Sam PalermoAnalog & Mixed-Signal Center

Texas A&M University

ECEN620: Network TheoryBroadband Circuit Design

Fall 2014

Lecture 3: PLL Analysis

Agenda & Reading

• PLL Overview & Applications• PLL Linear Model• Phase & Frequency Relationships• PLL Transfer Functions• PLL Order & Type

• Reading• Chapter 2, 3, 5, & 12 of Phaselock Techniques,

F. Gardner, John Wiley & Sons, 2005. 2

References

• M. Perrott, High Speed Communication Circuits and Systems Course, MIT Open Courseware

• Chapter 2 of Phase-Locked Loops, 3rd Ed., R. Best, McGraw-Hill, 1997.

• Chapter 2, 3, 5, & 12 of Phaselock Techniques, F. Gardner, John Wiley & Sons, 2005.

3

PLL Block Diagram

[Perrott]

• A phase-locked loop (PLL) is a negative feedback system where an oscillator-generated signal is phase AND frequency locked to a reference signal

4

PLL Applications

• PLLs applications• Frequency synthesis

• Multiplying a 100MHz reference clock to 10GHz

• Skew cancellation• Phase aligning an internal clock to an I/O clock

• Clock recovery• Extract from incoming data stream the clock frequency and

optimum phase of high-speed sampling clocks

• Modulation/De-modulation• Wireless systems• Spread-spectrum clocking

5

Forward Clock I/O Circuits

• TX PLL

• TX Clock Distribution

• Replica TX Clock Driver

• Channel

• Forward Clock Amplifier

• RX Clock Distribution

• De-Skew Circuit• DLL/PI• Injection-Locked Oscillator

6

Embedded Clock I/O Circuits

• TX PLL

• TX Clock Distribution

• CDR• Per-channel PLL-based• Dual-loop w/ Global PLL &

• Local DLL/PI• Local Phase-Rotator PLLs• Global PLL requires RX

clock distribution to individual channels

7

Linear PLL Model

• Phase is generally the key variable of interest• Linear “small-signal” analysis is useful for understand PLL dynamics if

• PLL is locked (or near lock)• Input phase deviation amplitude is small enough to maintain operation in

lock range8

Phase Detector

• Detects phase difference between feedback clock and reference clock• The loop filter will filter the phase detector output, thus to characterize

phase detector gain, extract average output voltage (or current for charge-pump PLLs)

φref

φfb

φe

9

Loop Filter

• Lowpass filter extracts average of phase detector error pulses

I

I

VCO ControlVoltage

C1

R

C2

Charging

Discharging

VDD

VSSF(s)

10

Voltage-Controlled Oscillator

• Time-domain phase relationship

VDDVDD/20

ω0 1KVCO

( ) ( ) ( )tvKtt cVCOoutout +=∆+= 00 ωωωω

( ) ( ) ( )∫ ∫=∆= dtdt tvKtt cVCOoutout ωφLaplace Domain Model

φout(t)

11

Loop Divider

• Time-domain model( ) ( )t

Nt outfb ωω 1=

( ) ( ) ( )∫ == tN

tN

t outoutfb φωφ 1dt1

[Perrott]

φout(t) φfb(t)

12

Phase & Frequency Relationships

• Phase Step

( ) ( )

( ) ( )∫=

=

t

o

t

tdt

ττωφ

ωφ

d

td

phase of time) vschange of (rate derivativefirst theisFrequency Angular

( ) ( ) ( )

( ) ( ) ( )( )ttu

tu

111

111

tsint

phase and tfrequency angular with t sinusoid aConsider

φω

φω

+=

( ) ( )

( ) ( ) ( )( )

frequencyin change No

sin 11

1

tuttu

tut

∆Φ+=

∆Φ=

ω

φ

∆Φ

[Best]

13

Phase & Frequency Relationships

• Frequency Step ( )

( ) ( ) ( )( )

( )

phasein ramp a produces stepfrequency A

where

sinsin

1

1001

01

tt

tttttu

t

ωφ

φωωω

ωωω

∆=

+=∆+=

∆+=

[Best]

( ) tt ωφ ∆=1

14

( )

( ) ( )( )

( )

phasein change quadratic a produces rampfrequency A

2 where

sin2

sinsin

21

102

00

01

01

tt

ttttdtu

tt

t

•

••

•

∆=

+=

∆+=

∆+=

∆+=

∫

ωφ

φωωωττωω

ωωω

Phase & Frequency Relationships

[Best]

