# Oscillators

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```					                  Lecture 3 Oscillator
• Introduction of Oscillator
• Linear Oscillator
– Wien Bridge Oscillator
– RC Phase-Shift Oscillator
– LC Oscillator
• Stability

Ref:06103104HKN          EE3110 Oscillator   1
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
Ref:06103104HKN     EE3110 Oscillator           2
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.

Ref:06103104HKN        EE3110 Oscillator               3
Linear Oscillators
1.    Wien Bridge Oscillators
2.    RC Phase-Shift Oscillators
3.    LC Oscillators
4.    Stability

Ref:06103104HKN         EE3110 Oscillator   4
Integrant of Linear Oscillators
+              V
Vs                           Amplifier (A)            Vo
+
Positive              Vf        Frequency-Selective
Feedback Network ()
Feedback

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

Ref:06103104HKN                 EE3110 Oscillator              5
Basic Linear Oscillator
+            V
Vs                             A(f)                     Vo
+

Vf    SelectiveNetwork
(f)

Vo  AV  A(Vs  V f )         and          V f   Vo
Vo    A
          
Vs 1  A
If Vs = 0, the only way that Vo can be nonzero
is that loop gain A=1 which implies that
| A | 1       (Barkhausen Criterion)
A  0
Ref:06103104HKN                   EE3110 Oscillator                  6
Wien Bridge Oscillator
1             1                Frequency Selection Network
Let X C1        and XC2 
C1           C2                          Z1
Z1  R1  jX C1                                        R1   C1             Z2
1
1      1         jR2 X C 2
Z2                                     Vi                   C2   R2        Vo
 R2  jX C 2    R2  jX C 2

Therefore, the feedback factor,

Vo   Z2             ( jR 2 X C 2 / R2  jX C 2 )
          
Vi Z1  Z 2 ( R1  jX C1 )  ( jR 2 X C 2 / R2  jX C 2 )

 jR 2 X C 2

( R1  jX C1 )( R2  jX C 2 )  jR 2 X C 2

Ref:06103104HKN                  EE3110 Oscillator                            7
 can be rewritten as:
R2 X C 2

R1 X C 2  R2 X C1  R2 X C 2  j ( R1 R2  X C1 X C 2 )
For Barkhausen Criterion, imaginary part = 0, i.e.,
0.34
R1 R2  X C1 X C 2  0                                 0.32

Feedback factor 
0.3
1 1                                             =1/3
or   R1 R2                                            0.28

C1 C2                                   0.26
0.24
   1 / R1 R2C1C2                                    0.22
0.2
f(R=Xc)
Supposing,                                                     1

R1=R2=R and XC1= XC2=XC,                           Phase      0.5    Phase=0

RX C                                                0

3RX C  j ( R 2  X C )
2
-0.5

-1
Ref:06103104HKN                 EE3110 Oscillator                             Frequency   8
1
Example
By setting   RC , we get
1                                    Rf
Imaginary part = 0 and  
3                    R1
Due to Barkhausen Criterion,                               

Loop gain Av=1                                            +
where                                                           C     R
Vo
Av : Gain of the amplifier
Rf                        R             Z1
Av   1  Av  3  1                           C         Z2
R1
Rf
Therefore,              2                            Wien Bridge Oscillator
R1

Ref:06103104HKN                  EE3110 Oscillator                            9
RC Phase-Shift Oscillator
Rf

R1
                C      C   C

+
R   R   R

    Using an inverting amplifier
    The additional 180o phase shift is provided by an RC
phase-shift network

Ref:06103104HKN              EE3110 Oscillator                 10
Applying KVL to the phase-shift network, we have
V1  I1 ( R  jX C )  I 2 R                                                       C    C        C
0   I1 R              I 2 (2 R  jX C )  I 3 R                      V1                                     Vo
0                      I2R                     I 3 (2 R  jX C )
R        R        R
I1       I2       I3
Solve for I3, we get

R  jX C      R        V1
R       2 R  jX C   0
R
I3        0                    0
R  jX C      R             0
R       2 R  jX C      R
0          R        2 R  jX C

V1R 2
Or          I3 
( R  jX C )[(2R  jX C ) 2  R 2 ]  R 2 (2R  jX C )

Ref:06103104HKN                              EE3110 Oscillator                                            11
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  5RXC )  j ( X C  6 R 2 X C )
2           3

For 180o phase shift, the imaginary part = 0, i.e.,
X C  6 R 2 X C  0 or X C  0 (Rejected)
3

