Automated Oscillator Macromodelling Techniques for Capturing Amplitude Variations and Injection Locking
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1 Automated Oscillator Macromodelling Techniques for Capturing Amplitude Variations and Injection Locking Xiaolue Lai, Jaijeet Roychowdhury ECE Dept., University of Minnesota, Minneapolis December 1, 24 Slide 1
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 1, 24 Slide 2
3 Periodic Input: Injection Locking If the oscillator is under periodic perturbation (eg, substrate/supply coupling from other ckts) i=f(v) 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 1, 24 Slide 3
4 Transient simulation of locking process 1.5 Oscillator waveform Injection signal T 1T 15T Tim e (s) x 1-8 Not locked in the beginning (note phase shifts) Tim e (s) Locked after 1 cycles (with phase shift) December 1, 24 Slide x 1-6
5 Conditions for Injection Locking.1 Frequency difference Max locking range Locking area Injection amplitude V inj V If NOT locked Large amplitude variations (periodic beat notes) December 1, 24 Slide 5
6 Amplitude Variations (unlocked driven oscillator) 1 i inj =.1A sin(2 π 1.6f t) (Full sim ulation).5 Voltage (v) Periodic beat notes t/t December 1, 24 Slide 6
7 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 December 1, 24 Slide 7
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 1, 24 Slide 8
9 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 2 automatically extracted from SPICE level circuit) Applicable to any kind of oscillator LC, ring, lasers,... Our method applies to ANY oscillator! Bonus: semi analytical equation for maximum locking range of oscillators Proof: linear models (LTI/LTV) cannot capture injection locking December 1, 24 Slide 9
10 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 2 December 1, 24 Slide 1
11 phase (radian) Phase slippage between oscillator and injection signal Phase slippage Phase of the oscillator.5 1 x 1-8 time (s) Phase of the injected signal Phase of the oscillator December 1, 24 Slide 11
12 Predicting Injection Locking If locked: phase error phase slippage should make up the Use nonlinear phase equation to predict Locking test: does phase error grow linearly with slope? December 1, 24 Slide 12
13 Macromodelling Amplitude Variations Simulate the oscillator to steady state Linearize the oscillator over steady state Calculate phase error Calculate the PPV Linearize the oscillator over Phase error / nonlinear time shift December 1, 24 Slide 13
14 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 1, 24 Slide 14
15 Macromodelling Amplitude Variations Steady state of the oscillator Phase Error Amplitude variations December 1, 24 Slide 15
16 Negative resistance LC oscillator i=f(v) b(t).1 Current > December 1, 24 Slide Voltage >
17 LC osc: Max locking range vs.15.1 injection strength Reference (full simulation) Nonlinear macromodel Adler eqn December 1, 24 Slide 17
18 LC osc: Amplitude variations Oscillation voltage (v) Phase deviation (s) -1 x Phase error t/t Macromodel t/t Oscillation voltage (v) Amplitude variation (v) December 1, 24 Slide Amplitude variations t/t Full simulation t/t
19 LC Osc: Amplitude variations (detail) 1.8 Macromodel Full simulation.6.4 Oscillation voltage (v) times speedup t/t December 1, 24 Slide 19
20 LC osc: alpha equation range of validity Full simulation Macromodel Good match Macromodel is not suitable Good match t /T December 1, t /T t /T Slide 2
21 3 stage ring oscillator: locking range.25 vs injection strength Reference (full simulation) Nonlinear macromodel.5 Adler equation does not apply to non LC oscillators December 1, 24 Slide 21
22 3 stage ring: range of validity.5 35 times speedup.5 Full simulation Macromodel 1 Good match Good match Macromodel is Not suitable December 1, 24 4 t /T t /T 6 8 Slide 22
23 Colpitts oscillator (LC) Courtesy: Madhavan Swaminathan, Georgia Institute of Technology Rp=5 1.4p L1=2.1n Cp=1p Rb=22k 3 Cm=.6p C1=1p Rl=2 Cb=1.5p Re=1 C2=2.3p December 1, 24 Slide 23
24 Colpitts: max locking range vs injection strength Reference (full simulation) Nonlinear macromodel Injection amplitude (mv) Adler eqn December 1, 24 Slide 24
25 Colpitts: Amplitude variations Phase shift (s) 5 x Phase error Amplitude variation (ma) 1 Amplitude variations 1 times speedup Oscillation current (ma) Macromodel time (t/t) Oscillation current (ma) 25 Full simulation time (t/t) December 1, 24 Slide 25
26 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 1 times speedup for a small oscillator circuit Accurate in its validity range Current work: Using Krylov subspace based method to reduce the LPTV system. December 1, 24 Slide 26
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