For analysis purposes, each transformer is described by the admittances of a general two-port network. When the current equations of Figure 1 are written as follows, the transformer admittances correspond to the coefficient matrix.

 $\displaystyle\mathbf{I_{p}}$ $\displaystyle=\mathbf{Y_{pp}V_{p}+Y_{ps}V_{s}}$ (13) $\displaystyle\mathbf{I_{s}}$ $\displaystyle=\mathbf{Y_{sp}V_{p}+Y_{ss}V_{s}}$ (14)

where

Ypp is the driving point admittance of the primary.

Yss is the driving point admittance of the secondary.

Yps is the transfer admittance from the primary to the secondary.

Ysp is the transfer admittance from the secondary to the primary.

Noting that

 $\mathbf{Y_{L}}=\frac{1}{\mathbf{Z_{L}}}$

and

 $\mathbf{Y_{M}}=\frac{1}{\mathbf{Z_{M}}}$

It is apparent from inspection of Figure 1 that

 $\mathbf{a^{\star}I_{p}=\left(\frac{V_{p}}{a}-{V}_{s}\right)Y_{L}+Y_{M}\frac{V_% {p}}{a}}$

and

 $\mathbf{I_{s}=\left(V_{s}-\frac{V_{p}}{a}\right)Y_{L}}$

Converting these equations to the desired form

 $\displaystyle\mathbf{I_{p}}$ $\displaystyle=\mathbf{\left(\frac{1}{aa^{\star}}\right)\left(Y_{L}+Y_{M}\right% )V_{p}-\left(\frac{Y_{L}}{a^{\star}}\right)V_{s}}$ (15) $\displaystyle\mathbf{I_{s}}$ $\displaystyle=\mathbf{-\left(\frac{Y_{L}}{a^{\star}}\right)V_{p}+Y_{L}V_{s}}$ (16)

Recalling that aa${}^{\star}$= a2 and equating the coefficients in Equation 13 and Equation 14 with their counterparts in Equation 15 and Equation 16, yields the transformer’s admittances:

 $\displaystyle\mathbf{Y_{pp}}$ $\displaystyle=\frac{\mathbf{Y_{L}+Y_{M}}}{a^{2}}$ (17) $\displaystyle\mathbf{Y_{ss}}$ $\displaystyle=\mathbf{Y_{L}}$ (18) $\displaystyle\mathbf{Y_{ps}}$ $\displaystyle=\mathbf{-\frac{Y_{L}}{a^{\star}}}$ (19) $\displaystyle\mathbf{Y_{sp}}$ $\displaystyle=\mathbf{-\frac{Y_{L}}{a}}$ (20)

In Equation 17 the real number a is the absolute value of the complex voltage ratio. When the magnetizing impedance is quite large, YM approaches zero and Equation 17 simplifies to

 $\mathbf{Y_{pp}}=\frac{\mathbf{Y_{L}}}{a^{2}}$

The following sections examine data and computations required to incorporate this model into balanced power system analyses.