Synchronous machines formulas

$n_s$ in rpm, $f$ = supply or generated frequency in Hz, $P$ = number of poles. In steady-state operation the rotor and the rotating armature field both turn at this speed. Unlike an induction machine, there is no slip.

$k_w$ = winding factor ($\le 1$), $\Phi$ = fundamental flux per pole (Wb), $T$ = series turns per phase. All quantities are per-phase RMS values of the fundamental, for sinusoidally distributed flux.

$\alpha$ = the angle, in electrical degrees, by which the coil span falls short of full pitch ($180^{\circ }$). For a full-pitch coil, $\alpha =0$ and $k_p=1$.

$m$ = slots per pole per phase. $\beta$ = slot pitch in electrical degrees $=180^{\circ }/(\text{slots per pole})$.

All quantities are per-phase phasors, and $j$ is the imaginary unit. $I$ lags $V$ by the power-factor angle $\theta$ for a lagging load and leads it for a leading load. The equation is valid for a round rotor, where a single $X_s$ applies whatever the rotor position.

Motor convention: $I$ is the current drawn from the supply and $E$ is the back emf. In motoring operation $E$ lags $V$ by the load angle $\delta$.

$E$ = the terminal voltage on no load, with the field current and speed unchanged, when that field current gives rated $V$ at the specified load. $V$ = rated terminal voltage. Both are per phase. Regulation is positive for lagging-pf loads (the terminal voltage falls on loading) and can be zero or negative for leading-pf loads.

Here $X_d$ is the saturated direct-axis (synchronous) reactance in per unit, taken from the OCC at rated voltage. A higher SCR means a smaller synchronous reactance, a larger maximum power ($EV/X_s$), better steady-state stability and a smaller voltage change with load, at the cost of a larger machine.

Per-phase power in W, with per-phase $E$ and $V$. Multiply by 3 for the total 3-phase power. Valid for round-rotor machines only. For a generator, $\delta >0$ when $E$ leads $V$.

Per phase. $E$ lies along the q-axis. $I_d$ and $I_q$ are the components of $I$ perpendicular to and along $E$.

Per phase. $\theta$ = power-factor angle (lagging). Method: form $E^{\prime }=V+jI{X}_q$, whose angle is $\delta$, then add $I_d(X_d-X_q)$ along the same direction to get $E$. For a leading pf, replace $\theta$ with $-\theta$.

Per phase. The first term is the excitation power: it has the same form as for a round rotor, with $X_d$ in place of $X_s$. The second term is the reluctance power. It arises only from saliency ($X_d\ne X_q$) and is present even with zero excitation ($E=0$). With $E=0$ it is the only term, and its maximum is at $\delta =45^{\circ }$. With $E>0$ and $X_d>X_q$, maximum power occurs at some $\delta <90^{\circ }$ that depends on the excitation.

Per phase. The rotor is driven slightly away from synchronous speed with the field winding open, and a reduced balanced 3-phase voltage is applied to the armature. The armature voltage and current swing slowly. $V_{max}$ and $I_{min}$ occur when the armature mmf lies along the d-axis, and $V_{min}$ and $I_{max}$ when it lies along the q-axis.

Per phase, cylindrical rotor, $R_a\approx 0$, in W per electrical radian at operating angle $\delta$. It is the restoring power per unit change in $\delta$ when synchronism is briefly disturbed. Stable operation needs $P_{sy}>0$, which is why a cylindrical rotor is limited to $\delta <90^{\circ }$.