12 - 1 Electronics(3), 2012Prof. Tai-Haur Kuo
Basic Principles of Sinusoidal OscillatorsLinear oscillatorLinear region of circuit : linear oscillationNonlinear region of circuit : amplitudes stabilization
Barkhausen criterion
loop gain L(s) = β (s)A(s)characteristic equation 1-L(s) = 0oscillation criterion L(j0= A(j0) β(j0) = 1
Amplifier ASXOX
fX
Frequency-selectivenetwork
12 - 2 Electronics(3), 2012Prof. Tai-Haur Kuo
Basic Principles of Sinusoidal Oscillators (Cont.)at 0, the phase of the loop should be zero and themagnitude of the loop gain should be unity.Oscillation frequency 0 is determined solely by
A steep phase response results in a small for agiven change in phase
0
dωdφ∆φ∆ω0
∆φ
0
φ
0ω ω
12 - 3 Electronics(3), 2012Prof. Tai-Haur Kuo
Nonlinear Amplitude ControlTo sustain oscillation : βA>1a.overdesign for βA variationsb.oscillation will grow in amplitudepoles are in the right half of the s-planec.Nonlinear network reduces βA to 1 when the desired
amplitude is reachedpoles will be pulled to j-axis
12 - 4 Electronics(3), 2012Prof. Tai-Haur Kuo
Nonlinear Amplitude Control (Cont.)Limiter circuit for amplitude control linear region
54
5O
54
4B
32
2O
32
3A
i1
fO
RRRV
RRRVV
RRRV
RRRVV
)VRR(V
2D
1D
R1
OVIVRf
R2
R3
R4
R5
A
B
V
-V
12 - 5 Electronics(3), 2012Prof. Tai-Haur Kuo
Nonlinear Amplitude Control (Cont.)nonlinear region
)RR(1V
RRVL similarly,
)RR(1V
RRV
)VRR
RV(R
RRL V
5
4D
5
4
2
3D
2
3
D32
3
2
32-VVO DA
1
3f
R)R(RSlope
L
L
01
4f
R)R(RSlope
OV
IV
1
f
RRSlope
12 - 6 Electronics(3), 2012Prof. Tai-Haur Kuo
OPAMP-RC Oscillator CircuitsWien-bridge oscillator
_
C R
C R
1R
2R
SZ
PZ
OV
dV
2RR1sL
RC1ω
RCω1RCω 0. phase For
ωRC1ωRCj3
RR1
jωL
SCR1SCR3
RR1
ZZZ
RR1sL
1
2
0
00
1
2
1
2
sp
p
1
2
12 - 7 Electronics(3), 2012Prof. Tai-Haur Kuo
OPAMP-RC Oscillator Circuits (Cont.)Wien-bridge oscillator with a limiter
15V
_
16nF 10k
20.3k
1V
16nF10k
2D
1D 3kR3
1kR4
1kR5
3kR6
10k
b
a
15V
OV
12 - 8 Electronics(3), 2012Prof. Tai-Haur Kuo
OPAMP-RC Oscillator Circuits (Cont.)
10k
-
16nF 10k
10k
16nF
50k b ap
Ov
12 - 9 Electronics(3), 2012Prof. Tai-Haur Kuo
Phase-Shift OscillatorWithout amplitude stabilization
------- phase shift of the RC network is 180 degrees.Total phase shift around the loop is 0 or 360 degrees.
