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Voltage-Controlled Oscillator (VCO)
fosc
Desirable characteristics:
• Monotonic fosc vs. VC characteristic
with adequate frequency range
• Well-defined Kvco
KPD
VD
F(s)
+
VC
slope = Kvco
fmin
fVC
fin
fmax
VC
^
Kvco
s
fout
Noise coupling from VC into PLL
output is directly proportional to Kvco.
¸N
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
1
Oscillator Design
Vin Þ 0
A(s)
Vout
Vout
A(s)
º HCL (s) =
Vin
1+ f × A(s)
loop gain
f
Barkhausen’s Criterion:
If a negative-feedback loop satisfies:
then the circuit will oscillate at frequency osc.
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
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 / Winter 2016
Prof. M. Green / U.C. Irvine
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
Let each inverter have transfer function Hinv ( jw) = Loop gain: Hloop ( jw) = éëHinv ( jw)ùû = 3
A03
(1+ jw p)
A0
1+ jw p
3
Applying Barkhausen’s criterion:
Hloop ( jwosc ) =
EECS 270C / Winter 2016
A03
é1+ 3ù
ë
û
Prof. M. Green / U.C. Irvine
3
> 1 Þ A0 > 2
2
4
Ring Oscillator Operation
tp
VA
tp
tp
VB
VC
VA
tp
VB
1
Tosc = 3t p
2
Þ Tosc = 6t p
tp
VC
tp
VA
1
Tosc
2
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
5
Variable Delay Inverters (1)
Inverter with variable load capacitance:
Vin
Current-starved inverter:
Vout
VC
Vin
Vout
VC
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
6
Variable Delay Inverters (2)
Interpolating inverter:
ISS
+
VC
_
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 / Winter 2016
Prof. M. Green / U.C. Irvine
7
Differential Ring Oscillator
+
−
+
−
VA
+
−
VB
VA
VB
VC
VC
+
−
VD
−
+
VA
additional inversion
(zero-delay)
tp
tp
1
Tosc = 4t p
2
Þ Tosc = 8t p
tp
tp
VD
Use of 4 inverters makes
quadrature signals available.
VA
EECS 270C / Winter 2016
1
Tosc
2
Prof. M. Green / U.C. Irvine
8
Resonance in Oscillation Loop
Hr ( jw)
Hr (s)
1
Hr (s)
+
p
ÐHr ( jw)
r
2
r
-
At dc:
Since Hr(0) < 1, latch-up does not occur,
even with positive feedback.
EECS 270C / Winter 2016


p
2
At resonance:
Hr ( jwr ) > 1 Þ w = w
osc
r
ÐHr ( jwr ) = 0
Prof. M. Green / U.C. Irvine
9
LC VCO (1)
L
Vin
Hr (s)
C
Vout
wr =
Vout
Vin
1
LC
Hr (s)
Þ
Þ
2L
C
C
Hr (s)
realizes negative
resistance
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
10
LC VCO (2)
Consider simplified parallel LCR connected to
a negative resistance element:
LC tank with loss
Negative
conductance
element:
Exhibits slope -gp at
origin of the i-v
characteristic
Necessary condition for oscillation:
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
11
LC VCO (3)
I
I
V
where a1 , a3 > 0
V
Negative conductance in
parallel with Rp = 200:
I’
I’
V
V
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
12
LC VCO (4)
VC
Case 1: gp < 1/Rp (stable)
IL
Transient waveforms:
IL
Iloss
+
t
VC
_
IL
Initial condition
Phase plot:
VC
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
13
LC VCO (5)
IL VC
Case 2: gp = 10m > 1/Rp (unstable)
IL
VC
Transient waveforms:
Iloss
Iloss
+
VC
_
t
Iloss
IL
Limit cycle
Phase plot:
Initial condition
Vc
VC
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
14
LC VCO (6)
IL VC
Case 2: gp = 10m > 1/Rp (unstable)
with different initial condition
IL
Iloss
Transient waveforms:
VC
+
Iloss
VC
_
t
IL
Initial condition
Phase plot:
Limit cycle
VC
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
15
LC VCO (7)
LC VCO schematic:
Cross-coupled characteristic:
I1
I2
DC biasing:
Vdm
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
16
LC VCO (8)
Voutn
Voutp
I1
I2
Voutp , Voutn
I1 , I2
t
“Voltage-Controlled” topology
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
17
LC VCO (9)
LC VCO with current source:
Cross-coupled characteristic:
I1
I2
DC biasing:
2Rcm
1
2 ISS
EECS 270C / Winter 2016
Vdm
Prof. M. Green / U.C. Irvine
18
LC VCO (9)
Voutn
Voutp
I1
I2
Voutp , Voutn
I1 , I2
= 2.6 mA
t
“Current-Controlled” topology
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
19
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
VBG
accumulation
region
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
inversion
region
20
LC VCO Variations (1)
IS
2L
C
C
C
C
2L
2L
C
IS
2L
C
C
C
ISS
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
21
LC VCO Variations (2)
2L
2L
C
C
C
C
ISS
Voltage-controlled topology
•
•
Current-controlled topology
DC biasing set by VDD dropped
directly across diode-connected
transistors
•
DC biasing set by ISS
•
Amplitude limited by ISS
Amplitude limited by VDD
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
22
Effect of CML Loading
1.
