Sinusoidal Oscillators• Signal generators: sinusoidal, rectangular, triangular, TLV, etc.

▪ Sine wave generation: frequency selective network in a

feedback loop of a PF amplifier: sinusoidal oscillator

-Oscillation frequency: f0

-Oscillation amplitude:

- Oscillation criterion

- Frequency stability

- Amplitude stability

- Distortion coefficient

oV̂

PF amplifier

rsi xxx +=

ar

aA

−=

1

❖ Special case of PF:

→==− Aarar ;1;0)1(

so Axx = xo finite only if xs=0

Oscillator feedback loop

0)()(1 00 =− jrja

In the complex domain

for a unique 00 2 f =

)()(1

)()(

jrja

jajA

−=

Barkhausen criterion

1)()( 00 = jrja

Signal reconstruction on the feedback loop

Frequency dependent

components (C, L)

fLjLjZ

fCjCjZc

L

2

2

11

==

==

Complex number - short review

Oscillation criteria

ajejaja

)()( =

rjejrjr

)()( =

1)()()()()(

0000 ==+ raj

ejrjajrja

kra 2=+

✓ magnitude condition:

✓ phase condition:

1)()( 00 = jrja

gives f0

gives a0

The loop gain is equal to unity in absolute magnitude

The phase shift around the loop is zero or an integer multiple of 2π

➢ Basic amplifier – frequency independent

• inverting

• noninverting

➢Frequency selective feedback network

0=a

o180=a

To fulfil the phase condition, there must be

a unique frequency, f0 where the phase shift is:

00 180 if ,180 == ar 00 0 if ,0 == ar

RC Oscillators

−+++

==

12

21

1

2

2

1 11

1

)(

)()(

CRCRj

C

C

R

Rjv

jvjr

o

r

( ) ( ) jvjrjv or )(=)(

)()(

jv

jvjr

o

r=

21

2)(ZZ

Zjr

+=

1

11

1

CjRZ

+=

2

22

1||

CjRZ

=

WIEN Bridge Frequency selective network

out in

Transfer function

complex number

voltage transfer

WIEN Bridge

rjejrjr

)()( =

2

12

21

2

1

2

2

1 11

1)(

−+

++

=

CRCR

C

C

R

R

jr

++

−

−=

1

2

2

1

12

21

1

1

C

C

R

R

CRCR

arctgr

out in

modulus

phase

WIEN Bridge

2

12

21

2

1

2

2

1 11

1)(

−+

++

=

CRCR

C

C

R

R

jr

0→ 0)( →jr

2121

1

CCRRo ==

0)( →jr→

1

2

2

10

1

1)(

C

C

R

Rjr

++

=

The maximum value (as a function of ω):

asymptote

asymptote

Modulus

WIEN Bridge

2

12

21

2

1

2

2

1 11

1)(

−+

++

=

CRCR

C

C

R

R

jr

0→ 0)( →jr

2121

1

CCRRo ==

0)( →jr→

1

2

2

100

1

1)(

C

C

R

Rjrr

++

==

The maximum value: asymptote

Modules

asymptote

ω0

WIEN Bridge

++

−

−=

1

2

2

1

12

21

1

1

C

C

R

R

CRCR

arctgr

0→090+→r

2121

1

CCRRo ==

→

intermediate

value

asymptote

asymptote090−→r

00=r

Phase

WIEN Bridge

++

−

−=

1

2

2

1

12

21

1

1

C

C

R

R

CRCR

arctgr

0→090+→r

2121

1

CCRRo ==

→

intermediate

value

asymptote

asymptote090−→r00=r

Phase

ω0

For only one unique

frequency, f0 we have

0=r

WIEN Bridge

For the phase condition:

noninverting amplifier

0=a

out in

Frequency

response

2121

00

2

1

2 CCRRf

==

WIEN BridgeSummary

−+++

==

12

21

1

2

2

1 11

1

)(

)()(

CRCRj

C

C

R

Rjv

jvjr

o

r

Barkhausen criterion: 1)()( 00 = jrjaReal number

Real number

01

120

210 =−CR

CR

𝜔0 =1

2𝜋 𝑅1𝑅2𝐶1𝐶2

WIEN BridgeSummary

−+++

==

12

21

1

2

2

1 11

1

)(

)()(

CRCRj

C

C

R

Rjv

jvjr

o

r

𝜑𝑟(𝑗𝜔0) = 0

2121

02

1

CCRRf

=

1

2

2

10

1

1)(

C

C

R

Rjr

++

=

RRR == 21

CCC == 21

If

RCf

2

10 =

3

1)( 0 =jr

Op amp and WIEN bridge oscillator

tfVtv oo 02sinˆ)( =

2121

02

1

CCRRf

=

1

2

2

10

1

1)(

C

C

R

Rjr

++

=

Op amp and WIEN bridge oscillator

RRR == 21

CCC == 21

RCf

2

10 =

3

1)( 0 =jr

3)(

1)(

0

0 ==

jr

ja3

41R

Ra += 31

3

4 =+R

R34 2RR =

?ˆ =oV Nonlinearity of the gain, close to saturation

2121

02

1

CCRRf

=

1

2

2

10

1

1)(

C

C

R

Rjr

++

=

For

1)()( jrja

1)()( jrja

oscillations are attenuated - zero

oscillations are amplified - saturation

tfVv oo 02sinˆ =

cstjr =)(

00ˆ,)(,ˆ VjaVo

Stability of the oscillation amplitude

Automatic gain control - depending on the output voltage magnitude

1)()( 00 = jrja oscillations are maintained - oscillate

Automatic gain control (AGC)

oV̂

AGC for WIEN bridge oscillator

3

41R

Ra +=

How an AGC can be implemented,

so that a will depend on vo value?

• R4 - dependent on vo

or

• R3 - dependent on vo

Diode revisited - as variable resistor

Static resistance

of a diode in the

operating point

D

DD

I

Vr =

=== k24.4mA 0.279

V495.0

1

11

D

DD

I

Vr

Q1 Q2

Q3

Q4

=== k599.0mA 0.926

V555.0

2

22

D

DD

I

Vr

=== k263.0mA 2.279

V6.0

3

33

D

DD

I

Vr === k115.0

mA 5.604

V645.0

4

44

D

DD

I

Vr

If the voltage drop VD increases, the equivalent static resistance rD decreases

AGC using Diodes - how?

D

DD

I

Vr =

If the voltage drop VD increases,

the equivalent static resistance rD

decreases 3

41R

Ra +=

• R4 - dependent on vo

or

• R3 - dependent on vo

echR ,4

1)()( 00 = jrjaon

1)()( 00 jrjaoff

1. AGC using diodes

for )(tvo small, D1, D2 – (off)

3

''

4

'

41R

RRaoff

++=

to maintain oscillations

)(tvo increases,

D1 – (on) on the

positive half-cycle

D2 – (on) on the

negative half-cycle

3

''

4

'

4 ||1

R

rRRa D

on

++=

oV̂ is given by the value of rD

3

,41

R

Ra

ech+=

Problema) How does the voltages vo(t)

and v+(t) look like in the steady-

state regime? What is the

oscillation frequency?

b) Size R4 so that the circuit will

maintain the oscillation. For the

maximum value (magnitude) of

the sinusoidal output, consider

rD1= rD2= 0.5 kΩ. Verify if the

oscillation can start.

c) What is the magnitude of vo(t)

in the conditions of question b),

if the voltage drop across one

diode is vD = 0.58 V for the

equivalent resistance rD= 0.5 kΩ

d) How does the voltage vo(t)

look like in the steady-state

regime if D2 diode is missing?

2. AGC using n-channel

depletion-type MOSFET

)(2

1

ThGS

DSVv

r−

DSrR

Ra

++=

'3

41

DSGS rv ,||

vGS<0

Op amp and RC ladder network oscillator

• High pass band

• Low pass band

• the phase-shift is in the

range of [0o; -90o]

• inverting basic amplifier

• how many identical RC cells

are necessary to build an

oscillator?

Low pass RC ladder

with 3 cells

( ) ( ) 32651

1)(

RCRCjRCjr

−+−=

0=r

( ) 063

00 =− RCRC

RCf

2

60 =

( )29

10 −=jr

The circuit of RC ladder network oscillator