C
RRR
CCK-
12 - 10 Electronics(3), 2012Prof. Tai-Haur Kuo
Phase-Shift Oscillator (Cont.)With amplitude stabilization
16nF
RRC C C
16nF16nF
10k 10k
100k
1R
2R
3R
4R
V
V
2D
1D
tR1P
50k
_x Ov
12 - 11 Electronics(3), 2012Prof. Tai-Haur Kuo
Quadrature Oscillator
O1V
_
_
X R 2R
2R
2R
1R
2R
3R
4R
C
-V
V
1OP
2OPO2V
fR2R) (NominallyC
12 - 12 Electronics(3), 2012Prof. Tai-Haur Kuo
Quadrature Oscillator (Cont.)-------Break the loop at X, loop gain
oscillation frequency
-------Vo2 is the integral of Vo1
90° phase difference between Vo1 and Vo2
“quadrature” oscillator
222X
02
RCS1
=VV
=L(s)
RC1
0
12 - 13 Electronics(3), 2012Prof. Tai-Haur Kuo
Active-Filter Tuned OscillatorBlock diagram
High-distortion v2
High-Q bandpass low-distortion v1
V-
2V 1V
t
1V
V 2V
0f
t
V
V-
12 - 14 Electronics(3), 2012Prof. Tai-Haur Kuo
Active-Filter Tuned Oscillator (Cont.)Practical implementation
-
CRQRR R
R
1V2V 1R
C
12 - 15 Electronics(3), 2012Prof. Tai-Haur Kuo
A General Form of LC-Tuned Oscillator Configuration
Many oscillator circuits fall into a general form shown below
Z1,Z2,Z3 :capacitive or inductive
-
OR 2
3Z
2Z
1Z
3Z
2Z
1Z
1
3
OV
LZLZ
1
32
OV
OR
A
0I
13V 13V
13V13vVA- ˆ
13V
12 - 16 Electronics(3), 2012Prof. Tai-Haur Kuo
A General Form of LC-Tuned Oscillator Configuration (Cont.)
2
1
31
1
312
21
321
312321
21
332211
312321
21
13
13
31
113
13
)(
0
)()(
1
)()(ˆ
ˆ
XXAT
XXXA
XXXXXAT
XXX
XXXXXXjRXXAT
CXLX
jXZjXZjXZZZZZZZR
ZZAVVTV
ZZZV
RZZVAV
v
vv
O
v
O
vO
OL
LvO
01T n,oscillatio for
ecapacitanc for inductance for
, , if
12 - 17 Electronics(3), 2012Prof. Tai-Haur Kuo
A General Form of LC-Tuned Oscillator Configuration (Cont.)
With oscillation|T| = 1 and T = 0, 360, 720, … degree.
X1 & X2 must have the same sign if Av is positiveX1 & X2 are L, X3 = -(X1 + X2) is C
or X1 & X2 are C, X3 = -(X1 + X2) is LTransistor oscillators1.Collpitts oscillator
-- X1 & X2 are Cs, X3 is L2.Hartley oscillator
-- X1 & X2 are Ls, X3 is C
)ωC1
-=X or ωL=(X 1=T i.e.
12 - 18 Electronics(3), 2012Prof. Tai-Haur Kuo
LC Tuned OscillatorsTwo commonly used configurations1.Collpitts (feedback is achieved by using a capacitive
divider)2.Harley (feedback is achieved by using an inductive
divider)Configuration
1.Collpitts 2.Hartley
1C
L
2L
1L
C
R
2C
R
)V(da.c. groun DD )V(da.c. groun DD
12 - 19 Electronics(3), 2012Prof. Tai-Haur Kuo
LC Tuned Oscillators (Cont.)Collpitts oscillatorEquivalent circuit
R = loss of inductor + load resistance of oscillator + output resistance of transistor
πV
2 πsC V 2O π 2V V (1 s LC )L
R 1CπGmV2C
2 πsC VC
12 - 20 Electronics(3), 2012Prof. Tai-Haur Kuo
LC Tuned Oscillators (Cont.)
