1

OSCILLATORS

An oscillator is a circuit, which produces a periodic signal withoutany input signal. It converts DC power (from the supply) to a periodic signal.

Oscillators are extensively used in both receive and transmit paths.They are used to provide the local oscillation for the mixers for upand down conversion.

On this topic, we cover:

1. Oscillator basics2. Oscillator topologies3. Phase noise issues4. Oscillator implementations and design

ECEN 665

2

Oscillator types:

1. Crystal oscillators2. Active-RC and Gm-C oscillators3. Ring oscillators4. LC timed oscillators5. Relaxation oscillators.

… etc.

Voltage Controlled Oscillators:

VCO’s are used in RF applications, usually, in a Phase locked loop to provide Different LO frequencies needed for channel selection. Besides frequency synthesis it is used for other applications.

3

OSCILLATOR BASICS

Most RF oscillators can be viewed as + ve feedback system.

+ A(s)

β(s)

+

+vovi

)()()(1

)()( svssA

sAsv io β−=

The Barkhausen Criterion states that sustained oscillation can be achieved.

if Loop gain equal to unity.

and

( ) ( ) 1=oo ssA β

( ) ( ) ooo ssA 0=β

For zero vi the output will be finite at a given frequency (ωo). For loop gain >1 oscillation will grow.

4

Two-Port vs One-Port ModelsThe previously shown model is known as the two-port model, because the A(s) and β(s) Networks are both two-port nets in a closed loop.

The one-port model of oscillators is shown below:

-Rp Rp L C

Active Network Resonator

The resonator (LC tank for example) has parasitic resistances (RP) which prevents the resonator from oscillating because the stored energy will leak through the resistance. To compensate for this loss a positive feedback negative parallel resistances (-Rp) will be added to the resonator so that the energy loss in Rp is replenished by the negative resistance. The negative resistance’s implementation is typically an active network.

5

What sets the Frequency and Amplitude of an Oscillator?

The frequency is usually set by Barkhausen Criterion. It is the frequency at which the loop gain is greater or equal to 1 and the phase is zero.

In many cases, as in LC tuned oscillators, the ideal oscillation frequency is determined by the LC tank.

LCfo π2

1=

The amplitude is, however, more a complicated parameter to set.

6

More on Oscillator Theory:Ideal Oscillator:

iL iC

CL

ωj

LCo1

=ω

σ

Oω−

Once the tank is excited (by a current pulse for example) a certain amount of energywill be conserved. This energy will alternate between magnetic and electric forms.

In the time domain:

LCv

dtd

dtdLCv

dtdvCi

iidtdLv

oov

v

c

cLiL

1 ere wh 0or

and

but

22

2

2

2

==+

−=⇒

=

−==

ωω

7

The Differential Equation of the Ideal Oscillator

022

2=+ v

dtd

ov ω

When solved yields:tojtoj eVeVtv

ωω −+= 21)(

Choosing the case where the phase angle is zero at t=0

tVtv oωsin)( =

In the frequency domain:

( )

22

2 )()(

)( )()( but )()()(

o

Lo

c

LcLL

soisLsV

svsCIsisIoisisLsV

ωω

+=⇒

⋅=−=+=

Both poles are at as expected !! ojω±

8

Real Oscillator:

Initial excitation well start some oscillation.This will die out after a while because theEnergy leaks through the parasitic resistances.

L

RL

C

RC

σ−a

ωj

2411Qo −ω

2411Qo −− ω

oω

oωCharacteristic equation

( )

C

L

C

CL

L

CLC

LCRR

LCRCRRLss

RCRRLsRLCs

++

+

+++

2

2

or

C

CL

CCo

LCRCRRL

RLRL

2

2

+=

=

τ

ω

9

If oscillation is to be maintained. The energy loss (due to parasitic resistance) is to becompensated for. This energy is usually provided from the power supply via a negativeresistance, a nonlinear conductance or regenerative feedback, which convert dc power to signal power.

CoC

L

o

s

cL

ccLL

CRQ

RL

CLR

RRRQRQ

G

ω

ω

1

Q where

ly.respective , and theof parasitic series

theare and //1

L

22

=

=

=

LLiC

ci ri

Ggmi

v

q p

flatter

steeper rgm ii +

slope = - a + G

ri

Slope = G

gmi

Slope = - a

v

v

v

NegativeResistor

10

The following model equation can be written

vr

c

L

gmrcL

GidtdvCi

vdTL

i

iiii

=

=

∫=

=+++

1

0

Differentiating both sides of the above equation:

012

2=+++

dtdi

dtdvG

dtvdCv

Lgm

[ ] 02

2=+++ viG

dtdL

dtvdLC gmv

The above DE describes the behaviour of many oscillator implementations.

