VOLTAGE-CONTROLLED OSCILLATOR (VCO)

Voltage-Controlled Oscillator (VCO) fosc

Desirable characteristics: •  Monotonic fosc vs. VC characteristic with adequate frequency range •  Well-defined Kvco

fmax

slope = Kvco

fmin

φVC φin

VD

KPD

F (s)

+

€

VC

VC

K^vco s

φout φout

€

€

€

€

€ €

K^vco = ⋅ φVC s + KPD K^vco F (s) / N

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

÷N € EECS 270C / Spring 2014

€

Prof. M. Green / U.C. Irvine

1

Oscillator Design

Vin ⇒ 0

A(s)

Vout

Vout A(s) ≡ HCL (s) = Vin 1+ f ⋅ A(s) loop gain

€

€

€

f

€

Barkhausen’s Criterion: If a negative-feedback loop satisfies:

( ) ∠A( jω ) = −180 f ⋅ A jω o ≥ 1

€

o

!

then the circuit will oscillate at frequency ω0.

€ EECS 270C / Spring 2014


2

Inverters with Feedback (1) 1 inverter: V1

V2

1 inverter

V2

feedback

1 stable equilibrium point

V1 V2

2 inverters: V1

feedback

V2

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

2 inverters V1 EECS 270C / Spring 2014


3

Inverters with Feedback (2) 3 inverters forming an oscillator: V1

V2 V2

1 unstable equilibrium point due to phase shift from 3 capacitors V1

A0 Let each inverter have transfer function Hinv ( jω ) = 1+ jω p A30 3 Loop gain: Hloop ( jω ) = [Hinv ( jω )] = 3 1+ j ω p ( ) € % ( −1 ω Applying Barkhausen’s criterion: ∠Hloop ( jω ) = −3 tan ' * = −180! ⇒ ω o = 3⋅ p

& p)

€

Hloop ( jω o ) =


A30

[1+ 3]


€

3

2

> 1 ⇒ A0 > 2 4

Ring Oscillator Operation tp

tp

VA

tp

VB

VA

VC

tp

VB

1 Tosc = 3t p 2 ⇒ Tosc = 6t p

tp

VC

tp

VA 1 Tosc 2 EECS 270C / Spring 2014

€


5

Variable Delay Inverters (1)

Inverter with variable load capacitance: Vin

Current-starved inverter:

Vout

VC Vin

Vout

VC

EECS 270C / Spring 2014


6

Variable Delay Inverters (2) Interpolating inverter: ISS + V _C

R Vout+

R Vout-

Vin+

Vin- Vin+

VinRG

Ifast

RG Islow

•  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. EECS 270C / Spring 2014


7

Differential Ring Oscillator

+ −

+ −

VA

+ −

VB

VA

VC

+ −

VD

− +

VA

additional inversion (zero-delay)

tp tp

VB VC

1 Tosc = 4t p 2 ⇒ Tosc = 8t p

tp tp

VD

Use of 4 inverters makes quadrature signals available.

VA EECS 270C / Spring 2014

€

1 Tosc 2


8

Resonance in Oscillation Loop H r ( jω )

€

Hr (s)

1

€ π + 2

Hr (s)

€

∠Hr ( jω )

€

ωr

€ −

At dc: € Since Hr(0) < 1, latch-up does not occur.

ωr

At resonance:

H r ( jω r ) > 1


€

ω

π 2

∠Hr ( jωr ) = 0 EECS 270C / Spring 2014

ω

⇒ ωo = ωr 9

€

LC VCO L Vin

Hr (s)

C

Vout

ωr = Vout

Vin

1 LC

€

€ Hr (s)

2L

⇒

€

⇒

C

C

Hr (s)

realizes negative resistance

€

€


€


10

Variable Capacitance varactor = variable reactance Cj

A. Reverse-biased p-n junction +

VR

–

VR

B. MOSFET accumulation capacitance

Cg

p-channel

– VBG +

n diffusion in n-well accumulation region EECS 270C / Spring 2014


inversion region

VBG 11

LC VCO Variations IS

2L C

C

C

C

2L

2L C

IS

2L

C

C

C ISS



12

Effect of CML Loading 1. 1. ideal capacitor load 1 nH

3.8 Ω

400 fF

400 fF

108 fF

108 fF

2. Cg = 108fF 1 nH

3.8 Ω

400 fF

400 fF

2. CML buffer load EECS 270C / Spring 2014


13

CML Buffer Input Admittance (1) Yin = jωCgs + jωCgd A0 ⋅

1+ jω / z 1+ jω / p

A0 = 1+ gm R

€

(

)

where: 1/ p = CL + Cgd R

€

1/ z =

(note p < z)