For oscillations to start, both the real and imaginary parts must be zero
Oscillation frequency
)c+ccc
L(
1=ω
21
210
22 m 1 2
3 2 21 2 1 2 m
232
m 1 2 1 2
1SC V+g v+( +SC )(1+S LC )V=0RLC 1S LC C +S ( )+S(C +C )+(g + )=0R R
1 w LC(g + )+j W(C +C ) W LC C =0R R
S=jw
12 - 21 Electronics(3), 2012Prof. Tai-Haur Kuo
LC Tuned Oscillators (Cont.)Gain
Oscillation amplitude1.LC tuned oscillators are known as self – limiting
oscillators. (As oscillations grown in amplitude, transistor gain is reduced below its small – signal value)
2.Output voltage signal will be a sinusoid of high purity because of the filtering action of the LC tuned circuit
Hartley oscillator can be similarity analyzed
)ccRg (Actually,
ccRg
1
2m
1
2m
12 - 22 Electronics(3), 2012Prof. Tai-Haur Kuo
Crystal oscillatorsSymbol of crystal
Circuit model of crystal
r
SCPC
L
12 - 23 Electronics(3), 2012Prof. Tai-Haur Kuo
Crystal oscillators (Cont.)Reactance of a crystal assuming r = 0
(Crystal is high – Q device)
pω
Sω ω0
inductive
capacitive
reactanceCrystal
SPSP
2P
2
2S
2
P
PS
2P
S
2S
PS
SP2
S
2
P
S
P
ωω then ,CC Ifωωωω
ωC1jjωZ
C1
C1
L1ω
LC1ω
Let
CLCCCs
LC1s
sC1
sC1sL
1sC
1Z(s)
12 - 24 Electronics(3), 2012Prof. Tai-Haur Kuo
Crystal oscillators (Cont.)The crystal reactance is inductive over the narrow frequency band between ws and wp
Collpitts crystal oscillatorConfiguration
R 1C
2C
crystal
)V(da.c. groun DD
12 - 25 Electronics(3), 2012Prof. Tai-Haur Kuo
Crystal oscillators (Cont.)Equivalent circuit
S P 1 2
0 SS
C <<C , C , C1ω =ω
LC
LPC
SC
-
2C πv 1CR
12 - 26 Electronics(3), 2012Prof. Tai-Haur Kuo
Crystal oscillators (Cont.)
2.2kΩ
-22V
C=300pF
gd strayC +C
1MHzXTAL
10M0.02μF
2N2608S
D
C -22V
CL
Bias circuit
(For AC,-22V and ground are the same=0)
uH~
L12261
12 - 27 Electronics(3), 2012Prof. Tai-Haur Kuo
Crystal oscillators (Cont.)
Since return ratio then X1 must be large for the loop gain to be greater than one.
X1 is very large when
PS
p
ωωωand ω to closes ω
1 2 3 3 1 2
2
2
1X +X +X =0 & X = X &X are inductiveωC
1 jZ =L//C= =1 1jω C+ ω C+jω L ω L
1 1 1X = >0 ωL> ω>1 ωC LCωC+ωL
v 1
2
A XT=X
For 1MHz crystal,
C=300pFL 84.4μH
1 1 1MHz2π LC
12 - 28 Electronics(3), 2012Prof. Tai-Haur Kuo
Bistable MultivibratorsMultivibrators bistable:two stable states
(3 types) monostable:one stable stateastable:no stable state
BistableHas two stable statesCan be obtained by connecting an amplifier in a
amplifier in a positive-feedback loop having a loop gain greater than unity. I.e. βA>1 where β=R1/(R1+R2)
12 - 29 Electronics(3), 2012Prof. Tai-Haur Kuo
Bistable Multivibrators (Cont.)Bistable circuit with clockwise hysteresis
Clockwise hysteresis (or inverting hysteresis)L+:positive saturation voltage of OPAMPL- :negative saturation voltage of OPAMP
-
-
1R 2R
Ov
Iv
THVTLV
-L
Lov
Iv0
+v
12 - 30 Electronics(3), 2012Prof. Tai-Haur Kuo
Bistable Multivibrators (Cont.)
