Automated Oscillator Macromodelling Techniques ... - Semantic Scholar
Automated Oscillator Macromodelling Techniques for Capturing Amplitude Variations and Injection Locking Xiaolue Lai, Jaijeet Roychowdhury ECE Dept., University of Minnesota, Minneapolis
December 10, 2004
Slide 1
Oscillators and Perturbation Oscillators are very important in RF and digital circuits Information carrier, clock generator, ...
Phase response to perturbation is the major concern Phase is important Phase is sensitive to perturbation
Two major phase responses Injection locking Timing jitter/phase noise December 10, 2004
Slide 2
Periodic Input: Injection Locking If the oscillator is under periodic perturbation
i=f(v)
(eg, substrate/supply coupling from other ckts) Periodic perturbation injected
The oscillator “forgets” its natural frequency Its frequency “locks” to external frequency Exploited in modern designs to improve phase/frequency stability and pulling performance December 10, 2004
Slide 3
Transient simulation of locking process 1.5
Oscillator waveform
Injection signal
0
-1.5
0
500T
1000T
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0
0.5 Tim e (s)
1
-8
x 10
Not locked in the beginning (note phase shifts) December 10, 2004
4.9
4.92
1500T
4.94
4.96
Tim e (s)
4.98
-6
x 10
Locked after 1000 cycles (with phase shift) Slide 4
Conditions for Injection Locking 0.1 0.09
Frequency difference Max locking range
0.08 0.07
0 0
0.06 0.05 0.04
Locking area
0.03 0.02
Injection amplitude
0.01 0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
V inj V0
If NOT locked Large amplitude variations (periodic beat notes) December 10, 2004
Slide 5
Amplitude Variations (unlocked driven oscillator) 1 i
inj
= 0.1A sin(2 π 1.06f t ) (Full sim ulat ion) 0 0
Volt age (v)
0.5
Periodic beat notes
0
-0.5
-1
0
20
40
60
80
100
t /T December 10, 2004
Slide 6
SPICElevel simulation: not ideal for oscillators Transient Simulation is Inefficient Many timesteps for each cycle (accuracy) Many (thousands/millions) cycles needed in simulation Transient Simulation is Inaccurate difficult to extract phase information Numerical integration errors
December 10, 2004
Slide 7
Previous Work on Injection Locking Adler's equation (1946) Analytical equation relates maximum locking range and injection amplitude applicable only to simple LC oscillator (with explicit Q factor)
Linear oscillator phase macromodels LTI models LPTV models Linear phase models cannot capture injection locking December 10, 2004
Slide 8
Contributions of this work Fast, accurate prediction of injection locking AND unlocked amplitude variations Via nonlinear oscillator macromodel Demir/Mehrotra/Roychowdhury: Phase Noise in Oscillators: ..., IEEE Trans CAS I 2000 automatically extracted from SPICElevel circuit)
Applicable to any kind of oscillator Our method applies to ANY oscillator! LC, ring, lasers, ... Bonus: semianalytical equation for maximum locking range of oscillators
Proof: linear models (LTI/LTV) cannot capture injection locking December 10, 2004
Slide 9
Nonlinear phase macromodel (PPV)
Nonlinear scalar differential equation
Phase error
perturbation projection vector (PPV)
Perturbation
Details/derivation: Demir/Mehrotra/Roychowdhury: Phase Noise in Oscillators: ..., IEEE Trans CAS I 2000 December 10, 2004
Slide 10
Phase slippage between oscillator and injection signal Phase of the injected signal
80
phase (radian)
70 60
Phase slippage
50 40 30 20 10 00
Ph ase of th e oscillator 0.5
time (s)
December 10, 2004
1
Phase of the oscillator
-8
x 10
Slide 11
Predicting Injection Locking If locked: phase error should make up the phase slippage