CL R A0

€

€

€

Re Yin = A0Cgd ω 2 ⋅

( )

1 p −1 z

(

1+ ω p

)

2

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



14

CML Buffer Input Admittance (2) Yin magnitude/phase:

Yin real part/imaginary part:

magnitude imaginary

phase real

Contributes 2kΩ additional parallel resistance



15

CML Buffer Input Admittance (3)

3. CML tuned buffer load

Cg = 108 fF 1 nH

imaginary

3.8 Ω

400 fF

400 fF

3.8 nH

real

Contributes negative parallel resistance



16

CML Buffer Input Admittance (4)

ideal capacitor load Cg = 108 fF 1 nH

400 fF

3.8 Ω

400 fF

3.8 nH

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



CML tuned buffer load

17

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



18

Oscillator Type Comparison

Ring Oscillator

LC Oscillator

– slower

+ faster

– low Q ⇒ more jitter generation

+ high Q ⇒ less jitter generation

+ Control voltage can be applied differentially

– Control voltage applied single-ended

+ Easier to design; behavior more predictable

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

+ Less chip area

– More chip area (inductor)



19

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.

PX(x) 1

Example 1: Random variable

X ∈ [−∞,+∞]

0.5


x Prof. M. Green / U.C. Irvine

20

Random Processes (2) Probability of X within a range is straightforward: PX(x) 1

0.5

(

)

P X ∈ [x 1, x 2 ] = P(x 2 ) − P(x 1)

x1 x2

x

€ If we let x2-x1 become very small …



21

Random Processes (3) Probability density function pX(x): Probability that random variable X lies within the range of x and x+dx. p X (x) ⋅ dx = PX (x + dx) − PX (x) ⇒ p X (x) =

dPX (x) dx

€ €

(

PX(x)

) ∫

P X ∈ [ x 1, x 2 ] =

€

1

x2 x1

p X (x) dx

pX(x)

0.5

dx EECS 270C / Spring 2014

x

x Prof. M. Green / U.C. Irvine

22

Random Processes (4) Expectation value E[X]: Expected (mean) value of random variable X over a large number of samples. +∞

∫ x ⋅p

E[X ] ≡ X =

X

(x)dx

−∞

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

€

+∞

E[X 2 ] =

∫x

2

⋅ p X (x)dx

−∞

Variance:

[

2

+∞

]

2

E (X − X ) ≡ σ =

€

∫ (x − X ) p

[


€

X

(x)dx

−∞

2 Standard deviation: σ = E (X − X )

€

2

]


23

Gaussian Function 1.  Provides a good model for the probability density functions of many random phenomena. 2.  Can be easily characterized mathematically σ , X . 3.  Combinations of Gaussian random variables are themselves Gaussian.

(

€

)

f (x)

1

σ 2π 0.607

% −(x − X )2 ( * f (x) = exp' 2 '& 2σ *) σ 2π 1

+∞

∫ f (x)dx = 1

€

€

€

€

σ 2π

€

X −σ

−∞


2σ


€

€

€

X

X +σ

x 24

Joint Probability (1)

Consider 2 random variables:

(

P(x, y) ≡ P X ≤ x and Y ≤ y

)

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

€

(

)

P X ∈ [ x, x + dx ] and Y ∈ [ y, y + dy ] = p X (x) ⋅ pY (y) ⋅dx dy

€



25

Joint Probability (2) Consider sum of 2 random variables:

Z = X +Y

(

) ∫∫

P Z ∈ [ z0 , z0 + dz ] =

y

strip

% =' &

€

p X (x)pY (y) dx dy

( p X (x)pY (z0 − x) dx * dz −∞ )

∫

∞

€

x + y = z0 + dz €

x + y = z0 € €

p Z (z0 )

dy = dz

dx


determined by convolution of pX and pY.

x

€


26

Joint Probability (3) Example: Consider the sum of 2 non-Gaussian random processes:

*



27

Joint Probability (4) 3 sources combined:

*



28

Joint Probability (5) 4 sources combined:

*



29

Joint Probability (6) Noise sources

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



30

Fourier transform of Gaussians: p X (x) =

% 2( −(x − X ) * exp' '& 2σ 2 *) 2π X

1

σX

% 1 2 2( PX (ω ) = exp'− σ X ω * & 2 )

F

Recall:

€

& P Z ∈ [ z0 , z0 + dz ] = ( '

