55:041 Electronic Circuits
Oscillators
Sections of Chapter 15 + Additional Material
A. Kruger
Oscillators 1
Stability
Recall definition of loop gain: T(jω) = βA
A( j )
Af ( j ) 
1  T ( j )
If T(jω) = -1, then
We can write
A( j )
Af ( j ) 

11
Instability
T ( j )  T ( j ) 
Equivalent conditions for stability
T ( j )  1  less than 180
Gain margin: when the amplifier phase shift is 180o , how much
headroom/margin before the gain is 1 and the amplifier becomes unstable?
Gain margin: when the amplifier gain is 1, how much more headroom/margin
before the phase shift is180o amplifier becomes unstable?
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Oscillators 2
Barkhausen Criterion
The condition 𝑇(𝑗𝜔) = −1 is called the Barkhausen criterion
Note that this formulation assumes negative feedback. In some
instances, we use explicit positive feedback and then the condition
is 𝑇 𝑗𝜔 = +1.
The total phase shift through the amplifier and feedback network
must be N×360o. This true for negative and positive feedback.
The magnitude of the loop gain must be exactly 1
Loop gain < 1 => oscillations die out
Loop gain > 1 => oscillations grow and clip
at supply rails
In practice, make loop gain > 1 and to start oscillation and then
use some automatic gain control to limit loop gain to 1 (not
covered well in textbook)
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Oscillators 3
RC Phase Shift Oscillator
60o Phase shift
60o Phase shift
3
v3  jRC 

   ( )
vI  1  jRC 
60o Phase shift
Gain + 180o Phase shift
 R  jRC   R2   jRC 
T ( j )   2 
   
3
R
1

j

RC
 
  R  1  jRC 
3
R
A 2
R
3
 jRC RC 
R 
T ( j )   2 
 1 T(jω) = -1 (Barkhausen criterion)
2 2 2
2 2 2
R
1