 XC  6R 2
2

1

6 RC                Note: The –ve sign mean the
and,                                      phase inversion from the
1
                   voltage
29
Ref:06103104HKN                   EE3110 Oscillator                      12
LC Oscillators
     The frequency selection
network (Z1, Z2 and Z3)                     
provides a phase shift of                   Av Ro
180o                                            ~
     The amplifier provides an                   +
2       Z1           Z2   1
Two well-known Oscillators:
• Colpitts Oscillator                                     Z3
Zp
• Harley Oscillator

Ref:06103104HKN          EE3110 Oscillator                     13
Av Ro                                          Z1
~                      V f   Vo                Vo
Z1  Z 3
+
Vf        Z1                 Z2      Vo         Z p  Z 2 //(Z1  Z 3 )
Z 2 ( Z1  Z 3 )
Z3                       
Zp              Z1  Z 2  Z 3
For the equivalent circuit from the output
Ro         Io
+            AvVi    Vo    Vo  Av Z p
    or   
+                    Zp Vo            Ro  Z p Z p    Vi Ro  Z p
 AvVi


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 )
Ref:06103104HKN                             EE3110 Oscillator                             14
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  jX 1 , Z 2  jX 2 and Z 3  jX 3
Av X 1 X 2
The loop gain becomes, A 
jR o ( 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
 Av X 1  AX
With imaginary part = 0, A                             v 1
X1  X 3   X2
X2
For Unit Gain &        180o     Phase-shift,            A  1    Av 
X1
Ref:06103104HKN                       EE3110 Oscillator                      15
Hartley Oscillator                       Colpitts Oscillator

C1
R     L1                          R
C                               L
C2
L2

1                           1
o                              o          CT 
C1C2
( L1  L2 )C                     LCT         C1  C2
L1                              C
gm                               gm  2
RL2                              RC1
Ref:06103104HKN           EE3110 Oscillator                   16
Colpitts Oscillator
Equivalent circuit
C1                                L
R
L              +
C2             C2     V                    R   C1
        gmV

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

Ref:06103104HKN           EE3110 Oscillator                   17
At node 1,                                           L        I1
V1  V  i1 ( jL)                                             node 1
I2                     V
where,                                     +                   I3
i1  jC2V                    C2     V              R                  C1
       gmV             I4
 V1  V (1   LC 2 )
2

Apply KCL at node 1, we have
V1
jC2V  g mV   jC1V1  0
R
1       
jC2V  g mV  V (1   2 LC 2 )  jC1   0
R       
For Oscillator V must not be zero, therefore it enforces,
     1  2 LC2 
 gm  
           R 

  j  (C1  C2 )   3 LC1C2  0
     R         
Ref:06103104HKN               EE3110 Oscillator                            18
     1  2 LC2 
 gm  
           R 
                         
  j  (C1  C2 )   3 LC1C2  0
     R         

Imaginary part = 0, we have
1             C1C2
o               CT 
LCT          C1  C2

Real part = 0, yields
C2
gm 
RC1

Ref:06103104HKN              EE3110 Oscillator        19
Frequency Stability
• The frequency stability of an oscillator is
defined as
1  d 
                 ppm/ o C
o  dT    o

• Use high stability capacitors, e.g. silver
mica, polystyrene, or teflon capacitors and
low temperature coefficient inductors for
high stable oscillators.

Ref:06103104HKN               EE3110 Oscillator   20
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
Ref:06103104HKN         EE3110 Oscillator       21
Wien-bridge oscillator with bulb stabilization

R                   C

+

R        C                                      irms
R2

Blub                       Operating
point

Vrms

Ref:06103104HKN                EE3110 Oscillator                             22
Wien-bridge oscillator with diode stabilization

Rf

R1

Vo
+
C        R

R
C

Ref:06103104HKN             EE3110 Oscillator        23
Twin-T Oscillator
low pass filter
Filter output


low pass region        high pass region

+

high pass filter

fr                           f

Ref:06103104HKN           EE3110 Oscillator                                               24
Bistable Circuit
vo
+Vcc
v+      +
Vth
vo                                            v1
v1           
-Vcc

vo                                               vo
+Vcc                                                        +Vcc

-Vth                                          -Vth               Vth
v1                                                    v1

-Vcc                                           -Vcc

Ref:06103104HKN                EE3110 Oscillator                                       25
A Square-wave Oscillator

vc


vo
vf    +
v
+ f

vc           v
¡Ð f

+vmax
vo
v
¡ Ð max

Ref:06103104HKN     EE3110 Oscillator    26

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