(

) ∫

p Z (z0 ) =

€

) € p X (x)pY (z0 − x) dx + dz −∞ *

∫

∞

−∞

∞

F

p X (x)pY (z0 − x) dx

PZ (ω ) = PX (ω ) ⋅PY (ω ) % 1 ( % 1 ( = exp'− σ 2X ω 2 * ⋅exp'− σ Y2 ω 2 * & 2 ) & 2 )

€ p Z (z) =

1

(

2 π σ 2X + σ Y2

)

% −(z − Z)2 exp' '2 σ 2 +σ 2 X Y &

(

)

( * * )

€

F -1

€

% 1 ( = exp'− (σ 2X + σ 2X )ω 2 * & 2 )

Variances of sum of random normal processes add. €

€



31

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

RX (t1,t2 ) = E [ X (t1) ⋅ X (t2 )]

€

For a stationary random process, RX depends only on the time difference τ = t1 − t 2

RX (τ ) = E [ X (t) ⋅ X (t + τ )] for any t 2 Note RX (0) = σ

€ €

€ Power spectral density SX(ω): 2* ' +∞ ) , SX (ω ) = E) X (t) ⋅ e − jωt dt , )( −∞ ,+

∫


€

SX(ω) given in units of [dBm/Hz]


32

Relationship between spectral density & autocorrelation function:

1 RX (τ ) = 2π

∫

∞

−∞

SX (ω ) ⋅e jωτ dω

1 ⇒ RX (0) = σ = 2π

∫

2

∞

−∞

SX (ω )dω

€

€

infinite variance (non-physical)

Example 1: white noise SX (ω )

RX (τ )

€

€

τ

ω

( )

SX ω = K

€


RX (τ ) =

€


K ⋅δ t 2π

()

33

Example 2: band-limited white noise RX (τ )

SX (ω )

σ2 =

1 Kω p 2

K € €

€

€ −ω p

( )

SX ω = €

€

ωp

ω

K ω2 1+ 2 ωp

RX (τ ) = σ 2e

€

−ω p τ

τ

p X (x)

For parallel RC circuit capacitor€voltage noise: € i n2 K= ⋅R2 = 2kBTR Δf

ωp =

σ V2C =

1 RC

kBT C

−σ


€

€


€

+σ

x 34

€

Random Jitter (Time Domain)

Experiment: CLK

data source


DATA

CDR (DUT)

RCK


analyzer

35

Jitter Accumulation (1)

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

NT Free-running oscillator output

Tosc =

1

τ4

fosc

σ1 €

τ3

τ2

τ1

σ2

trigger

EECS 270C€ / Spring 2014

€

σ3

€

Prof. M. Green€ / U.C. Irvine

σ4

Histogram plots

36


σ τ2

€ €

τ proportional to τ

proportional to τ2

Observation: As τ increases, rms jitter increases. EECS 270C / Spring 2014


€

37

Noise Spectral Density (Frequency Domain) Single-sideband spectral density:

Power spectral density of oscillation waveform:

( ) [dBc Hz]

Ltotal Δf

Sv(f)

[dBm Hz]

1/Δf3 region (-30dBc/Hz/decade)

€ 1/Δf2 region (-20dBc/Hz/decade)

€

fosc

fosc+Δf

" P f + Δf 1Hz osc Ltotal (Δf ) = 10 log$ $# Ptotal

(


Δf (log scale)

) %' '&

Ltotal(Δf) given in units of [dBc/Hz] Ltotal includes both amplitude and phase noise Prof. M. Green / U.C. Irvine

38

Noise Analysis of LC VCO (1) noise from resistor

+

C

L

R

vc

-R

_

C

L

active circuitry

ωr =

€

1 LC

Z( jω ) =

R Q= ωr L

Consider € frequencies near resonance:

[(

(

)]

Z j ωr + Δω = 1−

ωr L = €

2 r

( )

ω + 2ωr Δω + Δω

2

j ωL $ ω '2 1− & ) % ωr (

€

ωr2 ≈ jL ⋅ 2Δω

ωr2 R R ωr ⇒ Z j ωr + Δω ≈ j ⋅ Q 2Q Δω


€

)

j ωr + Δω L

inR

[(

ωr ωr + Δω

)]


€ €

39

Noise Analysis of LC VCO (2) +

vc _

Spot noise current from resistor: C

L

inR

2 i nR =

4kT ⋅ Δf R

2 v c2 = i nR ⋅| Z( jω ) |2

€

% ωr Δω (2 4kT = Δf ⋅ 'R * R 2Q ) & % ωr Δω (2 = 4kTR ⋅ ' * ⋅ Δf & 2Q )