3

R
C

j

RC
3


R
C
 
1
2 2 2


o
This means the imaginary part must be zero: 1  3o R C  0
3RC
2





At this frequency:
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 
 

j 3 1 3
R 
 R 1
T ( j )   2 
  2 
 R  0  j 3 1 33  1 3
 R 8
R2
8
R
Oscillators 4
RC Phase Shift Oscillator
180o Phase shift
Gain + 180o Phase shift
Same idea, analysis more difficult because phase shift networks load each other
o 
1
6 RC
R2
 29
R
Will this work too?
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Oscillators 5
Wien Bridge Oscillator
Notice positive feedback
Zp, and Zs provide frequency selection
Zp 
1  jRC
R
Zs 
jC
1  jRC
T ( j )  A
T ( j ) 
Zp
Z p  Zs
A  1
R2
R1
A
3  jRC  1 jRC
Use 𝑇 𝑗𝜔 = +1 because of explicit positive feedback
T ( jo ) 
A
 1 Imaginary part must be zero
3  jo RC  1 jo RC
jo RC 
1
0
jo RC
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o 
1
RC
Substitute into T(jω) = 1 to find A = 3 or
R2
2
R1
Oscillators 6
Wien Bridge Oscillator
vo
3
No explicit negative feedback, but explicit
positive feedback
Zp, and Zs provide frequency selection
vo
3
1
o 
RC
vx  v y 
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A3
R2
2
R1
vo
3
Oscillators 7
Gain Control
Lamp is a non-linear resistor
vo
3
Initially, lamp is cold, and R1= Rlamp is small. The gain A = 1 + 𝑅2 𝑅𝑙𝑎𝑚𝑝 > 3,
and the oscillation starts.
As output amplitude increases, current through lamp increases and Rlamp decreases,
and loop gain (1+R2/Rlamp) decreases.
Output amplitude stabilizes when loop gain (1+R2/Rlamp) = 3, and voltage across
lamp is vo/3
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Oscillators 8
Determine the amplitude for the output voltage at which the Wien bridge oscillator below
stabilizes. The graphs is the lamp resistance as a function of output voltage.
At startup, the lamp is cold and 𝑅𝑙𝑎𝑚𝑝 = 5 Ω. The amplifier
gain is
𝑅4
120
+1 =
+ 1 = 3.73
𝑅5 + 𝑅𝑙𝑎𝑚𝑝
39 + 5
This is more than 3, and oscillations start. As the output
voltage amplitude grows, the lamp heats up, and its
resistance increases It stabilizes when the gain is 3:
120
𝑅4
+ 1 = 3 ⇒ 𝑅𝑙𝑎𝑚𝑝 = 21 Ω
+1 =3 ⇒
39 + 𝑅𝑙𝑎𝑚𝑝
𝑅5 + 𝑅𝑙𝑎𝑚𝑝
From the graph, 𝑅𝑙𝑎𝑚𝑝 is 21 Ω when the lamp voltage is ≅ 1.25 V.
The current that flows through the lamp is 1.25 21 = 60 mA
The same current flows through 𝑅5 and 𝑅4 and the output voltage is
0.06 21 + 39 + 120 = 10.8 V
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Oscillators 9
Gain Control
𝑣𝐷 = 0.6 V
i
i
i
Model with D2 off
Estimate output voltage
Current through R1
𝑣𝑜 3 𝑅1
Same current flows through R2 , voltage across R3 is
Current through R3
𝑣𝑜 3 𝑅3
Current through R4
[ 𝑣𝑜 3 𝑅1 − 𝑣𝑜 3 𝑅3 )]𝑅4 + 𝑣𝐷 = 𝑣𝑜 3
𝑣𝑜 3
𝑣𝑜 3 𝑅1 − 𝑣𝑜 3 𝑅3
Solving for vo yields
𝑣𝑜 = 3 V
Previous Exam Question
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Oscillators 10
Practical Wien Bridge Oscillators
Output amplitude is quite sensitive to variation in diode
forward voltage drop
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Oscillators 11
Practical Wien Bridge Oscillators
R2
Figure 10.3 (F)
At power on, 1 uF cap is uncharged, and
gate ~ 0 V  low channel resistance, so
that R2 / R1 ~ 2.11  starts up.
R1
As voltage increases, FET progressively turns off more and more. In the
limit R2 / R1 = 20/11 = 1.8 < 2
Loop stabilizes when the JFET turns on just enough so that R2 / R1
Problem: JFET characteristics vary significantly…
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Oscillators 12
Practical Wien Bridge Oscillators
Figure 10.5 (F)
Use a limiter
Make sure you can figure out what the output amplitude is.
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Oscillators 13
Total Harmonic Distortion
THD is a term used to quantify the purity of a sine wave.
One can decompose a periodic signal into a fundamental sine wave and
harmonics (Fourier series).
THD(%)  100 D22  D32  D42  ...
Dk = ratio of amplitude of the k-th harmonic to the fundamental
Triangular wave: THD = 12% -crude approximation of a sine wave
Website: http://www.integracoustics.com/MUG/MUG/articles/phase/
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Oscillators 14
Wien Bridge Practical Considerations
Use good quality capacitors, e.g., polycarbonate—exceptional stability and
environmental performance
Use good quality resistors—metal-film
Practical Wien bridge oscillator have trimming elements and can achieve THD <
0.01 % (What is THD?)
Beware of slew-rate (SR) effects of op-amp. Make sure SR > 2π Vom fo
Assuming SR is OK, the finite GBP causes a downshift of the actual frequency
One can show that to keep downshift < 10%, GBP ≥ 43 fo
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Oscillators 15
Phase Shift Oscillator Gain Control
A small signal analysis of the oscillator below reveals that the loop gain is
greater than 29, the value required to sustain oscillation. This suggests that the
circuit will start oscillating with growing amplitude and will eventually be
clipped by the power supply, and the output will be close to a square wave. A
SPICE simulation and an actual circuit both show that the amplitude is
sinusoidal and stabilizes at about 1.8 V at node A, even though there is no
explicit amplitude limiting device. What is going on? What is the purpose of
the SPICE statement .IC V(D) = 0.001?
Previous Exam Question
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Oscillators 16
Colpitts Oscillator
RFC (Radio Frequency Choke) creates an open circuit at
the oscillation frequency but does not disturb dc biasing.
Equivalent
ac circuit
Small-signal model
Simple: no rπ, Cπ,…
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Oscillators 17
Colpitts Oscillator – Method A
Technique used thus far: Determine loop gain T. Then set T(jω) = 1
Vr
Vx
T ( j ) 
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Vr ( )
Vx ( )
Oscillators 18
Colpitts Oscillator Method B
KCL at node C:
1

sC2V  g mV    sC1  (1  s 2 LC2 )V  0
R

Assume oscillation has started: Vπ  0
Then we can eliminate Vπ (divide both sides
by Vπ) from the equation and it can be
rearranged:
s3 LC1C2  s 2 ( LC2 R)  s(C1  C2 )  ( g m 
1
)0
R
s  j



