Three-phase (EE) formulas

Phase b lags phase a by $120^{\circ }$ and phase c lags phase b by $120^{\circ }$. For negative sequence (acb) the angles of $V_{bn}$ and $V_{cn}$ are interchanged: $V_{bn}=V_p\angle +120^{\circ }$, $V_{cn}=V_p\angle -120^{\circ }$.

For abc: $V_{bn}=a^2{V}_{an}$ and $V_{cn}=a{V}_{an}$. Hence any balanced set sums to zero, $V_{an}+V_{bn}+V_{cn}=0$, and the same holds for a balanced set of currents.

Each line voltage leads the phase voltage of its first subscript by $30^{\circ }$ ($V_{bc}$ leads $V_{bn}$, $V_{ca}$ leads $V_{cn}$). For negative sequence the magnitudes are unchanged but $V_{ab}=\sqrt{3}{V}_{an}\angle -30^{\circ }$.

Phase currents $I_{ab}$, $I_{bc}$, $I_{ca}$ are taken in the direction of their subscripts; each line current lags, by $30^{\circ }$, the phase current leaving that line's corner ($I_b$ lags $I_{bc}$, $I_c$ lags $I_{ca}$). For negative sequence, $I_a=\sqrt{3}{I}_{ab}\angle +30^{\circ }$.

Identical impedances only. The division is by a real number, so the impedance angle $\varphi$, and hence the power factor, is unchanged.

Balanced load, star or delta; rms values; $\varphi$ = angle of the per-phase impedance. Pair the delta resistance with the delta phase current, or the star-equivalent resistance with the line current.

$Q>0$ (absorbed) for an inductive, lagging load; $Q<0$ for a capacitive, leading load.

$V_{ph}$ and $I_{ph}$ are the phasors of the same phase and ${}^{\ast }$ is the complex conjugate. $|S|$ is in VA (or kVA); a three-phase kVA figure means this total for all three phases, not the per-phase value.

Balanced sinusoidal voltages and a balanced load; $V_{ph}$, $I_{ph}$ rms. The total does not vary with time and equals the average power $P$.

Readings are taken with their signs. Valid for balanced and unbalanced three-wire loads.

Balanced sinusoidal system. The current coils, in lines a and c, carry the line currents towards the load; the pressure coils measure $V_{ab}$ and $V_{cb}$ (each coil's $\pm$ terminal on its own current-coil line, the other end on line b). For abc sequence the meter in line c reads $W_1$ and the meter in line a reads $W_2$. The formulas hold for either sign of $\varphi$ ($\varphi >0$ lagging, $\varphi <0$ leading), so a leading load, or a reversed sequence, swaps which meter reads more.

The same pair written with $P=\sqrt{3}{V}_L{I}_L\cos \varphi$ and $Q=\sqrt{3}{V}_L{I}_L\sin \varphi$; handy for predicting the readings when $P$ and $Q$ (or $P$ and the power factor) are given. $W_2<0$ exactly when $Q>\sqrt{3}P$, i.e. $\varphi >60^{\circ }$.

Balanced load only. Follows from $W_1+W_2=\sqrt{3}{V}_L{I}_L\cos \varphi$ and $W_1-W_2=V_L{I}_L\sin \varphi$. The two readings alone fix $|\varphi |$; whether the load lags or leads needs either the nature of the load or knowledge of which physical meter reads $W_1$, which depends on the connection and the phase sequence.

Balanced load; the same content as the $\tan \varphi$ form. Insert a negative reading with its sign.

Three-wattmeter method: one meter per line, each pressure coil from its line to the neutral, so each meter reads one phase power; valid for balanced and unbalanced loads. One-wattmeter method: for a balanced load, a single meter with its current coil in one line and its pressure coil from that line to the neutral (or to an artificial neutral formed by three equal impedances, one of them the pressure coil) reads one-third of the total.

Total for all three phases; ${\varphi }_1$ is the original angle and ${\varphi }_2$ the target angle, both lagging. For a unity target, $Q_C$ equals the original $Q$.

$V_L$ = line voltage (rms), $\omega =2\pi f$. Each star capacitor sees $V_L/\sqrt{3}$ and supplies $Q_C/3=\omega C_Y{V}_L^2/3$; each delta capacitor sees the full $V_L$ and supplies $Q_C/3=\omega C_{\Delta }{V}_L^2$.

Nodal equation at N', with the source phase voltages measured from N; any line impedance is included in $Z_k$. It is zero when $Y_a=Y_b=Y_c$, which recovers the balanced case. If N' and N are joined through a neutral impedance $Z_n$, add $Y_n=1/Z_n$ to the denominator.