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Frequency Domain Techniques for Simulation of Oscillators

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Time domain model: Frequency domain model: ... Two-level algorithm. Upper Level, solve: ... 1. The continuation algorithm is performed with only one harmonic. ... – PowerPoint PPT presentation

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Title: Frequency Domain Techniques for Simulation of Oscillators


1
Frequency Domain Techniques for Simulation of
Oscillators
Institute of Design Problems in Microelectronics
(IPPM RAS)
M. Gourary, S. Rusakov, A. Stempkovsky
S. Ulyanov, M. Zharov
2
Outline
  • Introduction. Harmonic Balance (HB)
  • Oscillator HB steady-state analysis
  • Probe Implementation
  • Continuation Algorithm with Frequency Adjusting
  • Additional Oscillations in Ring Oscillator
  • Parasitic Solutions (Images)
  • Explanation of Convergence Difficulties
  • Convergence Improvement
  • Conclusions

3
Introduction. Harmonic Balance
  • Harmonic Balance is the collocation method for
    solving boundary
  • value problem the solution is found in the
    terms of Fourier coefficients.

Time domain model
Waveforms can be represented as a truncated
Fourier series
( - Fourier transform)
Frequency domain model
Harmonics of currents charges
excitations
Harmonics indexes
4
Oscillator HB steady-state analysis
Harmonic Balance (HB) system for the oscillator
Basic HB system
additional variable
Equation to fix the phase
Harmonic
Node
Newton update at iteration step
Newton iterations converge to the degenerate
solution (DC point)
5
Probe Implementation
E. Ngoya, A. Suarez, R. Sommet, R. Quere, 1995
The proposal to avoid the convergence of HB
Newton iterations to the degenerate solution (DC
point)
Probe sinusoidal voltage source ideal filter
of the 1st harmonic
filter
oscillator
Probe
The aim obtain probe voltage (Vprobe) providing
Kurokawa conditions for free oscillations
6
Probe Implementation
  • Two-level algorithm
  • Upper Level, solve
  • Jacobi matrix
  • Lower Level solve conventional HB system with
    driving probe.

Advantage the analysis of an autonomous circuit
is converted into an analysis of a series of
closely related nonautonomous circuits, handled
using standard harmonic balance
7
Probe Implementation
  • The difficulty choosing a suitable starting
    frequency and amplitude for the probe to provide
    convergence.
  • Continuation algorithm
  • Initial probe frequency from small signal
    analysis.
  • Initial probe amplitude sweeping Vp from zero
    and looking for a minimum in the probe current
    magnitude.

Ip
  • However
  • No guarantee that there is a minimum
  • No guarantee that minimum is in domain of
    attraction of Newton method

8
HB Continuation Algorithm with Frequency Adjusting
Initial frequency ( ) (AC oscillation
conditions for input impedance at unstable DC
point)
Continuation process
For each Vprobe HB system for forced circuit is
solved
Probe frequency is defined by the condition
Vprobe is increasing until probe current is zero
Iprobe
Typical continuation curve
Iprobe is real, so Newton iterations are not
required
DC point
Periodical solution
Vprobe
9
HB Continuation Algorithm with Frequency Adjusting
Wien
TNT
CMOS Ring
Colpitts
Iprobe
Frequency
Vprobe
Vprobe
Vprobe
Vprobe
10
Additional Oscillations in Ring Oscillator
n is odd number
2
1
n
Self-oscillation condition for n-stages RO with
identical cells
is the cell delay, k is any integer
defines inversion of the output
signal
Ideal RO Infinite set of oscillations (for
each k) Real RO The number oscillations
depends on amplifying
properties of the cell at high frequencies
11
Additional Oscillations in Ring Oscillator
Additional oscillations in real RO are defined
by additional solutions of oscillation
conditions in DC-point
For 7-stages RO
Main oscillation (k1)
Additional oscillation (k2)
12
Parasitic Solutions (Images)
Multiple representation of periodic functions
Each periodic (with period T and frequency f
1/T) function can be also considered as the
function with period kT and frequency f/k, where
k is an integer
HB solution for the oscillation
Parasitic HB solutions (images of the oscillation)
13
Explanation of Convergence Difficulties
Effect of the image depends on the deviation of
its frequency from the frequency of the
solution to be found.
Frequencies of oscillations and images in
7-stages CMOS RO
Oscillation
Frequency GHz
Main
Additional
No simulation difficulties for additional
oscillation because its frequency is much higher
than any image frequency.
4th image of additional oscillation leads to
poor convergence in the simulation of the main
oscillation.
14
Convergence Improvement
Under truncated set of harmonics an image is not
the exact solution of truncated HB system
Solution of full HB system
Images
Solution of truncated HB system
Truncated images are not solutions of truncated
HB system
The number of harmonics is 1
15
Convergence Improvement
Modification of continuation method for RO
1. The continuation algorithm is performed with
only one harmonic. 2. The obtained truncated
solution is considered as initial guess of
Newton iterations for full solution. 3. Newton
iterations for HB system are performed with
required number of harmonics
  • Results
  • successful simulation of RO circuit up to 15
    stages
  • essential speed-up in oscillator simulations

16
Conclusions
  • the suggested special-purpose numerical
    continuation procedure based on frequency
    adjusting provides robust simulation of
    oscillator circuits
  • the convergence difficulties in HB simulations of
    ring oscillators are resulted from 2 sources
  • multiple oscillations in RO with sufficiently
    large number of stages
  • multiple representation of periodic functions
    creating parasitic solutions (images)
  • convergence properties of HB RO simulation can be
    improved by preliminary single-harmonic
    simulation for evaluating initial guess of Newton
    iterations
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