perspective components lecture 3 cmos phase locked loops · 2018-08-28 · lecture 3 – cmos phase...

Lecture 03 – (8/9/18) Page 3-1

CMOS Phase Locked Loops © P.E. Allen - 2018

LECTURE 3 – CMOS PHASE LOCKED LOOPS

Topics

• The acquisition process – unlocked state

• Noise in linear PLLs

Organization:

PLL

Applications and Examples

Systems

Perspective

Circuits

Perspective

140418-02

PLL

Components

Types of PLLs

and PLL Measurements

Technology

PerspectiveCMOS

Technology

Lecture 03 – (8/9/18) Page 3-2


THE ACQUISTION PROCESS – LPLL IN THE UNLOCKED STATE

Unlocked Operation

If the PLL is initially unlocked, the phase error, e, can take on arbitrarily large values

and as a result, the linear model is no longer valid.

The mathematics behind the unlocked state are beyond the scope of this presentation. In

the section we will attempt to answer the following questions from an intuitive

viewpoint:

1.) Under what conditions will the LPLL become locked?

2.) How much time does the lock-in process require?

3.) Under what conditions will the LPLL lose lock?

Lecture 03 – (8/9/18) -3


Some Definitions of Key Performance Parameters

1.) The hold range (H) is the frequency range over which an LPLL can statically

maintain phase tracking. A PLL is conditionally stable only within this range.

2.) The pull-in range (P) is the range within which an LPLL will always become

locked, but the process can be rather slow.

3.) The pull-out range (PO) is the dynamic limit for stable operation of a PLL. If

tracking is lost within this range, an LPLL normally will lock again, but this process can

be slow.

4.) The lock range (L) is the frequency range within which a PLL locks within one

single-beat note between reference frequency and output frequency. Normally, the

operating frequency range of an LPLL is restricted to the lock range.

wow

Hold-In Range (Static Limits of Stability)

wo+DwHwo-DwH 140421-01

wow

wo+DwPwo-DwP 140421-02

Pull-in Range

wow

(Dynamic Limits of Stability)

wo+DwPOwo-DwPO 140421-03

wow

wo+DwLwo-DwL 140421-04

Lock Range

Lecture 03 – (8/9/18) Page 3-4


Illustration of Static Ranges

Assume the frequency of the VCO is varied very slowly from a value below o-H to a

frequency above o+H.

The following pages will attempt to relate the key parameters of hold range, pull-in

range, pull-out range, and lock range to the time constants, 1 and 2 and the gain factors

Kd, Ko, and Ka.

wVCO

win

wo+DwP

wo-DwP

wo+DwH

wo-DwH

Pull-in RangeHold-in Range

140421-05

Lecture 03 – (8/9/18) Page 3-5


Hold Range (H)

The magnitude of the hold range is calculated by finding the frequency offset of the input

that causes a phase error of ±/2.

Let,

1 = o ± H → 1(t) = H t → 1(s) =

s2

e(s) =1(s) He(s) =

s2 s

s + KoKdF(s)

limt→

e(t) = lims→0

se(s) =

KoKdF(0) (valid for small values of e)

For large variations, we write

limt→

sine(t) = H

KoKdF(0) → H = ±KoKdF(0) when e = ±/2

For the various filters-

1.) Passive lag filter: H = ±KoKd

2.) Active lag filter: H = ±KoKdKa

3.) Active PI filter: H = ±

(If H = ± the actual hold range may be limited by the frequency range of the VCO)

Lecture 03 – (8/9/18) Page 3-6


Lock Range (L)

Assume the loop is unlocked and the reference frequency is 1 = o + . Therefore,

v1(t) = V10 sin(ot + t)

The VCO output is assumed to be

v2(t) = V20 sgn(ot)

vd(t) = Kd sin( t) + higher frequency terms

Assuming the higher frequency terms are filtered out, the filter output is

vf(t) Kd|F(j)| sin( t)

This signal causes a frequency modulation of the VCO output frequency as shown.

Note: No locking occurs in the above illustration because > KoKd |F(j)|.

