C 2 )  jR2 X C 2
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 can be rewritten as:
R2 X C 2

R1 X C 2  R2 X C1  R2 X C 2  j ( R1R2  X C1 X C 2 )
For Barkhausen Criterion, imaginary part = 0, i.e.,
0.34
Feedback factor 
R1R2  X C1 X C 2  0
1 1
or R1 R2 
C1 C2
   1 / R1 R2C1C 2
0.32
0.3
0.28
=1/3
0.26
0.24
0.22
0.2
f(R=Xc)
Supposing,
R1=R2=R and XC1= XC2=XC,
0.5
Phase
RX C

3RX C  j ( R 2  X C2 )
1
0
-0.5
-1
Ref:06103104HKN
Phase=0
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Frequency
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Example
1
By setting   RC , we get
1


Imaginary part = 0 and
3
Due to Barkhausen Criterion,
Loop gain Av=1
where
Av : Gain of the amplifier
Av   1  Av  3  1 
Therefore,
Ref:06103104HKN
Rf
R1
Rf
Rf
R1

+
C
C
R1
2
R
Vo
Z1
R
Z2
Wien Bridge Oscillator
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RC Phase-Shift Oscillator
Rf
R1

C
+


C
R
C
R
R
Using an inverting amplifier
The additional 180o phase shift is provided by an RC
phase-shift network
Ref:06103104HKN
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Applying KVL to the phase-shift network, we have
V1  I1 ( R  jX C )  I 2 R
0   I1 R
 I 2 (2 R  jX C )  I 3 R
0 
 I2R
C
I3 
Or
R
2 R  jX C
R
Vo
 I 3 (2 R  jX C )
I1
V1
0
0
R  jX C
R
0
C
V1
Solve for I3, we get
R  jX C
R
0
C
R
2 R  jX C
R
R
I2
R
I3
R
0
R
2 R  jX C
V1R 2
I3 
( R  jX C )[(2R  jX C ) 2  R 2 ]  R 2 (2R  jX C )
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The output voltage,
V1R3
Vo  I 3 R 
( R  jX C )[(2R  jX C ) 2  R 2 ]  R 2 (2R  jX C )
Hence the transfer function of the phase-shift network is given by,
Vo
R3
  3
V1 ( R  5RX C2 )  j ( X C3  6R 2 X C )
For 180o phase shift, the imaginary part = 0, i.e.,
X C3  6 R 2 X C  0 or X C  0 (Rejected)
 X C2  6 R 2
1

6 RC
and,
1
 
29
Ref:06103104HKN
Note: The –ve sign mean the
phase inversion from the
voltage
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LC Oscillators


The frequency selection
network (Z1, Z2 and Z3)
provides a phase shift of
180o
The amplifier provides an
addition shift of 180o

Av Ro
~
+
2
Z1
Two well-known Oscillators:
• Colpitts Oscillator
• Harley Oscillator
Ref:06103104HKN
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Z2
1
Z3
Zp
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Av Ro
Z1
V f  Vo 
Vo
Z1  Z 3
~
+
Z1
Vf
Z2
Z p  Z 2 //(Z1  Z 3 )
Vo
Z3
Zp

Z 2 ( Z1  Z 3 )
Z1  Z 2  Z 3
For the equivalent circuit from the output
Ro
Io
+
+
 AvVi
Zp Vo

 AvVi
Vo
Vo  Av Z p

or

Ro  Z p Z p
Vi Ro  Z p
Therefore, the amplifier gain is obtained,
Vo
 Av Z 2 ( Z1  Z 3 )
A 
Vi Ro ( Z1  Z 2  Z 3 )  Z 2 ( Z1  Z 3 )
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The loop gain,
 Av Z1Z 2
A 
Ro ( Z1  Z 2  Z 3 )  Z 2 ( Z1  Z 3 )
If the impedance are all pure reactances, i.e.,
Z1  jX1, Z2  jX 2 and Z3  jX 3
Av X 1 X 2
The loop gain becomes, A 
jRo ( X 1  X 2  X 3 )  X 2 ( X 1  X 3 )
The imaginary part = 0 only when X1+ X2+ X3=0


It indicates that at least one reactance must be –ve (capacitor)
X1 and X2 must be of same type and X3 must be of opposite type
With imaginary part = 0, A 
For Unit Gain &
Ref:06103104HKN
180o
 Av X 1
AX
 v 1
X1  X 3
X2
Phase-shift,
A  1 
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Av 
X2
X1
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Hartley Oscillator
R
Colpitts Oscillator
R
L1
C
C2
L2
1
( L1  L2 )C
L1
gm 
RL2
o 
Ref:06103104HKN
C1
1
o 
LCT
C
gm  2
RC1
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L
C1C2
CT 
C1  C2
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Colpitts Oscillator
Equivalent circuit
R
C1
C2
L
L
C2
+
V

gmV
R
C1
In the equivalent circuit, it is assumed that:
 Linear small signal model of transistor is used
 The transistor capacitances are neglected
 Input resistance of the transistor is large enough
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At node 1,
V1  V  i1 ( jL)
where,
i1  jC2V
C2
 V1  V (1   LC2 )
+
V

2
I1
L
I2
gmV
node 1
I3
R
I4
V
C1
Apply KCL at node 1, we have
V1
jC2V  g mV   jC1V1  0
R
1

jC2V  g mV  V (1   2 LC2 )  jC1   0
R

For Oscillator V must not be zero, therefore it enforces,

1  2 LC2 
 g m  
  j  (C1  C2 )   3 LC1C2  0
R
R 


Ref:06103104HKN

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
1  2 LC2 
 g m  
  j  (C1  C2 )   3 LC1C2  0
R
R 



Imaginary part = 0, we have
o 
1
LCT
C1C2
CT 
C1  C2
Real part = 0, yields
gm 
C2
RC1
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Frequency Stability
• The frequency stability of an oscillator is
defined as
1  d 


o  dT    o
ppm/o C
• Use high stability capacitors, e.g. silver
mica, polystyrene, or teflon capacitors and
low temperature coefficient inductors for
high stable oscillators.
Ref:06103104HKN
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Amplitude Stability
• In order to start the oscillation, the loop gain
is usually slightly greater than unity.
• LC oscillators in general do not require
amplitude stabilization circuits because of
the selectivity of the LC circuits.
• In RC oscillators, some non-linear devices,
e.g. NTC/PTC resistors, FET or zener
diodes can be used to stabilized the
amplitude
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Wien-bridge oscillator with bulb stabilization
R
C
+
R
C

irms
R2
Blub
Operating
point
Vrms
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Wien-bridge oscillator with diode stabilization
Rf
R1

Vo
+
C
C
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R
R
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Twin-T Oscillator
low pass filter
Filter output

low pass region
high pass region
+
high pass filter
f
fr
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Bistable Circuit
vo
+Vcc
v+
+
vo

v1
Vth
-Vcc
vo
vo
+Vcc
-Vth
+Vcc
v1
-Vcc
Ref:06103104HKN
v1
-Vth
Vth
v1
-Vcc
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A Square-wave Oscillator
vc

vo
vf
+
v
+ f
vc
vo
v
+vmax
¡Ð f
v
¡ Ð max
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