Fundamental Wave

Overview of the concept of fundamental waves

The exact magnetic field in the air gap of an electric machine is usually determined by an electro-magnetic finite element analysis. The waveform of the magnetic field, e.g., the magnetic potential difference, ${\stackrel{^}{V}}_{m}\left(\mathrm{\varphi }\right)$, consists of a spatial fundamental wave with respect to an equivalent two pole machine and additional harmonic waves of higher order. The fundamental wave is dominant in the air gap of an electric machine.

Fig. 1: Field lines of a four-pole induction machine

In the fundamental wave theory only a pure sinusoidal distribution of magnetic quantities is assumed, other harmonic wave effects are ignored.

Fig. 2: Magnetic potential difference of a four pole machine, where $\mathrm{\varphi }$ is the angle of the spatial domain with respect to one pole pair.

The waveforms of the magnetic field quantities, e.g., the magnetic potential difference, ${\stackrel{^}{V}}_{m}\left(\mathrm{\varphi }\right)$, can be represented by a complex phasor, ${\underset{¯}{V}}_{m}$, with

${\underset{¯}{V}}_{m}={V}_{m,r}+j{V}_{m,i}$

${V}_{m}\left(\mathrm{\varphi }\right)=\Re \left({\underset{¯}{V}}_{m}\mathrm{exp}\left(-j\mathrm{\varphi }\right)\right)={V}_{m,r}\mathrm{cos}\left(\mathrm{\varphi }\right)+{V}_{m,i}\mathrm{sin}\left(\mathrm{\varphi }\right)$

It is important to note that the magnetic potential used in this library always refers to an equivalent two-pole machine.

Fig. 3: Spatial distribution of the magnetic potential difference (red shade = positive sine wave, blue shade = negative sine wave) including complex phasor representing this spatial distribution

The potential and flow quantities of this library are the complex magnetic potential difference and the complex magnetic flux as defined in the basic magnetic port.  Due to the sinusoidal distribution of magnetic potential and flux, such a complex phasor representation can be used.  This way, the Fundamental Wave library can be seen as a spatial extension of the standard magnetic library.

The specific arrangement of windings in electric machines with pole pairs gives rise to sinusoidal dominant magnetic potential wave.  The spatial period of this wave is determined by one pole pair [Mueller70, Spaeth73].

The main components of an electric machine model based on the Fundamental Wave library are multi-phase and single-phase windings, air gap as well as symmetric or salient cage models. The electric machine models provided in this library are based on symmetrical windings in the stator and equivalent two or three phase windings in squirrel cage rotors.  Slip ring induction machines may have different phase numbers in the stator and rotor.

Assumptions

The machine models of the FundamentalWave library are currently based on the following assumptions

 • The number of stator phases is greater or equal to three [Eckhardt82]
 • The phase windings are assumed to be symmetrical; an extension to this approach can be considered
 • Only fundamental wave effects are taken into account
 • The magnetic potential difference refers to an equivalent two pole machine
 • There are no restrictions on the waveforms of voltages and currents
 • All resistances and inductances are modeled as constant quantities; saturation effects, cross coupling effects [Li07], temperature dependency of resistances and deep bar effects could be considered in an extension to this library
 • Hysteresis losses are currently not considered [Haumer09]
 • The losses dissipated in the electric machine models are
 • ohmic heat losses,
 • eddy current losses in the stator core,
 • friction losses.

The term fundamental-wave refers to spatial waves of the electro-magnetic quantities. This library has no limitations with respect to the waveforms of the time-domain signals of any voltages, currents, etc.

Libraries

 Library Description Electrical machines and their components Basic magnetic components Complex magnetic sensors Complex magnetic sources

References

 [Beuschel00] M. Beuschel, A uniform approach for modelling electrical machines, Modelica Workshop, pp. 101-108, October 23-24, 2000.
 [Eckhardt82] H. Eckhardt, "Grundzüge der elektrischen Maschinen" (in German), B. G. Teubner Verlag, Stuttgart, 1982.
 [Haumer09] A. Haumer, and C. Kral, The AdvancedMachines Library: Loss Models for Electric Machines, Modelica Conference, 2009.
 [Lang84] W. Lang, "Über die Bemessung verlustarmer Asynchronmotoren mit Käfigläufer für Pulsumrichterspeisung" (in German), Doctoral Thesis, Technical University of Vienna, 1984.
 [Laughton02] M.A. Laughton, D.F. Warne, "Electrical Engineer's Reference Book," Butterworth Heinemann, 16th edition, ISBN 978-0750646376, 2002
 [Li07] Y. Li, Z. Q. Zhu, D. Howe, and C. M. Bingham, "Modeling of Cross-Coupling Magnetic Saturation in Signal-Injection-Based Sensorless Control of Permanent-Magnet Brushless AC Motors," IEEE Transactions on Magnetics, vol. 43, no. 6, pp. 2552-2554, June 2007.
 [Mueller70] G, Müller, "Elektrische Maschinen -- Grundlagen, Aufbau und Wirkungsweise" (in German), VEB Verlag Technik Berlin, 4th edition, 1970.
 [Spaeth73] H. Späth, "Elektrische Maschinen -- Eine Einführung in die Theorie des Betriebsverhaltens" (in German), Springer-Verlag, Berlin, Heidelberg, New York, 1973.