1. 2 loop dynamics to keep track of deviations from the free-running frequency,

2

LOOP DYNAMICS

P h as eD e te c to r

K P

Lo w P a s sF ilte r F (s )

A m plifie rA

V o lta g e C o ntro lle dO s c illa to r

1V 2V 3V

oo ,

ii ,

,1 ssKsV oiP ,1 ssKssV oiP

dt

d ii

dt

d oo

tsj j Transform Laplace where

To keep track of deviations from the free-running frequency,

iFRi oFRo

ss

s iFR

i s

ss o

FRo

3

P ha s eD e te c to r

K P


A m plifie rA


K O

1V 2V 3V

oo ,

ii ,

,1 ssKssV oiP

ss

s iFR

i

ss

s oFR

o

,1 sss

KsV oi

P

,3 sss

KsAFsV oi

P

sVKs 300 oK

1

3V

oFR ,0 ss

s

KsAFKoi

P

sAFKKs

ssAFKKs

P

iPo

0

0

4


K P


A m plifie rA


K O

1V 2V 3V

oo ,

ii ,

sAFKKs

ssAFKKs

P

iPo

0

0

When frequency is the output variable,

sAFKKs

sAFKK

s

s

P

P

i

o

0

0

sFAs

sFA

s

s

LG

LG

i

o

When voltage is the output variable,

sAFKKs

sAFK

s

sV

P

P

i 0

3

sFAs

sFKA

s

sV

LG

LG

i

03 /

1secsec

V

V

V

rad

rad

VALG

5

L00P DYNAMICS - NO FILTER

sAs

KAsV i

LG

LG

03

/ 1sF

sFAs

sFKA

s

sV

LG

LG

i

03 /

Step change in input frequency.

s

si

tALGetVtVVtV 3333 0

= final value +initial value -final value ti

t

FR

t

tV3

0

3V

sec 1

constant timeLGA

6

ti t

t

tVi

FR

t

tV3

0

sAs

KAsV i

LG

LG

03

/ sAs

As i

LG

LGO

itt

tV i

t

t

tO

tVO

PLL OUTPUT WAVEFORMS

7

ti t

t

tVi

FR

t

tV3

0

itt

tV i

t

t

tO

tVO

FREQUENCY RESPONSE CONCLUSIONS

If the loop is being used as an FM demodulator, V3(t), the detected information (e.g. voice) waveform, has error due to inherent bandwidth limitations of the loop.

Increased loop gain increases bandwidth and decreases response time.

Adding a filter F(s) further changes the loop frequency response and time response.

If the loop uses the output frequency, e.g. in a frequency multiplier, the output waveform will have transient behavior caused by the loop dynamics.

LG

LG

i Aj

KA

j

jV

03 /

logLGA

i

V

3log20

8

SIMPLEST LOWPASS FILTER• Integrator

• Not a good idea, but simple to check out.


K P

F (s ) = 1 /sA m plifie r

A


1V 2V 3V

oo ,

ii ,

LG

LG

i

o

As

A

s

s

2

sFAs

sFKA

s

sV

LG

LG

i

03 /

LG

LG

i As

KA

s

sV

203 /

X

X

j

9


K P

A m plifie rA


1V 2V 3V

oo ,

ii ,

L

sSF

1

1

FIRST-ORDER LOWPASS FILTER

LGL

LG

i Ass

A

s

s

20

LGLL

LLG

Ass

A

2

22

0

2 nn

LLG

i ss

A

s

s

sFAs

sFA

s

s

LG

LG

i

o

LGLn A LG

L

A

2

1

R

C

10

LOOP GAIN, FILTER BANDWIDTH, AND SETTLING TIME

22

0

2 nn

LLG

i ss

A

s

s

LGLn A LG

L

A

2

1

22

0

3

2 nn

LLG

i ssK

A

s

sV

step response frequency response

11

PROBLEMS WITH FIRST ORDER LOWPASS FILTER


K P

A m plifie rA


1V 2V 3V

oo ,

ii ,

L

sSF

1

1

LGLn A LG

L

A

2

1

LPF bandwidth simultaneously changes bandwidth of PLL frequency response and

Not enough degrees of design freedom.

12

LEAD-LAG FILTER

P

Z

s

ssF

1

11R

C

2R

LGZLGP

ZLG

i AAss

s

K

A

s

sV

1

12

0

3

P

LG

P

ZLG

P

Z

P

LG

AAss

s

K

A

1

1

20

22

0 2

1

nn

Z

P

LG

ss

s

K

A

P

LGn

A

LGP

LGZ

A

A

2

1

2CRZ 21 RRCP

Design:

1. For a given ALG, set n.

2. Independently set

13

PHASE DETECTOR (EOR-TYPE)Exclusive OR gives logic-one output whenever input waveforms differ;

gives -5 V logic-zero output when waveforms are the same.

The average output is the VCO output voltage, V1.

The rest of the output must be eliminated by the PLL filter.

