4.2 Transfer Admittances

Equation 19 defines the transfer admittance from the transformer’s primary to its secondary as

$$\mathbf{{Y}_{ps}=-\frac{Y_{L}}{a^{\star}}}$$

(31)

Substituting Equation 25 and resolving the tap ratio into polar form

$$\mathbf{Y_{ps}=-t^{\star}Y_{L}}=-te^{j\delta}\mathbf{Y_{L}}$$

Expressing the tap ratio and leakage admittance in rectangular form

$$\mathbf{Y_{ps}}=-t(\cos\delta+j\sin\delta)(g_{L}+jb_{L})$$

Carrying out the multiplications then collecting real and imaginary terms

$$\mathbf{Y_{ps}}=-t((-{b_{L}}\sin\delta+{g_{L}}\cos\delta)+j({g_{L}}\sin\delta+{b_{L}}\cos\delta))$$

Therefore,

$$\mathbf{{Y}_{ps}}=-t({g}_{ps}+j{b}_{ps})$$

(32)

where

$$g_{ps}=-{b_{L}}\sin\delta+{g_{L}}\cos\delta$$

(33)

and

$$b_{ps}={g_{L}}\sin\delta+{b_{L}}\cos\delta$$

(34)

Equation 20 defines the transfer admittance from the transformer’s secondary to its primary as

$$\mathbf{Y_{sp}=-\frac{Y_{L}}{a}}$$

(35)

Substituting Equation 25 and resolving the tap ratio into polar form

$$\mathbf{Y_{sp}=-t^{\star}Y_{L}}=-te^{j\delta}\mathbf{Y_{L}}$$

Expressing the tap ratio and leakage admittance in rectangular form

$$\mathbf{Y_{sp}}=-t(\cos(-\delta)+j{\sin(-\delta)})(g_{L}+jb_{L})$$

Invoking the trigonometric identities $\cos(-\delta)=\cos\delta$ and $\sin(-\delta)=-\sin\delta$ the equation becomes

$$\mathbf{Y_{sp}}=-t(\cos\delta-j\sin\delta)(g_{L}+jb_{L})$$

Carrying out the multiplications then collecting real and imaginary terms

$$\mathbf{Y_{sp}}=-t(({b_{L}}\sin\delta+{g_{L}}\cos\delta)+j(-{g_{L}}\sin\delta+{b_{L}}\cos\delta))$$

Therefore,

$$\mathbf{Y_{sp}}=-t(g_{sp}+jb_{sp})$$

(36)

where

$$g_{sp}={b_{L}}\sin\delta+{g_{L}}\cos\delta$$

(37)

and

$$b_{sp}=-{g_{L}}\sin\delta+{b_{L}}\cos\delta$$

(38)

The preceding results suggest the computational sequence is not too crucial. The values sin $\delta$ and cos $\delta$ appearing in Equation 32 and Equation 36 should only be computed once. Marginal benefits may also accrue from a single computation of the terms g_L sin $\delta$, g_L cos $\delta$, b_L sin $\delta$, b_L cos $\delta$ and t².