Leeson’s formula (taken from measurements): € 2 +- % (2 /-% ω 3 (5 kT ω r L{ Δω} = 10 ⋅ log4F ⋅ ,1+ ' * 0''1+ 1/ f **7 4 Psig - & 2Q ⋅ Δω ) Δω )7 . 1& 3 6

spot noise relative to carrier power

dBc/Hz

Where F and ω1/f3 are empirical parameters. €



40

Oscillator Phase Disturbance ip(t)

ip(t)

Current impulse Δq/Δt

ip(t)

_

t

τ1

Vosc +

Vosc(t)

Vosc(t)

Δφ = 0

€

t

τ2

€

€

Δφ < 0

€

Vosc jumps by Δq/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 EECS 270C / Spring 2014


41

Impulse Sensitivity Function (1) The phase response for a particular noise source can be determined at each point τ over the oscillation waveform. Δφ (τ ) ⋅q max Impulse sensitivity function (ISF): Γ(τ ) ≡ Δq = C ⋅Vmax

change in phase charge in impulse Example 1: sine wave

€

Example 2: square wave

€

Vosc (t) Vmax €

€

Vosc (t)

t

t

Γ(τ )

Γ(τ )

€

€

(normalized to signal amplitude)

τ

τ

Note Γ has same period as Vosc. EECS 270C / Spring 2014


42

Impulse Sensitivity Function (2) Recall from network theory:

H(s) h(t)

i in

LaPlace transform:

φout

Φ out (s) = H(s) Iin (s) t

Impulse response: φout (t) =

∫ h(t,τ ) ⋅ i

in

(τ ) dτ

0

€

€

€ Recall: € Γ(τ ) ≡

Δφ (τ ) Γ(τ ) ⋅q max ⇒ Δφ (τ ) = ⋅ Δq Δq q max

ISF convolution integral: t Γ(τ ) φ (t) = ⋅u(t − τ ) ⋅ [i (τ ) ⋅dτ ] = € q max 0

∫

€

t

∫ qΓ(τ ) ⋅i (τ ) ⋅dτ 0

time-variant impulse response

max

Γ can be expressed in terms of Fourier coefficients: ∞

from Δq

Γ(τ ) =

= 1 for τ ∈ (0,t)

∑c

k

(

cos kωoscτ + θ k

k=0

€ € EECS 270C / Spring 2014


€

43

)

Impulse Sensitivity Function (3) Case 1: Disturbance is sinusoidal:

[(

)]

i (t) = I0 cos mωosc + Δω t , m = 0, 1, 2, …

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

I φ (t) = 0 q max

€

∞

t

∑ c ∫ {cos(kω k

k=0

0

osc

)

) ] } dτ

[(

τ + θ k ⋅ cos mωosc + Δω t

Γ(τ ) I = 0 2q max

€

& sin (k + m)ω + Δω t + θ sin [(k − m)ωosc + Δω ]t + θ k [ ] ( osc k ck ' + (k + m) ω + Δ ω (k − m)ωosc + Δω () osc k=0 ∞

∑

{

€

€

negligible

(

sin Δω t + θ k I0 ≈ ⋅cm ⋅ 2q max Δω EECS 270C / Spring 2014

€

}

{

} *(+ (,

significant only for m=k

) Prof. M. Green / U.C. Irvine

44

Impulse Sensitivity Function (4)

[(

)]

For i (t) = I0 cos mωosc + Δω t

€

(

)

sin Δω t + θ k I0 I02 cm2 2 φ (t) ≈ ⋅cm ⋅ ⇒φ = 2 ⋅ 2q max Δω 8q max Δω

( )

2

Current-to-phase frequency response: I

€

ωosc

ω1

×

€

I0 c0 2q max ω1

φ


×

I0 c1 2q max ω1

€

ω1

2ωosc

ωosc-ω1

ωosc+ω1

2ωosc-ω1 2ωosc+ω1

×

ω

I0 c2 2q max ω1

€ Δω


45

Impulse Sensitivity Function (5) Case 2: Disturbance is stochastic: 2 MOSFET current noise: i n (f ) = 4kTγgm + gm2 Kf Δf Cg f i n2 Δf cm2 2 φ Δf ≈ 2 ⋅ thermal 1/f 8q max Δω 2 noise noise

A2/Hz

in

( )

€

i n2 Δf

€ 1/f noise

gm2

2 π ⋅ Kf Cg ω

thermal noise

×c0

€

€

€

i n2 Δf

€4kTγgm 2ωosc

ωosc

€ Sφ Δω

×c0

ω €

×c2

×c1 2ωosc

ωosc

€

€

( )


Δω


46

ω

Impulse Sensitivity Function (6) 1 Total phase noise: Sφ (Δω ) = 2 8q max

∑

( )

×c0

€

( )

due to thermal noise

i n2 Δf

€

∞ * , ck2 , gm2 Kf c02 0 + 2π ⋅ , 4kTγgm ⋅ 2 C g Δω Δω , ,+

due to 1/f noise

×c1

ωn €

×c2 2ωosc

ωosc

€

/ / 3/ / /.

