CHAPTER 12 SIGNAL GENERATORS AND
WAVEFORM-SHAPING CIRCUITS
Chapter Outline
12.1 Basic Principles of Sinusoidal Oscillators
12.2 Op Amp-RC Oscillators
12.3 LC and Crystal Oscillators
12.4 Bistable Multivibrators
12.5 Generation of Square and Triangular Waveforms using Astable Multivibrators
12.6 Generation of a Standardized Pulse-The Monostable Multivibrators
12.7 Integrated-Circuit Timers
12.8 Nonlinear Waveform-Shaping Circuits
12.9 Precision Rectifier Circuits
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12.1 BASIC PRINCIPLES OF SINUSOIDAL OSCILLATORS
Types of Oscillators
 Linear oscillator:
 Employs a positive feedback loop consisting of an amplifier and a frequency-selective network.
 Some form of nonlinearity has to be employed to provide control of the amplitude of the output.
 Nonlinear oscillator:
 Generates square, triangular, pulse waveforms.
 Employs multivibrators: bistable, astable and monostable.
The Oscillator Feedback Loop and Oscillation Criterion
 Positive feedback loop analysis:
Af (s) 
xo
A( s)

xi 1  A( s)  ( s)
loop gain : L( s)  A( s)  ( s)
L( j0 )  A( j0 )  ( j0 )  1
 Barkhausen criterion:
 The phase of loop gain should be zero at 0 .
 The magnitude of the loop gain should be unity at 0 .
 The characteristic equation has roots at s =  j0 .
 Stability of oscillation frequency:
 0 is determined solely by the phase characteristics.
 A “steep” function  () results in a more stable frequency.
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Nonlinear Amplitude Control




Oscillation: loop gain A = 1
Growing output: loop gain A > 1
Decaying output: loop gain A < 1
Oscillation mechanism:
 Initiating oscillation: loop gain slightly larger than unity (poles in RHP).
 Gain control: nonlinear network reduces loop gain to unity (poles on j-axis).
Limiter Circuits for Amplitude Control
 For small amplitude (D1 off, D2 off)
 incremental gain (slope) = Rf /R1
 For large negative swing (D1 on, D2 off)
 incremental gain (slope) = (Rf ||R4) /R1
 For large positive swing (D1 off, D2 on)
 incremental gain (slope) = (Rf ||R3) /R1
vA 
R2
R3
vO
V
R2  R3
R2  R3
vB  
R5
R4
vO
V
R4  R5
R4  R5
L 
R4
R  R5
V 4
VD
R5
R5
L  
R3
R  R3
V 2
VD
R2
R2
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12.2 OP AMP-RC OSCILLATOR CIRCUITS
Wien-Bridge Oscillator
 R  Zp
1  R2 / R1
L( s )  1  2 
 L( s) 
3  sRC  1 / sRC
 R1  Z p  Z s
L( j ) 



1  R2 / R1
3  j (RC  1 / RC )
For L = 1 → 0=1/RC and R2/R1 = 2.
To initiate oscillation → R2/R1 = 2 + .
Limiter is used for amplitude control.
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Phase-Shift Oscillator






The circuit oscillates at the frequency for which the phase shift of the RC network is 180.
Only at this frequency will the total phase shift around the loop be 0 or 360.
The minimum number of RC sections is three.
K should be equal to the inverse of the magnitude of the RC network at oscillation frequency.
Slightly higher K is used to ensure that the oscillation starts.
Limiter is used for amplitude control.
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Quadrature Oscillator
 Based on the two-integrator loop without damping.
 R1, R2, R3, R4, D1 and D2 are used as limiter.
 Loop gain:
t
vO 2
V
2 v
1
 2v   O1 dt  o 2 
C 0 2R
Vo1 sRC
t
vO1 
V
1 vX
1
dt  o1 

C0 R
Vx sRC
L( s ) 
Vo 2
1
 2 2 2
Vx
s RC
 Poles are initially located in RHP (decreasing Rf ) to
ensure that oscillation starts.
 Too much positive feedback results in higher output
distortion.
 vO2 is purer than vO1 because of the filtering action
provided by the second integrator on the peak-limited
output of the first integrator.
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Active-Filter Tuned Oscillator
 The circuit consists of a high-Q bandpass
filter connected in a positive-feedback loop
with a hard limiter.
 Any filter circuit with positive gain can be
used to implement the bandpass filter.
 Can generate high-quality output sine waves.
 Have independent control of frequency,
amplitude and distortion of the output
sinusoid
sinusoid.
Final Remark
 Op amp-RF oscillators ~ 10 to 100kHz.
 Lower limit: passive components.
 Upper limit: frequency response and slew
rate of op amp.
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12.3 LC AND CRYSTAL OSCILLATORS
LC Tuned Oscillators