– Hysteresis width = VTH - VTL
LRR
RβLV
LRR
RβLV
21
1TL
21
1TH
12 - 31 Electronics(3), 2012Prof. Tai-Haur Kuo
Noninverting Bistable CircuitCounterclockwise hysteresisConfiguration
2 1+ I 0
1 2 1 2
10 + + I TL TL +
2
10 - + I TH TH
2
R Rv =v +vR +R R +R
RFor v =L , v =0,v =v v = L ( )RRFor v =L , v =0,v =v v = L ( )R
-
-
1R 2R
Ov
Iv
THVTLV
-L
L
ov
Iv0
+v
12 - 32 Electronics(3), 2012Prof. Tai-Haur Kuo
Noninverting Bistable Circuit (Cont.)Comparator characteristics with hysteresisCan reject interference
t
THV
TLV0RV
ceinterferenwithcorruptedSignal
crossings zeroMultiple
12 - 33 Electronics(3), 2012Prof. Tai-Haur Kuo
Generation of Square and Triangular Waveforms using Astable Multivibrators
Can be done by connecting a bistable multivibrator with a RC circuit in a feedback loop.
THVTLV
-L
L2v
1v0
βLVTH
βLVTL
t
C R
-L
L
1v2v
t
12 - 34 Electronics(3), 2012Prof. Tai-Haur Kuo
Generation of Square and Triangular Waveforms using Astable Multivibrators (Cont.)
-
-
1R 2R
Ov-v R
- 0
0
-L To
L To
-L
L tOv 1T 2T
Time constant = RC
0 t
t
t
+v
βLVTH
βLVTL
βLVTH
βLVTL
t-v
tv
12 - 35 Electronics(3), 2012Prof. Tai-Haur Kuo
Generation of Square and Triangular Waveforms using Astable Multivibrators (Cont.)
During T1
During T2
βLLβ
TTt at βL Vif
RRRβ RC, whereeβLLLV
11
21
1t
1
1lnτ
ττ
assumed is LL ;ββTTT
βLLβ
TTt at βL Vif
eβLLLV
21
22
t
11lnτ2
1
1lnτ
τ
12 - 36 Electronics(3), 2012Prof. Tai-Haur Kuo
Generation of Triangular Waveforms
THVTLV
-L
L
2v
1v0
_
tC
R
Bistable
t
1V
2V1V 2V
2V
2T1T
TL
L
0
1V
2T1T-LSlopeRC
THV
TLV0
RCLSlope
tt
12 - 37 Electronics(3), 2012Prof. Tai-Haur Kuo
Generation of Triangular Waveforms (Cont.)During T1
During T2
To obtain symmetrical waveforms
;
1T
1TL TH C C0
TH TL1
1 L T LV V i dt where iC RC R
V VT RCL
TH TL
2
SimilarilyV VT RC
L
LLTT 21
L
12 - 38 Electronics(3), 2012Prof. Tai-Haur Kuo
Monostable MultivibratorsIts alternative name is “ one shot “Has one stable stateCan be triggered to a quasi-state
_
2C 2D
2R
3R1D 1C
B
1RA
-βL
-βL
βL
-L
L
)t(Bv
)t(v
)t(Av
T
+ D2(βL V )
D1V L To
L To
4R
E
)t(Ev
12 - 39 Electronics(3), 2012Prof. Tai-Haur Kuo
Monostable Multivibrators (Cont.)During T1
βL+ is greater then VD1
Stable state is maintained
)β1
1(CRT|L|ForV
)LβLLVln( CRT
)eV(LLβLβL(T)V
)eV(LL(t)V
13D1
D113
CRT
D1B
CRt
D1B
13
13
12 - 40 Electronics(3), 2012Prof. Tai-Haur Kuo
Monostable Multivibrators(Cont.)Monostable multivibrator using NOR gates
C
RDDV
NOR NOR
O2VinV Xv
1T
O1v
1T
DDV
O2v
TVV
0 1T Tt
inv
t t0 0
XV
t1T0
DDVDD
TDD
V 23VV
2VV DDT
o1V
DDV
12 - 41 Electronics(3), 2012Prof. Tai-Haur Kuo
Mono-stable Multivibrator (Cont.)
