Oscillators
• Positive feedback is deliberately applied to an
amplifier in order to sustain instability.
• The feedback is frequency selective so that oscillation
occurs at a particular frequency only.
• Applications: Provide repetitive ac signals, clock
generators, transmitters, watches, digital/analog
processors, etc.
Common Types:
– RC oscillators, freq. up to a few hundred kHz,
– LC oscillators at much higher frequencies,
– Crystal controlled oscillators which provide very
stable oscillators.
Positive feedback system
A f (s) =
Vo
A(s)
A(s)
(s) =
=
Vi
1! A(s)! (s) 1! T (s)
loop gain
1
• The required condition for the feedback loop to
provide sinusoidal oscillations of frequency ωo is that:
1 − T ( jω o ) = 0
T ( jωo ) = +1
Barkhausen Criterion
• This requires that:
1) ∠T ( jωo ) = 0o or multiples of 360o.
2)
T ( jω o ) = 1
Phase-shift oscillator
v1
sRC
=
v I 1 + sRC
3
v3 ⎛ sRC ⎞
=⎜
⎟ = β (s)
v I ⎝ 1 + sRC ⎠
⎛ R ⎞⎛ sRC ⎞
T ( s ) = A( s ) β ( s ) = ⎜ − 2 ⎟⎜
⎟
R
1
+
sRC
⎠
⎝
⎠⎝
3
2
T ( jω ) =
( − R2 R )( − jω 3 R 3C 3 )
[1 − 3ω 2 R 2C 2 ] + jωRC [3 − ω 2 R 2C 2 ]
• If the condition T( jωo) = +1 is to be met, then the
real term in the denominator must be zero:
1 − 3ωo2 R 2C 2
=0
! !o =
! R $! 1 $
T ( j! o ) = # 2 &# &
" R %" 8 %
1
3RC
R2 R = 8
T ( jωo ) = 1
Phase-shift oscillator without buffers
(− R2 R )(sRC )3
T (s) =
1 + 5sRC + 6( sRC ) 2 + ( sRC )3
j ( R2 R )(ωRC )3
T ( jω ) =
[1 − 6(ωRC ) 2 ] + jωRC[5 − (ωRC ) 2 ]
3
• If the condition T(jωo) = +1 is to be met, then the
real term in the denominator must be zero:
⇒ ωo =
1 − 6(ωo RC ) = 0
2
T ( jω o ) = +
1
6 RC
R2
29 R
R2 R = 29
T ( jωo ) = 1
ELEC2002 – Academic Year 2011/12
Complementary
slide
ii)
(analysis of phase-shift oscillator without buffers
R2
C v1 C
R
vo ! v1 v1 v1 ! v2
= +
1
1
R
sC
sC
v1 ! v2 v2 v2 ! 0
= +
1
R 1 +R
sC
sC
v2
v
=! o
1
R
2
+R
sC
From (3), v2 =
"
" (vo ! v1 )sC =
v2 C
R
R
–
vO
+
v1
+ (v1 ! v2 )sC
R
(1)
v2 v2 (sC)
+
R 1+ sRC
(2)
" (v1 ! v2 )sC =
v2 (sC)
v
=! o
1+ sRC
R2
(3)
!vo
(1+ sRC)
sR2C
From (2),
!
1
sC $
v (1+ sRC) !
1
1 $
v1 (sC) = v2 # sC + +
) #1+
+
& ' v1 = ( o
&
"
" sRC 1+ sRC %
R 1+ sRC %
sR2C
From (1),
!
$
1
vo (sC) = v1 # sC + + sC & ' v2 (sC)
"
%
R
4
Wien-bridge oscillator
⎛ R ⎞⎛ Z p ⎞⎟
T ( s ) = ⎜⎜1 + 2 ⎟⎟⎜
⎝ R1 ⎠⎜⎝ Z p + Z s ⎟⎠
Zp =
Zs =
R
1 + sRC
1 + sRC
sC
⎛ R ⎞⎡
⎤
1
T ( s) = ⎜⎜1 + 2 ⎟⎟⎢
⎥
⎝ R1 ⎠⎣ 3 + sRC + (1 sRC) ⎦
⎛ R ⎞⎡
⎤
1
T ( jω ) = ⎜⎜1 + 2 ⎟⎟⎢
⎥
R
3
+
j
ω
RC
+
(
1
j
ω
RC
)
⎣
⎦
1 ⎠
⎝
• If the condition T(jωo) = +1 is to be met, then:
jω o RC +
1
=0
jω o RC
⇒ ωo =
1
RC
5
• Hence, at ω o = 1 RC
⎛ R ⎞⎛ 1 ⎞
T ( jω o ) = ⎜⎜1 + 2 ⎟⎟⎜ ⎟
⎝ R1 ⎠⎝ 3 ⎠
• If
R2
= 2 , then,
R1
T ( jω o ) = 1
The second Barkhaunsen criterion is met and sinusoidal
oscillations will be achieved.
