4 Computing Admittances

This section examines Equation 17 through Equation 20 with an eye for streamlining their computation. The resulting computational sequence explicitly decomposes the complex equations into real operations.

A transformer’s leakage admittance YL is derived from the leakage impedance ZL in the usual manner. Expressing the leakage impedance in rectangular form

𝐙𝐋=rL+jxL\mathbf{Z_{L}}=r_{L}+jx_{L} (21)


𝐘𝐋=1𝐙𝐋=1rL+jxL=gL+jbL\mathbf{Y_{L}}=\frac{1}{\mathbf{Z_{L}}}=\frac{1}{r_{L}+jx_{L}}=g_{L}+jb_{L} (22)


gL=rLrL2+xL2g_{L}=\frac{r_{L}}{r_{L}^{2}+x_{L}^{2}} (23)
bL=-xLrL2+xL2b_{L}=-\frac{x_{L}}{r_{L}^{2}+x_{L}^{2}} (24)

For the sake of convenience, define the tap ratio as

𝐭=1𝐚=tej-δ\mathbf{t}=\frac{1}{\mathbf{a}}=te^{j-\delta} (25)
Transformer Admittances in Rectangular Form