Post on 11-Mar-2018
A. Kruger Oscillators 1
55:041 Electronic Circuits
Oscillators
Sections of Chapter 15 + Additional Material
A. Kruger Oscillators 2
Stability
)(1
)()(
jT
jAjAf
Recall definition of loop gain: T(jω) = βA
11
)()(
jAjAf Instability If T(jω) = -1, then
We can write )()( jTjT
Equivalent conditions for stability 1)( jT 180thanless
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?
A. Kruger Oscillators 3
Barkhausen Criterion
The condition 𝑇(𝑗𝜔) = −1 is called the Barkhausen criterion
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)
Note that this formulation assumes negative feedback. In some
instances, we use explicit positive feedback and then the condition
is 𝑇 𝑗𝜔 = +1.
A. Kruger Oscillators 4
RC Phase Shift Oscillator
Gain + 180o Phase shift 60o Phase shift 60o Phase shift 60o Phase shift
)(1
3
3
RCj
RCj
v
v
IR
RA 2
3
3
2
3
2
11)(
RCj
RCj
R
R
RCj
RCj
R
RjT
T(jω) = -1 (Barkhausen criterion)
1
331)(
222222
2
2
CRRCjCR
RCRCj
R
RjT
This means the imaginary part must be zero: 031 222 CRo RCo
3
1
At this frequency: 8
1
3133130
313)( 22
R
R
j
j
R
RjT 82
R
R
A. Kruger Oscillators 5
RC Phase Shift Oscillator
Gain + 180o Phase shift 180o Phase shift
Same idea, analysis more difficult because phase shift networks load each other
RCo
6
1 292
R
R
Will this work too?
A. Kruger Oscillators 6
Wien Bridge Oscillator
sp
p
ZZ
ZAjT
)(
RCj
RZ p
1 Cj
RCjZs
1
RCjRCj
AjT
13)(
Notice positive feedback
Zp, and Zs provide frequency selection
113
)(
RCjRCj
AjT
oo
o
Imaginary part must be zero
01
RCj
RCjo
o
RC
o
1 2
1
2 R
RSubstitute into T(jω) = 1 to find A = 3 or
1
21R
RA
Use 𝑇 𝑗𝜔 = +1 because of explicit positive feedback
A. Kruger Oscillators 7
Wien Bridge Oscillator
No explicit negative feedback, but explicit
positive feedback
Zp, and Zs provide frequency selection
RCo
1 2
1
2 R
R3A
3
oyx
vvv
3
ov
3
ov
A. Kruger Oscillators 8
Gain Control
3
ov
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
Lamp is a non-linear resistor
A. Kruger Oscillators 9
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
𝑅4𝑅5 + 𝑅𝑙𝑎𝑚𝑝
+ 1 =120
39 + 5+ 1 = 3.73
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:
𝑅4𝑅5 + 𝑅𝑙𝑎𝑚𝑝
+ 1 = 3 120
39 + 𝑅𝑙𝑎𝑚𝑝+ 1 = 3 ⇒ 𝑅𝑙𝑎𝑚𝑝 = 21 Ω ⇒
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
A. Kruger Oscillators 10
Gain Control
Same current flows through R2 , voltage across R3 is
Model with D2 off
Current through R1
Current through R3 Current through R4
Solving for vo yields
Estimate output voltage
𝑣𝐷 = 0.6 V
i
i
i
𝑣𝑜 3 𝑣𝑜 3 𝑅1
𝑣𝑜 = 3 V
𝑣𝑜 3 𝑅3 𝑣𝑜 3 𝑅1 − 𝑣𝑜 3 𝑅3
[ 𝑣𝑜 3 𝑅1 − 𝑣𝑜 3 𝑅3)]𝑅4 + 𝑣𝐷 = 𝑣𝑜 3
Previous Exam Question
A. Kruger Oscillators 11
Practical Wien Bridge Oscillators
Output amplitude is quite sensitive to variation in diode
forward voltage drop
A. Kruger Oscillators 12
Practical Wien Bridge Oscillators Figure 10.3 (F)
As voltage increases, FET progressively turns off more and more. In the
limit R2 / R1 = 20/11 = 1.8 < 2
At power on, 1 uF cap is uncharged, and
gate ~ 0 V low channel resistance, so
that R2 / R1 ~ 2.11 starts up.
