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PW-C2 · Transmission and per-unit

Per-unit: the transformer ratio vanishes

The impedances of generators, transformers and lines are scattered over several voltage levels in a system, so in ohms they must be converted by the square of the turns ratio every time a transformer is crossed. Turning every quantity into a ratio to a base value (per-unit) makes that conversion disappear; drag the transformer ratio yourself and see the ohm value swing widely while the per-unit value does not budge.

Drag the ratio and compare the two

One and the same load-side impedance is referred to the source side. The upper bar is the ohm value, which grows with the square of the turns ratio a, and the lower bar is the per-unit value. Dragging the handle changes a, and the ohm bar swings widely while the per-unit bar does not budge from its place.

Drag the handle left and right to change the transformer ratio a between 1 and 10. The load side is 0.5 Ω at 22.9 kV and the base power is 100 MVA, so the per-unit value is fixed at 0.5 ÷ (22.9² ÷ 100) ≈ 0.095.
Impedance referred to the source side: ohms vs per-unit
Z = 0.095 pu (per-unit unchanged)
a ≈ 6.7 Z' ≈ 22.4 Ω (= a² × 0.5 Ω)

A ratio to a base

The per-unit system expresses voltage, current, impedance and power each as a ratio to its own base value. Choose one base power S_base for the whole system and a base voltage V_base for each voltage level, and the impedance base is Z_base = V_base²/S_base, while the per-unit value of any impedance is Z_pu = Z/Z_base. Every quantity becomes a tidy dimensionless number near 1, which makes comparison and calculation easy.

The same across a transformer

The key lies in choosing the base voltages according to the transformer turns ratio. Then the per-unit value of the same physical impedance is exactly the same whether seen from the source side or the load side. In ohms, referring multiplies by the square of the turns ratio, a², and changes the value greatly, but in per-unit the same a² multiplies the base impedance as well and cancels out. That is why the upper bar you dragged swung with a² while the lower bar did not budge.

ObserveZpu = Z / Zbase
Per-unit is an impedance divided by the base impedance.
ChooseZbase = Vbase² / ?
Base impedance is base voltage squared over base power.
Fill inZpu(1) ? Zpu(2)
With bases set by the turns ratio, the per-unit values on both sides are equal.
On your ownReferred: Ω × a², pu × ?
When referring, ohms change by a², per-unit by 1.

So the system unifies into one

Because the transformer ratio disappears, impedances at different voltage levels can simply be added and compared once they are in per-unit. That is also why generator and transformer nameplates give impedance in % or per-unit, and why load-flow and fault calculations no longer need tedious conversion at every voltage level: the whole system becomes one impedance diagram without transformers. The per-unit system is not mere notation but the language that makes system analysis possible.

Back to the first screen

The more the transformer ratio a was dragged, the more the upper ohm bar swung with a², yet the lower per-unit bar did not budge from its place. The same a² multiplies both the referred impedance and the base impedance, so it cancels out in per-unit. In the language of ohms the value jumps at every transformer, but in the language of per-unit the transformer becomes invisible, so the whole system can be handled as a single impedance diagram without transformers.

Per-unit system: every quantity is expressed as a ratio to a base value. With a common base power Sbase and a base voltage Vbase for each zone, Zbase = Vbase²/Sbase and Zpu = Z/Zbase. Choosing Vbase by the transformer turns ratio makes the per-unit value of the same impedance equal on both sides of the transformer, and the a² gap that opens in ohms cancels out in per-unit. So the system is unified into one impedance diagram without transformer ratios.
The next step

With impedances unified in per-unit, it is time to put numbers on what happens when current flows through that line impedance. The next unit (PW-C3) covers voltage drop and power loss: how the difference between sending-end and receiving-end voltage depends on the load current, the line Z and the power factor, and why the I²R loss that line resistance eats up shrinks as the voltage is raised for the same power.