Gen-Adler: The Generalized Adler's Equation for Injection Locking Analysis in Oscillators
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1 Gen-Adler: The Generalized Adler's Equation for Injection Locking Analysis in Oscillators Prateek Bhansali, Jaijeet Roychowdhury University of Minnesota, USA Slide 1
2 Outline Introduction Previous work Challenges involved Adler's equation Gen-Adler injection locking equations Ring oscillator Sinusoidal, exponential, square injection signal Experimental results Conclusion Slide 2
3 Introduction Injection signal!1 Injection locking Frequency and phase are locked Engineering Applications variable phase shifts, frequency multiplication, low power frequency dividers, precision quadrature generation Slide 3
4 SPICE-level Simulation of Injection Locking Inefficient and inaccurate Direct simulation of oscillators Extremely small time steps are required Accumulation of phase error Difficult to extract phase and frequency information Locking process can take several cycles Simulation for hundreds of cycles to conclusively declare oscillator locked or unlocked. Distinction between lock and quasi-lock, occurs when injection frequency is just outside the locking range Slide 4
5 SPICE-level Simulation of Injection Locking (Cont'd) Oscillator Waveform Oscillator appears to be locked Injection Signal Not Locked Several number of simulations required to determine the oscillator's locking range Alternative to SPICE-level simulation required Slide 5
6 Previous Work on Injection Locking 1. Adler, R., "A Study of Locking Phenomena in Oscillators," Proceedings of the IRE, vol.34, no.6, pp , June Razavi, B., "A study of injection locking and pulling in oscillators," SolidState Circuits, IEEE Journal of, vol.39, no.9, pp , Sept Xiaolue Lai; Roychowdhury, J., "Capturing oscillator injection locking via nonlinear phase-domain macromodels," Microwave Theory and Techniques, IEEE Transactions on, vol.52, no.9, pp , Sept Xiaolue Lai; Roychowdhury, J., "Analytical equations for predicting injection locking in LC and ring oscillators," Custom Integrated Circuits Conference, Proceedings of the IEEE 2005, vol., no., pp , Sept Gourary, M.M.; Rusakov, S.G.; Ulyanov, S.L.; Zharov, M.M.; Mulvaney, B.J.; Gullapalli, K.K., "Injection locking conditions under small periodic excitations," Circuits and Systems, ISCAS IEEE International Symposium on, vol., no., pp , May Gourary, M.M.; Rusakov, S.G.; Ulyanov, S.L.; Zharov, M.M.; Mulvaney, B.J.; Gullapalli, K.K., "Smoothed form of nonlinear phase macromodel for oscillators," Computer-Aided Design, ICCAD IEEE/ACM International Conference on, vol., no., pp , Nov Slide 6
7 Original Adler's Equation Slide 7
8 Adler's Equation Adler's Equation Phase difference Frequency difference Quality Factor Slide 8
9 Injection Locking Dynamics Adler's equation provides quick insight into locking dynamics Instantaneous phase difference, Instantaneous frequency of oscillator, In steady state, when oscillator is injection locked Analytical equation relating locking range and injection amplitude Slide 9
10 Locking Range Locking Range Applicable only to LC oscillator (Q explicitly required) with sinusoidal injection signal Slide 10
11 Review of Perturbation Projection Vector (PPV) Oscillator state variables Resistive components Perturbation to the oscillator Time PPV equation Injection coupling signal coupling Oscillator phase Slide 11
12 PPV Equation for LC Oscillator PPV equation Slide 12
13 PPV Equation and Phase Difference Slide 13
14 PPV Equation and Adler's Equation fast varying Slide 14
15 Adler's Equation from PPV Equation PPV equation Average over fast varying variable Adler's equation Slide 15
16 Gen-Adler: Generalized Adler's Equation Slide 16
17 Generalized Adler's Equation and PPV Equation PPV equation (1) Step 1: (2) Step 2: Modified phase equation Slide 17
18 Generalized Adler's Equation Step 3: Average over the fast varying variable where, slow fast Slide 18
19 Generalized Adler's Equation Contd.... Same form as of original Adler's equation Applicable for analysis of any oscillator unlike original Adler's Equation Any type of periodic injection signal: exponential, sinusoidal, square Obtained by averaging accurate PPV equation, but has Adler like simplicity Analytical formulation Slide 19
20 Analytical Formulation of Injection Locking Dynamics in Ring Oscillator Slide 20
21 Injection Locking in Ring Oscillator Three stage ring oscillator DEs Ideal switching characteristics Slide 21
22 Ring Oscillator's PPV Steady State Waveforms PPV Waveforms PPV at node V3 Slide 22
23 Gen-Adler for Ring Oscillator Ideal inverter Sinusoidal injection signal Exponential injection signal Square injection signal (with any duty cycle) Slide 23
24 Analytical Injection Locking Dynamic Equations Sinusoidal Injection to a ring oscillator where, Slide 24
25 Injection Locking Range In steady state, when oscillator is injection locked Lock Range Slide 25
26 Graphical Injection Locking Analysis Steady state phase Slide 26
27 Graphical Injection Locking Analysis Unstable and stable steady state phase Slide 27
28 Graphical Injection Locking Analysis Slide 28
29 Square Wave Injection Signal = duty cycle of square wave Slide 29
30 Square Wave Injection Signal S U Lock Range Slide 30
31 Exponential Injection Signal Lock Range Slide 31
32 Instantaneous Phase and Frequency Slide 32
33 Comparison with Full Simulation Sinusoidal Injection Square Wave Injection Excellent match with the full simulation Slide 33
34 Conclusion Simple analytical equations for injection locking analysis in ring oscillators maintain Adler like simplicity quick insight into injection locking process via graphical analysis hand analysis of injection locking range for variety of injection signals good match with the full simulation Gen-Adler is numerically applicable to any oscillator for injection locking analysis Slide 34
35 End Slide 35
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