Question 5.10: In the simplified theory of Section 5.2, the synchronous mac......
In the simplified theory of Section 5.2, the synchronous machine is assumed to be representable by a single reactance, the synchronous reactance X_s of Eq. 5.25. The question naturally arises: How serious an approximation is involved if a salient-pole machine is treated in this simple fashion? Suppose that a salient-pole machine were treated by cylindrical-rotor theory as if it had a single synchronous reactance equal to its direct-axis value X_d ? To investigate this question, we will repeat Example 5.9 under this assumption.
\begin{aligned}X_s &=ω_e L_s=ω_eL_{\mathrm{al}}+ω_e\left( \frac{3}{2}L_{aa0}\right)\\& = X_{\mathrm{al}}+X_φ \quad \quad \quad \quad \quad \quad (5.25)\end{aligned}
In this case, under the assumption that
X_q = X_d = X_s = 1.0 \text{per unit}
the generated voltage can be found simply as
\begin{aligned}\hat{E}_{af} & =V_a + jX_s\hat{I}_a \\& = 1.0 + j 1.0(1.0 e^{-j36.9°}) = 1.79 e^{j26.6°}\end{aligned} per unit
Comparing this result with that of Example 5.9 (in which we found that E_{af} = 1.77 e^{j19.4°}), we see that the magnitude of the predicted generated voltage is relatively close to the correct value. As a result, we see that the calculation of the field current required for this operating condition will be relatively accurate under the simplifying assumption that the effects of saliency can be neglected.
However, the calculation of the power angle δ (19.4° versus a value of 26.6° if saliency is neglected) shows a considerably larger error. In general, such errors in the calculation of generator steady-state power angles may be of significance when studying the transient behavior of a system including a number of synchronous machines. Thus, although saliency can perhaps be ignored when doing “back-of-the-envelope” system calculations, it is rarely ignored in large-scale, computer-based system studies.