Use nonlinear phase equation to predict Locking test: does phase error grow linearly with slope ? December 10, 2004
Slide 12
Macromodelling Amplitude Variations Simulate the oscillator to steady state Calculate phase error
Linearize the oscillator over steady state
Calculate the PPV
Linearize the oscillator over
December 10, 2004
Phase error / nonlinear time shift
Slide 13
Capture the amplitude variation Phase error / nonlinear time shift
Floquet decompose the new LPTV system
Reduce the system by dropping fast fading Floquet exponents
Rebuild the system equations for this smaller system December 10, 2004
Slide 14
Macromodelling Amplitude Variations
Steady state of the oscillator
December 10, 2004
Phase Error
Amplitude variations
Slide 15
i=f(v)
Negative resistance LC oscillator
0.01
Current >
b(t)
0.005
0
-0.005
-0.01 -1
-0.5
0
0.5
1
Voltage > December 10, 2004
Slide 16
LC osc: Max locking range vs injection strength Nonlinear macromodel
0.15
Reference (full simulation) 0.1
Adler eqn 0.05
0
December 10, 2004
0
5
10
15
20
25
Slide 17
LC osc: Amplitude variations -1 0
Phase error
8 6 4 2 0 -2
0
1
20
40
t/T
60
80
100
0
-0.1
0
1
Macromodel
0.5
0
-0.5
-1
Amplitude variations
December 10, 2004
20
40
t/T
40
60
80
100
60
80
100
Full simulation
0.5
0
-0.5
-1 0
20
t/T Oscillation voltage (v)
Phase deviation (s)
10
Oscillation voltage (v)
0.1
Amplitude variation (v)
x 10
12
0
20
40
t/T
60
80
100
Slide 18
LC Osc: Amplitude variations (detail) 1
Macromodel Full simulation
0.8
0.6
Oscillation voltage (v)
0.4
0.2
0
-0.2
-0.4
-0.6
29 times speedup
-0.8
-1
25
30
35
40
t/T December 10, 2004
Slide 19
LC osc: alpha equation range of validity Full simulation Macromodel
1
0.5
0
0.5
Good match
Macromodel is not suitable
Good match
0.5
0
-0.5
-1 0
20
40
60
t /T
December 10, 2004
80
0
20
40
60
t /T
80
0
20
40
60
80
t /T
Slide 20
3stage ring oscillator: locking range vs injection strength 0.25
0.2
Reference (full simulation)
0.15
Nonlinear macromodel
0.1
0.05
0
December 10, 2004
Adler equation does not apply to nonLC oscillators 0
0.05
0.1
0.15
0.2
0.25
Slide 21
3stage ring: range of validity 0.5
35 times speedup
0
0.5
Full simulation
Macromodel
1
Good match
Good match
Macromodel is Not suitable
0.5
0
-0.5
0
20
December 10, 2004
40
t /T
60
80
0
20
40
t /T
60
80
Slide 22
Colpitts oscillator (LC) Courtesy: Madhavan Swaminathan, Georgia Institute of Technology Rp=50 1
Cp=1p 0.4p
L1=2.1n
Rb=22k
Cm=0.6p 3
5
2
Rl=200
Cb=1.5p
December 10, 2004
4
C1=1p
Re=100
C2=2.3p
Slide 23
Colpitts: max locking range vs injection strength
Nonlinear macromodel
0.12
0.1
Reference (full simulation)
0.08
0.06
0.04
0.02
Adler eqn 0 0
10
20
30
40
50
Injection amplitude (mV) December 10, 2004
Slide 24
Colpitts: Amplitude variations -11
5
Phase error
Phase shift (s)
4 3
2 1 0 -1 0
50
150
Macromodel
20 15 10 5 0 -5
-10 0
December 10, 2004
50
100 time (t/T)
10 8 6
Amplitude variations
4 2 0 -2 -4
100 times speedup
-6 -8
0
50
25 Oscillation current (mA)
Oscillation current (mA)
25
100
Amplitude variation (mA)
x 10
150
100
150
200
150
200
Full simulation
20 15 10 5 0
-5 -10 200 0
50
100 time (t/T)
Slide 25
Conclusions Our oscillator macromodelling technique is ideal for capturing injection locking and amplitude variation in oscillators Injection locking prediction Efficient, semianalytical equation Applicable to any oscillator Amplitude variation Efficient, more than 100 times speedup for a small oscillator circuit Accurate in its validity range Current work: Using Krylovsubspacebased method to reduce the LPTV system. December 10, 2004
Slide 26
December 10, 2004
Slide 1
Oscillators and Perturbation Oscillators are very important in RF and digital circuits Information carrier, clock generator, ...