1  2 LC2 
3
 gm  
  j  (C1  C2 )   LC1C2  0
R
R 

This requires imaginary and real parts = 0
0  1
 CC 
L 1 2 
 C1  C2 
g m R  C2 C1
Condition for oscillation to start
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Oscillators 19
Colpitts Gain Control
Gain control
Resonant circuit
360o Phase Shift
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Oscillators 20
Quartz Crystal
Equivalent model
L ~ Henrys
C p ~ few pF
Cs ~ 0.001 pF
Cost?
Q ~ 104
Temperature Stabillity ~ 50  100 ppm
1
s 2  1 LCs 
Z ( s) 
sC p s 2  C p  Cs  / LCsC p 
Two resonant frequencies fp, and fs
fp, and fs are very close together
At fp Z → ∞, at fs Z = 0, in-between
Z is inductive
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Oscillators 21
Pierce Oscillator
CMOS Gate
Inductive
Inductive
microcontroller
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Application in
microcontrollers
Oscillators 22
Types of Oscillators
Sinusoidal Oscillators
THD(%)  100 D22  D32  D42  ...
Dk = ratio of amplitude of the k-th harmonic to the fundamental
Triangular wave, is a crude approximation of a sine wave, and has THD =
12%
SPICE has capabilities to estimate THD during simulations.
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Oscillators 23
Types of Oscillators
Relaxation Oscillators
Use bistable devices (Schmitt triggers, logic gates, flip-flops) to charge and
discharge a capacitor.
Waveforms are triangular, square, sawtooth, pulse, exponential
Waveforms are triangular, square, sawtooth, pulse, exponential
See Chapter 10 of the Franco text
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Oscillators 24
Review – Capacitor Charging
𝑑𝑣𝑐 𝑡
𝑖𝐶 = 𝐶
𝑑𝑡
Charged with a constant current 𝐼
𝐼
𝑣𝑐 (𝑡)
𝑑𝑣𝑐 𝑡
𝐼=𝐶
𝑑𝑡
𝐼Δ𝑡 = 𝐶Δ𝑣
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𝐼𝑑𝑡 = 𝐶𝑑𝑣𝑐 (𝑡)
𝐼Δ𝑡 = 𝐶Δ𝑣
Charged through a resistor
𝑅
𝑖(𝑡)
𝑣𝑐 (𝑡)
𝑣∞ − 𝑣0
Δ𝑡 = 𝜏ln
𝑣∞ − 𝑣
𝜏 is the time constant, 𝑣0 is the initial
voltage 𝑣∞ is the voltage if 𝑡 → ∞, Δ𝑡 is the
time to reach 𝑣.
Oscillators 25
Review - Inverting Schmitt Trigger
Positive feedback
Assume vI is low and Vo= VH
Now vI is high and Vo= VL
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 R1 
v  
VH
 R1  R2 
Increase vI and observe Vo
 R1 
v  
VL Decrease vI and observe Vo
 R1  R2 
Oscillators 26
Review – Open Collector
Open-collector or open-drain is a type of output
stage found in some Ics.
As the name implies, the collector or drain of the
output stage is not collected internally.
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Oscillators 27
LM311 Comparator with Open Collector
Pull-up resistor. Newbie mistake – forget to
add pull-up resistor.
The “amplifier” part of a comparator has
similarities with op-amps. However, they don’t
have internal frequency compensation. This
makes them fast, but potentially unstable.
Common op-amp structure
The purpose of 𝐶𝐹 is to create a dominant pole at a low
frequency, using the Miller effect. Comparators don’t
have 𝐶𝐹 .
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Oscillators 28
LM311 Comparator with Open Collector
Pull-up
Provides hysteresis (can you
calculate this?)
Comparator is configured as a Schmitt Trigger
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Oscillators 29
Inverting Schmitt Trigger
VTL  0.4 V  0 V
VTH 
3.6
15  10 V
1.8  3.6
< 0.4 V when BJT
is in saturation
Open collector
comparator
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Oscillators 30
Review: Comparators Open Collector
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Oscillators 31
Voltage-Controlled Oscillator
Figure 10.21 (F)
Inverting Schmitt trigger with
thresholds VTL = 0, VTH = 10 V
Voltage-controlled switch
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Oscillators 32
Voltage-Controlled Oscillator
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Oscillators 33
Voltage-Controlled Oscillator
iI  vI  vI / 2 /( 2R)  vI /( 4R)
Current through here is always
The Schmitt trigger and switch
determines if the current flows
here
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Oscillators 34
Voltage-Controlled Oscillator
Assume vSQ is low and switch is open
Current flows through here,
charging the capacitor
iI
This voltage drops until it reaches VTL ~ 0.
Then the Schmitt trigger snaps.
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Oscillators 35
Voltage-Controlled Oscillator
vI  vI / 2 /( 2R)  vI /( 4R)
Current through here is always
This means iI has to come from
here
 iI
The current here is 2iI
This voltage now rises until it
reaches VTH = 10 V when the
trigger snaps again.
Now the switch is closed
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Oscillators 36
Voltage-Controlled Oscillator
Capacitor current is iI  vI /( 4R) or
iI  vI /( 4R)
The time to charge/discharge the capacitor is one-half the period
iI t  Cv
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(vI /( 4R))t  C (VTH  VTL )
f0 
vI
8RC (VTH  VTL )
Oscillators 37
Basic Sawtooth Generator
Assume the switch is open
Capacitor charges through R and vST rises linearly until it reaches the trip voltage VT
Remember:
Cv  It
and here I = iI = |vI|/R, so
TCH  RCVT / | vI |
Once the trip voltage is reached, the Schmitt trigger snaps, and closes the switch,
which discharges the capacitor.
Now vST = 0, and the Schmitt trigger snaps back, the switch opens, etc.,…
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Oscillators 38
Basic Sawtooth Generator
Provides “one-shot” action, making sure the switch
(FET) is on long enough so C is fully discharged.
The delay TD is proportional to R1C1 , keep it much
smaller than TCH.
f0 
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1
1

