GATE Electrical Machines formula sheet

$E_{rms}$ in V, $f$ in Hz, $N$ = turns of that winding, ${\Phi }_m$ = peak core flux in Wb (${\Phi }_m=B_m{A}_i$, with $A_i$ the net iron cross-section). The derivation assumes the core flux is sinusoidal. It applies to the primary and secondary separately, each with its own $N$.

An ideal transformer has no winding resistance, no leakage flux, no core loss and an infinitely permeable core, so it needs no magnetising current. In a practical transformer, $V_1/V_2=a$ holds only approximately (closest at no load), and $I_2/I_1=a$ holds only when the no-load current $I_0$ is neglected.

Coenergy is written as a function of $i$ and $x$, and the partial derivative is taken with $i$ held fixed. This form is convenient when the excitation current is known.

Field energy is written as a function of $\lambda$ and $x$, and the partial derivative is taken with $\lambda$ held fixed. Note the minus sign. This gives the same force as the constant-current form.

$f$ = supply frequency (Hz), $P$ = number of poles. This is the speed of the rotating stator field. In rad/s: ${\omega }_s=\frac{2\pi n_s}{60}=\frac{4\pi f}{P}$.

$F_{max}$ is the peak MMF of one phase winding. Holds for a balanced three-phase winding carrying balanced three-phase currents. The resultant has constant amplitude and rotates at $n_s$.

$n_s$ 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.

$k_h$ is a material constant and $V$ is the core volume. The exponent (the Steinmetz coefficient) depends on the core material, and 1.6 is the classical value. A problem will state the exponent to use, or will say only that $P_h\propto f$ at constant $B_{max}$.

$t$ is the lamination thickness, and $k_e\propto 1/\rho$ (core resistivity). This form assumes sinusoidal flux and thin laminations. At the same $B_{max}$, $f$ and core volume, halving $t$ cuts $P_e$ to one quarter. This is why cores are laminated rather than solid.