• Frequency Ramp

0ω t•

∆+ ωω0

( ) 21 2

tt•

∆=

ωφ

15

Understanding PLL Frequency Response

• Linear “small-signal” analysis is useful for understand PLL dynamics if• PLL is locked (or near lock)• Input phase deviation amplitude is small enough to maintain operation in

lock range

• Frequency domain analysis can tell us how well the PLL tracks the input phase as it changes at a certain frequency

• PLL transfer function is different depending on which point in the loop the output is responding to

Input phase response VCO output response

[Fischette]16

Open-Loop PLL Transfer Function

( ) ( )( )

( )s

sFKKsssG VCOPD

e

out =ΦΦ

=

• Open-loop response generally decreases with frequency

17

Closed-Loop PLL Transfer Function

( ) ( )( )

( )( )

( )( )

NsFKKs

sFKK

NsG

sGsssH

VCOPD

VCOPD

ref

out

+=

+=

ΦΦ

=1

( )

( ) ( )

( ) ( )N

sGN

sG

NsG

sNsFKKl

sG

VCOPD

+=+

−−=∆

=−=∆

−=−=

=

101t Determinan System

101t DeterminanPath Forward

Gain Loop

GainPath Forward

1

1

• Low-pass response whose overall order is set by F(s)

18

PLL Error Transfer Function

( ) ( )( ) ( ) ( )

NsFKKs

s

NsGs

ssEVCOPDref

e

+=

+=

ΦΦ

=1

1

( ) ( )

( ) ( )N

sGN

sG

NsG

sNsFKKl VCOPD

+=+

−−=∆

=−=∆

−=−=

=

101t Determinan System

101t DeterminanPath Forward

Gain Loop

1GainPath Forward

1

1

• Ideally, we want this to be zero• Phase error generally increases with

frequency due to this high-pass response19

PLL Order and Type

• The PLL order refers to the number of poles in the closed-loop transfer function• This is typically one greater than the number of loop

filter poles

• The PLL type refers to the number of integrators within the loop• A PLL is always at lease Type 1 due to the VCO

integrator

• Note, the order can never be less than the type

20

First-Order PLL

• Simple first-order low-pass transfer function

• Closed-loop bandwidth is equal to the DC loop gain magnitude

( )

( )

( )

( )

( )DCdBVCOPD

DCVCOPD

dB

DC

DC

dB

dB

VCOPD

VCOPD

VCOPD

sDC

DCVCOPD

Kss

ss

NKKKs

ssE

KN

KKK

KsNK

sN

NKKKs

KKKsH

NKKK

NssGK

sNK

sKKKsG

KsF

+=

+=

+=

==

+=

+=

+=

=

=

==

=

→

31

13

3

3

1

1

1

0

1

1

:FunctionError

:Bandwidth Loop-Closed

:FunctionTransfer

lim :MagnitudeGain Loop DC

:GainPath Forward

ω

ω

ωω

21

First-Order PLL Tracking Response• The PLL’s tracking behavior, or how the phase error responds to an input

phase change, varies with the PLL type• Phase Step Response

( ) ( )

( ) ( ) ( )( )

frequencyin change No

sin 11

1

tuttu

tut

∆Φ+=

∆Φ=

ω

φ

[Best]

• The final value theorem can be used to find the steady-state phase error

( )( ) 0limlim00

=+∆Φ

=

∆Φ

→→DC

ss KssssE

s

• All PLLs should have no steady-state phase error with a phase step error• Note, this assumes that the frequency of operation is the same as the VCO

center frequency (Vctrl=0). Working at a frequency other than the VCO center frequency is considered having a frequency offset (step).

22

First-Order PLL Tracking Response

• Frequency Offset (Step)

• The final value theorem can be used to find the steady-state phase error

( )( )DCDC

ss KKsssE

sωωω ∆

=+∆

=

∆

→→ 020limlim

• With a frequency offset (step), a first-order PLL will lock with a steady-state phase error that is inversely proportional to the loop gain

( )

( ) ( ) ( )( )

( )

phasein ramp a produces stepfrequency A

where

sinsin

1

1001

01

tt

tttttu

t

ωφ

φωωω

ωωω

∆=

+=∆+=

∆+=

[Best]

23

First-Order PLL Issues

• The DC loop gain directly sets the PLL bandwidth• No degrees of freedom

• In order to have low phase error, a large loop gain is necessary, which implies a wide bandwidth• This may not be desired in applications where we would like to filter

input reference clock phase noise

• First-order PLLs offer no filtering of the phase detector output• Without this filtering, the PD may not be well approximated by a

simple KPD factor• Multiplier PDs have a “second-harmonic” term• Digital PDs output square pulses that need to be filtered