Frequency

t

w1

wo

Dw

w2(t)

KoKd|F(jDw)|

140421-06

Lecture 03 – (8/9/18) Page 3-7


Lock Range – Continued

Locking will take place if KoKd |F(j)| ≥ . Therefore, the lock range can be

expressed as,

L = ± KoKd |F(j)|

and is illustrated as,

Locks within one cycle or beat note.

Frequency

t

w1

wo

Dww2(t)

KoKd|F(jDw)|

140421-07

w2 = w1

Lecture 03 – (8/9/18) Page 3-8


Lock Range - Continued

If we assume that the lock range is greater than the filter frequencies, 1/1 and 1/2, the

lock range for the various filters can be expressed as,

1.) Passive lag filter: L = ± KoKd |F(j)| ± KoKd 2

1+2 ± KoKd

2

1

2.) Active lag filter: L = ± Ka |F(j)| ± Ka 2

1

3.) Active PI filter: L = ± |F(j)| ± 2

1

Previously, we found expressions for n and for each type of filter. Using these

expressions and assuming that the loop gain is large, we find for all three filters that

L ±2n

The lock-in time or settling time can be approximated as one cycle of oscillation,

TL 1

fn =

2

n

Lecture 03 – (8/9/18) Page 3-9


Pull-In Range (P)

Again assume the loop is unlocked and the reference frequency is 1 = o + and the

VCO initially operates at the center frequency of o.

Let us re-examine the previous considerations:

Since min is less than max, the frequency of the positive going sinusoid is less than

the frequency of the negative going sinusoid. As a consequence, the average value of the

VCO output, 2 __

, “pulls” toward 1.

Frequency

t

w1

wo

Dwmax

w2(t)

140421-08

Dwmin

w2

Pull-in Effect

Lecture 03 – (8/9/18) Page 3-10


The Pull-In Process

For an unlocked PLL with the frequency offset, , less than the pull-in range, P, the

VCO output frequency, 2 will approach the reference frequency, 1, over a time

interval called the pull-in time, TP.

Illustration:

Frequency

w1

wow2(t)

w2

t

Dw

140421-09

Pull-in Time, TP

Lecture 03 – (8/9/18) Page 3-11


Pull-In Range (P) for Various Types of Filters

The mathematical treatment of the pull-in process is beyond the scope of this

presentation†. The results are summarized below.

Type of Filter P (Low Loop Gains) P (High Loop Gains) Pull-In Time, TP

Passive Lag 4

2nKoKd - n

2 4 2

nKoKd =

2

16 o

2

n3

Active Lag

4

2nKoKd -

n2

Ka

4 2

nKoKd =

2

16 o

2Ka

n3

Active PI Lag → → = 2

16 o

2

n3

† R.M. Best, Phase-Locked Loops – Design, Simulation, and Applications, 4th ed., McGraw-Hill Book Co., 1999, Appendix A.

Lecture 03 – (8/9/18) Page 3-12


Example 3

A second-order PLL having a passive lag loop filter is assumed to operate at a center

frequency, fo, of 100kHz and has a natural frequency, fn, of 3 Hz which is a very narrow

band system. If = 0.7 and the loop gain, KoKd = 2·1000 sec.-1, find the lock-in time,

TL, and the pull-in time, TP, for an initial frequency offset of 30 Hz.

Solution

TL 1

fn =

1

3 = 0.333 secs.

TP = 2

16 o

2

n3 =

44

16·83 fo2

fn3 = 302

32(0.7)33 = 4.675 secs.

Lecture 03 – (8/9/18) Page 3-13


Pull-Out Range (PO)

The pull-out range is that frequency step which causes a lock-out if applied to the

reference input of the PLL.