TCH  TD RCVT / | vI | TD
Oscillators 39
Monolithic Waveform Generators
Sect 10.6 (F)
Figure 10.25 (F)
ICs designed to provide waveforms with minimum of external components
At core they have a triangular/square wave generator
Triangular output passed through a wave shaping circuit to provide a sine wave
Grounded-Capacitor VCOs
Voltagecontrolled
current
sources
Schmitt Trigger
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Oscillators 40
ICL8038/NTE864 Waveform Generator
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Oscillators 41
ICL8038/NTE864 Wave Shaper
Figure 10.27 (F)
a4  ??
a3  ??
a2 
10 || 27
 0.68
1  10 || 27
a1 
10
 0.909
1  10
a0  1
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Oscillators 42
ICL8038/NTE864 Application
Figure 10.28 (F)
ICL8038 is obsolete, but one can still
find old stock
NTE864 is a pin-for-pin replacement
but pricey ($50).
Output is centered around Vcc/2, sine TDH ~ 1%
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Oscillators 43
Emitter-Coupled VCO
Astable
Figure 10.30 (F)
Low
50 % Duty cycle, square and
triangle waveforms available
Low
Off
High
On
OffOn
VBE increases
Fixed
iI t  Cv
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v  2VBE
f0 
iI
4CVBE
Easy to convert into CurrentControlled Oscillator (CCO)
Oscillators 44
XR-2206 Function Generator
0.1 Hz  1 MHz
20
ppm/oC
0.5% THD
Figure 10.31 (F)
Much less
expensive than 8038
What type of
capacitor should this
be?
This is an emitter-coupled CCO
similar to the previous slide
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Oscillators 45
Frequency-Shift Key Modulation
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Oscillators 46
Sinusoidal FSK Generator
This adjusts iI  oscillation frequency
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f0 
Figure 10.32 (F)
iI
4CVBE
Oscillators 47
XR-2209 VCO
The XR-2209 Is a simplified version of the
XR-2209. It does not contain the triangle 
sine shaper. Provides square and triangle
wave.
It is cheaper than the XR-2206 and costs
about $2.80.
We will use the XR-2209 for the IR link
labs.
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Oscillators 48
Sect 10.7 (F)
V-F and F-V Converters (VFCs)
 Difference between V-F and VCO?
 Usually, VFCs have more stringent
requirements than VCOs
 VCOs are often designed to be used inside of
control loops, which corrects errors, etc.
 VFC have large dynamic range (4 decades or
more)
 Low linearity error (< 0.1%)
 Great temperature stability
Note the cost
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Oscillators 49
AD537 Voltage-to-Frequency Converter
30 ppm/oC
Linearity error: 0.1% typical
Note OC
Figure 10.33 (F)
What type of capacitor
should this be?
f0 
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vI
10 RC
Oscillators 50
AD537 Application
Figure 10.34 (F)
Note Open Emitter
Note Open Emitter
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Oscillators 51
Charge-Balancing VFCs
Supply a capacitor with continuous charge, by charging with a
voltage-controlled current source
Simultaneously pull out discrete charge packets at a rate f0
Control f0 such that the net charge flow is always zero
I packet
vI
C
Sense voltage and control switch
frequency so that net charge flow into C
is zero
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f 0  kvI
Note, in principle, the value of
C is not important
Oscillators 52
Charge-Balancing VFCs
VFC32 Voltage-to Frequency Converter
Figure 10.35 (F)
Choose R so that iI is less than 1 mA
TH 
f0 
7.5 V
C
1 mA
vI
7.5RC
D(%)  100
vI
R  1 mA
TL  C1v1 iI
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Oscillators 53
Charge-Balancing VFCs
Figure 10.35 (F)
VFC32 Voltage-to Frequency Converter
Choose R so that iI is less than 1 mA
TL  C1v1 iI
f0 
TH 
vI
7.5RC
7.5 V
C
1 mA
D(%)  100
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vI
R  1 mA
Oscillators 54
Frequency-to-Voltage Conversion
Voltage across
capacitor is now the
output
Figure 10.36 (F)
Drive Comparator
Some Ripple
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Oscillators 55
Basic Free-Running Multivibrator
 R1 
VT  
 Vsat
 R1  R2 
Figure 10.7 (F)
Steady state
voltage if t ∞
t   ln
V  V0
V  V1
Capacitor charged
through a resistor, see
equation 10.3 in the text.
Duty cycle?
50%
 R1 
 VT  
 ( Vsat )
 R1  R2 
Frequency?
T
V V
 RC ln sat T
2
Vsat  VT
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T
V V
 RC ln sat T
2
Vsat  VT
f0 
1
1

