In the context of modeling and analysis of electrical networks, the transfer admittances of a transformer can be thought of as the device’s impact on the off-diagonal elements of of the nodal admittance matrix Ybus.

## 4.2.1 Primary to Secondary

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)

## 4.2.2 Secondary to Primary

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)

## 4.2.3 Observations on Computational Sequence

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 gL sin $\delta$, gL cos $\delta$, bL sin $\delta$, bL cos $\delta$ and t2.