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Equivalent circuit + OC and SC tests

Two simple lab tests that pin down every parameter of the transformer's equivalent circuit.

Freshman ~9 min

Step 1 — The full equivalent circuit of a transformer

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Reference notes

Use Next → on the narrator above to step through six configurations: from the full equivalent circuit, to the approximate form, to the open-circuit and short-circuit lab tests that pin down every parameter. Sample numbers correspond to a 5 kVA, 240/120 V single-phase transformer.

The transformer's equivalent circuit

Each winding has resistance and leakage reactance. The iron core, in parallel, has a magnetising reactance X_m (which carries the no-load magnetising current) and a core-loss resistance R_c (which represents hysteresis + eddy-current losses). The full per-phase equivalent circuit, with the secondary already referred to the primary side, looks like:

V₁ — r₁ — jx₁ — [ R_c ‖ jX_m ] — ja²x₂ — a²r₂ — V₂′

where a = N₁ / N₂ is the turns ratio. Multiplying the secondary impedances by a² is the standard "impedance referral" — it lets us analyse the whole transformer on a single side.

The approximate equivalent circuit

Because the magnetising current is small (a few per cent of rated current), the voltage drop across r₁ and jx₁ due to just the magnetising current is also small. We push the shunt branch all the way to the input terminals and combine the remaining series elements:

R_eq = r₁ + a²·r₂ X_eq = x₁ + a²·x₂

So the approximate model has just three elements: a shunt branch R_c ‖ jX_m at the input terminals, and a series R_eq + jX_eq in front of the load. This is the form we use for almost all hand calculations.

The open-circuit (OC) test — measures the shunt branch

Setup: apply rated voltage on the LV side, leave the HV side open. Measure V_oc, I_oc, and P_oc.
Why LV side? A low test voltage is easier and safer to apply. With the HV open there's no current in the series branch — only the magnetising and core-loss currents flow through the shunt branch.
What it gives us: P_oc equals the core loss at rated voltage (essentially constant regardless of load). From V_oc, I_oc, P_oc we extract the shunt-branch parameters R_c and X_m.

R_c = V_oc² / P_oc X_m = V_oc² / Q_oc, where Q_oc = √(S_oc² − P_oc²)

The short-circuit (SC) test — measures the series branch

Setup: short-circuit the LV side (or in some conventions HV — either works), and apply a reduced voltage on the other side until rated current circulates. Measure V_sc, I_sc, P_sc.
Why a reduced voltage? Because the impedance is small, only a small fraction of rated voltage is needed to drive rated current. At this low voltage the magnetising current is negligible — the shunt branch is essentially "off" — so what we measure is purely the series impedance.
What it gives us: P_sc equals the full-load copper loss (because rated current is flowing). From V_sc, I_sc, P_sc we extract the series-branch parameters R_eq and X_eq.

R_eq = P_sc / I_sc² Z_eq = V_sc / I_sc X_eq = √(Z_eq² − R_eq²)

The big idea. The OC and SC tests are complementary: each one isolates one branch of the equivalent circuit by making the other branch carry essentially no current. OC sees only the shunt branch (because the series branch is broken by the open secondary). SC sees only the series branch (because the shunt branch is starved at the low test voltage). Two simple bench tests pin down every parameter of the model — and from there you can predict efficiency, regulation, and behaviour at any load.

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