Recalling that the secondary voltage of a transformer is a function of the tap ratio

 $V_{s}=tV_{p}$

implies that

 ${V}_{s}^{new}={t}^{new}{V}_{p}$

and

 ${V}_{s}^{old}={t}^{old}{V}_{p}$

Substituting these values into Equation 40 yields

 $t^{new}V_{p}=t^{old}V_{p}+\alpha\Delta vt^{old}V_{p}$

which simplifies to

 $t^{new}=t^{old}+\alpha\Delta vt^{old}$ (41)

To obtain a computational formula, substitute Equation 39

 $t^{new}=t^{old}+\alpha t^{old}\frac{V_{k}^{sp}-V_{k}}{V_{k}}$

and simplify as follows

 $t^{new}=t^{old}+\alpha t^{old}(\frac{V_{k}^{sp}}{V_{k}}-1)=t^{old}+\alpha t^{% old}\frac{V_{k}^{sp}}{V_{k}}-\alpha t^{old}$

or

 $t^{new}=t^{old}(1-\alpha+\alpha\frac{V_{k}^{sp}}{V_{k}})$ (42)

Normally, $\alpha$ is assumed to be one and Equation 42 reduces to

 $t^{new}=t^{old}\left(\frac{V_{k}^{sp}}{V_{k}}\right)$ (43)

Computation of $\alpha$, the sensitivity of the regulated voltage to transformer tap changes, is discussed in Section 5.3.