EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 1 Voltage-Controlled Oscillator (VCO) VCVC f osc f min f max slope = K vco Desirable characteristics:

EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 1

Voltage-Controlled Oscillator (VCO)

VC

fosc

fmin

fmax

slope = Kvco

Desirable characteristics:

• Monotonic fosc vs. VC characteristic with adequate frequency range

• Well-defined Kvco

^

^

Noise coupling from VC into PLL output is directly proportional to Kvco.

^


Oscillator Design

loop gain

Barkhausen’s Criterion:

If a negative-feedback loop satisfies:

then the circuit will oscillate at frequency 0.


Inverters with Feedback (1)

V1 V2

V1

V2 1 inverter

feedback

V1

V2

2 inverters

feedback

1 stable equilibrium point

3 equilibrium points: 2 stable, 1 unstable(latch)

1 inverter:

V1 V2

2 inverters:


Inverters with Feedback (2)

3 inverters forming an oscillator:

1 unstable equilibrium point due to phase shift from 3 capacitors

V1 V2

V1

V2

Let each inverter have transfer function

Loop gain:

Applying Barkhausen’s criterion:


Ring Oscillator Operation

VA VB VC

tp tp tp

VA

VB

VC

VA

tp

tp

tp


Variable Delay Inverters (1)

VC

Vin Vout

Current-starved inverter:Inverter with variable load capacitance:

Vin Vout

VC


Variable Delay Inverters (2)

R R

Vin+ Vin- Vin+ Vin-

Vout-Vout+

IfastIslow

RG RG

ISS

VC

+

_

Interpolating inverter:

• tp is varied by selecting weighted sum of fast and slow inverter.

• Differential inverter operation and differential control voltage

• Voltage swing maintained at ISSR independent of VC.

VA

VB

VC

VD

tp

tp

tp

tp

VA


Differential Ring Oscillator

additional inversion (zero-delay)

VA

+

−

Use of 4 inverters makes quadrature signals available.

VB

+

−VC

+

−VD

+

−VA

−

+


Resonance in Oscillation Loop

r

r

1

At dc:

Since Hr(0) < 1, latch-up does not occur.

At resonance:


LC VCO

Vin Vout

Vin

Vout

CL

realizes negative resistance

2L

CC


A. Reverse-biased p-n junction

+ –VR

VR

Cj

B. MOSFET accumulation capacitance

+

–

VBG

varactor = variable reactance

Variable Capacitance

VBG

Cg

accumulationregion

inversionregion

p-channel

n diffusion in n-well


LC VCO Variations

2L

CC

2L

CC

2L

CC

ISS

2L

CC

ISIS


1. ideal capacitor load

2. CML buffer load

Effect of CML Loading

1.

3.8 1 nH

400 fF 400 fF

Cg = 108fF

1 nH 3.8

400 fF 400 fF 108 fF108 fF

2.


Substantial parallel loss at high frequencies weakens VCO’s tendency to oscillate

(note p < z)where:

CML Buffer Input Admittance (1)


Yin magnitude/phase: Yin real part/imaginary part:

magnitude

phase

imaginary

real

Contributes 2k additional parallel resistance



imaginary

real

Contributes negative parallel resistance

Cg = 108 fF

3.8 nH

3.8 1 nH

400 fF 400 fF


3. CML tuned buffer load


Loading VCO with tuned CML buffer allows negative real part at high frequencies more robust oscillation!

ideal capacitor load

CML buffer load

CML tuned buffer load


Cg = 108 fF

3.8 nH

3.8 1 nH

400 fF 400 fF


Differential Control of LC VCO

Differential VCO control is preferred to reduce VC noise coupling into PLL output.


Ring Oscillator LC Oscillator

– slower

– low Q more jitter generation

+ Control voltage can be applied differentially

+ Easier to design; behavior more predictable

+ Less chip area

+ faster

+ high Q less jitter generation

– Control voltage applied single-ended

– Inductors & varactors make design more difficult and behavior less predictable

– More chip area (inductor)

Oscillator Type Comparison


Random Processes (1)

Random variable: A quantity X whose value is not exactly known.

Probability distribution function PX(x): The probability that a random variable X is less than or equal to a value x.

0.5

1

x

PX(x)

Example 1:

Random variable


0.5

1

x

PX(x)

x1 x2

Probability of X within a range is straightforward:

If we let x2-x1 become very small …



Probability density function pX(x): Probability that random variable X lies within the range of x and x+dx.

0.5

1

x

PX(x)

x

pX(x)

dx



Expectation value E[X]: Expected (mean) value of random variable X over a large number of samples.

Mean square value E[X2]: Mean value of the square of a random variable X2 over a large number of samples.

