24 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 14, NO. 1, JANUARY 1999
simulation are presented. Simulation and experimental results
are presented and discussed in Section V. A brief review
of recently published literature on the practical remedies is
given in Section VI. Section VII presents the summary and
conclusions.
II. DESCRIPTION OF THE PROBLEM
In order to estimate the voltage distribution among the coils
of the stator during a typical PWM wavefront, it is necessary
to use a high-frequency distributed-circuit model for the stator
winding [15]. This is mainly because of the high-frequency
content of the wavefront with sharp rise time. The machine
behavior at such high frequencies (which are in the order of
megahertz) corresponding to fast switching transients is totally
different than at power frequency, and, hence, calculating
the distributed parameters is very difficult. For example, at
megahertz-range frequencies corresponding to switching times
in the order of fractions of microseconds, the steel laminations
act like flux barriers, and, hence, most of the flux produced
by the windings will be leakage fluxes. The high-frequency
eddy currents induced in the rotor confine the flux to the
air gap itself, and, hence, there will be no flux penetration
across the air gap. Since the magnetic steel laminations act
as flux barriers, the flux lines are confined to the stator slot
portion itself, as will be shown later. It is also essential to
represent the winding by its distributed equivalent circuit since
the wavelength of the high-frequency traveling wave is very
small compared to that of the power frequency.
The distributed-circuit parameters to be calculated for the
individual turns are the self-inductance and resistance of each
turn, mutual inductances between turns within the same slot,
turn-to-ground capacitances, and turn-to-turn capacitances [3],
[4], [7], [8], [15], [16]. The distributed parameters of the
individual turns depend on the slot geometry and also on
their relative position within the slot. For example, a turn
located near the slot wall will have a higher turn-to-ground
capacitance compared to a turn located in the middle of
the slot. A turn which is located in the middle of the slot
will have a lesser turn-to-ground capacitance because of the
larger separation between the turn conductor and ground.
Furthermore, the conductors around the turn located in the
middle of the slot act as a shield between the turn and ground.
These factors complicate the computation of the parameters.
Another difficulty with random wound machines is that the
relative positioning of the individual turns in different slots
may not be the same.
In order to overcome these difficulties, finite-element anal-
ysis techniques are used to compute the circuit parameters of
the coils even at high frequencies. A single-slot model of the
machine will be enough to calculate all the parameters required
for simulation if it is assumed that the relative positioning of
the individual turns within the slots remains the same in a
given phase which is a reasonably good assumption.
The following are the advantages of using finite-element
analysis.
• Actual material properties can be defined for computing
the parameters.
• High-frequency field effects on the parameters can be
considered.
• The exact geometry of the slot can be modeled for
computing the parameters.
The details of the finite-element model along with the
procedure of computing the various parameters are presented
in the following section.
III. FINITE-ELEMENT ANALYSIS OF THE SINGLE-SLOT MODEL
For computing the equivalent circuit parameters of the coils,
a single-slot model of the cage induction motor was used. The
motor used for analysis in the present work has the following
specifications: 3 HP, 3 phase, 4 pole, 60 Hz, and 230/460 V.
The stator has 36 slots and has a single-layer random winding
with 6 coils per phase, each coil having 54 turns.
The Ansoft [24] package is used for performing finite-
element analysis of the single-slot model which basically
consists of one stator slot pitch with all the turns placed
randomly to simulate random winding. The slot wall insulation
and individual turn insulation are also modeled in order to
compute the capacitance values accurately. Although the flux
lines do not penetrate the air gap, the rotor slot portion is also
modeled. The package computes the field quantities per unit
length of the single-slot model assuming symmetry in the axial
direction. There are different solvers available in the package
for performing electric and magnetic field analysis. From the
finite-element analysis results, the impedance and capacitance
matrices are obtained which are then used for forming the
equivalent circuit. The details of the finite-element analysis
are explained in the following sections.
A. Computation of the Inductance and Resistance Matrices
For computing the inductance and resistance matrices, an
eddy-current analysis option available in the Ansoft package
is used. Once the geometry of the single-slot model is created,
an eddy-current analysis solver is used. The setting up of
the eddy-current analysis includes setting up the excitation,
defining boundary conditions, etc. When eddy-current analysis
is used, the package automatically defines each conductor as
a current source. The only boundary condition used is balloon
boundary condition in which a surface far away from the slot
model is set to zero magnetic vector potential. The frequency
for which the eddy-current analysis has to be computed is
also defined in the analysis.
The main advantage of using an eddy-current solver is
that the effects of time-varying currents in parallel conductors
are considered in the analysis. Furthermore, the eddy-current
effects caused in the conductors at high frequencies are taken
care of. Time-varying currents flowing in a conductor produce
a time-varying magnetic field in planes perpendicular to the
conductor. In turn, this magnetic field induces eddy currents
in the source conductor and in any other conductor parallel to
it. The eddy-current field solver calculates the eddy currents
by solving for and in the field equation
(1)