Oscillator
• Introduction of Oscillator • Linear Oscillator
– Wien Bridge Oscillator – RC Phase-Shift Oscillator – LC Oscillator
• Stability
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
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.
Linear Oscillators
1. Wien Bridge Oscillators 2. RC Phase-Shift Oscillators 3. LC Oscillators 4. Stability
Integrant of Linear Oscillators
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
Σ
+
+ Amplifier (A)
Frequency-SelectiveFeedback Network (β)
Vf
Vs VoVε
PositiveFeedback
Basic Linear Oscillator Σ
+
+
SelectiveNetwork β(f)
Vf
Vs VoVεA(f)
)( fso VVAAVV +== ε and of VV β=
βAA
VV
s
o
−=⇒1
If Vs = 0, the only way that Vo can be nonzero is that loop gain Aβ=1 which implies that
01||=∠
=
β
β
AA (Barkhausen Criterion)
Wien Bridge Oscillator Frequency Selection Network Let
11
1C
XC ω= and
111 CjXRZ −=2
21C
XC ω=
22
22
1
222
11
C
C
C jXRXjR
jXRZ
−
−=⎥
⎦
⎤⎢⎣
⎡
−+=
−
Therefore, the feedback factor,
)/()()/(
222211
2222
21
2
CCC
CC
i
o
jXRXjRjXRjXRXjR
ZZZ
VV
−−+−
−−=
+==β
222211
22
))(( CCC
C
XjRjXRjXRXjR
−−−
−=β
Vi Vo
R1 C1
R2C2
Z1Z2
β can be rewritten as:
)( 2121221221
22
CCCCC
C
XXRRjXRXRXRXR
−+++=β
For Barkhausen Criterion, imaginary part = 0, i.e.,
02121 =− CC XXRR
Supposing, R1=R2=R and XC1= XC2=XC,
2121
2121
/1
11or
CCRR
CCRR
=⇒
=
ω
ωω
)(3 22CC
C
XRjRXRX
−+=β
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
Feed
back
fact
or β
-1
-0.5
0
0.5
1
Phas
e
Frequency
β=1/3
Phase=0
f(R=Xc)
Example Rf
+
−
R
R
C
CZ1
Z2
R1
Vo
By setting , we get Imaginary part = 0 and
RC1
=ω
31
=β
Due to Barkhausen Criterion, Loop gain Avβ=1 where Av : Gain of the amplifier
1
131RR
AA fvv +==⇒=β
21
=RRfTherefore, Wien Bridge Oscillator
RC Phase-Shift Oscillator
+
−
Rf
R1
R R R
C C C
§ Using an inverting amplifier § The additional 180o phase shift is provided by an RC
phase-shift network
Applying KVL to the phase-shift network, we have
R R R
C C CV1 Vo
I1 I2 I3Solve for I3, we get
)2(0
)2( 0)(
32
321
211
C
C
C
jXRIRIRIjXRIRI
RIjXRIV
−+−=
−−+−=
−−=
C
C
C
C
C
jXRRRjXRR
RjXRRjXRR
VRjXR
I
−−
−−−
−−−
−−
−−
=
202
00002
3
1
)2(])2)[(( 222
21
3CCC jXRRRjXRjXR
RVI−−−−−
=Or
The output voltage,
)2(])2)[(( 222
31
3CCC
o jXRRRjXRjXRRV
RIV−−−−−
==
Hence the transfer function of the phase-shift network is given by,
)6()5( 2323
3
1 CCC
o
XRXjRXRR
VV
−+−==β
For 180o phase shift, the imaginary part = 0, i.e.,
RC
RXXRX CCC
61
6X
(Rejected) 0or 0622
C
23
=
=⇒
==−
ω
and,
291
−=β
Note: The –ve sign mean the phase inversion from the voltage
LC Oscillators
+
−
~Av Ro
Z1 Z2
Z3
12
Zp
§ The frequency selection network (Z1, Z2 and Z3) provides a phase shift of 180o
§ The amplifier provides an addition shift of 180o
Two well-known Oscillators: • Colpitts Oscillator • Harley Oscillator
+~Av Ro
Z1 Z2
Z3Zp
Vf Vo
321
312
312
)(
)//(
ZZZZZZZZZZ p
++
+=
+=
For the equivalent circuit from the output
po
pv
i
o
p
o
po
iv
ZRZA
VV
ZV
ZRVA
+
−==
+
−or
Therefore, the amplifier gain is obtained,
)()()(
312321
312
ZZZZZZRZZZA
VV
Ao
v
i
o
++++
+−==
oof VZZ
ZVV
31
1
+== β
Zp−AvVi
Ro
+−
−
+Vo
Io
The loop gain,
)()( 312321
21
ZZZZZZRZZA
Ao
v
++++
−=β
If the impedance are all pure reactances, i.e., 332211 and , jXZjXZjXZ ===
The loop gain becomes, )()( 312321
21
XXXXXXjRXXA
Ao
v
+−++=β
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
2
1
31
1
XXA
XXXAA vv =
+
−=βWith imaginary part = 0,
For Unit Gain & 180o Phase-shift, 1
2 1XXAA v =⇒=β
R L1
L2C
RC1
C2L
Hartley Oscillator Colpitts Oscillator
To LC
1=ω
1
2
RCCgm =
21
21
CCCCCT +
=CLLo )(1
21 +=ω
2
1
RLLgm =
Colpitts Oscillator
RC1
C2L
Equivalent circuit
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
+−Vπ gmVπ
R C1C2
L
)(11 LjiVV ωπ +=
At node 1,
where,
πω VCji 21 =
)1( 22
1 LCVV ωπ −=⇒
Apply KCL at node 1, we have
0111
2 =+++ VCjRVVgVCj m ωω ππ
01)1( 122
2 =⎟⎠
⎞⎜⎝
⎛ +−++ CjR
LCVVgVCj m ωωω πππ
[ ] 0)(121
321
22
=−++⎟⎟⎠
⎞⎜⎜⎝
⎛−+ CLCCCj
RLC
Rgm ωω
ω
For Oscillator Vπ must not be zero, therefore it enforces,
+−Vπ
gmVπR C1C2
Lnode 1
I1
I2I3
I4
V1
Imaginary part = 0, we have
[ ] 0)(121
321
22
=−++⎟⎟⎠
⎞⎜⎜⎝
⎛−+ CLCCCj
RLC
Rgm ωω
ω
To LC
1=ω
1
2
RCCgm =
21
21
CCCCCT +
=
Real part = 0, yields
Frequency Stability • The frequency stability of an oscillator is
defined as • Use high stability capacitors, e.g. silver
mica, polystyrene, or teflon capacitors and low temperature coefficient inductors for high stable oscillators.
Cppm/ T
o
oo dd
ωωω
ω =⎟⎠
⎞⎜⎝
⎛⋅1
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
Wien-bridge oscillator with bulb stabilization
Vrms
irms
Operating point
+
−
R
R
C
C
R2
Blub
Wien-bridge oscillator with diode stabilization
Rf
+
−
R
R
C
C
R1
Vo
Twin-T Oscillator
+
−
low pass filter
high pass filter
low pass region high pass region
fr f
Filter output
Bistable Circuit
+
−vo
v1
v+Vth
+Vcc
-Vcc
vo
v1
-Vth
+Vcc
-Vcc
vo
v1 Vth
+Vcc
-Vcc
vo
v1-Vth
A Square-wave Oscillator
+
−
vo
vc
vf
vc
vo
+vf¡Ðvf+vmax¡Ðvmax