3-Phase Voltage Drop: Formula and Worked Example

For a balanced three-phase circuit the voltage drop is Vd = √3 × I × L × (R cos φ + X sin φ). I is the current in each line, L is the one-way length of the cable, R and X are the resistance and reactance of the conductor per unit length, and φ is the phase angle of the load. The result is the drop between two lines, so it is compared with the line-to-line voltage. For a 100 A load 50 m away on 35 mm² copper at 400 V, the formula gives about 4.98 V, which is 1.25%.

Where the √3 comes from

In a single-phase circuit the current goes out along one conductor and returns along another, so the length that counts is twice the route length, and the formula has a 2 in it. In a balanced three-phase circuit there is no return conductor carrying the full current: the three line currents cancel at the load. The drop along one line is I × L × (R cos φ + X sin φ), and the voltage between two lines is the difference of two such drops that are 120° apart, which comes to √3 times the drop along one line. That is why √3 (about 1.732) replaces the 2. The length is still entered once.

The steps

1. Find R at the operating temperature. Cable tables give resistance at 20 °C. Correct it with R_t = R₂₀ × [1 + α (t − 20)], where α is 0.00393 per °C for copper.
2. Find the impedance term z = R cos φ + X sin φ. R and X are per kilometre here, so the length is in kilometres in the next step.
3. Multiply: Vd = √3 × I × L × z.
4. Turn it into a percentage of the line-to-line supply voltage: Vd ÷ V × 100.

Worked example: 100 A, 50 m of 35 mm² copper, 400 V, power factor 0.85

The cable is at 70 °C. Its resistance at 20 °C is the IEC 60228 maximum for 35 mm² copper, 0.524 Ω/km, and the reactance is taken as 0.08 Ω/km, a typical figure for multicore cable. These are the numbers thevoltage drop calculator uses.

What the power factor does here

With a power factor of 1 the same cable gives z = 0.627, so Vd = √3 × 100 × 0.050 × 0.627 = 5.43 V, or 1.36%. A power factor of 0.85 gives a slightly lower drop, 1.25%, because for this cable the resistance is much larger than the reactance, so the small reactive term adds less than the resistive term loses. For cables with a larger reactance, or much larger conductors, the effect can go the other way, which is why the calculator takes the power factor and the reactance as inputs.

Common mistakes

• Doubling the length. The √3 already stands in for the return path. Enter the one-way route length.
• Using 230 V for a three-phase percentage. The √3 formula gives a line-to-line drop, so divide by the line-to-line voltage, such as 400 V.
• Using the 20 °C resistance for a loaded cable. A cable at its working temperature has a higher resistance, so the drop is larger. The calculator uses the maximum operating temperature for the insulation, which errs on the high side.
• Treating the answer as a cable size. The result is the voltage drop only. It does not say that the cable can carry the current without overheating, which depends on the installation and is a separate check.

Checking the result

The voltage drop calculator does these steps for copper or aluminium in metric or imperial sizes and shows the working. For what the percentage is compared with, see the guide onvoltage drop limits. The results are estimates, so check them against your own data and a qualified person before relying on them.