3 Transformer Admittance Model

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.

𝐈𝐩\displaystyle\mathbf{I_{p}} =𝐘𝐩𝐩𝐕𝐩+𝐘𝐩𝐬𝐕𝐬\displaystyle=\mathbf{Y_{pp}V_{p}+Y_{ps}V_{s}} (13)
𝐈𝐬\displaystyle\mathbf{I_{s}} =𝐘𝐬𝐩𝐕𝐩+𝐘𝐬𝐬𝐕𝐬\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

𝐘𝐋=1𝐙𝐋\mathbf{Y_{L}}=\frac{1}{\mathbf{Z_{L}}}

and

𝐘𝐌=1𝐙𝐌\mathbf{Y_{M}}=\frac{1}{\mathbf{Z_{M}}}

It is apparent from inspection of Figure 1 that

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

and

𝐈𝐬=(𝐕𝐬-𝐕𝐩𝐚)𝐘𝐋\mathbf{I_{s}=\left(V_{s}-\frac{V_{p}}{a}\right)Y_{L}}

Converting these equations to the desired form

𝐈𝐩\displaystyle\mathbf{I_{p}} =(𝟏𝐚𝐚)(𝐘𝐋+𝐘𝐌)𝐕𝐩-(𝐘𝐋𝐚)𝐕𝐬\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)
𝐈𝐬\displaystyle\mathbf{I_{s}} =-(𝐘𝐋𝐚)𝐕𝐩+𝐘𝐋𝐕𝐬\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:

𝐘𝐩𝐩\displaystyle\mathbf{Y_{pp}} =𝐘𝐋+𝐘𝐌a2\displaystyle=\frac{\mathbf{Y_{L}+Y_{M}}}{a^{2}} (17)
𝐘𝐬𝐬\displaystyle\mathbf{Y_{ss}} =𝐘𝐋\displaystyle=\mathbf{Y_{L}} (18)
𝐘𝐩𝐬\displaystyle\mathbf{Y_{ps}} =-𝐘𝐋𝐚\displaystyle=\mathbf{-\frac{Y_{L}}{a^{\star}}} (19)
𝐘𝐬𝐩\displaystyle\mathbf{Y_{sp}} =-𝐘𝐋𝐚\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

𝐘𝐩𝐩=𝐘𝐋a2\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.