T 2 RC ln(1  R1 R2 )
Oscillators 56
Adjustable Square-Wave Generator
Figure 10.8 (F)
What should Vz be for a ± 5 V
output?
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Provides a well-defined
Vsat and output voltage
Oscillators 57
Single-Supply Multivibrator
Figure 10.9 (F)
Note the open collector on the
comparator
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Oscillators 58
CMOS Gates
Figure 10.11 (F)
What are these for? What type
of diodes are these?
Very high input impedance, VT ~ VDD/2
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Oscillators 59
CMOS-Gate Free-Running Multivibrator
Figure 10.12 (F)
VDD
VT
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0
0 VDD
Oscillators 60
CMOS-Gate Free-Running Multivibrator
Figure 10.12 (F)
What is the purpose of this?
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Oscillators 61
Monostable Multivibrator
Figure 10.14 (F)
Self Study
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Oscillators 62
CMOS-Gate With Feedback
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Oscillators 63
CMOS Crystal Oscillator
Figure 10.13 (F)
Bias at VDD/2
180o phase shift
180o phase shift at
resonant frequency
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Oscillators 64
555 Timer
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Sect. 10.3 (F)
Oscillators 65
555 Timer Astable
Charge via RA and RB
Charge via RA and RB
Discharge via RB
Discharge via RB
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Oscillators 66
555 Timer Astable
During TL the time constant is RBC so that
TL  RB C ln
0  VTH
 RB C ln 2
0  VTL
During TH the time constant is (RA+RB)C
TH  RA  RB C ln
T  RA  RB C ln
Steady state
voltage if t ∞
t   ln
V  V0
V  V1
Capacitor charged through a resistor,
see equation 10.3 in the text.
A. Kruger
T  RA  RB C ln
VCC  VTL
VCC  VTH
VCC  VTL
 R B C ln 2
VCC  VTH
VCC  VCC 3
 R B C ln 2
VCC  2VCC 3
T  RA  RB C ln 2  R B C ln 2  RA  2RB C ln 2
fo 
1.44
RA  2 RB
D(%)  100
RA  RB
RA  2 RB
Oscillators 67
555 Timer Monostable
Trigger occurs when TRIG pins falls below 1/3 of 𝑉𝐶𝐶
The trigger pulse must be shorter than the output pulse
A. Kruger
Oscillators 68
Making Trigger Pulses
The 𝑅𝐶 circuit approximates differentiation
𝑑𝑉𝑠 (𝑡)
𝑉𝑜 (𝑡) ≅ 𝑅𝐶
𝑑𝑡
Note that the output goes
below 0 V.
A. Kruger
Oscillators 69
Making Trigger Pulses
Note that the output goes below
above power supply rail.
555
A. Kruger
Oscillators 70
Making Trigger Pulses
Diodes clamps 𝑉𝑡𝑟𝑖𝑔𝑔 to 𝑉𝑐𝑐
Don’t use a rectifier, use a switching diode.
A. Kruger
Oscillators 71
Making Trigger Pulses
More reliable circuit - can drive
low impedance loads.
A. Kruger
Oscillators 72
PWM Generation
A. Kruger
Oscillators 73
A. Kruger
Oscillators 74