Variance:

Standard deviation:



Gaussian Function

x

2

1. Provides a good model for the probability density functions of many random phenomena.

2. Can be easily characterized mathematically .

3. Combinations of Gaussian random variables are themselves Gaussian.


Joint Probability (1)

If X and Y are statistically independent (i.e., uncorrelated):

Consider 2 random variables:


Consider sum of 2 random variables:

x

y

dx

dy = dz

determined by convolutionof pX and pY.



*

Example: Consider the sum of 2 non-Gaussian random processes:



3 sources combined:

*



4 sources combined:

*



Central Limit Theorem:Superposition of random variables tends toward normality.

Noise sources



Fourier transform of Gaussians:

F

Recall:

F

F -1

Variances of sum of random normal processes add.


Autocorrelation function RX(t1,t2): Expected value of the product of 2 samples of a random variable at times t1 & t2.

For a stationary random process, RX depends only on the time difference

for any t

Note

Power spectral density SX():

SX() given in units of [dBm/Hz]


Relationship between spectral density & autocorrelation function:

Example 1: white noise

infinite variance(non-physical)


Example 2: band-limited white noise

x

For parallel RC circuitcapacitor voltage noise:


Random Jitter (Time Domain)

Experiment:

datasource

CDR(DUT) analyzer

CLK

DATA RCK


Jitter Accumulation (1)

Free-runningoscillator output

Histogram plots

Experiment:Observe N cycles of a free-running VCO on an oscilloscope over a long measurement interval using infinite persistence.

NT

1 2 3 4

trigger


Observation:As increases, rms jitter increases.

proportionalto 2

proportional to



Noise Spectral Density (Frequency Domain)

foscfosc+f

Sv(f)

f (log scale)

1/f2 region (-20dBc/Hz/decade)

Power spectral densityof oscillation waveform:

Single-sideband spectral density:

Ltotal includes both amplitude and phase noise

Ltotal(f) given in units of [dBc/Hz]

1/f3 region (-30dBc/Hz/decade)


Noise Analysis of LC VCO (1)

active circuitry

C L R -R C L

+

_

vcinR

Consider frequencies near resonance:

noise from resistor


Spot noise current from resistor: C L

+

_

vcinR

Noise Analysis of LC VCO (2)

Leeson’s formula (taken from measurements):

Where F and1/f3 are empirical parameters.

dBc/Hz

spot noise relative to carrier power


Oscillator Phase Disturbance

Current impulse q/t

_+Vosc

t t

ip(t)

Vosc(t) Vosc(t)

Vosc jumps by q/C

• Effect of electrical noise on oscillator phase noise is time-variant.• Current impulse results in step phase change (i.e., an integration).

current-to-phase transfer function is proportional to 1/s

ip(t)

ip(t)


Impulse Sensitivity Function (1)The phase response for a particular noise source can be determined at each point over the oscillation waveform.

Impulse sensitivity function (ISF):

(normalized to signal amplitude)

change in phasecharge in impulse

t

Example 1: sine wave

t

Example 2: square wave

Note has same period as Vosc.


Impulse Sensitivity Function (2)

Recall from network theory:

LaPlace transform:

Impulse response:

time-variant impulse response

Recall:

ISF convolution integral:

from q

can be expressed in terms of Fourier coefficients:


Case 1: Disturbance is sinusoidal:

, m = 0, 1, 2, …

negligible significant only form = k

(Any frequency can be expressed in terms of m and .)



I

2osc


Current-to-phase frequency response:

oscosc

osc 2osc 2osc

For


osc 2osc

Case 2: Disturbance is stochastic:


MOSFET current noise:

thermalnoise

1/fnoise

A2/Hz

osc 2osc

thermal noise

1/f noise



osc 2osc

due to 1/f noise

due to thermal noise

Total phase noise:

n



noise corner frequency n

(log scale)

(dBc/Hz)

1/(3 region: −30 dBc/Hz/decade



t

t

Example 1: sine wave Example 2: square wave


Example 3: asymmetric square wave

t

will generate more 1/(3 phase noise

is higher will generate more 1/(2 phase noise



Effect of current source in LC VCO:

Vosc+ _

Due to symmetry, ISF of this noise source contains only even-order coefficients − c0 and c2 are dominant.

Noise from current source will contribute to phase noise of differential waveform.



