Three-phase circuit: star and delta

Enter the source voltage and phase impedances Za, Zb, Zc from R, XL, XC — get the neutral shift, phase and line currents, neutral current, P, Q and S per phase and in total, plus the circuit, topographic and phasor diagrams. The load may be unbalanced; a phase can be opened or shorted.

U, V

Rn, ΩXn, Ω

Phase impedances Z = R + j(XL − XC); for delta — branches AB, BC, CA
	R, Ω	X_L, Ω	X_C, Ω	State
Z_A
Z_B
Z_C

Source: positive sequence ABC (Ub lags Ua by 120°). Rn, Xn is the neutral wire impedance; empty means an ideal wire. A short circuit is modelled by 1 mΩ.

Calculate

Star and delta

In a star connection the line current equals the phase current and the line voltage is √3 times the phase voltage: Ul = √3·Uph. In a delta connection the phase voltage equals the line voltage and a line current is the difference of the adjacent phase currents; for a balanced load Il = √3·Iph.

Neutral shift

Without a neutral wire and with an unbalanced load the load neutral potential shifts: U(N'n) = (Ya·Ea + Yb·Eb + Yc·Ec) / (Ya + Yb + Yc), where Y = 1/Z. The phase voltages are no longer equal: some phases are overloaded, others underloaded. A low-impedance neutral wire fixes the neutral but carries In = Ia + Ib + Ic.

Open and short circuit

An open phase in a star without neutral leaves the other two in series across the line voltage. A shorted phase in a star without neutral raises the voltage on the other phases to the line voltage. In a delta an open branch does not change the other branch currents.

Power

For each phase S = U·I* = P + jQ, the total is the sum over phases. For a balanced load P = √3·Ul·Il·cos φ. For an unbalanced load the overall cos φ = P/S is only a figure of merit: it equals the cosine of none of the phase angles.

Topographic diagram

Every circuit point maps to a point on the complex plane, and the potential difference between two points is the vector between them. The diagram shows the line voltages (sides of triangle ABC), load phase voltages (from N' to the vertices) and the neutral shift (segment n–N').

FAQ

How do I calculate an unbalanced three-phase load?

Find the complex phase impedances, admittances Y = 1/Z and the neutral shift, then currents Ik = Yk·(Ek − Un). The calculator does it all and shows the steps.

What is the neutral wire current?

The phasor sum of the line currents: In = Ia + Ib + Ic. It is zero for a balanced load.

What happens when a phase opens?

In a star with neutral the other phases are unchanged and the neutral carries the open phase current with the opposite sign. In a star without neutral the two remaining phases end up in series across the line voltage.

How do I convert line voltage to phase voltage?

Uph = Ul / √3. For example 380 V line is 220 V phase.

How do I enter a capacitive load?

Enter XC (in ohms) — the net phase reactance is XL − XC; a negative value means a capacitive load.