Magnetic QuasiStatic Fundamental Wave

Overview

Reference frames

 • Quasistatic magnetic ports contain the complex magnetic flux (flow variable) and the complex magnetic potential difference (potential variable) and a reference angle. The relationship between the different complex phasors with respect to different references will be explained by means of the complex magnetic flux. The same transformation relationships also apply to the complex magnetic potential difference. These relationships are important for handling connectors in the air gap model, transforming equations into the rotor fixed reference frame, etc.
 • Let us assume that the air gap model contains stator and rotor magnetic ports which relate to the different sides of the machine. The angle relationship between these ports is

${\mathrm{\gamma }}_{s}=\mathrm{\gamma }+{\mathrm{\gamma }}_{r}$

where

 – ${\mathrm{\gamma }}_{s}$ is the connector reference angle of the stator ports,
 – ${\mathrm{\gamma }}_{r}$ is the connector reference angle of the rotor ports, and
 – $\mathrm{\gamma }=p\left({\mathrm{\phi }}_{a}-{\mathrm{\phi }}_{\mathrm{support}}\right)$  is the difference of the mechanical angles of the flange and the support, respectively, multiplied by the number of pole pairs, p.
 • The stator and rotor reference angles are directly related with the electrical frequencies of the electric circuits of the stator, ${f}_{s}$, and rotor, ${f}_{r}$, respectively, by means of

$2\pi {f}_{s}=\frac{d{\mathrm{\gamma }}_{s}}{\mathrm{dt}},\phantom{\rule[-0.0ex]{2.0ex}{0.0ex}}2\pi {f}_{r}=\frac{d{\mathrm{\gamma }}_{s}}{\mathrm{dt}}$

This is a strict consequence of the electromagnetic coupling between the quasi-static electric and the quasi-static magnetic domains.

$2\pi {f}_{s}=\frac{d{\mathrm{\gamma }}_{s}}{\mathrm{dt}}$

 • The complex magnetic flux with respect to stator and rotor magnetic port are equal,

${\overline{\mathrm{\Phi }}}_{\left(\mathrm{ref}\right)}={\mathrm{\Phi }}_{\mathrm{re}}+j{\mathrm{\Phi }}_{\mathrm{im}}$

 • but the reference phase angles are different according to the relationship explained above. The stator and rotor reference angles refer to quasi-static magnetic connectors. The complex magnetic flux of the (stator) port with respect to the stator fixed reference frame is then calculated by

${\overline{\mathrm{\Phi }}}_{\left(s\right)}={\overline{\mathrm{\Phi }}}_{\left(\mathrm{ref}\right)}\mathrm{exp}\left(j{\mathrm{\gamma }}_{s}\right)\cdot$

 • The complex magnetic flux of the (rotor) magnetic port with respect to the rotor fixed reference frame is then calculated by

${\overline{\mathrm{\Phi }}}_{\left(r\right)}={\overline{\mathrm{\Phi }}}_{\left(\mathrm{ref}\right)}\mathrm{exp}\left(j{\mathrm{\gamma }}_{r}\right)\cdot$

 • The stator- and rotor-fixed complex fluxes are related by

${\overline{\mathrm{\Phi }}}_{\left(r\right)}={\overline{\mathrm{\Phi }}}_{\left(s\right)}\mathrm{exp}\left(-j\mathrm{\gamma }\right)\cdot$

Libraries

 Library Description Basic quasistatic machine models Basic fundamental wave components Loss models Sensors to measure variables in magnetic networks Sources to supply magnetic networks Utilities for quasistatic fundamental wave machines