ω

€

( )

Sφ Δω

2 0

( )

c = Γ

2

∞


Δω

∑ k=0


€

ck2 = Γrms

( )

2

47

Impulse Sensitivity Function (7) Sφ (Δω ) =

1 2 8q max

4kTγgm ⋅

€

) Γrms + 4kT γ g ⋅ m + Δω +*

( ) ( )

( ) (Δω) Γrms

2

2

= 2π

2 m

2

2

, g Kf . + 2π ⋅ 3. Cg Δω .-

( )

2 m

Γ

2

( )

( ) Γ

g Kf ⋅ Cg Δω

2

( )

3

⇒

Δωn,phase

2 ( + π g Γ = ⋅ m ⋅ ** 2kT γCg ) Γrms -,

noise corner frequency ωn

€ Sφ Δω (dBc/Hz)

( )

€ 1/(Δω)3

€

region: −30 dBc/Hz/decade € 1/(Δω)2 region: −20 dBc/Hz/decade

Δωn,phase

Δω (log scale)



€

48

Impulse Sensitivity Function (8) Example 1: sine wave Vosc (t)

Example 2: square wave Vosc (t)

t

€

t

€

Γ(τ )

Γ(τ )

τ

€

τ

€

Γrms is higher ⇒ will generate more 1/(Δω)2 phase noise

Example 3: asymmetric square wave Vosc (t)

€

t

€

Γ(τ )

τ

€

Γ > 0 ⇒ will generate more 1/(Δω)3 phase noise Prof. M. Green / U.C. Irvine


€

49

Impulse Sensitivity Function (9) Effect of current source in LC VCO:

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

Vosc


_

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


50

Impulse Sensitivity Function (10) ID varies over oscillation waveform

Same period as oscillation

i n2 = 4kTγgm (t) Δf & ) W = (4kTγ ) ⋅ ( µCox ⋅ VGS (t) −Vt + L ' *

(

2 & ) i n0 W = (4kTγ ) ⋅ ( µCox ⋅ VGS(DC ) −Vt + Let Δf L ' *

(

€

2 2 i i n n0 Then = ⋅ α (t) Δf Δf €

We can use

VGS (t) −Vt VGS(DC ) −Vt

€ Prof. M. Green / U.C. Irvine

€

where α (t) =

)

Γeff (τ ) = Γ(τ ) ⋅ α (τ )


)

51

ISF Example: 3-Stage Ring Oscillator R1A

R1B

M1A

M1B MS1

R2A

R2B

M2A

M2B MS2

R3A

R3B

M3A

+ Vout −

M3B MS3

fosc = 1.08 GHz PD = 11 mW



52

ISF of Diff. Pairs ISF by tx1 for 3stage differential ring osc

ΓM1A

3

3

2

2

2

1

1

-1

0

1

2

3

4

5

6

-2

7

-1

0

1

4

5

6

7

0 -1

-3

-4

-4

-4

-5

-5

3

2

2

1

1

€

0 -1

2

3

4

5

-2

6

7

ΓM2B

7

5

6

7

1

€ 0

1

2

3

4

5

-2

6

7

0 -1

-4

-4

-5

0

1

2

3

4

-2

-4

Γ = −0.26

6

2

-3

Γrms = 1.86

5

ISF by tx6 for differential ring osc

ΓM3B

-3

-5 Radian

Radian

4

3

-3

-5

3

Radian


0 -1

2

-5

ISF by tx6


1

1

Radian

3

0

0

-2

-3

ISF by tx4

ISF by tx2

3

-3

ΓM1B

Radian

for each diff. pair transistor


€

2

-2

Radian

€

1

€

0

ISF by tx5

€

0


ΓM3A

3

ISF by tx3

ISF by tx1

€


ΓM2A


53

ISF of Resistors

ΓR1A

ΓR2A

€

€

Γrms = 1.72 Γ = −0.16

€

ΓR3A

€

for each resistor



54

ISF of Current Sources

ISF by tail tx1 for differential ring osc

ΓMS1

2

2

1.5

1.5

1.5

ISF by tail tx1

0.5 0 -0.5

0

1

2

3

4

5

€

1

6

7

1

0.5 0 -0.5

0

1

2

3

4

5

6

ISF by tail tx3

€

1

7

0.5 0 -0.5

-1

-1

-1

-1.5

-1.5

-1.5

-2

-2 Radian

Γrms = 1.00 Γ = −0.12


ΓMS3

2

ISF by tail tx2

€


ΓMS2

0

1

2

3

4

5

6

-2 Radian

Radian

for each current source transistor ISF shows double frequency due to source-coupled node connection.



