ECE 453 Wireless Communication Systems Oscillatorsemlab.illinois.edu/ece453/Oscillators.pdf ·...

ECE 453 – Jose Schutt‐Aine 1

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]

ECE 453Wireless Communication Systems

Oscillators


Feedback – Basic Concept


1. The closed-loop transfer function is a function of frequency

2. The manner in which the loop gain varies with frequency determines the stability or instability of the feedback amplifier

3. The frequency at which the phase of the transfer function is equal to 180o will be unstable if the magnitude is greater than unity

Feedback and Frequency Dependence


1 ( )fA s

A sA s s

Feedback and Stability

When loop gain A(j)(j) has 180o phase, we have positive feedback

1 ( )fA j

A jA j j


Nyquist Plot

- Radial distance is |A|- Angle is phase of - Intersects negative real axis at 180


Stability and Pole Location ( ) 2 coso ot tj t j t nv t e e e e t


Oscillator

• Closed‐Loop Transfer function:

–

• Barkhausen’s criteria for oscillation:

• 𝜔 = oscillation‐frequency.


Oscillator Topologies• Common‐base topology is usually preferred becauseFeedback within transistor is minimized (low Miller)External feedback is dominantCurrent gain has little phase shift and is constant with f

• Oscillator built around 2 major requirements:Oscillation frequency foPower Po that must be delivered to a load

• Other criteriaEfficiencySpectral purityFrequency stability


BJT – High-Frequency Model


Common-Collector Colpitts

• Minimize circuit dependence on transistor internal parameters

• Choose1C C

2 oC C


rightZ

leftZ

For Oscillation, we want Zleft = ‐Zright

Incremental Model


2 2 2 2 4 2 2 21 2 1 2 1 2

1 1 1 1 0me e e

r gC r C R C C r R C C R

2 2 2 2 21 2 1 2

1 0m me

g grC C R C C

2 11 2

2

2

1

1e

ms

CC C rC Rg C

C

Oscillation Criteria

Term in ‐4 can be neglected. This gives

Gives transconductance to set net resistance to zero

Condition gives


2 11 2

2

1ms

e

Cg C C rC R

1 2 12

1oms

L e

C C CgQ C R

Oscillation Conditions

If gm


CC-Colpitts - Observation• To implement tunable oscillatorUse variable capacitor in parallel or is series with inductor

• To extract power from oscillatorPlace resistance in shunt with inductorAccount for loading effect in analysis and design

1. Oscillation amplitude builds up and vbe increases

2. Transistor moves out of linear range and operation becomes nonlinear. Transconductance decreases.

3. When transconductance reaches gms, steady state operation is achieved.


L pt L p

L p

R RR R R

R R

i e eR R r

Common-Base ColpittsCf: tuning capacitorC1 & C2: feedback ratioRFC: RF choke to prevent power dissipation in RERP is equivalent resistance of coil Lt.

1/i ig R

1/t tg R

DEFINITIONS


1 2 1

1 11

0

iiS

i t o f ot

g s C C sCVI

g sC g s C C C VsL

Setting the determinant to zero provides the criterion for the onset of oscillation.

CB-Colpitts - Incremental Model


21

3 21 0

i t t i a t i b t t a t i

t a b t

s g s L g g C s L g C L g C L C g

s L C C L C

1 2aC C C 1b o fC C C C

2 1Re 0i t b i a t ij g L C g C g C g

CB-Colpitts - Oscillation ConditionsSetting the determinant to zero gives.

Defining and

we get a complex conjugate pair of roots. This leads to 2 equations for real and imaginary parts that must be equal to zero separately

2 21Im 0t t i a t a bj L g g C L C C C


22

/t i a to

a b t

g g s C LC C C

2

1 2 1 2 1 2 1 2

1 1/o t o f t i f o

LC tank circuit transistor and load

L C C C C C C R R C C C C C C

CB-Colpitts - Oscillation ConditionsFrom the equation for the imaginary part, we get

or

First term should predominate


1 2t t iL R R C C

2

1 2 1 2

1/o t o fL C C C C C C

CB-Colpitts - Oscillation Conditions

In that case

which leads to

Circuit oscillates at frequency determined by Ltand equivalent capacitance.


2min 2

1 1 1

11 1f o it o t

C C R CC R C L C

2 2 2

min1 2 1 2 1 1

11 1 11 /

i i

t t

C R C R CC C R C C C R C

CB-Colpitts - Oscillation Conditions

The solution of the real part using gives

For oscillations to start, of the transistor should be greater than min.


