PW-C2 · Transmission and per-unit

Per-unit: the transformer ratio vanishes

A power system scatters the impedances of generators, transformers and lines across voltage levels, so in ohms you must convert by the square of the turns ratio every time you cross a transformer. Express every quantity as a ratio to a base (per-unit) and that conversion disappears. Drag the transformer ratio and watch the ohms swing while the per-unit value does not budge.

Drag the ratio and compare the two

One and the same secondary impedance, referred to the primary. The top bar is the ohm value (it grows as the square of the turns ratio a), the bottom bar is the per-unit value. Drag the handle to change a. The ohm bar swings, but the per-unit bar does not move from its place.

Drag the handle left and right to set the transformer ratio a.

Impedance referred to primary: 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 as ratios to their own base values. One base power S_base is fixed for the whole system, and a base voltage V_base for each voltage level. Then the base impedance is Z_base = V_base²/S_base, and the per-unit value of any impedance is Z_pu = Z/Z_base. Every quantity becomes a clean dimensionless number near 1, easy to compare and compute with.

The same across a transformer

The key is to set the base voltages in the transformer’s turns ratio. Then the per-unit value of one physical impedance is exactly the same seen from the primary or the secondary. In ohms the conversion multiplies by the square of the turns ratio a², so the value changes a lot, but in per-unit that a² multiplies the base impedance equally and cancels out. That is why the top bar swung with a² while the bottom bar held still.

ObserveZ_pu = Z / Z_base

Per-unit is impedance divided by base impedance.

ChooseZ_base = V_base² / ?

Base impedance is base voltage squared over base power.

Fill inZ_pu(1) ? Z_pu(2)

With bases in the turns ratio, per-unit matches on both sides.

On your own환산: Ω × a², pu × ?

On referral, ohms scale by a², per-unit by 1.

So the system unifies into one

Because the transformer ratio vanishes, impedances at different voltage levels can simply be added and compared once put in per-unit. That is why generator and transformer nameplates state impedance in percent or per-unit. The chore of converting at each voltage level in power-flow and fault studies disappears, and the whole system becomes one impedance diagram with no transformers. Per-unit is not mere notation but the language that makes system analysis possible.

Back to the first screen

The more you dragged the ratio a, the more the top ohm bar swung with a², yet the bottom per-unit bar never moved. The same a² multiplies both the referred impedance and the base impedance, so it cancels and disappears in per-unit. In the language of ohms the value jumps at every transformer, but in the language of per-unit the transformer is invisible. That is how the whole system can be handled as a single impedance diagram with no transformers.

The per-unit system — every quantity as a ratio to a base. From a common base power S_base and a per-zone base voltage V_base, Z_base = V_base²/S_base, Z_pu = Z/Z_base. Set V_base in the transformer turns ratio and the per-unit value of one impedance is the same on both sides of the transformer (an a² difference in ohms cancels in per-unit). So the system unifies into one impedance diagram with no transformer ratios.

The next step

With impedances unified in per-unit, it is time to quantify what happens when current flows through that line impedance. The next unit (PW-C3) looks at voltage drop and power loss: how the difference between sending and receiving voltage depends on the load current, the line Z and the power factor, and why the I²R loss the line resistance soaks up falls as the voltage is raised.