;
1
tRC
x DDT
RCx 1 T DD
DD1
DD T
DDT T
v V (1 e )
v (T ) V V (1 e )VT RCln RCln2 0.693RC
V VVwhere V V is NOR gate threshold voltage2
RCt
TDDx eVVv
0(0)vC
-
-O1V 0
DDV
R
C
-
XvcV
DDV
T1C V)(Tv
-
-O1 DDV V
R
C
-
XvcV
12 - 42 Electronics(3), 2012Prof. Tai-Haur Kuo
Mono-stable Multivibrator (Cont.)Monostable multivibrator with catching diode
DDV
RCconstant Time
C
R D
NOR NOR
O2VinV o1V
xV
XV
t1T0
5.6V
5VDV CRconstant Time f
forward resistance of diode
12 - 43 Electronics(3), 2012Prof. Tai-Haur Kuo
Astable Multivibrator Using NOR(or Inverter) Gates
Transient behavior(1)0<t<T1
:
o 1 D D
o 2 D D- t
R Cx D D T
- tR C
c x O 2 D D D D T
( i ) v :V 0 w h e n t= 0( i i ) v 0 V w h e n t= 0
( i i i ) v = ( V + V ) e
( iv ) v = v - V = - V + ( V + V ) e
C
Cv _
O2VO1VXV
R
o 1v
T 10 t
o 2v
2 T 1
D DV
2 T 1
T 10 t2 T 1
D DV
TV
TV
T 1
0 t
XvDD TV V
2D D
TVV
TVCv
T 10 t2 T 1
D DV
12 - 44 Electronics(3), 2012Prof. Tai-Haur Kuo
Astable Multivibrator Using NOR(or Inverter) Gates (Cont.)
(2)T1<t<(T1+T2)
1
1
o 1 D D 1
o 2 D D 1
( t-T )R C
x D D D D T
( t-T )R C
c x O 2 x D D D D T
( i) v :0 V w h e n t= T( ii) v :V 0 w h e n t= T
( ii i) v = V - (V + V )e
( iv )v = v -V = v = V - (V + V )e
12 - 45 Electronics(3), 2012Prof. Tai-Haur Kuo
Astable Multivibrator Using NOR(or Inverter) Gates (Cont.)
Oscillation frequency
x 1 Tt
RCDD T T
DD T1
T
DDT 1 2
0
v (T )=V
(V +V )e =VV +VT =RCln
VVIf V = , then T =RCln3 and T =RCln32
1 0.455oscillation frequency f = 2RCln3 RC
12 - 46 Electronics(3), 2012Prof. Tai-Haur Kuo
Astable Multivibrator Using NOR(or Inverter) Gates (Cont.)
With catching diode at
Asymmetrical square wave
RC0.721
2RCln21f
RCln2TT
0
21
21
DDT
R(ii)R2
V(i)V
C
Cv _
O2VO1VXV
R
0 DDV
C
Cv _
O2VO1VXV
2R1R
1D 2D
XV
12 - 47 Electronics(3), 2012Prof. Tai-Haur Kuo
The 555 IC TimerWidely used as both a monostable and astable multivibratorUsed as monostable multivibrator
)t(tv
TLt Vv
0
v(t)
1T0
)t(Ov
nS0011
nR0101
n+1QnQ
10
N/A
nQ 1
2
32
32 1
3
cc
cc
ccc n
VV
VV
VV R
CCV
_
_
R
S
Q
Q
1R
1R
1R
transistor Discharge
ComparatorThreshold
Trigger
1Q
OVTHV
Comparator
1
2
Totem-poleoutput stage
TLV
C cV
tV
)t(cv
0 1T
)e(1V RCtCC
32V V
CC
TH
12 - 48 Electronics(3), 2012Prof. Tai-Haur Kuo
The 555 IC Timer (Cont.)For 0 t T1
For t = T1,
0)V(V(0) eV(0)VVv CE(sat)RC
t
CCCCx
32VV)(Tv CC
TH1C
0)(V(0) RCln3
3V
V(0)VRClnTCC
CC1
12 - 49 Electronics(3), 2012Prof. Tai-Haur Kuo
