Transformers: Equivalent Circuit, Testing and Regulation

A transformer transfers AC power between two circuits through magnetic coupling, changing voltage up or down. Its equivalent circuit — resistors and inductors modelling winding resistance, leakage flux and magnetising behaviour — is what engineers use for testing and for computing voltage regulation.

Transformers matter because the whole power grid depends on them: voltage steps up to hundreds of kilovolts for transmission and back down to 230 V for homes. The equivalent circuit predicts output sag under load before the transformer is ever built.

Key Concepts

• Turns ratio (n): n = N1/N2; V1/V2 = n and I1/I2 = 1/n for an ideal transformer.
• Referring to primary: secondary quantities scale as R2′ = n²·R2, X2′ = n²·X2, V2′ = n·V2, so the transformer looks like one primary-side circuit.
• Leakage reactance (X1, X2): flux linking only one winding; causes voltage drop under load.
• Magnetising branch (Rc, Xm): the shunt branch drawing no-load current — Rc models core loss, Xm the magnetising current.
• Open-circuit (OC) test: secondary open, rated voltage applied — measures core loss and no-load current.
• Short-circuit (SC) test: secondary shorted, reduced voltage applied until rated current flows — measures copper loss and leakage impedance.
• Voltage regulation: (V2_nl − V2_fl)/V2_fl × 100% — the percentage rise from full load to no load.

Formulas

OC test: core loss Pc = P0; no-load current splits into Ic = V0/Rc and Im = V0/Xm. SC test: full-load copper loss Pcu = Psc; Zeq = Vsc/Isc, Req = Psc/Isc², Xeq = √(Zeq² − Req²), all in ohms. Approximate regulation: %Reg = I1·(Req·cos φ ± Xeq·sin φ)/V1 × 100%, with + for lagging and − for leading power factor. Efficiency: η = output/(output + Pc + Pcu) × 100%, maximum when core loss equals copper loss. The approximate circuit neglects the shunt branch under load, valid since no-load current is only 2–5% of full-load current.

Worked Example

A 50 kVA, 11000/415 V transformer: OC test on the 415 V side reads 415 V, 2.5 A, 400 W; SC test on the 11000 V side reads 500 V, 4.55 A, 900 W. Core loss Pc = 400 W. Full-load HV current = 50000/11000 = 4.55 A — the SC test ran at rated current, so Pcu = 900 W. Zeq = 500/4.55 = 109.9 Ω, Req = 900/4.55² = 43.5 Ω, Xeq = √(109.9² − 43.5²) = 100.9 Ω. At full load, 0.8 pf lagging: %Reg = 4.55 × (43.5 × 0.8 + 100.9 × 0.6)/11000 × 100 = 3.94%. Efficiency at 0.8 pf: output = 40000 W; η = 40000/(40000 + 400 + 900) = 96.85%.

Common Misconceptions

• "The OC test measures copper loss." Reversed — with the secondary open, negligible winding current flows, so the wattmeter reads core loss. The SC test at rated current reads copper loss.
• "Regulation is always positive." With a leading load the −Xeq·sin φ term can make it negative — full-load voltage rises above no-load voltage.
• "Transformers work on DC." They do not — steady DC makes no changing flux, so nothing is induced in the secondary, and the winding's low resistance draws a damaging current.

Real-World Applications

Distribution transformers step 11 kV to 415/230 V for neighbourhoods. Furnace and welding transformers deliver huge low-voltage currents. Instrument transformers (CTs, PTs) scale current and voltage for meters and relays. Audio output transformers match amplifiers to speakers, and phone chargers hold tiny high-frequency transformers for isolation and step-down.

The equivalent circuit turns magnetic complexity into a few resistors and inductors — test it open, test it shorted, and regulation and efficiency follow for any load.