LC oscillators
Z L = [(Z1 Ri ) + Z3 ] Z 2
A(s) = Av
β ( s) =
ZL
Z L + Ro
Z1 Ri
( Z1 Ri ) + Z 3
6
T (s) =
Av ( Z1 Ri ) Z 2
Ro [(Z1 Ri ) + Z 2 + Z 3 ] + Z 2 [(Z1 Ri ) + Z 3 ]
• To simplify, assume that Ri >> Z1 . Thus:
T (s) =
Av Z1Z 2
Ro [ Z1 + Z 2 + Z 3 ] + Z 2 [ Z1 + Z 3 ]
• If the impedances are purely reactive (inductive or
capacitive), then
Z1 = jX1, Z2 = jX2, and Z3 = jX3, where X = ωL for
an inductance and X = –1/ωC for a capacitance.
T ( jω ) =
− Av X 1 X 2
jRo [ X 1 + X 2 + X 3 ] − X 2 [ X 1 + X 3 ]
• For T(jω) to be real with no phase shift, then:
X1 + X 2 + X 3 = 0
which provides the frequency of oscillation. Thus:
T ( jω ) =
Av X 1 X 2
AX
− Av X 1
= v 1 =
X 2[ X1 + X 3 ] X1 + X 3
X2
Since T(jω) must be +ve, X1 and X2 must have same sign.
(Note: the amplifier gain is assumed negative, e.g., CS stage)
7
• If X1 and X2 are inductors, and X3 is a capacitor, the
circuit is called a Hartley oscillator.
• If X1 and X2 are capacitors, and X3 is an inductor, the
circuit is called a Colpitts oscillator.
• In the case of a Hartley oscillator (X1 = ωL1, X2 = ωL2,
X3 = –1/ωC3):
1
ωo =
( L1 + L2 )C3
and for the oscillation to be maintained at ωo:
( L2 L1 ) = Av
• In the case of a Colpitts oscillator (X1 = –1/ωC1,
X2 = –1/ωC2, X3 = ωL3):
ωo =
1
⎛ CC ⎞
L1 ⎜⎜ 1 2 ⎟⎟
⎝ C1 + C2 ⎠
and for the oscillation to be maintained at ωo:
(C1 C2 ) = Av
8
FET-based Hartley oscillator:
ac-coupling
At resonance, the gain is:
Av = − g m (rd RD )
g m (rd RD ) ≥ ( L2 L1 )
FET-based Colpitts oscillator:
At resonance, the gain is:
Av = − g m (rd RD )
g m (rd RD ) ≥ (C1 C2 )
9
Crystal oscillator:
• X1, X2 and X3 can use more complex reactance
elements in the Hartley and Colpitts oscillators.
• Consider:
XL = ωL
XC = -1/ωC
• Now consider reactance of a series LC circuit:
X = ωL −
1
ωC
at high-frequencies is
inductive
at low-frequencies is
capacitive
10
• Oscillators with very high frequency accuracy and
stability can be formed using quartz crystals.
• The crystal is a piezoelectric device that vibrates in
response to electrical stimulus.
• The crystal can be modelled by a very high Q
resonant circuit:
Not pure reactance because of Rs, but since Q is very large Rs may be ignored
• Typical parameters for a 10-MHz crystal are:
Rs
15 Ω
Cs
25 fF
Ls
10.132 mH
Cp
6 pF
Q
42440
fs
10 MHz
fp
10.021 MHz
11
Crystal reactance versus frequency:
• Assuming that the operating frequency is chosen in
the range ωs < ω < ωp, the crystal can be used to
replace the inductor in a Colpitts oscillator.
FET-based Colpitts crystal oscillator:
L3 replaced by
crystal
At frequencies above 20-MHz, LN is placed as shown
to neutralize the package capacitance of the crystal
LN ≈
1
ω o2Co
12
CMOS crystal oscillator:
Vo
13