Balanced three-phase transformers increase or decrease alternating current (AC) voltage using three wires or "lines" denoted as phase "a," "b" and "c." "Balanced" indicates that each line carries the same voltage magnitude, but their "phases," i.e. their locations along the characteristic sine wave of AC power, are equally spaced. This spacing provides for smooth power. There are certain calculations in transformer science when you want to know the difference in voltage from one line to another: the line-to-line voltage. We write the line-to-line voltages as Vab, Vbc and Vac, respectively. Depending on your given information, there are two ways to calculate them.

## Given Ia, Ib, Ic and Z

Subtract Ib from Ia. For example, 8 amps minus 6 amps equals 2 amps.

Multiply the result from Step 1 by the internal impedance Z. Using the previous example, 2 amps times 5 ohms (example) equals 10 volts.

Repeat steps 1 and 2 with (Ia - Ic) and (Ib - Ic). The products from repeating step 2 are the magnitudes of the line-to-line voltages Vab, Vac, and Vbc, respectively.

## Given Line-to-Neutral Voltages

Note that section 1's results only give magnitudes, omitting phases. This problem requires college-level understanding of power engineering.

Multiply the line-to-neutral voltages by the square root of 3 (approximately 1.73). Using an example Van value of 10 volts, Vab = 10 volts times 1.73 which equals 17.3 volts. This is Vab's magnitude.

Subtract 120 degrees from the phase of the line-to-neutral voltage. For example, if Van's phase angle is 150 degrees, then Vab's phase angle will be 30 degrees.

Repeat steps 1 and 2 for Vab, Vbc, and Vca, respectively. Combine the magnitudes from steps 1 with the phase angles from steps 2 to get the line-to-line voltages' magnitudes and phases.

#### Warnings

#### References

- "Power Systems Analysis, Second Edition"; Arthur J. Bergen and Vijay Vittal, 2000

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