ELECTROMECHANICAL MOTION DEVICES
Rotating Magnetic Field Based Analysis
3rd Edition

Animation E: Synchronous Machines – Chapter 7

Paul Krause Oleg Wasynczuk Steven Pekarek Timothy O'Connell

Elementary Synchronous Machine

It is beyond the scope of this course to consider the synchronous machine shown on the previous page. Instead, the principle of synchronous machine operation can be established by considering the two-phase round-rotor machine with only a field winding on the rotor as shown below. This, of course, neglects the reluctance and damping torques but it is sufficient to introduce the steady-state main torque due to the interaction of $\text{mmf}_s$ and $\text{mmf}_r$. This is appropriate for the first look at a synchronous machine by an undergraduate electrical engineer.

Circuit

By a derivation similar to that conducted for the permanent-magnet ac machine, the steady-state voltage equations become $$\tilde{V}_{as} = (r_s+jX_{ss})\tilde{I}_{as}+\tilde{E}_a~~~~~~~~~~\text{(1)}$$ where $$\tilde{E}_a = \frac{1}{\sqrt{2}}\omega_eL_{ms}I'^r_{fd}e^{j\theta(0)}~~~~~~~~~~\text{(2)}$$ In the above equations $\tilde{V}_{as}$ is the terminal applied voltage, generally the power system voltage at the point of connection to the system, $\tilde{I}_{as}$ is the stator current, $r_s$ is the stator phase resistance, $L_{ss}$ is the self inductance at the stator phase, $\tilde{E}_a$ is the voltage due to the field winding, and $L_{ms}$ is the magnetizing inductances, that is $$L_{ss} = L_{ls}+L_{ms}~~~~~~~~~~\text{(3)}$$ The electromagnetic torque is $$\begin{split}T_e & = \frac{P}{2}L_{ms}I'^r_{fd}I^r_{qs}\\ & = \frac{P}{2}L_{ms}I'^r_{fd}\sqrt{2}I_s\cos[\theta_{esi}(0)+\theta_r(0)]\end{split}~~~~~~~~~~\text{(4)}$$ The load torque $T_e$ are related as $$T_e = J\left(\frac{2}{P}\right)\frac{d\omega_r}{dt}+B_m\left(\frac{2}{P}\right)\omega_r+T_L~~~~~~~~~~\text{(5)}$$ Where $J$ is the inertia in kg$\cdot$m$^2$ and $B_m$ is a damping coefficient. Note that similarity of $L_{ms}I^r_{fd}$ and $\lambda'_m$ of the permanent-magnet ac machine.


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