Crystal Oscillator

• Quartz exhibits piezoelectric effectReciprocal relationship between mechanical deformation and appearance of electrical potentialDeforming crystal produces voltageApplying voltage produces deformation

• When applied voltage is sinusoidalMechanical oscillations that exhibit resonances at multiple frequenciesExtremely high Q that can reach 105 and 106

Do not work above 250 MHz


Crystal Resonator - Circuit Model Rq, Lq and Cq describe mechanical resonance behavior C0 is capacitance due to external contacting

- Cq/C0 can be as high as 1000- Lq ranges from 0.1 mH to 100H


01

1 /q q qY j C G jB

R j L C

0 00 22

0 0

1 /0

1 /

q qo

q q q

L CC

R L C

Crystal Resonance FrequenciesAdmittance from terminals is

Resonance condition is expressed by


20

0 0 1 2q

S Sq

R CL

20

0 0 1 2q

P Pq

R CL

0 1 /S q qL C 0 0 0/P q q qC C L C C

Crystal Resonance FrequenciesSeries resonance frequency

Parallel resonance frequency

where


Crystal is characterized by Lq = 0.1H, Rq = 25, Cq= 0.3pF, and C0 = 1 pF. Find series and parallel resonance frequencies and compare them against the susceptance formula.

Crystal Oscillator - Example

2 20 0

011 1 0.919

2 22q q

S Sq qq q

R RC Cf f MHzL LL C

2 200 0

00

11 1 1.0482 2 2q q q

P Pq q q q

R C C RC Cf f MHzL L C C L


Voltage Controlled Oscillator

Varactor diode exhibits large change in capacitance in response to an applied bias voltage

Varactordiode


1 2 1 2 0IN IN C IN C B C B Cv i X i X i X i X

11 1 1 0B B C IN Ch i i X i X

Voltage Controlled OscillatorINv


11 1 2 1 211 1

1 1IN C C C CC

Z h X X X Xh X

21 2 11 1 2

1 1 1 1INZ j C C h C C

21 2

mIN

gRC C

1

ININ

Xj C

Voltage Controlled OscillatorFinding input impedance

Re-write as (assuming h11>>XC1 and ~ )

Using gm = /h11

Negative resistance


0 30 3 0 1 2

1 1 1 1 0j LC j C C

03 3 2 1

1 1 1 1 12

fL C C C

1 2

1 2IN

C CCC C

Resonance frequency follows from condition:

IN VARACTORX X

which gives

and

VCO – Resonance Frequency

Combined resistance of varactor diode must be equal to or less than |RIN| to create sustained oscillations

where 0 30 3

1VARACTORX j L C


• Most important performance characteristicFrequency‐domain equivalent of jitterOriginates from thermal, shot and 1/f noiseRandom variation in phase angle of oscillatorAffects frequency stability

• Phase noise quantificationCompare phase noise power to carrier powerDetermine phase noise spectral densityCan be characterized in time or frequency domain

Oscillator Phase Noise


Phase Jitter in Time Domain

If the phase varies, the waveform V(t) shifts back and forth along the time axis and this creates phase jitter


( ) cos 2 ( )o oV t V f t t (t) is a random noise process. The instantaneous frequency is

1 ( )2inst o

d tf fdt

1 ( )( )2

d tf tdt

( ) 1 ( )( )2o o

f t d ty tf f dt

Oscillator Phase NoiseOscillator operates as

Instantaneous frequency variation is

Fractional deviation in instantaneous frequency is


Phase, Period and CTC Jitter


Phase Noise in Spectral Domain

Phase noise appears as sidebands centered around the carrier frequency


Phase Noise Specification

( )( ) no

P fL fP f

: phase noise power (in watts) ( )nP f: carrier’s power (in watts) oP

: phase noise bandwidth (in hertz) f

1( ) ( )2

L f S f

Phase noise magnitude is specified relative to the carrier’s power on a per-hertz basis

: PSD of phase noise( )S f

10( )( ) 10log2

S fL f

or


Phase Noise to Phase Jitter

From the phase noise PSD, random jitter and deterministic jitter can be identified

Need: convert phase noise measured in the frequency domain to phase jitter for PLLs, clocks and oscillators


Phase Noise Impact on PLL

Phase noise is the key metric for evaluating the performance of a PLL system

ECE 453 Wireless Communication Systems Oscillatorsemlab.illinois.edu/ece453/Oscillators.pdf ·...

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