Phase response to perturbation is the major concern Phase is important Phase is sensitive to perturbation
Two major phase responses Injection locking Timing jitter/phase noise December 10, 2004
Slide 2
Periodic Input: Injection Locking If the oscillator is under periodic perturbation
i=f(v)
(eg, substrate/supply coupling from other ckts) Periodic perturbation injected
The oscillator “forgets” its natural frequency Its frequency “locks” to external frequency Exploited in modern designs to improve phase/frequency stability and pulling performance December 10, 2004
Slide 3
Transient simulation of locking process 1.5
Oscillator waveform
Injection signal
0
-1.5
0
500T
1000T
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0
0.5 Tim e (s)
1
-8
x 10
Not locked in the beginning (note phase shifts) December 10, 2004
4.9
4.92
1500T
4.94
4.96
Tim e (s)
4.98
-6
x 10
Locked after 1000 cycles (with phase shift) Slide 4
Conditions for Injection Locking 0.1 0.09
Frequency difference Max locking range
0.08 0.07
0 0
0.06 0.05 0.04
Locking area
0.03 0.02
Injection amplitude
0.01 0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
V inj V0
If NOT locked Large amplitude variations (periodic beat notes) December 10, 2004
Slide 5
Amplitude Variations (unlocked driven oscillator) 1 i
inj
= 0.1A sin(2 π 1.06f t ) (Full sim ulat ion) 0 0
Volt age (v)
0.5
Periodic beat notes
0
-0.5
-1
0
20
40
60
80
100
t /T December 10, 2004
Slide 6
SPICElevel simulation: not ideal for oscillators Transient Simulation is Inefficient Many timesteps for each cycle (accuracy) Many (thousands/millions) cycles needed in simulation Transient Simulation is Inaccurate difficult to extract phase information Numerical integration errors
December 10, 2004
Slide 7
Previous Work on Injection Locking Adler's equation (1946) Analytical equation relates maximum locking range and injection amplitude applicable only to simple LC oscillator (with explicit Q factor)
Linear oscillator phase macromodels LTI models LPTV models Linear phase models cannot capture injection locking December 10, 2004
Slide 8
Contributions of this work Fast, accurate prediction of injection locking AND unlocked amplitude variations Via nonlinear oscillator macromodel Demir/Mehrotra/Roychowdhury: Phase Noise in Oscillators: ..., IEEE Trans CAS I 2000 automatically extracted from SPICElevel circuit)
Applicable to any kind of oscillator Our method applies to ANY oscillator! LC, ring, lasers, ... Bonus: semianalytical equation for maximum locking range of oscillators
Proof: linear models (LTI/LTV) cannot capture injection locking December 10, 2004
Slide 9
Nonlinear phase macromodel (PPV)
Nonlinear scalar differential equation
Phase error
perturbation projection vector (PPV)
Perturbation
Details/derivation: Demir/Mehrotra/Roychowdhury: Phase Noise in Oscillators: ..., IEEE Trans CAS I 2000 December 10, 2004
Slide 10
Phase slippage between oscillator and injection signal Phase of the injected signal
80
phase (radian)
70 60
Phase slippage
50 40 30 20 10 00
Ph ase of th e oscillator 0.5
time (s)
December 10, 2004
1
Phase of the oscillator
-8
x 10
Slide 11
Predicting Injection Locking If locked: phase error should make up the phase slippage
Use nonlinear phase equation to predict Locking test: does phase error grow linearly with slope ? December 10, 2004
Slide 12
