Automated Oscillator Macromodelling Techniques for Capturing - - PowerPoint PPT Presentation

December 10, 2004 Slide 1

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 2

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 3

Periodic Input: Injection Locking

The oscillator “forgets” its natural frequency Its frequency “locks” to external frequency Exploited in modern designs to improve phase/frequency stability and pulling performance

i=f(v)

• Periodic

perturbation injected

If the oscillator is under periodic perturbation

(eg, substrate/supply coupling from other ckts)

December 10, 2004 Slide 4

Transient simulation of locking process

Locked after 1000 cycles (with phase shift)

0.5 1

x 10

• 8
• 1.5
• 1
• 0.5

0.5 1 1.5 2 Time (s)

4.9 4.92 4.94 4.96 4.98 x 10

• 6
• 1.5
• 1
• 0.5

0.5 1 1.5 2

Time (s) 500T 1000T 1500T

1.5

• 1.5

Not locked in the beginning (note phase shifts) Oscillator waveform Injection signal

December 10, 2004 Slide 5

Conditions for Injection Locking

If NOT locked Large amplitude variations (periodic beat notes)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 0 V inj V 0

Locking area Frequency difference Injection amplitude Max locking range

December 10, 2004 Slide 6

Amplitude Variations (unlocked driven oscillator)

20 40 60 80 100

• 1
• 0.5

0.5 1 t/T Voltage (v) iinj= 0.1A 0sin(2 π1.06f 0t) (Full simulation)

Periodic beat notes

December 10, 2004 Slide 7

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

SPICE-level simulation: not ideal for

• scillators

December 10, 2004 Slide 8

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 9

Our method applies to ANY oscillator!

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 SPICE-level circuit)

Applicable to any kind of oscillator

LC, ring, lasers, ...

Bonus: semi-analytical equation for maximum locking range of oscillators

Proof: linear models (LTI/LTV) cannot capture injection locking

December 10, 2004 Slide 10

Nonlinear phase macromodel (PPV)

perturbation projection vector (PPV)

Nonlinear scalar differential equation

Details/derivation: Demir/Mehrotra/Roychowdhury: Phase Noise in Oscillators: ..., IEEE Trans CAS I 2000

Phase error

Perturbation

December 10, 2004 Slide 11

Phase slippage between oscillator and injection signal

0.5 1 x 10-8 10 20 30 40 50 60 70 80

time (s)

phase (radian)

Phase of the oscillator

Phase slippage

Phase of the injected signal Phase of the

• scillator

December 10, 2004 Slide 12

If locked: phase error should make up the phase slippage

Predicting Injection Locking

Use nonlinear phase equation to predict Locking test: does phase error grow linearly with slope ?

December 10, 2004 Slide 13

Calculate the PPV Calculate phase error Linearize the oscillator

• ver steady state

Simulate the oscillator to steady state Linearize the oscillator

• ver

Macromodelling Amplitude Variations

Phase error / nonlinear time shift

December 10, 2004 Slide 14

Capture the amplitude variation

Floquet decompose the new LPTV system Reduce the system by dropping fast fading Floquet exponents Rebuild the system equations for this smaller system Phase error / nonlinear time shift

December 10, 2004 Slide 15

Macromodelling Amplitude Variations

Phase Error Amplitude variations Steady state of the oscillator

December 10, 2004 Slide 16

Negative resistance LC oscillator

b(t)

i=f(v)

• 1
• 0.5

0.5 1

• 0.01
• 0.005

0.005 0.01

Voltage --> Current -->

December 10, 2004 Slide 17

LC osc: Max locking range vs injection strength

5 10 15 20 25 0.05 0.1 0.15

Adler eqn

Nonlinear macromodel Reference (full simulation)

December 10, 2004 Slide 18

LC osc: Amplitude variations

20 40 60 80 100

• 2

2 4 6 8 10 12 x 10

• 10

t/T Phase deviation (s)

20 40 60 80 100

• 0.1

0.1

t/T Amplitude variation (v)

20 40 60 80 100

• 1
• 0.5

0.5 1

t/T Oscillation voltage (v)

20 40 60 80 100

• 1
• 0.5

0.5 1

t/T Oscillation voltage (v)

Phase error Amplitude variations Macromodel Full simulation

December 10, 2004 Slide 19

LC Osc: Amplitude variations (detail)

25 30 35 40

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

t/T Oscillation voltage (v)

Macromodel Full simulation

29 times speedup

December 10, 2004 Slide 20

LC osc: alpha equation range of validity

20 40 60 80

• 1
• 0.5

0.5

t/T

• 0.5

0.5 1 20 40 60 80

t/T

20 40 60 80

t/T

F u l l s i m u l a t i

• n

M a c r

• m
• d

e l

Good match Macromodel is not suitable Good match

December 10, 2004 Slide 21

3-stage ring oscillator: locking range vs injection strength

0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

Adler equation does not apply to non-LC oscillators

Reference (full simulation) Nonlinear macromodel

December 10, 2004 Slide 22

3-stage ring: range of validity

20 40 60 80

• 0.5

0.5

t/T

F u l l s i m u l a t i

• n

M a c r

• m
• d

e l

• 0.5

0.5 1

20 40 60 80 t/T

Good match Macromodel is Not suitable

35 times speedup

Good match

December 10, 2004 Slide 23

Colpitts oscillator (LC)

Rp=50 Cp=1p Rb=22k 0.4p L1=2.1n Cb=1.5p Re=100 C2=2.3p C1=1p Cm=0.6p

1 2 3 4 5

Rl=200

Courtesy: Madhavan Swaminathan, Georgia Institute of Technology

December 10, 2004 Slide 24

Colpitts: max locking range vs injection strength

10 20 30 40 50 0.02 0.04 0.06 0.08 0.1 0.12

Injection amplitude (mV)

Nonlinear macromodel Reference (full simulation)

Adler eqn

December 10, 2004 Slide 25

Colpitts: Amplitude variations

50 100 150 200

• 10
• 5

5 10 15 20 25 time (t/T) Oscillation current (mA) 50 100 150

• 1

1 2

3

4 5

x 10

• 11

time (t/T) Phase shift (s) 50

100

150 200

• 8
• 6
• 4
• 2

2 4 6 8 10

Amplitude variation (mA)

Full simulation

50 100 150 200

• 5

5 10 15 20 25

Oscillation current (mA)

• 10

Macromodel

100 times speedup

Phase error Amplitude variations

December 10, 2004 Slide 26

Conclusions

Our oscillator macromodelling technique is ideal for capturing injection locking and amplitude variation in

• scillators

Injection locking prediction Efficient, semi-analytical equation Applicable to any oscillator Amplitude variation Efficient, more than 100 times speedup for a small

• scillator circuit

Accurate in its validity range Current work: Using Krylov-subspace-based method to reduce the LPTV system.