Colpitts oscillator: capacitive divider.
Hartley oscillator: inductive divider.
Utilize a parallel LC circuit between base and collector.
R models the overall losses.
Analysis of Colpitts Oscillators
0  1 / L1 / C1  1 / C2 1
0  1 / ( L1  L2 )C
sC2V  g mV  ( sC1  1 / R)(1  s 2 LC2 )V  0
s 3 LC1C2  s 2 LC2 / R  s (C1  C2 )  ( g m  1 / R)  0
1  2 LC2
(gm  
)  j  (C1  C2 )   3 LC1C2  0
R
R


 0  1 / L1 / C1  1 / C2 
1
 g m R  C2 / C1
 LC-tuned oscillators utilize the nonlinear transistor I-V characteristics for amplitude control (self-limiting).
 Collector (drain) current waveforms are distorted due to the nonlinear characteristics.
 Output voltage is a sinusoid with high purity because of the filtering action of the LC tuned circuit.
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Complete Circuit for a Colpitts Oscillator
DC Analysis
RE
AC Analysis
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Crystal Oscillators
 Crystal impedance:


1
Z ( s)  1 /  sC p 

sL  1 / sCs 

s 2  1 / LCs
1
Z (s) 
sC p s 2  [(C p  Cs ) / LC p Cs ]
s  1 / LCs
 p  1 / L(1 / Cs  1 / C p ) 1
Z ( j )   j




1
C p
  2  s2 


 2 2 
p 

Crystal reactance is inductive over very narrow frequency ( s to p ).
The frequency band is well defined for a given crystal.
Use the crystal to replace the inductor of the Colpitts oscillators.
Oscillation frequency is dominated by Cs (much smaller than other C’s).
0  1 / LCs  s
 Crystals are available with resonance frequencies KHz ~ hundred MHz.
 The oscillation frequency is fixed (tuning is not possible).
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12.4 BISTABLE MULTIVIBRATORS
Bistable Characteristics
 Positive feedback is used for bistable multivibrator.
 Stable states:
(1) vO = L+ and v+ = L+R1/(R1+R2).
(2) vO = L and v+ = LR1/(R1+R2).
 Metastable state: vO = 0 and v+ = 0.
Transfer Characteristics of the Inverting Bistable Circuit




Initially vO = L+ and v+ = L+R1/(R1+R2) → vO change stage to L when vI increases to a value of L+R1/(R1+R2).
Initially vO = L and v+ = L R1/(R1+R2) → vO change stage to L+ when vI decreases to a value of L R1/(R1+R2).
The circuit exhibits hysteresis with a width of (VTH  VTL).
Input vI is referred to as a trigger signal which merely initiates or triggers regeneration.
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Transfer Characteristics of the Noninverting Bistable Circuit
 Initially vO = L+ and v+ = vI R2/(R1+R2) + L+ R1/(R1+R2) > 0
→ vO change stage to L when vI decreases to a value (VTL) that causes v+ = 0
→ VTL = L+(R1/R2)
 Initially vO = L and v+ = vI R2/(R1+R2) + L R1/(R1+R2) < 0
→ vO change stage to L+ when vI increases to a value (VTH) that causes v+ = 0
→ VTL = L(R1/R2)
Application of the Bistable Circuit as a Comparator
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Limiter Circuits for Precise Output Levels
L  VZ 1  VD
L  (VZ 1  VD )
L  VZ  VD1  VD 2
L  (VZ  VD 3  VD 4 )
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12.5 GENERATION OF SQUARE AND TRIANGULAR WAVEFORMS
USING ASTABLE MULTIVIBRATORS
Operation of the Astable Multivibrator
 F
For vO = L+ andd v+ = vO R1/(R1+R2) > 0
→ v is charged toward L+ through RC
→ vO change stage to L when v = v+
 For vO = L and v+ = vOR1/(R1+R2) < 0
→ v is discharged toward L through RC
→ vO change stage to L+ when v = v+
1   ( L / L )
1 
1   ( L / L )
 T1   ln
1 
v  L  ( L    L )e t / RC  L  ( L    L )e t /  T1   ln
v  L  ( L    L )e t / RC  L  ( L    L )e t /
T  2 ln
1 
1 
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Generation of Triangular Waveforms
Triangular can be obtained by replacing the low-pass RC circuit with an integrator.
The bistable circuit required is of the noninverting type.
VTH  VTL L
V  VTL