55

7

€

Phase Noise Calculation Using: Cout = 1.13 pF Vout = 601 mV p-p qmax = 679 fC 2 2 2 Γrms(dp) 4kTγ gm(dp) Γrms(res) Γrms(cs) 4kTγ gm(cs) 4kT R L{Δf } = 6 ⋅ 2 2 ⋅ + 6⋅ 2 2 ⋅ 2 + 3⋅ 2 2 ⋅ 2 2 8π Δf qmax 8π Δf qmax 8π Δf qmax

322 Δf 2

€ 514 €Δf 2

⇒ L{Δf } =

122 Δf 2

= −112 dBc/Hz @ Δf = 10 MHz


€


70 Δf 2

€

56

Phase Noise vs. Amplitude Noise (1)

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

[(

)]

Vosc (t) = [Vc + v (t)] ⋅exp j ωosc t + φ (t)

€

vφ φ ωosct


v

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


57

Phase Noise vs. Amplitude Noise (2) Recall that an input current impulse causes an enduring phase perturbation and a momentary change in amplitude: i(t)

i(t)

t Vc(t)

t Vc(t)

Δt = 0


Δt =

Δq ω osc


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

58

Phase Noise vs. Amplitude Noise (3) ( )

( )

Lφ Δω

Lamp Δω

€

€

+

ωc

Δω

Δω

Q

( )

Ltotal Δω Δω

€

Phase noise dominates € at low offset frequencies.

Δω



59

€

Phase Noise vs. Amplitude Noise (4) Sv(ω)

( ≈ (V

) ( + v (t)) ⋅ [cos(ω

Vosc (t) = Vc + v (t) ⋅ cos ωosc t + φ (t) c

osc

)

t) − φ (t) ⋅ sin(ωosc t)]

= Vc cos(ωosc t) − φ (t) ⋅Vc sin(ωosc t) + v (t) ⋅ cos(ωosc t) noiseless oscillation waveform

phase noise component

amplitude noise component

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



phase noise

amplitude noise

ωosc

ω

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

60

Sideband Noise/Phase Spectral Density (

)

Vosc (t) = Vc ⋅ cos ωosc t + φ (t)

≈ Vc ⋅ [cos(ωosc t) − φ (t) ⋅ sin(ωosc t)]

Vc ⋅ cos(ωosc t) −Vc ⋅ φ (t) ⋅ sin(ωosc t)

€

noiseless oscillation waveform

€

Pphase noise Psignal

phase noise component

1 2 2 Vc ⋅ φ 2 = = φ2 1 2 Vc 2

( )

Lphase Δω =

1 ⋅Sφ Δω 2

( )

€ € EECS 270C / Spring 2014


61

Jitter/Phase Noise Relationship (1) NT

σ τ2 ≡ =

2, 1 ) ⋅E φ (t + τ ) − φ (t) * [ ] 2 + . ωosc

1 2 2 ⋅ E φ (t + τ ) + E φ (t) − 2E [φ (t) ⋅ φ (t + τ )] 2 ωosc

{ [

autocorrelation functions

€

€

] [

Rφ (0)

}

]

Rφ (0)

2Rφ (τ )

2 ⋅ [Rφ (0) − Rφ (τ )] 2 ω € osc € €

⇒ σ τ2 =

Recall Rφ and Sφ(Δω) are a Fourier transform pair:

1 Rφ (τ ) = 2π

∫

∞

−∞


€

Sϕ (Δ€ ω ) ⋅e j ( Δω )τ d(Δω ) Prof. M. Green / U.C. Irvine

62

Jitter/Phase Noise Relationship (2) 1 Rφ (0) = 2π 1 Rφ (τ ) = 2π

€

∞

∫ S (Δω)d(Δω) φ

−∞ ∞

∫ S (Δω) ⋅e

j ( Δω )τ

φ

d(Δω )

−∞

∞

1 j ( Δω )τ σ = ⋅ S (Δ ω ) 1− e d(Δω ) φ 2 πωosc −∞

∫

2 τ

(

)

∞

1 = ⋅ Sφ (Δω ) [1− cos(Δω τ ) − j sin(Δω τ )] d(Δω ) 2 πωosc −∞

∫

4 = ⋅ 2 πωosc

€


∞

∫ 0

, Δω τ / Sφ (Δω ) ⋅sin . 1 d(Δω ) 2 0 2


63

Jitter/Phase Noise Relationship (3) 3

2

Jitter from 1/(Δω) noise:

Jitter from 1/(Δω) noise:

a^ Let

Sφ (Δω ) = (Δω )2

€

€

4 σ τ2 = ⋅ 2 πωosc

€

∫ 0

( + a^ 2 (Δω )τ ⋅sin * - d(Δω ) 2 2 (Δω ) ) ,

4 a^ πτ = ⋅ 2 πωosc 4

= €

∞

a^ 2 ωosc

€

⋅τ

Let

Sφ (Δω ) =

€

4 σ τ2 = ⋅ 2 πωosc

∞

∫ ε

b (Δω )3

( + b 2 (Δω )τ ⋅sin * - d(Δω ) 3 (Δω ) ) 2 ,

=ζ ⋅τ2

€ a = 2 ⋅ τ where a^ ≡ (2π )2 ⋅a € fosc

Consistent with jitter accumulation measurements!



64

Jitter/Phase Noise Relationship (4) ( )

Sφ Δf

€

(dBc/Hz)

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

-20dBc/Hz per decade -100

( )

Sφ Δf =

a (Δf )2

Setting Δf = 2 X 106 and Sφ =10-10:

(

Δf€

2 MHz

)

Sφ 2 ⋅106 =

a

(2 ⋅10 ) 6

2

= 10−10 ⇒ a = 400

Accumulated jitter:

σ τ2 =

€

a 400 ⋅ τ = fc2 10 ⋅109

(


)

2

[

]

⋅ τ€= 4 ⋅10−18 ⋅ τ

€

[

]

σ τ = 2 ⋅10−9 ⋅ τ Let τ = 100 ps (cycle-to-cycle jitter): ⇒ στ = 0.02ps rms (0.2 mUI rms)


65

Jitter/Phase Noise Relationship (5) More generally:

( )

Sφ Δf

€

a (Δfm )2 ⋅10Nm 10 Sφ Δf = = (Δf )2 (Δf )2

( )

(dBc/Hz)

σ τ2 =

-20 dBc/Hz per decade€

Nm

Δf

2 fosc

% f (2 ⋅ τ = ' m * ⋅10Nm 10 ⋅ τ & fosc )

$f ' σ τ = & m ) ⋅10Nm 20 ⋅ τ % fosc (

€

Δfm

a

στ = fm ⋅10Nm 20 ⋅ τ Tosc

[ps]

[UI]

€ Let phase noise increase by 10 dBc/Hz:

στ € ( Nm+10) → fm ⋅10 Tosc

20

& ) ⋅ τ = (fm ⋅10Nm 20 ⋅ τ + ⋅100.5 ' *

⇒ rms jitter increases by a factor of 3.2

€



66

Jitter Accumulation (1) φin φfb

phase detector

loop filter

VCO

Kpd

F (s)

Kvco

€

÷N

Open-loop characteristic: €

φvco

+

φout

€

φout K 1 = G(s) = K pd ⋅F (s) ⋅ vco ⋅ φε 2πs N

NG(s) 1 φ = ⋅ φ + ⋅ φvco Closed-loop characteristic: out in 1+G(s) 1+G(s) €



67

Jitter Accumulation (2) G(s) =

Recall from Type-2 PLL:

Ich Kvco 1 1+ sCR ⋅ 2 ⋅ N s (C + Cp ) 1+ sCeq R

-40 dB/decade

( )

Sφ Δω (dBc/Hz)

€

|1 + G| |G| z

1

φout jΔω φvco

(

ω0

p

Δω

1/(Δω)3 region: −30 dBc/Hz/decade € 1/(Δω)2 region: −20 dBc/Hz/decade

2

)

Δωn,phase

Δω

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

€

80 dB/decade

ω0


Δω


68


( )

Sφ Δf

€

-100

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

(dBc/Hz)

+ a , Δf << Δf0 2 2 Δf0 - Δf0 Sφ Δf = ≈ , $ Δf '2 - a , Δf >> Δf 0 1+ & ) - Δf 2 % Δf0 ( . a

-20dBc/Hz per decade

( )

( )

( ) ( )

∞

€ Δf0 = 2 MHz

Rφ (τ ) =

⇒ σ τ2 = =

€

(2π ⋅ Δf )