Loop stabilizes when the JFET turns on just enough so that R2 / R1
Problem: JFET characteristics vary significantly…
R1
R2
A. Kruger Oscillators 13
Practical Wien Bridge Oscillators Figure 10.5 (F)
Use a limiter
Make sure you can figure out what the output amplitude is.
A. Kruger Oscillators 14
Total Harmonic Distortion
Dk = ratio of amplitude of the k-th harmonic to the fundamental
Triangular wave: THD = 12% -crude approximation of a sine wave
...100(%) 2
4
2
3
2
2 DDDTHD
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).
Website: http://www.integracoustics.com/MUG/MUG/articles/phase/
A. Kruger Oscillators 15
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
A. Kruger Oscillators 16
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
A. Kruger Oscillators 17
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,…
A. Kruger Oscillators 18
Colpitts Oscillator – Method A
Technique used thus far: Determine loop gain T. Then set T(jω) = 1
Vr
Vx
)(
)()(
x
r
V
VjT
A. Kruger Oscillators 19
Colpitts Oscillator Method B
0)1(1
2
2
12
VLCssC
RVgVsC m
0)1
()()( 212
2
21
3 R
gCCsRLCsCLCs m
0)(1
21
3
212
2
CLCCCj
R
LC
Rgm
21
210 1
CC
CCL 12 CCRgm
Assume oscillation has started: Vπ 0
Then we can eliminate Vπ (divide both sides
by Vπ) from the equation and it can be
rearranged:
KCL at node C:
js
This requires imaginary and real parts = 0
Condition for oscillation to start
A. Kruger Oscillators 20
Colpitts Gain Control
Gain control
Resonant circuit
360o Phase Shift
A. Kruger Oscillators 21
Quartz Crystal
pssp
s
p CLCCCs
LCs
sCsZ
/
11)(
2
2
pFfew ~pC
pF001.0~sC
Henrys ~L
410~Q
ppm10050~Stabillity eTemperatur
Equivalent model
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
Cost?
A. Kruger Oscillators 22
Pierce Oscillator
Inductive
CMOS Gate
Inductive
Application in
microcontrollers microcontroller
A. Kruger Oscillators 23
Types of Oscillators
...100(%) 2
4
2
3
2
2 DDDTHD
Sinusoidal Oscillators
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.
A. Kruger Oscillators 24
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
A. Kruger Oscillators 25
Review – Capacitor Charging
𝑖𝐶 = 𝐶𝑑𝑣𝑐 𝑡
𝑑𝑡
Charged with a constant current 𝐼
𝐼
𝑣𝑐(𝑡)
𝐼 = 𝐶𝑑𝑣𝑐 𝑡
𝑑𝑡 𝐼𝑑𝑡 = 𝐶𝑑𝑣𝑐(𝑡)
𝐼Δ𝑡 = 𝐶Δ𝑣 𝐼Δ𝑡 = 𝐶Δ𝑣
Charged through a resistor
Δ𝑡 = 𝜏ln𝑣∞ − 𝑣0
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.
𝑖(𝑡)
𝑣𝑐(𝑡)
𝑅
A. Kruger Oscillators 26
Review - Inverting Schmitt Trigger
HVRR
Rv
21
1Assume vI is low and Vo= VH
LVRR
Rv
21
1
Increase vI and observe Vo
Now vI is high and Vo= VL Decrease vI and observe Vo
Positive feedback
A. Kruger Oscillators 27
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.
A. Kruger Oscillators 28
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.
A. Kruger Oscillators 29
LM311 Comparator with Open Collector
Pull-up
Provides hysteresis (can you
calculate this?)