An exact calculation is not possible but simulations show that,

PO ≈ 1.8n ( +1)

At any rate, the pull-out range for most systems is between the pull-in range and the lock-

range,

L < PO < P

Lecture 03 – (8/9/18) Page 3-14


Steady-State Error of the PLL

The steady-state error is the deviation of the controlled variable from the set point after

the transient response has died out. We have called this error, e().

e() = lims→0

se(s) = lims→0

s1(s) s

s + KoKdF(s)

Let us consider a generalized filter given as,

F(s) = P(s)

Q(s)sN

where P(s) and Q(s) can be any polynomials in s, and N is the number of poles at s = 0.

e() = lims→0

s2sNQ(s)1(s)

s·sNQ(s) + KoKdP(s)

Comments:

• Note that for the active PI filter, N = 1.

• For N >1, it becomes difficult to maintain stability.

• In most cases, P(s) is a first-order polynomial and Q(s) is a polynomial of order 0 or 1.

To find the steady-state error, the input, (s) must be known. We will consider several

inputs on the following slide.

Lecture 03 – (8/9/18) Page 3-15


Steady-State Error for Various Inputs

1.) A phase step, .

1(s) = s

2.) A frequency step, .

1(s) =

s2

(The LPLL must have one pole at s = 0 for the steady-state error to be zero.)

3.) A frequency ramp, · .

1(s) = ·

s3

For N = 2 and Q(s) =1, the order of the LPLL becomes 3 permitting a phase shift of

nearly 270° which must be compensated for by zeros to maintain stability.

e() = lims→0

s2sNQ(s)

s[s·sNQ(s) + KoKdP(s)] = 0 for any value of N.

e() = lims→0

s2sNQ(s)

s2[s·sNQ(s) + KoKdP(s)] = 0 if N ≥1

e() = lims→0

s2sNQ(s) ·

s3[s·sNQ(s) + KoKdP(s)] = 0 if N ≥ 2

Lecture 03 – (8/9/18) Page 3-16


NOISE IN LINEAR PLL SYSTEMS

Phase Noise

Illustration:

v1(t)

t

t

qn1(t)

v1(t)+qn1(t)

t

Phase error140421-10

Lecture 03 – (8/9/18) Page 3-17


PLL for Noise Analysis

Assume that the input is band limited as shown below.

Bi = Bandwidth of the prefilter (or system)

Some terminology:

• Power spectral density is the measure of power in a given frequency range (Watts/Hz)

or (V2/Hz). It is found by dividing the rms power by the bandwidth.

• Consider all noise signals as white noise which means the power spectrum is flat.

• Ps = input signal rms power (V1(rms)2/Rin)

• Pn = rms power of the input noise

Phase

DetectorFilter

VCO

fof

Att

enu

atio

n (

dB

)

Bi

Prefilter

Input

Signal Output

140421-11

Lecture 03 – (8/9/18) Page 3-18


Power Spectra of a PLL

Illustration of how input noise

becomes phase noise in the

frequency spectrum:

Power spectra of the reference

signal, v1(t), and the superimposed

noise signal, vn(t).

Spectrum of the phase noise at the

input of the PLL.

Frequency response of the phase-

transfer function, H(j).

Spectrum of the phase noise at the

output of the PLL.

Sp

ectr

al P

ow

er

Den

sity

Frequency

Wi

BiArea = Pn

= WiBi

fo

Area = Ps

Frequency

qn1(jw)2

Bi/2

Area = vn12

F

FrequencyBL

Function of z

|H(jw)|

Frequency

qn2(jw)2

BL

Area = vn22

140421-12

Lecture 03 – (8/9/18) Page 3-19


Noise Relationships for a PLL

Spectral density of the input noise signal:

Wi = Pn

Bi (W/Hz)

Input rms phase noise jitter (or the square of the rms phase noise):

n1(t) → n1 2 = Pn

2Ps (Comes from the assumption of white noise)

Signal-to-Noise Ratio (SNR):

SNR at the input = (SNR)i Ps

Pn → n1 2 =

Pn

2Ps =

1

2(SNR)i (radians2)

Input phase jitter (noise) spectrum:

n1 2 (j) = = n1 2

Bi/2 (radians2/Hz)

Output phase jitter (noise) spectrum:

n2 2 (j) = |H(j)|2 n1 2 (j) = |H(j)|2

Lecture 03 – (8/9/18) Page 3-20


RMS Value of the Output Phase Noise

The output phase noise is found by integrating n2(j) over the bandwidth of the PLL.

n2 2 =

0

n2 2 (j2f) df

where n2 2 is the area under the output phase noise plot in a previous slide.

n2 2 =

0

|H(j)|2df =

2

0

|H(j)|2d

The integral

0

|H(j2f)|2df = BL is called the noise bandwidth.

