Automated Oscillator Macromodelling Techniques for Capturing - PowerPoint PPT Presentation

Automated Oscillator Macromodelling Techniques for Capturing Amplitude Variations and Injection Locking Xiaolue Lai, Jaijeet Roychowdhury ECE Dept., University of Minnesota, Minneapolis Slide 1 December 10, 2004

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 Slide 2 December 10, 2004

Periodic Input: Injection Locking If the oscillator is under periodic perturbation (eg, substrate/supply coupling from other ckts) Periodic i=f(v) 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 Slide 3 December 10, 2004

Transient simulation of locking process 1.5 Injection Oscillator 0 signal waveform -1.5 0 500T 1000T 1500T 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 1 4.9 4.92 4.94 4.96 4.98 Time (s) -8 Time (s) -6 x 10 x 10 Locked after 1000 cycles Not locked in the beginning (with phase shift) (note phase shifts) Slide 4 December 10, 2004

Conditions for Injection Locking 0.1 0.09 Frequency 0.08 difference 0.07  0 0.06 0.05  0 Max locking 0.04 range 0.03 Locking area 0.02 0.01 Injection 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 V inj amplitude V 0 If NOT locked Large amplitude variations (periodic beat notes) Slide 5 December 10, 2004

Amplitude Variations (unlocked driven oscillator) 1 i inj = 0.1A 0 sin(2 π 1.06f 0 t) (Full simulation) 0.5 Voltage (v) Periodic beat 0 notes -0.5 -1 0 20 40 60 80 100 t/T Slide 6 December 10, 2004

SPICE-level 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 Slide 7 December 10, 2004

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 Slide 8 December 10, 2004

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 Our method applies to ANY oscillator! LC, ring, lasers, ... Bonus: semi-analytical equation for maximum locking range of oscillators Proof: linear models (LTI/LTV) cannot capture injection locking Slide 9 December 10, 2004

Nonlinear phase macromodel (PPV) Nonlinear scalar differential equation perturbation projection Phase Perturbation vector (PPV) error Details/derivation: Demir/Mehrotra/Roychowdhury: Phase Noise in Oscillators: ..., IEEE Trans CAS I 2000 Slide 10 December 10, 2004

Phase slippage between oscillator and injection signal 80 Phase of the 70 injected signal Phase slippage phase (radian) 60 50 40 30 20 Phase of the 10 Phase of the oscillator oscillator 0 0 0.5 1 x 10 -8 time (s) Slide 11 December 10, 2004

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 ? Slide 12 December 10, 2004

Macromodelling Amplitude Variations Linearize the oscillator Simulate the oscillator to over steady state steady state Calculate phase Calculate the PPV error Phase error / Linearize the oscillator nonlinear time over shift Slide 13 December 10, 2004

Capture the amplitude variation Phase error / nonlinear time shift Reduce the system by Floquet decompose the dropping fast fading new LPTV system Floquet exponents Rebuild the system equations for this smaller system Slide 14 December 10, 2004

Macromodelling Amplitude Variations Steady state of Amplitude Phase Error the oscillator variations Slide 15 December 10, 2004

Negative resistance LC oscillator i=f(v) - b(t) 0.01 Current --> 0.005 0 -0.005 -0.01 -1 -0.5 0 0.5 1 Voltage --> Slide 16 December 10, 2004

LC osc: Max locking range vs injection strength Nonlinear macromodel 0.15 Reference (full simulation) 0.1 Adler eqn 0.05 0 0 5 10 15 20 25 Slide 17 December 10, 2004

LC osc: Amplitude variations -10 12 x 10 0.1 Amplitude variations Phase error Amplitude variation (v) 10 Phase deviation (s) 8 6 0 4 2 0 -2 -0.1 0 20 40 60 80 100 0 20 40 60 80 100 t/T t/T 1 1 Full simulation Macromodel Oscillation voltage (v) Oscillation voltage (v) 0.5 0.5 0 0 -0.5 -0.5 -1 -1 0 20 40 60 80 100 0 20 40 60 80 100 t/T t/T Slide 18 December 10, 2004

LC Osc: Amplitude variations (detail) 1 Macromodel 0.8 Full simulation 0.6 0.4 Oscillation voltage (v) 0.2 0 -0.2 -0.4 -0.6 -0.8 29 times speedup -1 25 30 35 40 t/T Slide 19 December 10, 2004

LC osc: alpha equation range of validity 1 l e d 0.5 o m 0 o r c -0.5 a M Macromodel is Good match Good match not suitable n o i 0.5 t a l u 0 m i s -0.5 l l u F -1 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 t/T t/T t/T Slide 20 December 10, 2004

3-stage ring oscillator: locking range vs injection strength 0.25 0.2 Reference (full simulation) 0.15 Nonlinear macromodel 0.1 0.05 Adler equation does not apply to non-LC oscillators 0 0 0.05 0.1 0.15 0.2 0.25 Slide 21 December 10, 2004

3-stage ring: range of validity 1 l e 0.5 d o 35 times speedup m 0 o r c a -0.5 M Macromodel is Good match Good match Not suitable n o i 0.5 t a l u 0 m i s -0.5 l l u F 0 20 40 60 80 0 20 40 60 80 t/T t/T Slide 22 December 10, 2004

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 C1=1p 4 Cb=1.5p Re=100 C2=2.3p Slide 23 December 10, 2004

Colpitts: max locking range vs injection strength Nonlinear macromodel 0.12 Reference 0.1 (full simulation) 0.08 0.06 0.04 0.02 Adler eqn 0 0 10 20 30 40 50 Injection amplitude (mV) Slide 24 December 10, 2004

Colpitts: Amplitude variations -11 x 10 10 5 Amplitude variation (mA) Phase error Amplitude variations 8 4 6 Phase shift (s) 3 4 2 2 0 1 -2 -4 0 100 times speedup -6 -1 -8 200 0 50 100 150 0 50 100 150 25 25 Macromodel Full simulation Oscillation current (mA) Oscillation current (mA) 20 20 15 15 10 10 5 5 0 0 -5 -5 -10 -10 0 50 100 150 200 0 50 100 150 200 time (t/T) time (t/T) Slide 25 December 10, 2004

Conclusions Our oscillator macromodelling technique is ideal for capturing injection locking and amplitude variation in oscillators Injection locking prediction Efficient, semi-analytical 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 Krylov-subspace-based method to reduce the LPTV system. Slide 26 December 10, 2004