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f osc. f max. slope = K vco. f min. V C. ^. ^. ^. Voltage-Controlled Oscillator (VCO). Desirable characteristics: Monotonic f osc vs. V C characteristic with adequate frequency range Well-defined K vco. +. - PowerPoint PPT Presentation
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EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 1
Voltage-Controlled Oscillator (VCO)
VC
fosc
fmin
fmax
slope = Kvco
Desirable characteristics:
• Monotonic fosc vs. VC characteristic with adequate frequency range
• Well-defined Kvco
^
^
Noise coupling from VC into PLL output is directly proportional to Kvco.
^
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 2
Oscillator Design
loop gain
Barkhausen’s Criterion:
If a negative-feedback loop satisfies:
then the circuit will oscillate at frequency 0.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 3
Inverters with Feedback (1)
V1 V2
V1
V2 1 inverter
feedback
V1
V2
2 inverters
feedback
1 stable equilibrium point
3 equilibrium points: 2 stable, 1 unstable(latch)
1 inverter:
V1 V2
2 inverters:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 4
Inverters with Feedback (2)
3 inverters forming an oscillator:
1 unstable equilibrium point due to phase shift from 3 capacitors
V1 V2
V1
V2
Let each inverter have transfer function
Loop gain:
Applying Barkhausen’s criterion:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 5
Ring Oscillator Operation
VA VB VC
tp tp tp
VA
VB
VC
VA
tp
tp
tp
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 6
Variable Delay Inverters (1)
VC
Vin Vout
Current-starved inverter:Inverter with variable load capacitance:
Vin Vout
VC
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 7
Variable Delay Inverters (2)
R R
Vin+ Vin- Vin+ Vin-
Vout-Vout+
IfastIslow
RG RG
ISS
VC
+
_
Interpolating inverter:
• 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.
VA
VB
VC
VD
tp
tp
tp
tp
VA
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 8
Differential Ring Oscillator
additional inversion (zero-delay)
VA
+
−
Use of 4 inverters makes quadrature signals available.
VB
+
−VC
+
−VD
+
−VA
−
+
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 9
Resonance in Oscillation Loop
r
r
1
At dc:
Since Hr(0) < 1, latch-up does not occur.
At resonance:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 10
LC VCO
Vin Vout
Vin
Vout
CL
realizes negative resistance
2L
CC
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 11
A. Reverse-biased p-n junction
+ –VR
VR
Cj
B. MOSFET accumulation capacitance
+
–
VBG
varactor = variable reactance
Variable Capacitance
VBG
Cg
accumulationregion
inversionregion
p-channel
n diffusion in n-well
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 12
LC VCO Variations
2L
CC
2L
CC
2L
CC
ISS
2L
CC
ISIS
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 13
1. ideal capacitor load
2. CML buffer load
Effect of CML Loading
1.
3.8 1 nH
400 fF 400 fF
Cg = 108fF
1 nH 3.8
400 fF 400 fF 108 fF108 fF
2.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 14
Substantial parallel loss at high frequencies weakens VCO’s tendency to oscillate
(note p < z)where:
CML Buffer Input Admittance (1)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 15
Yin magnitude/phase: Yin real part/imaginary part:
magnitude
phase
imaginary
real
Contributes 2k additional parallel resistance
CML Buffer Input Admittance (2)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 16
imaginary
real
Contributes negative parallel resistance
Cg = 108 fF
3.8 nH
3.8 1 nH
400 fF 400 fF
CML Buffer Input Admittance (3)
3. CML tuned buffer load
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 17
Loading VCO with tuned CML buffer allows negative real part at high frequencies more robust oscillation!
ideal capacitor load
CML buffer load
CML tuned buffer load
CML Buffer Input Admittance (4)
Cg = 108 fF
3.8 nH
3.8 1 nH
400 fF 400 fF
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 18
Differential Control of LC VCO
Differential VCO control is preferred to reduce VC noise coupling into PLL output.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 19
Ring Oscillator LC Oscillator
– slower
– low Q more jitter generation
+ Control voltage can be applied differentially
+ Easier to design; behavior more predictable
+ Less chip area
+ faster
+ high Q less jitter generation
– Control voltage applied single-ended
– Inductors & varactors make design more difficult and behavior less predictable
– More chip area (inductor)
Oscillator Type Comparison
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 20
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.