Macromodelling Amplitude Variations Simulate the oscillator to steady state Calculate phase error
Linearize the oscillator over steady state
Calculate the PPV
Linearize the oscillator over
December 10, 2004
Phase error / nonlinear time shift
Slide 13
Capture the amplitude variation Phase error / nonlinear time shift
Floquet decompose the new LPTV system
Reduce the system by dropping fast fading Floquet exponents
Rebuild the system equations for this smaller system December 10, 2004
Slide 14
Macromodelling Amplitude Variations
Steady state of the oscillator
December 10, 2004
Phase Error
Amplitude variations
Slide 15
i=f(v)
Negative resistance LC oscillator
0.01
Current >
b(t)
0.005
0
-0.005
-0.01 -1
-0.5
0
0.5
1
Voltage > December 10, 2004
Slide 16
LC osc: Max locking range vs injection strength Nonlinear macromodel
0.15
Reference (full simulation) 0.1
Adler eqn 0.05
0
December 10, 2004
0
5
10
15
20
25
Slide 17
LC osc: Amplitude variations -1 0
Phase error
8 6 4 2 0 -2
0
1
20
40
t/T
60
80
100
0
-0.1
0
1
Macromodel
0.5
0
-0.5
-1
Amplitude variations
December 10, 2004
20
40
t/T
40
60
80
100
60
80
100
Full simulation
0.5
0
-0.5
-1 0
20
t/T Oscillation voltage (v)
Phase deviation (s)
10
Oscillation voltage (v)
0.1
Amplitude variation (v)
x 10
12
0
20
40
t/T
60
80
100
Slide 18
LC Osc: Amplitude variations (detail) 1
Macromodel Full simulation
0.8
0.6
Oscillation voltage (v)
0.4
0.2
0
-0.2
-0.4
-0.6
29 times speedup
-0.8
-1
25
30
35
40
t/T December 10, 2004
Slide 19
LC osc: alpha equation range of validity Full simulation Macromodel
1
0.5
0
0.5
Good match
Macromodel is not suitable
Good match
0.5
0
-0.5
-1 0
20
40
60
t /T
December 10, 2004
80
0
20
40
60
t /T
80
0
20
40
60
80
t /T
Slide 20
3stage ring oscillator: locking range vs injection strength 0.25
0.2
Reference (full simulation)
0.15
Nonlinear macromodel
0.1
0.05
0
December 10, 2004
Adler equation does not apply to nonLC oscillators 0
0.05
0.1
0.15
0.2
0.25
Slide 21
3stage ring: range of validity 0.5
35 times speedup
0
0.5
Full simulation
Macromodel
1
Good match
Good match
Macromodel is Not suitable
0.5
0
-0.5
0
20
December 10, 2004
40
t /T
60
80
0
20
40
t /T
60
80
Slide 22
Colpitts oscillator (LC) Courtesy: Madhavan Swaminathan, Georgia Institute of Technology Rp=50 1
Cp=1p 0.4p
L1=2.1n
Rb=22k
Cm=0.6p 3
5
2
Rl=200
Cb=1.5p
December 10, 2004
4
C1=1p
Re=100
C2=2.3p
Slide 23
Colpitts: max locking range vs injection strength
Nonlinear macromodel
0.12
0.1
Reference (full simulation)
0.08
0.06
0.04
0.02
Adler eqn 0 0
10
20
30
40
50
Injection amplitude (mV) December 10, 2004
Slide 24
Colpitts: Amplitude variations -11
5
Phase error
Phase shift (s)
4 3
2 1 0 -1 0
50
150
Macromodel
20 15 10 5 0 -5
-10 0
December 10, 2004
50
100 time (t/T)
10 8 6
Amplitude variations
4 2 0 -2 -4
100 times speedup
-6 -8
0
50
25 Oscillation current (mA)
Oscillation current (mA)
25
100
Amplitude variation (mA)
x 10
150
100
150
200
150
200
Full simulation
20 15 10 5 0
-5 -10 200 0
50
100 time (t/T)
Slide 25
Conclusions Our oscillator macromodelling technique is ideal for capturing injection locking and amplitude variation in oscillators Injection locking prediction Efficient, semianalytical equation Applicable to any oscillator Amplitude variation Efficient, more than 100 times speedup for a small oscillator circuit Accurate in its validity range Current work: Using Krylovsubspacebased method to reduce the LPTV system. December 10, 2004
Slide 26
