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  1

Non Sinusoidal

Oscillator (Part II)

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  2

Non sinusoidal oscillator 

Known as relaxation oscillator 

Typically use RC circuit or integrator and schmitt

comparator The capacitor charge and discharge. The comparator

compare the capacitor voltage with a reference voltage and

switch between high and low voltage

Lecture will covers

Suare wave generator  Triangular wave generator 

Sawtooth wave generator 

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  !

Square wave generator 

Composed of 2 components Bistable multivibrator (or Schmitt trigger)  "enerate output voltage vo # or $ depending

on vd

RC circuit Charge towards %&' voltage

  and %(' voltage depending on

  vo

satV +

satV −

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  )

*on +nverting Schmitt trigger  ,d - v& ( v( 

if vd /  v0 - &,sat

if vd 1 (  v0 - (,sat

Schmitt trigger will becovered in more detail

in chapter

δ 

(

&,sat

On oscillator circuit

V is connected to RC

networ! and com"ared to a

re#erence voltage

V$ is used as Vre# 

δ 

δ 

3or simplicity we

assume -0 in

the analysis

δ 

δ 

δ 

(,sat

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  4

Circuit O"eration

The circuit operation can be visuali5ed by loo6ing at the chainreaction in the circuit.

The comparator compare the capacitor voltage with thereference voltage and switch between #&,sat and (,sat$

3irst assume that ,o initial value is high #&,sat$ then analyses theeffect of this setting on the threshold #reference voltage$ and thecomparator output

7xplanation will be based on the notes

vC

vo

(,th

&,th

&,sat

(,sat

t

+nitial operation

Steady(state operation

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Revision% Ste" Res"onse o# &

RC Circuit (#rom circuit theor')

The detail derivation for the oscillation freuency f is

given in the note

 

+nitially at time t-08 ,0 - v#0$

The current through theresistor R is eual to current

flowing through the capacitor 

ic - #,s 9 v$: R - C dv:dt

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  ;

erivation o# the voltage vc(t)

(when ca"acitor is charging)*ote that in the oscillator circuit8 the out"ut voltage

Vo is used to charge the capacitor 

Charging starts when ,o - &,sat is applied

 ,s - ,o and ,#t$ - ,c#t$ ,#0&$- ,0 - (,th

 

( ) ( )   RC t c   eV V V t v  −

+−=∴thsatsat

,o

Vc

,#0&$

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  <

Oscillation requenc' # * + ,-

3irst find t= #the charging time of the

capacitor$

The detail derivation for f is given in the note

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  >

Com"uting the ca"acitor

charging time t+

 ?ssume currently at "oint

+ and moving to "oint .

@sing

 ?pplied voltage is &,sat 8

,#t$ - ,c#t$

,#0&$- ,0 - (,th

vC

vo

Vth

&,th

$Vsat

Vsat

t

+nitial operation

Steady(state operation

t=t2 +

.

( ) ( )

  RC t 

c  eV V V t v  −

+−=∴thsatsat

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  =0

Com"uting the ca"acitor

charging time t+

 

  vC

vo

Vth

&,th

$Vsat

Vsat

t

+nitial operation

Steady(state operation

t=t2

+

.

 ?t point 2 8 t - t=

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  ==

Square /ave Oscillation

requenc'

+n the circuit t= - t2

121   2t t t T    =+=

   

  

    += F 

 F 

 R

 R R RC    1

2ln2

   

 

 

 

+=  F  R R

 RC 

  12

1ln2

 

    

   +

==

 F 

o

 R R RC T 

 f 

121ln2

11

1164.1   R R

 F 

  ≈

  RC 

  f  o2

1=

.

I# ln(+ $ .R+ ,R

) * +

8

.

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  =2

-riangular /ave 0enerator Consist of comparator ?= and integrator ?2

Comparator continuously compares the non(inverting

input at point A with the inverting input #- 0,$.

+f ,& / 0 ,o' - &,sat

+f ,& 1 0 ,o' - (,sat

The integrator

integrate the

suare wave to

produce atriangular wave at

,o

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  =!