11

At the quiescent bias point (which is the equilibrium point)

( ) aidtd

vgm −==0

( ) 02

=+−+ vaGLdt

vdLC

In the frequency domain, the characteristic equation at the equilibrium point is given by:

( ) 012 =+−+ saGLLCsSolving for the poles of the system:

βα jC

aGLC

jC

aGs

±=

⎟⎠⎞

⎜⎝⎛ −

−±⎟⎠⎞

⎜⎝⎛ −

−=

2

12

2

2,1

In the time domain:

teAtv t βα cos)( =

Oscillationenvelope

Oscillation

12

CGa

teAtv t

2 where

cos)(−

=

=

α

βα

From the above equ., the following observations can be made:

1. The oscillator frequency is determined by

2. if

3. if

LC1

≅β

( )aGo <>α

Decaying oscillation

G < a G > a

RHP LHP

t

t

( )aGo >< α

growing oscillation.

13

In practical oscillators, the slope of the negative resistance or nonlinear element (a) is made

greater than G . This results in growing oscillations as long as the swing is limited

between points p and q.

( )Ga 3≅

p q

( )Ga −−

If the oscillation swing grows beyond p or q, the

slope (a-G) or α become negative and the poles

move the LHP causing the oscillation to decay

tempe. Until the swing drops within below p

and/or q. This is how a sustained oscillation is

produced and a steady sate can be reached.

Note that the oscillations will not be pure sinusoid. The harmonics are

Usually far out and can be easily filtered.

Also note that it is hard to determine the oscillation swing exactly!

14

The van der Pol Approximation

In the 1920’s , Van der Pol proposed to model the total I-V Chcis the simplest possible manner. He assumed a cubic approximation.

( )[ ]vvsii rgm +

( ) 311 vbvavi +−=

where( )vi

xv−

xv v21

1

xvGab

Gaa−

=

−=

Van der Pol’s analysis leads to the following time domain expression for the Oscillation voltage.

( )

CL where

cos1 34)(

1

21

1

1

a

LCte

batv o

LCott

=

⎟⎠⎞

⎜⎝⎛ +⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=

−−−

ε

φε

Φο is introduced to provide the proper phase in relation to the constant to

15

At steady state, the zero-to-peak voltage amplitude reaches a maximumvalue of

xvbav 15.1

34

1

1max ==

16

Basic LC Oscillator Topologies(a) Feedback model:

ii

Σ

-Gm+-

vGi mf =

if+ii vL C R

( ) ( )

( ) ( ) 1/

11 when

1

)1(

222

22

+−+=

+−+=

++=

++=

−=

=−

+=+=

mi

omi

o

mi

im

miif

GGsLLCssLi

cGGsscsiv

CGss

cssL

sCGZ

ZGZiv

ZiZGv

ZvGiZiiv

ω

ω

( ) 01s

:sticcharacteri The2 =+−+ sGGLLC m NOTE: Similarly to that on pp 11

17

(b) Negative Resistance Model:

L C R

vGm−

ri

+

-mG

vgm

v

negative resistance

Negative Resistance Realization:

v

vα

( )vgi m α−=

( ) 1 ,11

<−

−=−

= αα

αα mm g

viG

A MOS device can also be used.

v

vgmmg1

−

v

18

See the sustained on the next page. The frequency LC1

ggain=-100.0uv

v

Tvpairs=3Periodic:FALSE

Delay:0V2

G0C15C:1e-12 oL12

I:1e-9

VOR22R=10000

gnd

Rgm

1=

A small narrow pulse to start simulations.The pulse (to produce freq. at µi. o

p fT 1

<

19

20

21

22

23

Full Oscillator Realization:

To avoid loading the LC tank (to prevent reducing its Q), an impedance transformerto up-convert the impedance . ( )12 ZZ >

Different realization for the impedance transformation are shown below:

LCTank

ImpedanceTransformer

CCV

v2Z

1Zvα

CCV1:n

CCVCCV

1C

2CL 1L

2LC

12

2 ZnZ = ( ) 12

212 1 ZCCZ += ( ) LZLLZ 2122 1+=

COLPITTS OSC. HARTLEY OSC.