Notice these special features:

1. Output is zero for 90o phase difference, not zero phase difference.

2. Wraparound effect limits output range of the phase detector - in this case to +5V.

14

PHASE DETECTOR - CONTINUED

nVolt/radia 10

PK



A m plifie rA


1V 2V 3V

oo ,

ii ,

“Normal” phase difference is 90o.

Feedback corrections occur if angle deviates toward 0o or toward 180o.

Notice that every phase detector output is a periodic function of

15

LOCK RANGEP ha s e

D e te c to rLo w P a s sF ilte r F (s )

A m plifie rA

V o ltag e C o n tro lle dO s c illato r

K O

1V 2V 3V

oo ,

ii ,

A loop in lock remains in lock as long as the loop is capable of making suitable frequency corrections.

The lock range, L,MAX - L,MIN, is defined by the phase detector limits and the loop gain.

,When ,21

PKV

,2

)1( 12

PKVV

,23

PAKV ,

2,

POMAXO AKK

oK

3V

oFR

2,

POFRMAXO AKK

10

PK

16

LOCK RANGE CONTINUED



A m plifie rA

V o ltag e C o n tro lle dO s c illato r

K O

1V 2V 3V

oo ,

ii ,

oK

3V

oFR

,0When

2,

POFRMAXO KAK

Lock range = LGPOLOCK AAKK

17

EXAMPLE 15.3

Razavi’s example suggests the possibility of a nonlinear phase detector.

Given phase lock, as long as correction occurs, the steady-state local oscillator frequency will equal the input frequency.

Since the VCO curve is linear, the steady-state output voltage is a linear function of input frequency.

Since KP is not constant for all in, expect distortion in the time-varying output voltage waveform unless small-signal operation applies.

18

NONLINEAR PHASE DETECTORThe Gilbert cell (Lecture 5, pp17 & 18) can function as a phase detector.

incontDout VVKRV

If Vin and Vcont are two 0 to 1 V “square waves,” the product is the same as the Exclusive OR.

If Vin and Vcont are two sine waves, a different kind of phase detector characteristic is obtained.

tAtAKV iiout coscos

sinsincoscoscos2 tttKAV iiiout

2

cos average 2

1

KAV

1V

2

19

CAPTURE MECHANISM AND CAPTURE RANGE

“Capture is the complex nonlinear mechanism by which a PLL comes into lock.

To illustrate the main principles we use the Gilbert Cell Multiplier.

incontout VKVV

ttKV oiout coscos

Assume the loop is not in lock. Then the output is

oioiout

KV coscos

2All phase detectors produce such sum and difference frequencies.

The sum-frequency term is rejected by the low pass filter F(s)

The difference-frequency term eventually brings the loop into lock.

20

CAPTURE MECHANISMP h as e

D e te c to rK P


A m plifie rA


1V 2V 3V

oo ,

ii ,i

oPK jF tV2

When difference frequency is high, filter output and feedback are negligible.

dt

d ii

dt

d oo

oi

oi

dt

d

tk oioi

o

FR i

oi

jF oi

21

SIGNAL ACQUISITION

FR

jF

i

o

Because decreases linearly, V1(t) is periodic.

Note coefficient of t decreases as difference decreases.

Feedback signals go through the loop, initially small and fast - then increasing and becoming slower.

tk oioi

3V

o

FR

22

CAPTURE

jF

When the lower swing of the feedback signal pulls the instantaneous local oscillator frequency down to i, the loop comes into lock.

3V

o

FRi

This capture waveform shows the amplitude of V3(t) increasing as its frequency decreases during the capture process.

23

CAPTURE CONDITION

At the instant of capture,

iFRoiFRP AKFK

2


K P

L o w P a s sF ilte r F (s )

A m plifie rA


1V 2V 3V

oo ,

ii ,

FAKK oP 2

FAKK Po 2

X

When i approaches FR , either from above or from below, lock occurs when i come within X of FR

3V

o

FR

24

CAPTURE RANGE AND LOCK RANGE

POLOCK AKK 16, page From

XCAPTURE 2

XPoXX FAKKwhere 2 satisfies 2

Because the filter gain is less than one, LOCK > CAPTURE

FR

O

25

That’s all folks

X

26

Equations

22

1THGSOXnD VV

L

WCI

THGSOXnm VVL

WCg

DOXnm IL

WCg 2

THGS

Dm VV

Ig

2

FSBFTHTH VVV 220

ox

subsi

C

Nq

2 m

SBF

mmb gV

gg

22

SB

TH

V

VDSV

L

L L

1

ooxoxn AtXC 90 1034.1 4

ooxox AtmfFC 50 /9.6 2

ooxoxp AtXC 90 10835.3 5 K / J 10 38 . 123

X k

f

1 2

ker, WLC

KV

oxflicn

3

242

, TgkI mthermaln

2

2

1DSDSTHGSOXnD VVVV

L

WCI

42, kRTI thermaln

THGSOXn

triode

VVLW

CR

1OD

Dm V

Ig

2

Do Ir

1

ODom Vrg

2

22

1ODOXnD V

L

WCI

V

QC

SP RQR 2

S

o

R

LQ

L

RQ

o

P

1. 2 loop dynamics to keep track of deviations from the free-running frequency,

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