ELECTROMECHANICAL MOTION DEVICES
Rotating Magnetic Field Based Analysis
3^rd Edition

The induction machine is the workhorse of the power industry. It is used to convert electrical energy to mechanical motion. In our analysis we will assume that the induction machine has symmetrical stator and rotor windings. Torque is produced due to the interaction of mmf$_s$ and mmf$_r$ which travel about the air-gap at the same angular velocity regardless of the rotor speed. This results due to the fact that in most cases the rotor windings are short-circuited. Although there are some cases where the rotor windings are supplied from a voltage source we will not consider a doubly-fed machine. In most cases the rotor of an induction machine has short-circuited die casted windings that are referred to as a squirrel-cage rotor.

Using the voltage equations in the arbitrary reference frame, we will animate free acceleration and load torque switching. We will then animate a simplified version of Field Orientation.

Voltage and Torque Equations

We will consider the two-pole two-phase symmetrical induction machine shown below.

The voltage equations in the arbitrary reference frame are $$v_{qs} = r_si_{qs}+\omega\lambda_{ds}+p\lambda_{qs}~~~~~~~~~~~~~~~~~~~\text{(1)}$$ $$v_{ds} = r_si_{ds}-\omega\lambda_{qs}+p\lambda_{ds}~~~~~~~~~~~~~~~~~~~\text{(2)}$$ $$v'_{qr} = r'_ri'_{qr}+(\omega-\omega_r)\lambda'_{dr}+p\lambda'_{qr}~~~~~~~~\text{(3)}$$ $$v'_{dr} = r'_ri'_{dr}-(\omega-\omega_r)\lambda'_{qr}+p\lambda'_{dr}~~~~~~~~\text{(4)}$$ where for a linear magnetic system $$\lambda_{qs} = L_{ls}i_{qs}+L_{ms}(i_{qs}+i'_{qr})~~~~~~~~~~\text{(5)}$$ $$\lambda_{ds} = L_{ls}i_{ds}+L_{ms}(i_{ds}+i'_{dr})~~~~~~~~~~\text{(6)}$$ $$\lambda'_{qr} = L'_{lr}i'_{qr}+L_{ms}(i_{qs}+i'_{qr})~~~~~~~~~~\text{(7)}$$ $$\lambda'_{dr} = L'_{lr}i'_{dr}+L_{ms}(i_{qs}+i'_{dr})~~~~~~~~~~\text{(8)}$$ The torque equations expressed in substitute variables may be written $$T_e = \frac{P}{2}L_{ms}\left(i_{qs}i'_{dr}-i_{ds}i'_{qr}\right)~~~~~~~~~~\text{(9)}$$ where the torque $T_e$ is positive for motor action. The equivalent circuits in the arbitrary reference frame are shown below.

The torque and rotor speed are related by $$T_e = J\frac{2}{P}\frac{d\omega_r}{dt}+B_m\frac{2}{P}\omega_r+T_L~~~~~~~~~~\text{(10)}$$