24

Note that a resonance, the phase shift of the loop is supposed to be zero (according toThe Barkhausan Criterion). That is why the output signal is fed back to the emitter. TheZero-phase condition may be satisfied if the output signal is inverted and then fed back to the base as shown below:

This topology is widely used and is known as the –v Gm oscillator

inverter

Load

CCV

LC

2Q 1Q

C

Lv v−

2Q 1Q

25

It is interesting to note that the cross-coupled pair (BJT an MOS) presents a negativeresistance to the tank. That is why it can be classified as a negative resistance oscillator.

Since it is a fully differential circuit we may model the cross-coupled pair as follows:

a’av

v−vgm−

vv−

vgm−

The resistance between a and a’ can be expressed as:

( )mm gvg

vvR 2−=

−−−

=

26

1 – BJT Realization:CCV

L LvpC vpC

C′ C′

BBV

The capacitor divider is used to reduce the voltage swing at the base compared tothat at the collector to avoid saturating the BJTs.

vpvp

collectorbase CCCC

Cvv >′

′+×=

The AC coupling resistors add noise.

27

2 – NMOS (or PMOS) Realization

It is similar to the BJT except that it does not need capacitive division.

3 – CMOS Realization:

L

C

CCV

1P 2p

1v 2v

1N 2N

oI

28

Current Limited Oscillation

In this case the current of the tail current sources is fully switched from one side of thepair to the other at a frequency Assuming that the BJT or MOS switches

are fast enough, the current waveform in each branch is a square wave with 50% dutycycle. Assuming that the tail current is Io , the current wave form in one side of thepair has the following shape.

LCo1

=ω

oI

00 T 2T 3T

This can be described by its power series as

( ) ⎟⎠⎞

⎜⎝⎛ +++= LttIi ooof ω

πω

π3sin

34sin41

21

NOTE: If the current wave form deviate from a square wave which is typically the

π4

case the factor will change to some other number depending on the

on the current waveform shape!!

29

Consider the NMOS oscillator:

LCo1

=ω

2LR 2LR2L2LCCV

C2 C21v

2v

oIoI

2LC2

1v

222 RRQ LL =

L

oL R

LQ ω=

2//

2QR 2

sinsin2

2sin2)(

22L2112

1

CC

Loppp

oo

oooooo

RQRRIvvv

tRItRIZtItv

=⋅=−=

==×⎟⎠⎞

⎜⎝⎛= ==

π

ωπ

ωπ

ωπ ωωωω

NOTE: The current components at 3ωο, 5ωο, 7ωο … will produce voltage componentswith small power because the impedance at these frequencies is small.

30

C

QQ

QQ

GHzLC

LCC

LL

o

2092R

2.314 25

3.628 50

56.121

2

=

==

==

==ω

Switching of square wave current (pulses)

31

32

Now consider the CMOS oscillator with a similar tank as that used for the NMOSOscillator (pp 24).

In one half cycle, the devices P1 and N2 are on while N1 and P2 are off, so the oscillator can be modeled as follows:

On the second half cycle, the current through R is still Io but flows in the oppositedirection. So the current waveform (assuming that the PMOS and NMOS devicesswitch enough) looks as follows:

RIo

Io

Io

L

v1v1

C

+Io

-Io

33

This is different from the wave form previously shown, because it is bipolar. TheFourier expansion for the CMOS oscillator is:

⎟⎠⎞

⎜⎝⎛ ++−= LtttIti oooo ω

πω

πω

π5sin

543sin

34sin4)(

The differential voltage of the CMOS oscillator is hence, given by:

RIv

tRItv

op

oo

o

π

ωπωω

4

sin4)(

12

12

=

⋅==

• Note that swing of the CMOS oscillator is double that of the NMOS (or PMOS)oscillator as seen before.

34

Some Important Points:

1. In the oscillators cases we considered so far we have assumed that the swing issmall enough so that the swing is determined by the current and the tank parallel resistance. If this is the case, we consider the oscillator to be current limited. IfThe swing grows too much, the swing will be limited by the supply voltage and in the oscillator is known to be voltage limited.

2. In current limited cases, the swing increases by increasing the current (tail current)or by using high Q tank (which increases the resistance.).

3. The switching speed of the core devices will determine the current waveform shapewhich in turn determines magnitude of I at ωo (fundamental component).

35

Voltage Controlled Oscillators

This is one of the most critical blocks in any RF frequency synthesizers. Adjustableoscillators are used for tuning to desired frequencies. A VCO is used as part of aPLL to obtain precise frequencies.

The frequency tuning is done by changing the LC tank capacitance value. This is morepractical than varying the inductance. Different types of variations have been used toobtain a voltage controlled capacitance. Among them are pm junction varactors andMOS varactors and MOS varactors.

