Transformer Fundamentals

The ideal transformer is a ratio: volts scale by the turns, amps by the inverse, and impedance by the square. The real machine adds a series impedance and two loss mechanisms — and that one nameplate number, %Z, ends up setting both how far the voltage droops under load and how hard the secondary can hit a bolted fault. Load the transformer below and watch regulation, efficiency, and fault duty move together.

Nameplate

13 800 Δ — 480Y/277 V · 60 Hz
a = N₁/N₂ = 28.75

Impedance %Z
X/R ratio
No-load (core) loss

Core loss burns whenever the primary is energized — load or no load. Copper loss is I²R and grows with the square of load.

Load
Loading
Power factor
PF sense
Equivalent Circuit

Approximate equivalent circuit in per-unit. The shunt branch models the core — Rc carries the no-load loss, jXm the magnetizing current — and the series R + jX is the nameplate %Z split by the X/R ratio (colored to match the drop vectors in the phasor diagram). The ideal 28.75 : 1 transformer on the right is pure ratio arithmetic.

Secondary Phasor Diagram

VR (received voltage) is the reference. The I·R and j·I·X drop vectors are drawn ×5 actual size so the drop triangle is visible — real drops are only a few percent of V.

Efficiency & Regulation vs Load
Efficiency η (%) Voltage regulation (% · plotted ×5 for visibility) Operating load Max-η load (Pcu = Pfe)
Readout

The Physics

The ideal transformer is pure ratio arithmetic. With a = N₁/N₂ turns, voltage scales down by a, current scales up by a, and an impedance seen through the transformer scales by a²:

a = N₁/N₂  ·  V₂ = V₁/a  ·  I₂ = a·I₁  ·  Z′ = a²·Z

The real machine adds a series impedance — winding resistance plus leakage reactance — quoted on the nameplate as %Z (the voltage needed to push rated current through a shorted secondary). Split it with the X/R ratio: R = Z·cos θZ, X = Z·sin θZ. Load current through that impedance is what makes the secondary voltage move:

reg ≈ I·(R·cos φ ± X·sin φ)   (+ lagging, − leading; this page iterates the exact phasor sum instead)

With lagging current the I·X drop lines up with the voltage and the secondary droops. With leading current the reactive drop reverses and the secondary can rise above its no-load value — negative regulation, the same mechanism as Ferranti rise on a lightly-loaded cable.

Losses live in two places. Hysteresis and eddy currents in the core steel cost a fixed Pfe from the moment the primary is energized; I²R heating in the copper costs Pcu ∝ load². Efficiency peaks exactly where the two are equal:

η = Pout / (Pout + Pfe + Pcu)  ·  max η at load = √(Pfe / Pcu,100%)

%Z is the designer's tradeoff. The same impedance that softens the voltage under load is all that limits a bolted secondary fault (with a stiff primary source):

ISC ≈ Irated · 100 / %Z   (infinite primary bus)
⚡ Low %Z buys stiff voltage but brutal fault current; high %Z is the reverse. Slide %Z from 2% to 12% and watch: regulation improves ~6× at one end while ISC swells ~6× at the other. That is why a 5–6% impedance is such a common compromise on unit substation transformers — and why breaker interrupting ratings and transformer impedance get chosen together.