Comparator is configured as a Schmitt Trigger
A. Kruger Oscillators 30
Inverting Schmitt Trigger
V 10156.38.1
6.3
THV
< 0.4 V when BJT
is in saturation
V 0V 4.0 TLV
Open collector
comparator
A. Kruger Oscillators 31
Review: Comparators Open Collector
A. Kruger Oscillators 32
Voltage-Controlled Oscillator Figure 10.21 (F)
Inverting Schmitt trigger with
thresholds VTL = 0, VTH = 10 V
Voltage-controlled switch
A. Kruger Oscillators 33
Voltage-Controlled Oscillator
A. Kruger Oscillators 34
Voltage-Controlled Oscillator
Current through here is always )4/()2/(2/ RvRvvi IIII
The Schmitt trigger and switch
determines if the current flows
here
A. Kruger Oscillators 35
Voltage-Controlled Oscillator
Assume vSQ is low and switch is open
Ii
Current flows through here,
charging the capacitor
This voltage drops until it reaches VTL ~ 0.
Then the Schmitt trigger snaps.
A. Kruger Oscillators 36
Voltage-Controlled Oscillator
Now the switch is closed
Ii
This means iI has to come from
here
The current here is 2iI
This voltage now rises until it
reaches VTH = 10 V when the
trigger snaps again.
Current through here is always )4/()2/(2/ RvRvv III
A. Kruger Oscillators 37
Voltage-Controlled Oscillator
Capacitor current is )4/( Rvi II
The time to charge/discharge the capacitor is one-half the period
vCtiI )())4/(( TLTHI VVCtRv )(80
TLTH
I
VVRC
vf
)4/( Rvi II or
A. Kruger Oscillators 38
Basic Sawtooth Generator
Capacitor charges through R and vST rises linearly until it reaches the trip voltage VT
Assume the switch is open
tIvC Remember: and here I = iI = |vI|/R, so ||/ ITCH vRCVT
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.,…
A. Kruger Oscillators 39
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.
DITDCH TvRCVTTf
||/
110
A. Kruger Oscillators 40
Monolithic Waveform Generators Figure 10.25 (F) Sect 10.6 (F)
Grounded-Capacitor VCOs
Schmitt Trigger
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
Voltage-
controlled
current
sources
A. Kruger Oscillators 41
ICL8038/NTE864 Waveform Generator
A. Kruger Oscillators 42
ICL8038/NTE864 Wave Shaper Figure 10.27 (F)
909.0
101
101
a
10 a
68.0
27||101
27||102
a
??3 a
??4 a
A. Kruger Oscillators 43
ICL8038/NTE864 Application Figure 10.28 (F)
Output is centered around Vcc/2, sine TDH ~ 1%
ICL8038 is obsolete, but one can still
find old stock
NTE864 is a pin-for-pin replacement
but pricey ($50).
A. Kruger Oscillators 44
Emitter-Coupled VCO Figure 10.30 (F)
BE
I
CV
if
40 vCtiI BEVv 2
Easy to convert into Current-
Controlled Oscillator (CCO)
Astable
On
High
Off
Fixed
VBE increases
On
Low
Low
Off
50 % Duty cycle, square and
triangle waveforms available
A. Kruger Oscillators 45
XR-2206 Function Generator Figure 10.31 (F)
This is an emitter-coupled CCO
similar to the previous slide
0.1 Hz 1 MHz 20 ppm/oC 0.5% THD
What type of
capacitor should this
be?
Much less
expensive than 8038
A. Kruger Oscillators 46
Frequency-Shift Key Modulation
A. Kruger Oscillators 47
Sinusoidal FSK Generator Figure 10.32 (F)
BE
I
CV
if
40 This adjusts iI oscillation frequency
A. Kruger Oscillators 48
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.
A. Kruger Oscillators 49
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
Sect 10.7 (F)
Note the cost
A. Kruger Oscillators 50
AD537 Voltage-to-Frequency Converter Figure 10.33 (F)
RC
vf I
100
30 ppm/oC Linearity error: 0.1% typical
What type of capacitor
should this be?