The 555 IC Timer (Cont.)Used as an astable multivibrator
Oscillation frequency1 B 2 1 A BT R Cln2 ,T T (R R )Cln2
)Cln22R(R1
T1f
BA2
OV
Comparator
( )A BR R C
2 0, 13
1, 03
2 03 3
ccc
ccc
cc ccc
VV S R
VV S R
V VV S R
_
_
R
S
Q
Q
1R
1R
1R
transistor Discharge
ComparatorThreshold
Trigger
1Q
THV
1
2
Totem-poleoutput stage
TLV
C
cV
tV
R A
R B
555 timer chip
0 1T
CCVcc
T H2VV =
3
2Tt
ccTL
VV =3
VCC
12 - 50 Electronics(3), 2012Prof. Tai-Haur Kuo
Sine-Wave ShaperShape a triangular waveform into a sinusoidExtensively used in function generatorsNote:linear oscillators are not cost-effective for low
frequency application not easy to time over wide frequency ranges
0
Ov
fv tT2
T0
0
2T
Tt
12 - 51 Electronics(3), 2012Prof. Tai-Haur Kuo
Sine-Wave Shaper (Cont.)Nonlinear-amplification methodFor various input values, their corresponding output
values can be calculatedTransfer curve can be obtained and is similar to
CRCR
1Q 2Q
R
II
CCV
EEV-
Ov _
wave)(Sine
wave)r(Triangula
iv
12 - 52 Electronics(3), 2012Prof. Tai-Haur Kuo
Sine-Wave Shaper (Cont.)Breakpoint methodPiecewise linear transfercurveLow-valued R is assumed V1 and V2 are constant
D2 01in2
45
51DinD10
D22in1
inout1in1
V Vto Vlimit on is DV VRR
R)VV(VVVV
) Vdrop (voltage on is DVVVVVVVV
12 - 53 Electronics(3), 2012Prof. Tai-Haur Kuo
Sine-Wave Shaper (Cont.)
1R
V
3R
2R
3R
3R
1R
V
3D
1D
2D
4D
4RIn
2V
1V
1V
2V
5R
5R
Inout
2V
1V1V
2V0
1
2
3
4
5
out
12 - 54 Electronics(3), 2012Prof. Tai-Haur Kuo
Precision Rectifier CircuitsPrecision half-wave rectifier --- “superdiode”
An alternate circuit
_
R
OV
AV
"Superdiode"
iV
OV
iV
11
_OV1D
2D 2R
1R
iV1
1
OV
0
0 iV
12 - 55 Electronics(3), 2012Prof. Tai-Haur Kuo
Precision Rectifier Circuits (Cont.)Application:Measure AC voltages
Average ;where Vp is the peak amplitude of an input sinusoid
1
2P1 R
RπVV
_
OV
1D
2D
2R
1R
iV
1V
_3R
4R
C
1A2A
12 - 56 Electronics(3), 2012Prof. Tai-Haur Kuo
Precision Rectifier Circuits (Cont.)
If ;ωmin is the lowest expected frequency of the input sine wave
min4
ωCR
1
3
4
1
2P2 R
RRR
πVV
12 - 57 Electronics(3), 2012Prof. Tai-Haur Kuo
Precision Full-Wave Rectifiers
1
LR
AD
BD
A
B
C
or
A
B
12 - 58 Electronics(3), 2012Prof. Tai-Haur Kuo
Precision Full-Wave Rectifiers (Cont.)
1A
_
2A
_
2R
AiV
1D
2D
LR
OV
OV
iVP
1R
E
F
C
12 - 59 Electronics(3), 2012Prof. Tai-Haur Kuo
Peak RectifierWith load
buffered
1D1A
2A
2D
Civ
Ov
R
-
-
__
iv
-
_
Ov
-LRC
diode Super