The solution of this integral is,

BL = n

2

+ 1

4 →

dBL

d =

n

2

1 - 1

2 = 0 → = 0.5 → BL(min) = 0.5n

n2 2 = BL = n1 2

Bi/2 BL =

Pn

2Ps 2BL

Bi =

Pn

Ps ·

BL

Bi =

BL

(SNR)iBi

Lecture 03 – (8/9/18) Page 3-21


RMS Value of the Output Phase Noise – Continued

We noted previously that,

n1 2 = 1

2(SNR)i =

Pn

2Ps → (SNR)i =

Ps

Pn

A dual relationship holds for the output,

where (SNR)L is the signal-to-noise ratio at the output.

This equation suggests that the PLL improves the SNR of the input signal by a factor of Bi/2BL. Thus, the narrower the noise PLL bandwidth, BL, the greater the improvement.

Some experimental observations:

• For (SNR)L = 1, a lock-in process will not occur because the output phase noise is

excessive (0.707 radians or 40.4°).

• At an (SNR)L = 2, lock-in is eventually possible (0.5 radians or 28.6°).

• For (SNR)L = 4, stable operation is generally possible.

n2 2 = 1

2(SNR)L =

Pn

Ps ·

BL

Bi → (SNR)L =

Ps

Pn

Bi

2BL

(SNR)L = (SNR)i Bi

2BL

Note: (SNR)L = 4, n2 2 becomes 0.125 radians2. n2 2 = 0.353 radians 20° and the

limit of dynamic stability (180°) is rarely exceeded.

Lecture 03 – (8/9/18) Page 3-22


Summary of Noise Analysis of the LPLL

• Stable operation of the LPLL is possible if (SNR)L ≥ 4

• (SNR)L is calculated from

(SNR)L = Ps

Pn

Bi

2BL

where Ps = signal power at the reference input

Pn = noise power at the reference point

Bi = bandwidth of the system at the input

BL = noise bandwidth of the PLL

• The noise bandwidth, BL, is a function of n and . For = 0.7, BL = 0.53n

• The average time interval between two unlocking events gets longer as the (SNR)L

increases.

Lecture 03 – (8/9/18) Page 3-23


Pull-In Techniques for Noisy Signals

1.) The sweep technique.

When the noise bandwidth is made small, the SNR of the loop is sufficiently large to

provide stable operation. However, the lock range can become smaller than the

frequency interval within which the input signal is expected to be. The following

circuit solves this problem by providing a direct VCO sweep.

(1.) LPLL not locked.

(2.) RUN mode starts

a positive sweep.

(3.) When the VCO

frequency

approaches the

input frequency

the loop locks.

(4.) The “In-Lock”

detector switches

the sweep switch

to the “HOLD”

position.

Lecture 03 – (8/9/18) Page 3-24


Pull-In Techniques for Noisy Signals

2.) Switched filter technique.

In the unlocked state, the filter bandwidth is large so that lock range exceeds the

frequency range within which the input is expected.

In the locked state, the filter bandwidth is reduced in order to reduce the noise.

Phase

Detector

VCO

v2(t)

v1(t)Rsmall(not locked)

Rlarge(locked)

Switched Loop Filter

"In-Lock"

Detector

vf(t)

140421-14

Lecture 03 – (8/9/18) Page 3-25


SUMMARY

• Acquisition process – the PLL in the unlocked state

• Influence of noise on the linear PLL

• Pull-in techniques for noisy signals

perspective components lecture 3 cmos phase locked loops · 2018-08-28 · lecture 3 – cmos phase...

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