1. ideal capacitor load
1 nH
3.8 
400 fF
400 fF
108 fF
108 fF
2.
Cg = 108fF
1 nH
3.8 
400 fF
400 fF
2. CML buffer load
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
23
CML Buffer Input Admittance (1)
Yin = jwCgs + jwCgd A0 ×
A0 = 1+ gm R
(
1+ jw / z
1+ jw / p
)
where: 1/ p = CL +Cgd R
1/ z =
( )
(note p < z)
CL R
A0
Re Yin = A0Cgd w 2 ×
1 p -1 z
(
1+ w p
)
2
Substantial parallel loss at high
frequencies  weakens VCO’s
tendency to oscillate
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
24
CML Buffer Input Admittance (2)
Yin magnitude/phase:
Yin real part/imaginary part:
magnitude
imaginary
phase
real
Contributes 2k additional parallel resistance
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
25
CML Buffer Input Admittance (3)
3. CML tuned buffer load
Cg = 108 fF
1 nH
imaginary
3.8 
400 fF
400 fF
3.8 nH
real
Contributes negative parallel resistance
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
26
CML Buffer Input Admittance (4)
ideal capacitor load
Cg = 108 fF
1 nH
3.8 
400 fF
400 fF
3.8 nH
CML buffer load
Loading VCO with tuned CML buffer
allows negative real part at high
frequencies  more robust oscillation!
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
CML tuned buffer load
27
Differential Control of LC VCO
Differential VCO control is preferred to reduce VC noise coupling into PLL output.
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
28
Achieving Wide LC VCO Frequency Range
VCO characteristic:
Banded VCO characteristic:
fosc
fosc
freq.
range
freq.
range
slope = Kvco
VC
VC
For a given Kvco how can we
increase the VCO’s frequency
range?
EECS 270C / Winter 2016
Coarse tuning selects the individual band;
fine tuning is set by VC.
Prof. M. Green / U.C. Irvine
29
LC VCO with Banding
coarse tuning
fine tuning
EECS 270C / Winter 2016
VC
Prof. M. Green / U.C. Irvine
30
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)
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
31
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 Î [-¥,+¥]
0.5
x
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
32
Random Processes (2)
Probability of X within a range is straightforward:
PX(x)
1
(
0.5
)
P X Î [x1, x2 ] = P(x 2 ) - P(x1)
x1 x2
x
If we let x2-x1 become very small …
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
33
Random Processes (3)
Probability density function pX(x):
Probability that random variable X lies within the range of x and x+dx.
pX (x) ×dx = PX (x + dx) - PX (x)
Þ pX (x) =
(
) ò
P X Î [ x 1, x 2 ] =
dPX (x)
dx
PX(x)
x2
x1
pX (x) dx
pX(x)
1
0.5
dx
EECS 270C / Winter 2016
x
x
Prof. M. Green / U.C. Irvine
34
Random Processes (4)
Expectation value E[X]: Expected (mean) value of random variable X
over a large number of samples.
+¥
ò x ×p
E[X ] º 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 ] =
òx
2
× p X (x)dx
-¥
Variance:
[
+¥
]
E (X - X ) º s =
2
2
ò (x - X ) p
X
(x)dx
-¥
[
Standard deviation: s = E (X - X )2
EECS 270C / Winter 2016
2
]
Prof. M. Green / U.C. Irvine
35
Gaussian Function
1. Provides a good model for the probability density functions of many
random phenomena.