24

Second-Order Type-1 PLL w/ Passive Lag-Lead Filter

( ) ( )

CRCR

sssF

2211

21

2

11

==

+++

=

ττ

τττ

Passive Lag-Lead Loop Filter[Allen]

25

( ) ( )

CRCR

sssF

2211

21

2

11

==

+++

=

ττ

τττ

Second-Order Type-1 PLL w/ Passive Lag-Lead Filter

( ) ( )( )( )

( )

( )

( )

( )

( ) 22

2

2

21

2121

22

221

2

22

2

2121

22

221

2

0

21

221

2

21

2

2 :FunctionError

2 :Factor Damping

:Frequency Natural

1

1

2

2

1

1

:FunctionTransfer

lim :MagnitudeGain Loop DC

1

1

11 :GainPath Forward

nn

VCOPD

n

VCOPD

n

VCOPDn

DCDC

DC

nn

nVCOPD

nn

VCOPDVCOPD

VCOPD

VCOPD

sDC

DCVCOPD

ssKK

NsssE

KKN

NKK

KsKs

sKN

ss

sKK

N

N

NKKsNKKs

sKK

sH

NKK

NssGK

ss

sNK

sssKKsG

ωζω

ω

τωζ

ττω

τττττ

ττττ

ωζω

ωωζω

τττττ

ττττ

ττ

ττττ

τττ

++

+

=

+=

+=

++

+

++

+

+

=

++

+

−

=

++

+

++

+

+=

=

=

+

+

+

+

=+++

=

→

26

• Phase Step Response

( )( )

step phase a with zero be shoulderror phase Again,

02

limlim 22

2

00=

++

+∆Φ

=

∆Φ

→→nn

VCOPD

n

ss ssKK

NssssE

s ωζω

ω

• A second-order type-1 PLL will still lock with a phase error if there is a frequency offset!

Second-Order Type-1 PLL Tracking Response

• Frequency Offset (Step)

( )( )DCnn

VCOPD

n

ss KssKK

NsssE

sω

ωζω

ωωω ∆

=++

+∆

=

∆

→→ 22

2

020 2limlim

27

Second-Order Type-1 PLL Properties

• While the second-order type-1 PLL will still lock with a phase error with a frequency offset, it is much more useful than a first-order PLL

• There are sufficient design parameters (degrees of freedom) to independently set ωn, ζ, and KDC

• The loop filter conditions the phase detector output for proper VCO control

• Loop stability needs to be considered for the second-order system

28

• Note, this type of loop filter is typically used with a charge-pump driving it. Thus, the filter transfer function is equal to the impedance.

Second-Order Type-2 PLL w/ Passive Series-RC Lag-Lead Filter

( )s

RCsR

sF

+

=

1

Passive Series-RC Loop Filter

29

( )

( )

( ) 22

2

222

2

2 :FunctionError

2 :Factor Damping

:Frequency Natural

22

21

:FunctionTransfer

1

:GainPath Forward

(ideally) :MagnitudeGain Loop DC

nn

n

VCOPDn

nn

nn

VCOPDVCOPD

VCOPD

VCOPD

DC

ssssE

RC

NCKK

ss

sN

NCKKs

NRKKs

RCsRKK

sH

sRC

sRKKsG

K

ωζω

ωζ

ω

ωζωζωζω

++=

=

=

++

+=

+

+

+

=

+

=

∞=

Second-Order Type-2 PLL w/ Passive Series-RC Lag-Lead Filter

( )s

RCsR

sF

+

=

1

30

• Phase Step Response

( )( )

step phase a with zero be shoulderror phase Again,

02

limlim 22

2

00=

++∆Φ

=

∆Φ

→→nn

ss sssssE

s ωζω

• A second-order type-2 PLL will lock with no phase error with a frequency offset!

Second-Order Type-2 PLL Tracking Response

• Frequency Offset (Step)

( )( ) 02

limlim 22020=

++∆

=

∆

→→nn

ss sssssE

s ωζωωω

31

Second-Order Type-2 PLL Properties

• A big advantage of the type-2 PLL is that it has zero phase error even with a frequency offset• This is why type-2 PLLs are very popular

• A type-2 PLL requires a zero in the loop filter for stability.• Note, this is not required in a type-1 PLL

• This zero can cause extra peaking in the frequency response• Important to minimize this in some applications, such

as cascaded CDR systems

32

Next Time

• PLL System Analysis• PLL Stability• Noise Transfer Functions• Transient Response

33