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.

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

where

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, Y_M 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.