1. 2 loop dynamics to keep track of deviations from the free-running frequency,
TRANSCRIPT
![Page 1: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/1.jpg)
1
![Page 2: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/2.jpg)
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
![Page 3: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/3.jpg)
3
P ha s 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
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
![Page 4: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/4.jpg)
4
P ha s 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
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
![Page 5: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/5.jpg)
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
![Page 6: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/6.jpg)
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
![Page 7: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/7.jpg)
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
![Page 8: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/8.jpg)
8
SIMPLEST LOWPASS FILTER• Integrator
• Not a good idea, but simple to check out.
P h as eD e te c to r
K P
F (s ) = 1 /sA m plifie r
A
V o lta g e C o ntro lle dO s c illa to r
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
![Page 9: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/9.jpg)
9
P h as eD e te c to r
K P
A m plifie rA
V o lta g e C o ntro lle dO s c illa to r
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
![Page 10: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/10.jpg)
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
![Page 11: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/11.jpg)
11
PROBLEMS WITH FIRST ORDER LOWPASS FILTER
P h as eD e te c to r
K P
A m plifie rA
V o lta g e C o ntro lle dO s c illa to r
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.
![Page 12: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/12.jpg)
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
![Page 13: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/13.jpg)
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.
![Page 14: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/14.jpg)
14
PHASE DETECTOR - CONTINUED
nVolt/radia 10
PK
P ha s eD e te c to r
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 ,
“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
![Page 15: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/15.jpg)
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
![Page 16: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/16.jpg)
16
LOCK RANGE CONTINUED
P ha s eD e te c to r
Lo 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 ,
oK
3V
oFR
,0When
2,
POFRMAXO KAK
Lock range = LGPOLOCK AAKK
![Page 17: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/17.jpg)
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.
![Page 18: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/18.jpg)
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
![Page 19: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/19.jpg)
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.
![Page 20: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/20.jpg)
20
CAPTURE MECHANISMP h as e
D e te c to rK 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 ,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
![Page 21: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/21.jpg)
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
![Page 22: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/22.jpg)
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.
![Page 23: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/23.jpg)
23
CAPTURE CONDITION
At the instant of capture,
iFRoiFRP AKFK
2
P h as eD e te c to r
K P
L o 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 ,
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
![Page 24: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/24.jpg)
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
![Page 25: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/25.jpg)
25
That’s all folks
X
![Page 26: 1. 2 LOOP DYNAMICS To keep track of deviations from the free-running frequency,](https://reader036.vdocuments.site/reader036/viewer/2022062800/56649de45503460f94ada8ef/html5/thumbnails/26.jpg)
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