-ransient voltage current

relationshi" in a ca"acitor 

c

cc

dt 

dvC i   =

,/0

)(

)0(

∫ ∫    ==

t 

c

t v

t vc

c   dt C idV c

∫ =−   +t 

ccc   d ivt v0

)0()(   τ  

,c#t$

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  =)

Integrator 

3rom chapter = notes8 ,c - ,o

  -

 ?s value of t increases8 the area under ,i increases Thus integrating a rectangle wave give a linear increasing negative

value

 

∫ −

=t 

i

 F 

o   d V C  R

t V 01

)(1

)(   τ  τ  

C3

(

&

,i

,o

R=

,(

,&

∫ −

=−   +t 

i

 F 

oo   d V C  R

V t V 01

)(1

)0()(   τ  τ  

t

,i

 ?rea

under

curve

,i

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  =4

Com"arator out"ut Vo1 switch between $Vsat and 2

Vsat de"ending on oscillator out"ut Vo 

,o is connected to ,& through resistor R=8 Bhen ,o is more negative than 9,th /// the

comparator output will switch to 9,sat Bhen ,o is more positive than &,th /// the

comparator output will switch to &,sat

vo

vo1

Vth

$Vth

$Vsat

Vsat

t

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  =

Circuit O"eration ?ssume that when the

supply voltage areswitched on8 ,& slightly above 0 , 8

,c#t-0$ - 0 ,   ,o' - &,sat

etail description is

given in the note Knowing ,o' we can

derive ,o which turns outto be alternating

negative:positive ramp

vo

vo1

Vth

$Vth

$Vsat

Vsat

t=

=

2

!

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  =;

eriving the oscillation #requenc' Be are interested to find the time it

ta6es for the capacitor to chargefrom an initial voltage

,c - (,th to ,c - &,th Considering point 2 !

 ?ssume at point 28 t-08 at point ! t-t=-T:2 ic - C d,c:dt 8 ic - (iR - (,o':R - &,sat :R

  ,o - ,c -

,o - ,sat :RC$ t & ,c#t-0$

,0 - ,sat :RC$ t ( ,th

vo

vo1

Vth

$Vth

$V sat

Vsat

t=

2

!

t*3

)0(/0=+

∫   t V dt C i

c

t 

c

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  =<

eriving the oscillation #requenc' ,o - ,sat :RC$ t ( ,th

 ?t half period8 t-t= - T:28

T - period8

vo

vo1

Vth

$Vth

$V sat

Vsat

t=

2

!

t-0

( )  th1

  V t t vc   ==

th1sat V t 

 RC 

V −=

th

sat

2V 

T 

 RC 

V − 

  

  =

satsat

th   44V  RC 

V  RCV T    ==   ( )sat

1 V  R R

 F 

+

 F  R

 RCRT    1

4=

  ( )1t t vc   =

#=$

#2$

#=$ - #2$

#requenc' o# oscillation4      == 1411

 R R

 RC T   f     F o

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  =>

Sawtooth(wave "enerator 

 ? sawtooth wave#orm has uneual rise and

fall times.

The rise time tr  may be longer than the fall

time tf  or vice versa.

tr  tf 

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  20

Sawtooth(wave "enerator 

Circuit The triangular(wave generator can be

converted to a sawtooth generator by adding

a variable dc voltage to the noninverting

terminal of the op(amp.

,ref  acts as a reference signal for the

integrator(,sat 1 ,ref 1 ,sat

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  2=

Circuit O"eration

+n the triangle oscillator ,= - 0,

Dy changing ,= to ,ref 8 we can change the slope

for the decreasing and increasing ramp function.

That is the main idea of the circuit.

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  22

eriving the out"ut voltage Vo

ecreasing ram" #unction Vo(t) * (Vre# Vsat ) t,RC $ Vth

Increasing ram" #unction

Vo(t) * (Vre# $Vsat ) t,RC Vth

Alease see the

note for detail

explanation

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  2!

Oscillation requenc' # * +,-

Escillation period T - tf  & tr  

ref sat

th2

V V 

 RCV t  f  −

=ref sat

th2

V V 

 RCV t r  +

=

ref sat

th

ref sat

th  22

V V 

 RCV 

V V 

 RCV t t T  r  f 

+

+

−

=+=

2

ref 

2

sat

satth4V V V  RCV 

−=

   

  

 +==

sat

ref 12

1/

V 

V T t cycleduty   f  

(,sat 1 ,ref 1 ,sat

Alease see the

note for detail

explanation