ID varies over oscillation waveform Same period as

oscillation

Let

Then where

We can use


ISF Example: 3-Stage Ring Oscillator

M1A M1B M2A M2B M3A M3B

MS1 MS2 MS3

R1A R1B R2A R2B R3A R3B+

Vout

−

fosc = 1.08 GHzPD = 11 mW


ISF of Diff. Pairs

for each diff. pair transistor


ISF of Resistors

for each resistor


ISF of Current Sources

ISF shows double frequency due to source-coupled node connection.

for each current source transistor


Phase Noise Calculation

Using: Cout = 1.13 pF

Vout = 601 mV p-p

qmax = 679 fC

= −112 dBc/Hz @ f = 10 MHz


Phase Noise vs. Amplitude Noise (1)

osct

v

v Spectrum of Vosc would include effects of both amplitude noise v(t) and phase noise (t).

How are the single-sideband noise spectrum Ltotal() and phase spectral density S() related?



t t

i(t) i(t)

Vc(t) Vc(t)

Recall that an input current impulse causes an enduring phase perturbation and a momentary change in amplitude:

Amplitude impulse response exhibits an exponential decay due to the natural amplitude limiting of an oscillator ...


+

Phase noise dominates at low offset frequencies.



osc

Phase & amplitude noise can’t be distinguished in a signal.

Sv()

Amplitude limiting will decrease amplitude noisebut will not affect phase noise.


noiseless oscillation waveform

phase noise

component

amplitude noise

component

phase noise

amplitude noise


Sideband Noise/Phase Spectral Density

noiseless oscillation waveform

phase noise

component


Jitter/Phase Noise Relationship (1)

autocorrelation functions

Recall R and S() are a Fourier transform pair:

NT




Let Let

Consistent with jitter accumulation measurements!


Jitter from 1/( noise:2

Jitter from 1/( noise:3

^

^

^

^^



f

(dBc/Hz)

-100

-20dBc/Hzper decade

• Let fosc = 10 GHz• Assume phase noise dominated by 1/()2

Setting f = 2 X 106 and S =10-10:

Let = 100 ps (cycle-to-cycle jitter):

= 0.02ps rms (0.2 mUI rms)

Accumulated jitter:

2 MHz


More generally:

f

(dBc/Hz)

fm

Nm

-20 dBc/Hzper decade

rms jitter increases by a factor of 3.2


Let phase noise increase by 10 dBc/Hz:



Kpd

phasedetector

loopfilter

Kvco

VCOin out

vco

fb

Open-loop characteristic:

Closed-loop characteristic:



Recall from Type-2 PLL:

|G|

z p

|1 + G|

-40 dB/decade

(dBc/Hz)



1

80 dB/decade

As a result, the phase noise at low offset frequencies is determined by input noise...


• fosc = 10 GHz• Assume 1-pole closed-loop PLL characteristic


f

(dBc/Hz)

f0 = 2 MHz

-100-20dBc/Hzper decade


For large :

= 0.02 ps rms cycle-to-cycle jitter


f0 = 2 MHz

fosc = 10 GHz

For small :

(log scale)

= 1.4 ps rms Total accumulated jitter


The primary function of a PLL is to place a bound on cumulative jitter:

(log scale)

(log scale)

proportional to (due to thermal noise)

proportional to

(due to 1/f noise)



L() for OC-192 SONET transmitter

Closed-Loop PLL Phase Noise Measurement


Other Sources of Jitter in PLL

• Clock divider

• Phase detectorRipple on phase detector output can cause high-frequency jitter. This affects primarily the jitter tolerance of CDR.


Jitter/Bit Error Rate (1)

Histogram showing Gaussian distribution

near sampling point

1UI

Bit error rate (BER) determined by and UI …

L R

Eye diagram fromsampling oscilloscope


R

0 T

Probability of sample at t > t0 from left-hand transition:

Probability of sample at t < t0 from right-hand transition:



Total Bit Error Rate (BER) given by:



t0 (ps)

log BER

Example: T = 100ps

(64 ps eye opening)

(38 ps eye opening)

log(0.5)



Bathtub Curves (1)

The bit error-rate vs. sampling time can be measured directly using a bit error-rate tester (BERT) at various sampling points.

Note: The inherent jitter of the analyzer trigger should be considered.


Bathtub Curves (2)

Bathtub curve can easily be numerically extrapolated to very low BERs (corresponding to random jitter), allowing much lower measurement times.

Example: 10-12 BER with T = 100ps is equivalent to an average of 1 error per 100s. To verify this over a sample of 100 errors would require almost 3 hours!

t0 (ps)


Equivalent Peak-to-Peak Total Jitter

BER

10-10

10-11

10-12

10-13

10-14

, T determine BERBER determines effectiveTotal jitter given by:

Areas sumto BER

Documents

EECS 270C / Spring 2014Prof. M. Green / U.C. Irvine 1 Voltage-Controlled Oscillator (VCO) VCVC f osc f min f max slope = K vco Desirable characteristics: