Power Quality in Power Systems and Electrical Machines, 2nd Edition

4.3.4.2 Synchronous Machine Harmonic Model Based on abc Coordinates

To interconnect the synchronous machine model of Eq. 4-66 with the rest of the system, the dq0 voltages and currents must be converted to the corresponding phase abc quantities using Park’s transformation. Assume that the machine voltage [v_abc(t)] vector consists of a single harmonic component of order h:

$\left[v_{\mathit{abc}}\left(t\right)\right]=\mathrm{Real}\left\{\sqrt{2}\left[V_{\mathit{abc}}\left(h\right)\right]e^{\mathit{ih\omega t}}\right\}.$

  (4-67)

The corresponding dq0-voltage vector is obtained by Park’s transformation (Eq. 4-58) as

$\begin{array}{ll}\left[v_{\mathit{dqo}}\left(t\right)\right]& ={\left[P\right]}^{-1}\left[V_{\mathit{abc}}\left(t\right)\right]=\mathrm{Real}\left\{{\left[P\right]}^{–1}\sqrt{2}\left[V_{\mathit{abc}}\left(h\right)\right]e^{\mathit{jh\omega t}}\right\}\\ =\sqrt{2}\mathrm{Real}\{\left[D\right]\left[V_{\mathit{abc}}\left(h\right)\right]e^{j\left(h+1\right)\omega t}+{\left[D\right]}^C.\\ \left[V_{\mathit{abc}}\left(h\right)\right]e^{j\left(h-1\right)\omega t}+\left[D_o\right]\left[V_{\mathit{abc}}\left(h\right)\right]e^{j\omega t}\}\end{array}$

  (4-68)

Therefore, one harmonic abc voltage introduces three harmonics ...