⋅e −2 π ⋅f0τ

2 ⋅ [Rφ (0) − Rφ (τ )] 2 2π ⋅fosc a 2 fosc

1− e −2 π ⋅f0 τ ⋅ 2π ⋅ Δf0



a 0

−∞

Δf €

∫

Sφ (Δf ) ⋅e j (2 πΔf )τ ⋅d(Δf ) =

69

Jitter Accumulation (4) 2 τ

σ =

a 2 fosc

1− e −2 π ⋅f0 τ ⋅ 2π ⋅ Δf0

) a 1 ⋅ τ , τ << + 2 + fosc 2 π (Δf0 ) ≈* 1 1 + a ⋅ , τ >> 2 +, fosc 2 π (Δf0 ) 2 π (Δf0 )

a = 4 × 102 Δf0 = 2 MHz

€

fosc = 10 GHz

€

σ τ2 (log scale)

€

slope =

€

For small τ: στ = 0.02 ps rms cycle-to-cycle jitter For large τ: στ = 1.4 ps rms Total accumulated jitter

a 2 fosc

τ

1 (2π ) ⋅(2 MHz)

€


€


70

Jitter Accumulation (5) σ τ2 (log scale) proportional to τ2 (due to 1/f noise)

€ proportional to τ (due to thermal noise)

τ

The primary function of a PLL is to place a bound on cumulative jitter: σ τ2 (log scale)

€

τ



71

Closed-Loop PLL Phase Noise Measurement

L(Δω) for OC-192 SONET transmitter EECS 270C / Spring 2014


72

Other Sources of Jitter in PLL

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



73

Jitter/Bit Error Rate (1) Eye diagram from sampling oscilloscope

Histogram showing Gaussian distribution near sampling point 2σ L

€

2σ R

L

€

R

1UI

Bit error rate (BER) determined by σ and UI … EECS 270C / Spring 2014


74

Jitter/Bit Error Rate (2)

& 1 ( T −t pR (t) = ⋅exp(− 2σ 2 σ 2π ('

(

& t ) 1 pL (t) = ⋅exp(− 2 + ' 2σ * σ 2π 2

2σ

2σ

€

€

€

0

t0

T 2

T

T − t0 €

R

& x2 ) PL = ⋅ exp(− 2 + dx Probability of sample at t > t0 from left-€ t0 € ' 2σ * σ 2π € hand transition: 2) & T − x ∞ Probability of sample at t < t0 from right1 ( + P = ⋅ exp − dx R 2 ( + hand transition: t0 2σ σ 2π (' +* € EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine

1

∫

∫

€

∞

(

)

75

)

2

) + + +*

Jitter/Bit Error Rate (3) & x2 ) PL = ⋅ exp(− 2 + dx t0 ' 2σ * σ 2π 1

∫

∞

& 1 ( T −x PR = ⋅ exp(− t0 2σ 2 σ 2π ('

∫

€

∞

(

)

) 1 + dx = ⋅ + σ 2π +*

2

& x2 ) exp(− 2 + T−t 0 ' 2σ *

∫

∞

Total Bit Error Rate (BER) given by:

€

& x2 ) 1 BER = PL + PU = ⋅ exp(− 2 + dx + ⋅ t0 2 σ ' * σ 2π σ 2π 1

∞

∫

& x2 ) exp(− 2 + dx T−t 0 ' 2σ *

∫

∞

# t & #T − t &1* 0 0 = ,erfc%% (( + erfc%% ((/ 2 ,+ $ 2σ ' $ 2σ '/.

€

where erfc(t) ≡

2

π


€

⋅

∫

t

∞

( )

exp −x 2 dx Prof. M. Green / U.C. Irvine

76

Jitter/Bit Error Rate (4) Example: T = 100ps log(0.5)

log BER σ = 2.5 ps σ = 5 ps € €

•

•

•

•

t0 (ps)

σ = 2.5 ps : € €

€ € BER ≤ 10−12 for t0 ∈ [18ps, 82ps] (64 ps eye opening)

σ = 5 ps : BER ≤ 10−12 for t0 ∈ [36ps, 74ps] (38 ps eye opening)

€ €


€ €


77

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.

( ) RJ Jrms

2 measured


€

( )

RJ = Jrms

2 actual

( )

RJ + Jrms

2 trigger


78

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) Prof. M. Green / U.C. Irvine

79

Equivalent Peak-to-Peak Total Jitter p(t) BER

RJ JPP

10-10

12.7 ⋅ σ

10-11

13.4 ⋅ σ

€€ 10-12

14.1⋅ σ

10-13€

14.7 ⋅ σ

10-14€

15.3 ⋅ σ

Areas sum to BER

€

€

1 nσ 2

€

σ, T determine BER

RJ BER determines effective JPP Total jitter given by:

€

1 nσ 2

€

DJ J TJ = n ⋅ σ + JPP €

(

)


€


80

VOLTAGE-CONTROLLED OSCILLATOR (VCO)

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