Per Unit Quantities

The performance of a line of machines can be readily compared, and the analyses of complex power systems involving transformers of different ratios can be facilitated, by means of per unit quantities. Per unit quantities are decimal fractions that relate current, voltage, impedance, admittance, and power to certain base values of these respective quantities, i.e.

[6-97]

[6-98]

[6-99]

[6-100]

[6-101]

The values of the base quantities are dictated by convenience. In dealing with a specific piece of apparatus by itself, as in the case of a transformer, the rating of the transformer is generally taken as the base. In the transformer of Example 6-2, the following are the base quantities on the low-voltage side

Similarly, the base quantities on the high-voltage side are 150000 v, 2400 v, 62.5 amp, 38.4 ohms, and 0.02604 mho.

The leakage impedance of this transformer was found in Example 6-4 to be 1.008 ohm referred to the high-voltage side, and the per unit impedance from Eq. 6-99 is

On the other hand, when the analysis is applied to a number of equipments in combination, a larger kva base is used.

The leakage impedance of the transformer referred to the low side is that referred to the high side divided by the impedance ratio, and, in this case, is 1.008 / (10)² = 0.01008 ohm. The per unit impedance of the transformer referred to the low side is, therefore, 0.01008 ohm divided by the transformer leakage impedance referred to the low side, i.e., 0.384 ohm, which results in the same value as before, namely, 0.0263 per unit. This brings out an important advantage of the per unit system with regard to transformers, in that the per unit impedance has the same value whether referred to the high side or to the low side, regardless of the turns ratio of the transformer.

In the analysis of power systems encompassing several pieces of equipment of similar or of different ratings, it is generally more convenient to use a relatively large kva base, which may be several times the rating of any one of the generators or transformers in the system. The per unit values of generators and transformers are generally known in terms of their own rating and can be conveniently converted to any other base on the principle that the per unit value of an impedance, for a given base voltage, is proportional to the volt-ampere base. Thus, if an impedance has a per unit value of Z_pu1 for a base of VA₁ va, then its per unit value for a base of VA₂ va for the same voltage base, is expressed by

[6-102]

Thus, the per unit impedance of the transformer in Example 6-2 for a base of 10000 kva would be