Numerical determination of equivalent circuit parameters of synchronous machines

First, a brief overview of the calculation tasks in which the equivalent circuit elements are used in the development of permanent magnet synchronous machines (PMSM) and electrically excited synchronous machines (ESM) is given. Examples of data sheets and other steady-state and transient machine characteristics are presented, and acceptable assumptions and simplifications are discussed. Then, a fast numerical method for calculating winding resistances considering temperature and skin effect is presented. To calculate the equivalent circuit reactances for PMSM and ESM, the frozen permeability method is first discussed. It is shown how the complete inductance matrix can be created, the effective turns ratio can be determined, and the main and leakage inductances can be derived using simple transformations. Using calculation examples, it is demonstrated how a strong saturation in a load operation influences the inductances Ld and Lq for PMSM and ESM and how the coupling inductances between the d- and q-axes Ldq appear. Ignoring these effects when calculating voltages and torques in the high load operation can lead to significant calculation errors. For the large synchronous generators, the calculation of characteristic reactances and time constants is of great importance. The calculation of transient and subtransient reactances Xd', Xd" and Xq" as well as time constants Td0', Td', Td0", Tq0", Td", Tq" and Ta directly from eigenvalues and eigenvectors of the inductance matrix is presented and compared to conventional analytical methods. For the subtransient reactances and time constants, the influence of the leakage field saturation in a short circuit is evaluated. For ESM with a damper cage, the importance of the Canay reactance for the correct determination of the characteristic reactances is discussed, and a simple method for its calculation from the inductance matrix is demonstrated. Another influencing factor for the calculation of subtransient values is the frequency-dependent skin effect in the bars of a damper cage or the field penetration depth in a solid pole, for which description the extended equivalent circuits with chain conductors are introduced. Furthermore, a calculation method for calculation of the zero-sequence and negative-sequence system impedances is presented.