 T1  RC TH
T1
RC
L
VTH  VTL  L
V  VTL

 T2  RC TH
T2
RC
 L
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12.6 GENERATION OF A STANDARDIZED PULSE
– THE MONOSTABLE MULTIVIBRATORS
Op-Amp Monostable Multivibrators
 Circuit components:
 Trigger: C2 , R4 and D2
 Clamping diode: D1
 R4 >> R1 → iD4  0
 The circuit has one stable state:
 vO = L+
 vB = VD1  0
 D1 and D2 on
 Operation of monostable multivibrator
 Negative step as the trigger input
 D2 conducts heavily
 vC is pulled below vB
 vO changes state to L and vC becomes negative
 D1 and D2 off and C1 is discharged toward L
 vO changes state to L as vB = vC
 Stays in the stable state
 Positive trigger step turns off D2 (invalid trigger)
vB (t )  L  ( L  VD1 )e  t / R3C1
vB (T )  L  ( L  VD1 )e T / R3C1    L
 V L 
 1 
 T  C1 R3 ln D1    C1 R3 ln

L
L




1




 

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12.7 INTEGRATED-CIRCUIT TIMERS
Monostable Multivibrator using 555 Timer Circuit
S
R
 Stable state: S = R = 0 and Q = 0
 Q1 on and vC = 0
 Trigger (vtrigger < VTL): S = 1 and Q = 1
 Q1 off and vC is charged toward VCC
 Trigger pulse removal (vtrigger > VTL): S = R = 0 and Q = 1
 Q1 off and vC is charged toward VCC
 End of recovery period (vC = VTH): R = 1 and Q = 0
 Q1 on and vC is discharged toward GND
 Stable state: vC drops to 0 and S = R = 0 and Q = 0
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0
0
1
0
0
0
00
10
vC (t )  VCC (1  e  t / RC )
T  RC ln 3  1.1RC
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Astable Multivibrator using 555 Timer Circuit
vC (t )  VCC  (VCC  VTL )e  t / C ( RA  RB )
TH  C ( RA  RB ) ln 2  0.69C ( RA  RB )
vC  VTH e  t / CRB
TL  CRB ln 3  0.69CRB
T  TH  TL  0.69CRB
Duty cycle 
TH
R  RB
 A
TH  TL RA  2 RB
 Operation of astable multivibrator
 Initially vC = 0: S/R = 1/0 and Q = 1  Q1 off and vC is charged toward VCC thru RA and RB
 vC reaches VTH: S/R = 0/1 and Q = 0  Q1 on and vC is discharged toward GND thru RB
 vC reaches VTL: S/R = 1/0 and Q = 1  Q1 off and vC is charged toward VCC thru RA and RB
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12.8 NONLINEAR WAVEFORM-SHAPING CIRCUITS
Nonlinear Amplification Method
 Use amplifiers with nonlinear transfer characteristics
to convert triangular wave to sine wave.
 Differential pair with an emitter degeneration
resistance can be used as sine-wave shaper.
Breakpoint Method
 R4 , R5 >> R1 , R2 and R3 to avoid loading effect.
 –V1 < vIN < V1 :
 vO = vIN
 –V2 < vIN < –V1 or V1 < vIN < V2
 vO = V1 + (vIN – V1) R5 / (R4 + R5)
 vIN < –V2 or V2 < vIN
 vO = V2
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12.9 PRECISION RECTIFIER CIRCUITS
Precision Half-Wave Rectifier (Superdiode)
 Operation of super diode:
 vO = vI for vI > 0
 vO = 0 for vI < 0
 The offset voltage (~0.5V) can be eliminated.
 Nonideal characteristics are masked by the loop gain.
 Disadvantages:
 Reverse bias may damage the input terminals.
 Op saturation (open loop) degrades the speed.
Improved Precision Half-Wave Rectifier
 Rectifier operation:
 vI > 0 : D1 off, D2 on  vO = 0
 vI < 0 : D1 on, D2 off  vO = (R2 / R1) vI
 The feedback loop remains closed at all times.
 The op amp remains in its linear operation region.
 Can prevent the delay due to saturated op amp.
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AC Voltage Measurement
 The circuit consists of a half-wave rectifier and
a first-order low-pass filter.
 Dc component of v1 is (Vp/)(R2/R1).
 The LPF corner frequency should be much smaller than
the input sine wave to reduce error by the harmonics.
Precision Full-Wave Rectifier
 Full-wave rectifier is implemented by combining two rectifiers with a common load.
 Diode A is replaced by a superdiode.
 Diode B is replaced by the inverting precision half-wave rectifier.
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Precision Peak Rectifiers
Buffered Precision Peak Detector
Precision Clamping Circuit
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