# Synchronous machine power formula

An expression found in most texts on synchronous machines relates power to the power angle δ. δ is the angle between the terminal voltage and the generated EMF (under the applicable load current).

$$P=\frac{|E|\cdot|V|}{X_s}sin \delta\\$$

V and E are RMS values, albeit sometimes with phase.

This article gives a derivation that exposes the underlying assumptions that are not usually mentioned.

Above is the phasor diagram of the machine in generator mode.

$$sin \delta=\frac{I_a X_s}{|E|}$$

Substituting for sin δ:

$$P=\frac{|E|\cdot|V|}{X_s}\frac{I_a X_s}{|E|}\\$$

Simplifying:

$$P=|V|I_a\\$$

If Ia is purely real, this is valid, if and ONLY if.

So, this expression assume that the current Ia is in phase with the terminal voltage V. Perhaps an unusual scenario!

Whilst the formula includes the synchronous reactance Xs, it does not factor in the resistance of the armature ra.  ra is usually much smaller than Xs, so the error is small in that case… but there it is, another assumption ra<<Xs.