In a pn junction varactor the oscillator output node is connected to either the p or n sideand the controlling voltage is applied to the opposite terminal

orVosc.VC VC

Vosc.

36

(VC)

MOS varactor come in two types. Let us consider NMOS varactors.

Conventional MOS

p+ p+

p-well

VoscVW

On this device evenat high frequenciesthe capitance willreturn to Cox at some point.

n+ n+p-well

Vosc

VW(VC)

MOS Cap

For highfrequency

VW

37

For MOS Cap:

OXCC

modeon Accumulati0

=

⇒<GWV

dVdQ

C

CCC

VV

depdep

depox

dep

GWT

=

+=

⇒>>

oxCC

modeDepletion 0

M O S

Cox

Qm

Qssm QQV ∆∆∆ &→

depQV ∆∆ →

Cox

Qm

Qdep

Cdep

38

Qm

Qdep

Qinv

0=→ depinv QQV ∆∆∆

VGW > VT

At low frequency:

Since the frequency is low, the minoritycharges have time to be pulled from thebulk to the surface to make up the inv.layer and the depletion layer does notchange any more so

oxinv C

dVdQC ==

At high frequency:

There is no time to attract minority changesto the surface. So the change comes form the dep layer

depox

depox

CCCC

C+

=

39

An ideal VCO has the following law

CVCOoosc VKff +=

Free runningfrequency

VCO gain(Hz/V)

Control

fosc

fo

oVC

40

The spectrum of is up-converted to )(tnφ Cω±

Cω

Ideal Oscillator

Qn

Cω

Real Oscillator

The phase noise is measured by dividing the power in 1Hz bandwidth at an offset ∆fby the power of the carrier (fc).

( ) HzdBffL c⎟⎠

⎞⎜⎝

⎛==

carrier theofpower offset at 1Hzin power 40log noise Phase ∆∆

41

~

Implications of Phase Noise

In the case of receiver:

If wantedsignal

Interfer

Signal

Interfer

RFIF

LOOscillator

LOωc

Overlap of theinterference withthe wanted signalafter down conversion

42

Wantedsignal

ωTX

TXRX

TXω~

In the case of transmitter:

If the phase noise of the LO at the transmitters side is large, the skirtof the RF transmitted signal overlaps nearby wanted signals and hencereduces the SNR.

43

f1 f2f∆

Wantedsignal

Pint)( fS

Interferencewith skirtdue to phase noise

BWch

Assume that, from the blocking specs, that

Pint - Psig = YdB

Psig = the signal power integrated over

channel BW.

Assume also that a given signal to noise ratio (SNR) is to be met.

( )( ) ( )

( )

( )( ) HzdBYSNRBW

HzdBYSNRBW

PP

PP

BWPBWP

PBWP

PP

BWPffPPff

PfS

dffSP

cchn

cchn

sig

sig

n

chch

nchnnon

chnonon

non

nffn

++−=−−−=

×=×===

×=−=∴

∫==

log10log10

11

.&between constant is theassume simplicityfor

power noise The

intintintint

12

21

21

φφ

φ

44

LEESON’S MODEL

Based on a linear time in variant approach for timed LC tank oscillators, one canderive an expression for the phase noise.

In(w) L C GL -GactiveEffective noise source(white noise)

It can be shown that impedance seen by the effective noise source at ωo + (where is given by:

( )

tank theof Q loaded theis tank theof econductanc parasitic parallel theis

21

L

L

L

o

Loo

QG

QGjZ

ω∆ωω∆ω −=+

oω∆

oω∆oωω∆ <<

45

The phase noise (for the case of white noise) can, hence be given by:

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⋅+=

=

ω∆ω

ω

L

o

Lsig

n

sig

no

sig

noise

QGV

i

V

iBWZ

vvBWL

21

ˆ2121

log10

ˆ21

21

log10

log 10)(

2

2

2

22

2

2

2ni can be written as 4 F KT GL this expression for the effective noise curent density is

not physical. (Just a model!) F is difficult to derive and is just a fitting parameter!

Lsigos

L

o

sig

GViP

QPFKTL

ˆ21

22log10)(

2

==

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

ω∆ωω∆

The rms power dissipated in GL.

46

Even though the LTI model is not accurate, it yields important facts for oscillator design:

( )

( )

( ) .oscillator theinside noise reduce 3.

and increasingby increase1 2.

increase1 1. 2

⇒

⇒

⇒

FL

IQPP

L

QQ

L

Lss

LL

αω∆

αω∆

αω∆

Leeson’s model is not accurate and fails top account for the large signalBehavior of the oscillator. Yet it gives insight for oscillator design.