Parameter Finding Methods for Oscillators with a Specified Oscillation Frequency

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Parameter Finding Methods for Oscillatorswith a
Specified Oscillation Frequency
Igor Vytyaz, David C. Lee, Suihua Lu, Amit
Mehrotra, Un-Ku Moon, and Kartikeya Mayaram
Part of this work was performed while I. Vytyaz
was at Berkeley Design Automation
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Outline

• Background and motivation
• Periodic steady-state (PSS) analysis
• With a Specified oscillation Frequency (PSS-SF)
• Finite difference
• Shooting
• Harmonic balance
• Conventional search-based approach
• Examples and results

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Background and motivation

• Oscillator applications
• Data converters (ADC, DAC)
• Digital systems (memory, microprocessors)
• Communication (CDR, SERDES, RF transceivers)
• Oscillation frequency is a design specification
• Conventional PSS analysis
• Finds PSS waveforms and oscillation frequency
• Frequency may be different than in specification
• Alternative PSS analysis is needed
• Tunes oscillator design and finds PSS waveforms
• Frequency same as in specification

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PSS analysis for voltage controlled osc

• Phase-locked loop (PLL) design and analysis
• Oscillation frequency is known
• This analysis finds the control voltage for the
  circuit to oscillate at a specified frequency

fref
f0
Vctrl
VCO
LF
CP
PD
N
f0 N fref
A. Mehrotra, et al., Steady-state analysis of
voltage and current controlled oscillators,
ICCAD 2005, pp. 618-623, Nov 2005.
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PSS with Specified Frequency (PSS-SF)

• VCO design and analysis
• PSS-SF handles any frequency-tuning parameter
• MOSFET width
• Bias current
• Tank capacitor,

known
VCO
fixed
f0
Ctank
Ibias
W
Vctrl
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Frequency-tuning parameter

• Design freedom
• Many ways to achieve a specified frequency f0
• Unique solution
• Choose a single frequency-tuning parameter gf0

gf0 is W
W
gf0 is W
a unique solution
Ctank
W
f0
f0target
initial design
Ctank
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Equations for oscillators PSS

• Circuit DAEs
• Periodicity
• Initial phase

T
Ctank
Ibias
W
x(t)
gf0
reactive
resistive
sources
, gf0
, gf0
gf0
Osc. Eq.
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Conventional PSS analysis vs PSS-SF

• Conventional PSS
• PSS - specified frequency

Known parameter
T
x(t)
x(t), T
gf0
gf0
Osc. Eq.
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Finite difference method for PSS-SF

• Equations
  Unknowns
• Eliminate using periodicity
  constraint eqn.
• Discretize time
• Approximate by a finite difference
  formula

, gf0
, gf0
gf0
gf0
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Finite difference PSS-SF Jacobian

• Newton-Raphson iteration
• Jacobian matrix
• Compute sensitivities of device contributions to
  gf0

gf0
gf0
gf0
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Shooting method for PSS-SF

• Equations
  Unknowns
• Compute using transient analysis
• Initial condition

, gf0
, gf0
gf0
gf0
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Shooting PSS-SF Jacobian

• Jacobian matrix
• Compute using transient
  sensitivity analysis
• Initial condition

gf0
gf0
gf0
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Harmonic balance method for PSS-SF

• Equations
  Unknowns
• Transform DAEs to frequency domain by DTFT
• State unknowns are Fourier coefficients of

, gf0
gf0
, gf0
, gf0
gf0
, gf0
gf0
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Harmonic balance PSS-SF Jacobian

• Jacobian matrix
• Compute sensitivities of device contributions to
  gf0

gf0
gf0
gf0
gf0
gf0
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Search-based methods
gf0

• The problem of parameter finding
• Solved by a sequence of conventional PSS analyses
• Update rule for parameter value
• Bisection method is the simplest
• Newton method requires the sensitivity
• Search-based methods are slower than PSS-SF

gf0
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Example a ring oscillator

• Frequency-tuning parameters
• Capacitors, gf0 is C
• MOSFET sizes, gf0 is WMp

Mp1
Mp2
Mp3
C1
C2
C3
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PSS-SF convergence process

• MOSFET sizes tune the frequency
• Initial guess gf0 19.5 mm
  T 2.51 ns
• PSS-SF iterations
• Solution gf0 13.8 mm T
  3.00 ns

5
3
3
Output voltage Volts
0
Ttarget
0
1
2
0.5
1.5
2.5
Time ns
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Newton search convergence process

• MOSFET sizes tune the frequency
• Initial guess gf0 19.5 mm
  T 2.51 ns
• Search iterations
• Solution gf0 13.8 mm T
  3.00 ns

Conventional PSS iterations
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3
3
Output voltage Volts
0
Ttarget
0
1
2
0.5
1.5
2.5
3.5
Time ns
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Comparison

• PSS-SF
• PSS-SF iterations
• Newton-Raphson search
• Conventional PSS iterations
  (28654)

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gf0 13.8 mm
25
gf0 13.8 mm
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Summary of simulation results

• Iteration count
• PSS-SF
• Bisection
• Newton
• PSS-SF is 6x faster than Bisection
• PSS-SF is 3x faster than Newton
• PSS-SF successfully used to tune many large LC
  and ring oscillators with hundreds of transistors

gf0 is
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Summary

• PSS-SF PSS with a Specified Frequency
• New design-oriented analysis
• Finds the value of a circuit parameter
• Oscillation frequency is a design specification
• General and handles any frequency-tuning
  parameter
• More efficient than conventional search-based
  approaches
• Simulation results in good agreement with
  conventional PSS analysis