Lecture6 - Lecture 6 Induction Machine Equations in a-b-c...

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Lecture 6 Induction Machine Equations in a-b-c frame Having developed the 6x6 inductance matrix of the 3 stator windings magnetically coupled to the 3 rotor windings, they are assembled here with the 3x3 matrices of the stator and rotor resistances. The symbol p represents p=d. ./dt and it differentiates, with respect to time, all quantities on the right-hand side of it. The super-scripts s and r denote stator and rotor respectively. The subscripts abc signify that the voltage vector e and the current vector i belong to a-b-c the transformation frame. Power engineers are familiar with different transformation frames: 0- α - β , 0-d-q, 0- γ - δ . Kirchhoff’s Voltage Law in circuit theory takes the form: In the induction motor, the equations relating the 6-voltages at the terminals of the 6 windings to the 6 currents flowing through them, yield:
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The voltages and currents are represented as 3-tuple vectors and the 6x6 matrix is best described by 4 sets of 3x3 sub-matrices. In Lecture 4, the inductance matrix has been derived for a 2-pole ( n=1 ) machine. The equations below are written for a n=2p pole machine. Newton’s Law of Motion is applied to model the mechanical motion. The mechanical parameters are: J 1 to represent the combined moments of inertia of the rotor, the load and the gears in the shaft, f 1 to represent the friction in the bearings and windage. The load torque is T L and the machine output torque is T em . It is computed from: i L i i W T m m T m m em = = θ )] ( [ 2 1 1
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Note that the last equation relates the speed ω m with the rotor position θ m . Nature of equations System Order : The order of the equations is N=8. For the purpose of this course, the order is identified by the number of dx/dt terms. There are 6 dx/dt terms where the x s are currents (electrical quantities). There are 2 dx/dt terms the x s being mechanical quantities m and m . Electromechanical Coupling . The mechanical state-variable m is in the electrical equations. The electrical currents are appear in the torque T em term in the mechanical equations. Thus the electrical and mechanical equations are inter-coupled. Nonlinearity of the equations. In this course, a system of equations is linear when it can be put in the form: u B x A dt dx ] [ ] [ + = where x and are vectors an [A] and [B] are constant matrices.
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Lecture6 - Lecture 6 Induction Machine Equations in a-b-c...

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