Lecture 3 Synchronous Machine Modeling Inductance Calculation and

Lecture 3 Synchronous Machine Modeling Inductance Calculation and Turn-ratio Transformation Professor: Ali Keyhani

Inductance Calculation and Turn-ratio Transformation All the inductances can be calculated from physical dimension of the machine. The magnetizing inductance for phase as is: where, = axial length, = inner radius of stator From Fig. 1, we can define: The as-as’ winding distribution Fig. 1 2

Inductance Calculation and Turn-ratio Transformation The air gap distribution where, Hence, when And when , , Substituting (2) and (3) into (1) and integrating, we will have 3

Inductance Calculation and Turn-ratio Transformation Let us define: In terms of physical dimensions, the d-axis mutual and magnetizing inductances are: 4

Inductance Calculation and Turn-ratio Transformation It can also be shown 5

Inductance Calculation and Turn-ratio Transformation Let us define the quantity as , the direct-axis magnetizing inductance. From the above equations, it is apparent that all of the following quantities are equal: We should remember that by definition = the direct-axis magnetizing inductance 6

Inductance Calculation and Turn-ratio Transformation Note also that the and are The d-axis flux linkage equations, that is, the equations (47), (50) and (51) of Lecture 2 a, can be manipulated into the form: 7

Inductance Calculation and Turn-ratio Transformation Multiply (50) by and manipulate the equation. Define: 8

Inductance Calculation and Turn-ratio Transformation Define: Utilizing the above definitions and noting equation (14), the above flux linkage equations can be written as: Similarly, the q-axis flux linkage equations can be written as: 9

Inductance Calculation and Turn-ratio Transformation where, and = the quadrature axis magnetizing inductance 10

Inductance Calculation and Turn-ratio Transformation Voltage equations: The d-axis equations are: No change is needed in the following way: . and can be transformed in the 11

Inductance Calculation and Turn-ratio Transformation Defining: Using the above and previous defined quantities, the voltage equations referred to the stator are: Similarly, the q-axis voltage equations referred to the stator are: 12

Inductance Calculation and Turn-ratio Transformation where, Finally, it should be noted that the 0 -axis flux linkage and voltage equations are not affected by turn ratio transformation and remain the same. 13