A
M
C
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Bergische Universit¨at Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Institute of Mathematical Modelling, Analysis and Computational
Mathematics (IMACM)
Preprint BUW-IMACM 11/07
J. Virtanen, J. ter Maten, T. Beelen,
M. Honkala, and M. Hulkkonen
Initial conditions and robust Newton-Raphson for
Harmonic Balance analysis of free-running oscillators
April 2011
http://www.math.uni-wuppertal.de
Initial conditions and robust Newton-Raphson
for Harmonic Balance analysis of free-running
oscillators
J. Virtanen, J. ter Maten, T. Beelen, M. Honkala, and M. Hulkkonen
Abstract Poor initial conditions for Harmonic Balance (HB) analysis of free-
running oscillators may lead to divergence of the direct Newton-Raphson method
or may prevent to find the solution within an optimization approach. We exploit
time integration to obtain estimates for the oscillation frequency and for the oscilla-
tor solution. It also provides an initialization of the probe voltage. Next we describe
new techniques from bordered matrices and eigenvalue methods to improve Newton
methods for Finite Difference techniques in the time domain as well as for Harmonic
Balance. The method gauges the phase shift automatically. No assumption about the
range of values of the Periodic Steady State solution is needed.
1 Introduction
A free-running oscillator is an autonomous circuit, which has only DC bias sources
connected to the circuit and, thus, no periodic excitation. During the time-domain
transient analysis of an oscillator, the oscillation starts by itself due to noise or
instability. Long start-up time implies long simulation time to get the Periodic
Jarmo Virtanen, Mikko Honkala, Mikko Hulkkonen
Aalto University School of Electrical Engineering, Department of Radio Science and Engineer-
ing, P.O. Box 13000, FI-00076 AALTO, Finland, e-mail: {jarmo.virtanen,mikko.a.
honkala,mikko.hulkkonen}@aalto.fi
T.G.J Beelen
NXP Semiconductors, High Tech Campus 46, 5656 AE Eindhoven, the Netherlands, e-mail:
{Theo.G.J.Beelen}@nxp.com
E.J.W. ter Maten
Eindhoven University of Technology, Dep. Mathematics and Computer Science, CASA, P.O. Box
513, 5600 MB Eindhoven, the Netherlands, e-mail: {E.J.W.ter.Maten}@tue.nl
Bergische Universit¨at Wuppertal, FB C, Applied Mathematics / Numerical Analysis, Ben-
dahler Str. 29, Zi-503, D-42285 Wuppertal, Germany, e-mail: {Jan.ter.Maten}@math.
uni-wuppertal.de
1
2 J. Virtanen, J. ter Maten, T. Beelen, M. Honkala, and M. Hulkkonen
Steady State (PSS) solution. Harmonic balance (HB) analysis is a frequency-domain
PSS method. HB is needed for (phase) noise simulations and is more suitable for
frequency-dependent linear devices. It may converge faster to the PSS solution of a
free oscillator than the transient analysis. To enhance convergence one either mod-
ifies the HB equations or one applies artificial excitation. In addition, the oscilla-
tion frequency (the fundamental HB frequency), is unknown and one needs a gauge
equation and an initial estimate. Frequency domain methods to estimate these can
be found in [1,3,6–8] (and their references).
We present two algorithms for oscillation frequency detection from transient data
and improve by (vector) extrapolation [10]. The initialization of the probe voltage
amplitude and of the HB solution are considered. Finally we describe new tech-
niques from bordered matrices and eigenvalue methods to improve the Newton
method for HB analysis.
2 Initializing HB Oscillator Analysis
The oscillator analysis in the APLAC simulator [2] utilizes a probe element and op-
timization techniques. Inside an optimization loop HB analysis is performed with
new values of the optimization variables, being the oscillation frequency, f
osc
, and
the oscillation amplitude, v
osc
. An artificial excitation probe, being a voltage source
in series with a non-zero resistor (to prevent an increase of the DAE-index), is con-
nected to the circuit. The goal of the optimization is to have a zero current through
the probe element. For a related procedure see [8]. The initial conditions for the op-
timization of f
osc
and v
osc
are obtained from transient analysis as described next.
Initially a (limited) transient analysis is run, followed by a Fourier transform (FFT)
to get an impression of the spectrum of the oscillator and of the solution. A spec-
tral line having the largest magnitude indicates the oscillation frequency. Depending
on the sampling rate, the actual oscillation frequency may be situated between the
sampled frequency points. Therefore, quadratic interpolation with three frequency
points around the maximum is used to determine a more accurate estimate for the
6.703M 8.250M 9.797M 11.344M 12.891M
0.00
0.47
0.95
1.42
1.90
U/V
freq/Hz
FFT
parabola
fosc
fapprox= 9.880MHz
400n 442.5n 485n 527.5n 570n
-1.7
-0.9
-0.1
0.7
1.5
amplitude/V
time / s
Fig. 1 Left: Quadratic interpolation of the frequency from the spectrum. Right: Zero-crossing:
the x-markers connected with lines show the points used for interpolation.