0.5
1
x
PX(x)
Example 1:
Random variable
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 21
0.5
1
x
PX(x)
x1 x2
Probability of X within a range is straightforward:
If we let x2-x1 become very small …
Random Processes (2)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 22
Probability density function pX(x): Probability that random variable X lies within the range of x and x+dx.
0.5
1
x
PX(x)
x
pX(x)
dx
Random Processes (3)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 23
Expectation value E[X]: Expected (mean) value of random variable X over a large number of samples.
Mean square value E[X2]: Mean value of the square of a random variable X2 over a large number of samples.
Variance:
Standard deviation:
Random Processes (4)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 24
Gaussian Function
x
2
1. Provides a good model for the probability density functions of many random phenomena.
2. Can be easily characterized mathematically .
3. Combinations of Gaussian random variables are themselves Gaussian.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 25
Joint Probability (1)
If X and Y are statistically independent (i.e., uncorrelated):
Consider 2 random variables:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 26
Consider sum of 2 random variables:
x
y
dx
dy = dz
determined by convolutionof pX and pY.
Joint Probability (2)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 27
*
Example: Consider the sum of 2 non-Gaussian random processes:
Joint Probability (3)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 28
3 sources combined:
*
Joint Probability (4)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 29
4 sources combined:
*
Joint Probability (5)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 30
Central Limit Theorem:Superposition of random variables tends toward normality.
Noise sources
Joint Probability (6)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 31
Fourier transform of Gaussians:
F
Recall:
F
F -1
Variances of sum of random normal processes add.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 32
Autocorrelation function RX(t1,t2): Expected value of the product of 2 samples of a random variable at times t1 & t2.
For a stationary random process, RX depends only on the time difference
for any t
Note
Power spectral density SX():
SX() given in units of [dBm/Hz]
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 33
Relationship between spectral density & autocorrelation function:
Example 1: white noise
infinite variance(non-physical)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 34
Example 2: band-limited white noise
x
For parallel RC circuitcapacitor voltage noise:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 35
Random Jitter (Time Domain)
Experiment:
datasource
CDR(DUT) analyzer
CLK
DATA RCK
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 36
Jitter Accumulation (1)
Free-runningoscillator output
Histogram plots
Experiment:Observe N cycles of a free-running VCO on an oscilloscope over a long measurement interval using infinite persistence.
NT
1 2 3 4
trigger
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 37
Observation:As increases, rms jitter increases.
proportionalto 2
proportional to
Jitter Accumulation (2)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 38
Noise Spectral Density (Frequency Domain)
foscfosc+f
Sv(f)
f (log scale)
1/f2 region (-20dBc/Hz/decade)
Power spectral densityof oscillation waveform:
Single-sideband spectral density:
Ltotal includes both amplitude and phase noise
Ltotal(f) given in units of [dBc/Hz]
1/f3 region (-30dBc/Hz/decade)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 39
Noise Analysis of LC VCO (1)
active circuitry
C L R -R C L
+
_
vcinR
Consider frequencies near resonance:
noise from resistor
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 40
Spot noise current from resistor: C L
+
_
vcinR
Noise Analysis of LC VCO (2)
Leeson’s formula (taken from measurements):
Where F and1/f3 are empirical parameters.
dBc/Hz
spot noise relative to carrier power
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 41
Oscillator Phase Disturbance
Current impulse q/t
_+Vosc
t t
ip(t)
Vosc(t) Vosc(t)
Vosc jumps by q/C
• Effect of electrical noise on oscillator phase noise is time-variant.• Current impulse results in step phase change (i.e., an integration).
current-to-phase transfer function is proportional to 1/s
ip(t)
ip(t)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 42
Impulse Sensitivity Function (1)The phase response for a particular noise source can be determined at each point over the oscillation waveform.