2. Can be easily characterized mathematically s , X .
3. Combinations of Gaussian random variables are themselves
Gaussian.
(
)
f (x)
1
s 2p
é -(x - X )2 ù
ú
f (x) =
expê
2
êë 2s
úû
s 2p
1
0.607
s 2p
2
+¥
ò f (x)dx = 1
X -s
-¥
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
X
X +s
x
36
Joint Probability (1)
Consider 2 random variables:
(
P(x, y) º P X £ x and Y £ y
)
If X and Y are statistically independent (i.e., uncorrelated):
(
)
P X Î [ x, x + dx ] and Y Î [ y, y + dy ] = pX (x) × pY (y) ×dx dy
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
37
Joint Probability (2)
Consider sum of 2 random variables:
Z = X +Y
(
) òò
é
=ê ò
ë
P Z Î [ z0, z0 + dz] =
y
strip
pX (x)pY (y) dx dy
ù
p X (x)pY (z0 - x) dx ú dz
-¥
û
¥
x + y = z0 + dz
pZ (z0 )
dy = dz
x + y = z0
dx
EECS 270C / Winter 2016
determined by convolution
of pX and pY.
x
Prof. M. Green / U.C. Irvine
38
Joint Probability (3)
Example: Consider the sum of 2 non-Gaussian random processes:
*
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
39
Joint Probability (4)
3 sources combined:
*
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
40
Joint Probability (5)
4 sources combined:
*
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
41
Joint Probability (6)
Noise sources
Central Limit Theorem:
Superposition of random variables tends toward normality.
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
42
Fourier transform of Gaussians:
pX (x) =
é
2ù
-(x
X
)
ú
expê
ê 2s 2
ú
2p
ë
û
X
1
sX
æ 1 2 2ö
PX (w) = expç- s X w ÷
è 2
ø
F
Recall:
é
P Z Î [ z0, z0 + dz] = ê
ë
(
ù
p X (x)pY (z0 - x) dx ú dz
-¥
û
) ò
pZ (z0 ) =
ò
¥
-¥
¥
pX (x)pY (z0 - x) dx
F
PZ (w) = PX (w) ×PY (w)
æ 1 2 2ö
æ 1 2 2ö
= expç- s X w ÷ ×expç- sY w ÷
è 2
ø
è 2
ø
pZ (z) =
1
(
2p s 2X + s Y2
)
é
-(z - Z)2
expê
ê2 s 2 +s 2
X
Y
ë
(
)
ù
ú
ú
û
F -1
æ 1
ö
= expç- (s 2X + s 2X )w 2 ÷
è 2
ø
Variances of sum of random normal processes add.
EECS 270C / Winter 2016
Prof. M. Green / U.C. Irvine
43
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 t = t1 - t 2
RX (t ) = E [ X (t) × X (t + t )] for any t
2
Note RX (0) = s
Power spectral density SX():
2ù
é +¥
ê
ú
SX (w ) = Eê X (t) ×e - jwt dt ú
êë -¥
úû
ò
EECS 270C / Winter 2016
SX() given in units of [dBm/Hz]
Prof. M. Green / U.C. Irvine
44
Relationship between spectral density & autocorrelation function:
1
RX (t ) =
2p
ò
¥
-¥
SX (w) ×e jwt dw
ò
1
Þ RX (0) = s =
2p
2
¥
-¥
SX (w)dw
infinite variance
(non-physical)
Example 1: white noise
SX (w)
( )
SX w = K
EECS 270C / Winter 2016
RX (t )


RX (t ) =
Prof. M. Green / U.C. Irvine
K
×d t
2p
()
45
Example 2: band-limited white noise
RX (t )
SX (w)
1
2
s 2 = Kw p
K
-w p
( )
SX w =
wp

K
RX (t ) = s 2e
-w p t

w2
1+ 2
wp
pX (x)
For parallel RC circuit
capacitor voltage noise:
wp =
1
RC
EECS 270C / Winter 2016
s V2C =
kBT
C
-s
Prof. M. Green / U.C. Irvine
+s
x
46