**
The mathematics of electric machine
modeling**

abc or phase-variable modeling:

The first step in the mathematical modeling of an induction machine is by
describing it as coupled stator and rotor three-phase circuits using phase variables,
namely stator currents i_{as}, i_{bs}, i_{cs }and rotor
currents i_{ar}, i_{br}, i_{cr} ; in addition to the rotor speed ω_{m} and the angular displacement θ between stator and rotor windings. The machine
electrical parameters are expressed in terms of a resistance matrix
R
[6x6] and an
inductance matrix L [6x6] in which the magnetic
mutual coupling elements are functions
of position θ. The electrical variables V,
I, λ appear as 6-element column vectors (in the
matrix analysis connotation); so that, for instance, the current vector is
I =
[i_{as} i_{bs} i_{cs} i_{ar} i_{br} i_{cr}]^{t},
representing stator and rotor currents expressed in their respective stator and
rotor frames. While the matrix analysis of three-phase stator and rotor circuits
in relative motion is easy to formulate mathematically (in particular using
Matlab), it nevertheless obscures an understanding of the underlying physical
interactions and does not directly lead to the introduction of control
strategies. To view details of this matrix analysis of an induction machine,
press (or open in new window) the following link:

dq transformations:

The next step is to transform the original stator and rotor abc frames of
reference into a common ω_{k} or dq frame in which the new variables for
voltages, currents, and fluxes can be viewed as space vectors (in a 2-D
geometric sense) so that currents are now defined as i_{s} = [i_{ds} i_{qs}]
and i_{r} = [i_{dr} i_{qr}]. To view the transformation of variable
procedure, press the following link:

dq or space-vector modeling:

In the dq frame, the inductance parameters become constant,
independent of position. Among possible choices of dq frames are the following:
a) Stator frame where ω_{k }= 0; b) Rotor frame where ω_{k} = ω_{m};
c) Synchronous frame associated with the frequency ω_{s} (possibly time
varying); d) Rotor flux frame in which the d-axis lines up with the direction of
the rotor flux vector. Because it utilizes space vectors, the dq model of the
machine provides a powerful physical interpretation of the interactions taking
place in the production of voltages and torques, and, more importantly, it leads
to the ready adaptation of positional- or speed-control strategies such as
vector control and direct torque control. To view details of the space vector modeling,
click the
following link:

dq-model of the induction machine

dq modeling of the synchronous machine:

Because of the asymmetry of the synchronous machine created by rotor saliency and field excitation, the corresponding dq model must use the rotor coordinates as reference frame. To view details of the modeling, click the following link:

© M. Riaz