Impulse sensitivity function (ISF):
(normalized to signal amplitude)
change in phasecharge in impulse
t
Example 1: sine wave
t
Example 2: square wave
Note has same period as Vosc.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 43
Impulse Sensitivity Function (2)
Recall from network theory:
LaPlace transform:
Impulse response:
time-variant impulse response
Recall:
ISF convolution integral:
from q
can be expressed in terms of Fourier coefficients:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 44
Case 1: Disturbance is sinusoidal:
, m = 0, 1, 2, …
negligible significant only form = k
(Any frequency can be expressed in terms of m and .)
Impulse Sensitivity Function (3)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 45
I
2osc
Impulse Sensitivity Function (4)
Current-to-phase frequency response:
oscosc
osc 2osc 2osc
For
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 46
osc 2osc
Case 2: Disturbance is stochastic:
Impulse Sensitivity Function (5)
MOSFET current noise:
thermalnoise
1/fnoise
A2/Hz
osc 2osc
thermal noise
1/f noise
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 47
Impulse Sensitivity Function (6)
osc 2osc
due to 1/f noise
due to thermal noise
Total phase noise:
n
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 48
Impulse Sensitivity Function (7)
noise corner frequency n
(log scale)
(dBc/Hz)
1/(3 region: −30 dBc/Hz/decade
1/(2 region: −20 dBc/Hz/decade
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 49
t
t
Example 1: sine wave Example 2: square wave
Impulse Sensitivity Function (8)
Example 3: asymmetric square wave
t
will generate more 1/(3 phase noise
is higher will generate more 1/(2 phase noise
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 50
Impulse Sensitivity Function (9)
Effect of current source in LC VCO:
Vosc+ _
Due to symmetry, ISF of this noise source contains only even-order coefficients − c0 and c2 are dominant.
Noise from current source will contribute to phase noise of differential waveform.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 51
Impulse Sensitivity Function (10)
ID varies over oscillation waveform Same period as
oscillation
Let
Then where
We can use
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 52
ISF Example: 3-Stage Ring Oscillator
M1A M1B M2A M2B M3A M3B
MS1 MS2 MS3
R1A R1B R2A R2B R3A R3B+
Vout
−
fosc = 1.08 GHzPD = 11 mW
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 53
ISF of Diff. Pairs
for each diff. pair transistor
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 54
ISF of Resistors
for each resistor
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 55
ISF of Current Sources
ISF shows double frequency due to source-coupled node connection.
for each current source transistor
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 56
Phase Noise Calculation
Using: Cout = 1.13 pF
Vout = 601 mV p-p
qmax = 679 fC
= −112 dBc/Hz @ f = 10 MHz
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 57
Phase Noise vs. Amplitude Noise (1)
osct
v
v Spectrum of Vosc would include effects of both amplitude noise v(t) and phase noise (t).
How are the single-sideband noise spectrum Ltotal() and phase spectral density S() related?
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 58
Phase Noise vs. Amplitude Noise (2)
t t
i(t) i(t)
Vc(t) Vc(t)
Recall that an input current impulse causes an enduring phase perturbation and a momentary change in amplitude:
Amplitude impulse response exhibits an exponential decay due to the natural amplitude limiting of an oscillator ...
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 59
+
Phase noise dominates at low offset frequencies.
Phase Noise vs. Amplitude Noise (3)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 60
osc
Phase & amplitude noise can’t be distinguished in a signal.
Sv()
Amplitude limiting will decrease amplitude noisebut will not affect phase noise.
Phase Noise vs. Amplitude Noise (4)
noiseless oscillation waveform
phase noise
component
amplitude noise
component
phase noise
amplitude noise
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 61
Sideband Noise/Phase Spectral Density
noiseless oscillation waveform
phase noise
component
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 62
Jitter/Phase Noise Relationship (1)
autocorrelation functions
Recall R and S() are a Fourier transform pair:
NT
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 63
Jitter/Phase Noise Relationship (2)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 64
Let Let
Consistent with jitter accumulation measurements!