Note OC
A. Kruger Oscillators 51
AD537 Application Figure 10.34 (F)
Note Open Emitter
Note Open Emitter
A. Kruger Oscillators 52
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
Ikvf 0
Iv
C
Sense voltage and control switch
frequency so that net charge flow into C
is zero
packetI
Note, in principle, the value of
C is not important
A. Kruger Oscillators 53
Charge-Balancing VFCs Figure 10.35 (F)
VFC32 Voltage-to Frequency Converter
CTHmA 1
V 7.5
IL ivCT 11
RC
vf I
5.70
mA 1100(%)
R
vD I
Choose R so that iI is less than 1 mA
A. Kruger Oscillators 54
Charge-Balancing VFCs Figure 10.35 (F)
VFC32 Voltage-to Frequency Converter
CTHmA 1
V 7.5
IL ivCT 11
RC
vf I
5.70
mA 1100(%)
R
vD I
Choose R so that iI is less than 1 mA
A. Kruger Oscillators 55
Frequency-to-Voltage Conversion Figure 10.36 (F)
Drive Comparator
Voltage across
capacitor is now the
output
Some Ripple
A. Kruger Oscillators 56
Basic Free-Running Multivibrator Figure 10.7 (F)
Duty cycle?
1
0lnVV
VVt
Capacitor charged
through a resistor, see
equation 10.3 in the text.
Steady state
voltage if t ∞
Tsat
Tsat
VV
VVRC
T
ln
2Tsat
Tsat
VV
VVRC
T
ln
2
satT VRR
RV
21
1
)(21
1satT V
RR
RV
Frequency?
50%
)1ln(2
11
21
0RRRCT
f
A. Kruger Oscillators 57
Adjustable Square-Wave Generator Figure 10.8 (F)
Provides a well-defined
Vsat and output voltage
What should Vz be for a ± 5 V
output?
A. Kruger Oscillators 58
Single-Supply Multivibrator Figure 10.9 (F)
Note the open collector on the
comparator
A. Kruger Oscillators 59
CMOS Gates Figure 10.11 (F)
Very high input impedance, VT ~ VDD/2
What are these for? What type
of diodes are these?
A. Kruger Oscillators 60
CMOS-Gate Free-Running Multivibrator Figure 10.12 (F)
0 VDD
VT 0
VDD
A. Kruger Oscillators 61
CMOS-Gate Free-Running Multivibrator Figure 10.12 (F)
What is the purpose of this?
A. Kruger Oscillators 62
Monostable Multivibrator Figure 10.14 (F)
Self Study
A. Kruger Oscillators 63
CMOS-Gate With Feedback
A. Kruger Oscillators 64
CMOS Crystal Oscillator Figure 10.13 (F)
180o phase shift
Bias at VDD/2
180o phase shift at
resonant frequency
A. Kruger Oscillators 65
555 Timer Sect. 10.3 (F)
A. Kruger Oscillators 66
555 Timer Astable
Charge via RA and RB
Discharge via RB
Discharge via RB
Charge via RA and RB
A. Kruger Oscillators 67
555 Timer Astable
1
0lnVV
VVt
Capacitor charged through a resistor,
see equation 10.3 in the text.
Steady state
voltage if t ∞
During TL the time constant is RBC so that
2ln0
0ln CR
V
VCRT B
TL
THBL
During TH the time constant is (RA+RB)C
THCC
TLCCBAH
VV
VVCRRT
ln
2lnln CRVV
VVCRRT B
THCC
TLCCBA
2ln32
3ln CR
VV
VVCRRT B
CCCC
CCCCBA
2ln22ln2ln CRRCRCRRT BABBA
BA
BA
BA
oRR
RRD
RRf
2100(%)
2
44.1
A. Kruger Oscillators 68
555 Timer Monostable
The trigger pulse must be shorter than the output pulse
Trigger occurs when TRIG pins falls below 1/3 of 𝑉𝐶𝐶
A. Kruger Oscillators 69
Making Trigger Pulses
Note that the output goes
below 0 V.
The 𝑅𝐶 circuit approximates differentiation
𝑉𝑜(𝑡) ≅ 𝑅𝐶𝑑𝑉𝑠(𝑡)
𝑑𝑡
A. Kruger Oscillators 70
Making Trigger Pulses
Note that the output goes below
above power supply rail.
555
A. Kruger Oscillators 71
Making Trigger Pulses
Diodes clamps 𝑉𝑡𝑟𝑖𝑔𝑔 to 𝑉𝑐𝑐
Don’t use a rectifier, use a switching diode.
A. Kruger Oscillators 72
Making Trigger Pulses
More reliable circuit - can drive
low impedance loads.
A. Kruger Oscillators 73
PWM Generation
A. Kruger Oscillators 74