55:041 Electronic Circuits - University of...

A. Kruger Sinusoidal Oscillators 1

55:041 Electronic Circuits

Sinusoidal Oscillators

Sections of Chapter 15


Stability

)(1)()(

jT

jAjAf

Recall definition of loop gain: T(jω) = βARecall definition of loop gain: T(jω) = βA

11

)()( jAjAf InstabilityInstabilityIf T(jω) = -1, then If T(jω) = -1, then

We can writeWe can write )()( jTjT

Equivalent conditions for stabilityEquivalent 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 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?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 criterionThe condition T(jω) = -1 is called the Barkhausen criterion

The total phase shift through the amplifier and feedback network must be N×360oThe total phase shift through the amplifier and feedback network must be N×360o

The magnitude of the loop gain must be exactly 1The magnitude of the loop gain must be exactly 1

Loop gain < 1 => oscillations die outLoop gain < 1 => oscillations die out

Loop gain > 1 => oscillations grow and clip at supply railsLoop 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)

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)


RC Phase Shift Oscillator

Gain + 180o Phase shiftGain + 180o Phase shift60o Phase shift60o Phase shift60o Phase shift60o Phase shift 60o Phase shift60o Phase shift

)(1

33

RCj

RCjvv

I RRA 2

33

23

2

11)(

RCjRCj

RR

RCjRCj

RRjT

T(jω) = -1 (Barkhausen criterion)T(jω) = -1 (Barkhausen criterion)

1331

)( 222222

22

CRRCjCRRCRCj

RRjT

This means the imaginary part must be zero:This means the imaginary part must be zero: 031 222 CRo RCo 31

At this frequency:At this frequency: 8

13133130

313)( 22

RR

jj

RRjT 82

RR


RC Phase Shift Oscillator

Gain + 180o Phase shiftGain + 180o Phase shift180o Phase shift180o Phase shift

Same idea, analysis more difficult because phase shift networks load each other Same idea, analysis more difficult because phase shift networks load each other

RCo 61

292 RR

Will this work too?Will this work too?


Wien Bridge Oscillator

sp

p

ZZZ

AjT

)(

RCjRZ p

1 Cj

RCjZs

1

RCjRCjAjT

13)(

Notice positive feedback. This modifies the Barkhausen Criterion to 1Notice positive feedback. This modifies the Barkhausen Criterion to 1

Zp, and Zs provide frequency selectionZp, and Zs provide frequency selection

113

)(

RCjRCj

AjToo

o Imaginary part must be zeroImaginary part must be zero

01

RCjRCj

oo

RCo1

21

2 RR

Substitute into T(jω) = 1 to find A = 3 or Substitute into T(jω) = 1 to find A = 3 or

1

21RRA


Wien Bridge Oscillator Summary

No explicit negative feedback, but explicit positive feedbackNo explicit negative feedback, but explicit positive feedback

Zp, and Zs provide frequency selectionZp, and Zs provide frequency selection

RCo1

21

2 RR

3A

3o

yxvvv

3ov

3ov


Gain Control

3ov

Initially, lamp is cold, and R1= Rlamp is small. The gain A 1 ⁄ 3, and the oscillation starts. Initially, lamp is cold, and R1= Rlamp is small. The gain A 1 ⁄ 3, and the oscillation starts.

As output amplitude increases, current through lamp increases and Rlamp decreases, and loop gain (1+R2/Rlamp) decreases.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/3Output amplitude stabilizes when loop gain (1+R2/Rlamp) = 3, and voltage across lamp is vo/3

Lamp is a non-linear resistorLamp 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.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 5Ω. The amplifier gain isAt startup, the lamp is cold and 5Ω. The amplifier gain is

112039 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:

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:

1 3120

39 1 3⇒ 21Ω⇒

From the graph, is 21Ω when the lamp voltage is ≅ 1.25 V.From the graph, is 21Ω when the lamp voltage is ≅ 1.25 V.

The current that flows through the lamp is 1.25 21⁄ 60mAThe current that flows through the lamp is 1.25 21⁄ 60mA

The same current flows through and and the output voltage isThe same current flows through and and the output voltage is

0.06 21 39 120 10.8V

Note that the op-amp must supply 60 mA which is high for general purpose amplifiers.Note that the op-amp must supply 60 mA which is high for general purpose amplifiers.


Gain Control

Same current flows through R2 , voltage across R3 isSame current flows through R2 , voltage across R3 is

Model with D2 offModel with D2 off

Current through R1Current through R1

Current through R3Current through R3Current through R4Current through R4

Solving for vo yieldsSolving for vo yields

Estimate output voltageEstimate output voltage

0.6V

i

i

i

3⁄3⁄ ⁄

3V

3⁄ ⁄ 3⁄ 3⁄ ⁄⁄

3⁄ 3⁄ 3⁄⁄⁄

Previous Exam QuestionPrevious Exam Question


Gain ControlA 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 QuestionPrevious Exam Question

What is this?What is this?


Gain Control

Gain controlGain control

Resonant circuitResonant circuit

360o Phase Shift360o Phase Shift


Colpitts Oscillator

Can replace with crystalCan replace with crystal


Quartz Crystal

pssp

s

p CLCCCsLCs

sCsZ

/11)( 2

2

pFfew ~pCpF001.0~sC

Henrys hundredfew ~L

410~Q

ppm10050~Stabillity eTemperatur

Equivalent modelEquivalent model

Two resonant frequencies fp, and fsTwo resonant frequencies fp, and fs

fp, and fs are very close togetherfp, and fs are very close together

At fp Z → ∞, at fs Z =0, in-between Z is inductiveAt fp Z → ∞, at fs Z =0, in-between Z is inductive

Cost?Cost?


Pierce Oscillator

InductiveInductive

CMOS GateCMOS Gate

InductiveInductive

Application in microcontrollersApplication in microcontrollers

microcontrollermicrocontroller

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