Jitter/Phase Noise Relationship (3)
Jitter from 1/( noise:2
Jitter from 1/( noise:3
^
^
^
^^
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 65
Jitter/Phase Noise Relationship (4)
f
(dBc/Hz)
-100
-20dBc/Hzper decade
• Let fosc = 10 GHz• Assume phase noise dominated by 1/()2
Setting f = 2 X 106 and S =10-10:
Let = 100 ps (cycle-to-cycle jitter):
= 0.02ps rms (0.2 mUI rms)
Accumulated jitter:
2 MHz
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 66
More generally:
f
(dBc/Hz)
fm
Nm
-20 dBc/Hzper decade
rms jitter increases by a factor of 3.2
Jitter/Phase Noise Relationship (5)
Let phase noise increase by 10 dBc/Hz:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 67
Jitter Accumulation (1)
Kpd
phasedetector
loopfilter
Kvco
VCOin out
vco
fb
Open-loop characteristic:
Closed-loop characteristic:
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 68
Jitter Accumulation (2)
Recall from Type-2 PLL:
|G|
z p
|1 + G|
-40 dB/decade
(dBc/Hz)
1/(3 region: −30 dBc/Hz/decade
1/(2 region: −20 dBc/Hz/decade
1
80 dB/decade
As a result, the phase noise at low offset frequencies is determined by input noise...
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 69
• fosc = 10 GHz• Assume 1-pole closed-loop PLL characteristic
Jitter Accumulation (3)
f
(dBc/Hz)
f0 = 2 MHz
-100-20dBc/Hzper decade
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 70
For large :
= 0.02 ps rms cycle-to-cycle jitter
Jitter Accumulation (4)
f0 = 2 MHz
fosc = 10 GHz
For small :
(log scale)
= 1.4 ps rms Total accumulated jitter
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 71
The primary function of a PLL is to place a bound on cumulative jitter:
(log scale)
(log scale)
proportional to (due to thermal noise)
proportional to
(due to 1/f noise)
Jitter Accumulation (5)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 72
L() for OC-192 SONET transmitter
Closed-Loop PLL Phase Noise Measurement
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 73
Other Sources of Jitter in PLL
• Clock divider
• Phase detectorRipple on phase detector output can cause high-frequency jitter. This affects primarily the jitter tolerance of CDR.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 74
Jitter/Bit Error Rate (1)
Histogram showing Gaussian distribution
near sampling point
1UI
Bit error rate (BER) determined by and UI …
L R
Eye diagram fromsampling oscilloscope
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 75
R
0 T
Probability of sample at t > t0 from left-hand transition:
Probability of sample at t < t0 from right-hand transition:
Jitter/Bit Error Rate (2)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 76
Total Bit Error Rate (BER) given by:
Jitter/Bit Error Rate (3)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 77
t0 (ps)
log BER
Example: T = 100ps
(64 ps eye opening)
(38 ps eye opening)
log(0.5)
Jitter/Bit Error Rate (4)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 78
Bathtub Curves (1)
The bit error-rate vs. sampling time can be measured directly using a bit error-rate tester (BERT) at various sampling points.
Note: The inherent jitter of the analyzer trigger should be considered.
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 79
Bathtub Curves (2)
Bathtub curve can easily be numerically extrapolated to very low BERs (corresponding to random jitter), allowing much lower measurement times.
Example: 10-12 BER with T = 100ps is equivalent to an average of 1 error per 100s. To verify this over a sample of 100 errors would require almost 3 hours!
t0 (ps)
EECS 270C / Spring 2014 Prof. M. Green / U.C. Irvine 80
Equivalent Peak-to-Peak Total Jitter
BER
10-10
10-11
10-12
10-13
10-14
, T determine BERBER determines effectiveTotal jitter given by:
Areas sumto BER