55:041 Electronic Circuits - University of...

A. Kruger Oscillators 1

55:041 Electronic Circuits

Oscillators

Sections of Chapter 15 + Additional Material


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?


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.


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


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?


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


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


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


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


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


Practical Wien Bridge Oscillators

Output amplitude is quite sensitive to variation in diode

forward voltage drop


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


Practical Wien Bridge Oscillators Figure 10.5 (F)

Use a limiter

Make sure you can figure out what the output amplitude is.


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/


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


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


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,…


Colpitts Oscillator – Method A

Technique used thus far: Determine loop gain T. Then set T(jω) = 1

Vr

Vx

)(

)()(

x

r

V

VjT


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


Colpitts Gain Control

Gain control

Resonant circuit

360o Phase Shift


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?


Pierce Oscillator

Inductive

CMOS Gate

Inductive

Application in

microcontrollers microcontroller


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.


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


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


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


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.


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.


LM311 Comparator with Open Collector

Pull-up

Provides hysteresis (can you

calculate this?)

Comparator is configured as a Schmitt Trigger


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


Review: Comparators Open Collector


Voltage-Controlled Oscillator Figure 10.21 (F)

Inverting Schmitt trigger with

thresholds VTL = 0, VTH = 10 V

Voltage-controlled switch


Voltage-Controlled Oscillator



Current through here is always )4/()2/(2/ RvRvvi IIII

The Schmitt trigger and switch

determines if the current flows

here



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.



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



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


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.,…


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


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


ICL8038/NTE864 Waveform Generator


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


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).


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


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


Frequency-Shift Key Modulation


Sinusoidal FSK Generator Figure 10.32 (F)

BE

I

CV

if

40 This adjusts iI oscillation frequency


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.


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


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


AD537 Application Figure 10.34 (F)

Note Open Emitter

Note Open Emitter


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


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


Frequency-to-Voltage Conversion Figure 10.36 (F)

Drive Comparator

Voltage across

capacitor is now the

output

Some Ripple


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


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?


Single-Supply Multivibrator Figure 10.9 (F)

Note the open collector on the

comparator


CMOS Gates Figure 10.11 (F)

Very high input impedance, VT ~ VDD/2

What are these for? What type

of diodes are these?


CMOS-Gate Free-Running Multivibrator Figure 10.12 (F)

0 VDD

VT 0

VDD


CMOS-Gate Free-Running Multivibrator Figure 10.12 (F)

What is the purpose of this?


Monostable Multivibrator Figure 10.14 (F)

Self Study


CMOS-Gate With Feedback


CMOS Crystal Oscillator Figure 10.13 (F)

180o phase shift

Bias at VDD/2

180o phase shift at

resonant frequency


555 Timer Sect. 10.3 (F)


555 Timer Astable

Charge via RA and RB

Discharge via RB

Discharge via RB

Charge via RA and RB


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


555 Timer Monostable

The trigger pulse must be shorter than the output pulse

Trigger occurs when TRIG pins falls below 1/3 of 𝑉𝐶𝐶


Making Trigger Pulses

Note that the output goes

below 0 V.

The 𝑅𝐶 circuit approximates differentiation

𝑉𝑜(𝑡) ≅ 𝑅𝐶𝑑𝑉𝑠(𝑡)

𝑑𝑡



Note that the output goes below

above power supply rail.

555



Diodes clamps 𝑉𝑡𝑟𝑖𝑔𝑔 to 𝑉𝑐𝑐

Don’t use a rectifier, use a switching diode.



More reliable circuit - can drive

low impedance loads.


PWM Generation

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