Transformer Equivalent Circuit
Theory: two tests define the circuit
Overview
I've seen transformer equivalent circuits in textbooks. But how are the actual values determined?
Two simple tests. The open-circuit test applies rated voltage with the secondary open; current is small, so winding loss is negligible and measured power is essentially core loss. The short-circuit test shorts the secondary and applies the low voltage that drives rated current; voltage is low, so core loss is negligible and measured power is essentially copper loss. That separates the core branch from the winding branch. Let's derive the parameters of a 50 kVA distribution transformer and go on to regulation and efficiency.
Formulas from the tests
Open-circuit values refer to the side measured (here LV) and short-circuit values to theirs (HV). Multiply by the turns ratio squared to refer them to the other side.
Regulation and efficiency
$p, q$ are per-unit resistive and reactive drops, $x$ the load fraction, $P_0$ core loss and $P_{cu}$ rated copper loss. Lagging loads add reactive drop and lower voltage; leading loads can offset it and even raise voltage.
The transformer that made AC transmission possible
The practical transformer is usually traced to the closed-core design built in 1885 by Zipernowsky, Bláthy and Déri at Ganz in Hungary. Around the same time in America, Stanley improved the transformer for Westinghouse and demonstrated AC distribution in 1886. Stepping voltage up for transmission and down for use won the 'war of the currents' against Edison's DC. The equivalent circuit became the common language for subsequent design and testing.
Worked example: 50 kVA, 6600/210 V single-phase
Test results and parameters
| Test | Measured | Derived parameters |
|---|---|---|
| Open circuit (LV, 210 V) | 4.8 A, 250 W | Rc 176 Ω, Xm 45 Ω (LV) |
| Short circuit (HV, 7.58 A) | 264 V, 700 W | Req 12.2 Ω, Xeq 32.6 Ω (HV) |
Exciting current is 2.0% of the 238 A rating, with a low no-load power factor of 0.25. In per unit, resistive drop is 1.40%, reactive 3.75% and impedance 4.0%.
Voltage regulation (rated load)
| Power factor | Phasor calculation | Approximation |
|---|---|---|
| 1.0 | 1.47% | 1.40% |
| 0.8 lagging | 3.39% | 3.37% |
| 0.8 leading | −1.05% | −1.13% |
Load and efficiency
| Load | Efficiency (pf 1) | Efficiency (pf 0.8) |
|---|---|---|
| 25% | 97.70% | 97.15% |
| 50% | 98.33% | 97.92% |
| 60% (peak) | 98.35% | 97.95% |
| 100% | 98.14% | 97.68% |
| 125% | 97.90% | 97.38% |
Efficiency peaks at 60% load, not at rating. Why design it that way on purpose?
Distribution transformers don't run at rating all day. With 8 h at full load and 16 h at 20%, the daily energy efficiency is 97.89%, and core loss makes up half of the losses, because it runs 24 hours regardless of load. So for transformers spending long hours lightly loaded, trading slightly more copper loss for less core loss pays off — exactly why amorphous-alloy cores are used. Generator transformers, always heavily loaded, put peak efficiency near rating.
Accuracy of the approximate circuit
Comparing the approximate circuit (magnetising branch moved to the primary terminals) with the exact one (branch between the windings), with 6,600 V applied and rated 0.8-lagging load, the HV-referred secondary voltage was 6,383.6 V versus 6,381.0 V — 0.04% apart. With exciting current only 2% of rating, the approximate circuit suffices for routine work.
Analysis and design tips
- Run the open-circuit test on LV and the short-circuit test on HV to suit instrument ranges.
- Before use, refer all values to one side (turns ratio squared).
- Winding resistance varies with temperature; correct to a reference temperature (e.g. 75 °C).
- With harmonics or DC bias, core loss and exciting current change from the circuit values.
- For leakage reactance distribution or frequency behaviour, use electromagnetic field analysis.
“Voltage rose with a leading load”
A factory added capacitors for power-factor correction, but at night when machines stopped and load fell, supply voltage rose above rating and shortened lamp life. With capacitors still connected, leading current dominated at light load and the transformer's leakage reactance lifted the voltage — the same effect as the −1.05% regulation at 0.8 leading in the example. Switching capacitors with the load solved it.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Mixing referral sides | Values off by about a thousand | Refer with turns ratio squared |
| Taking no-load power as copper loss | Loss separation wrong | Understand each test |
| No temperature correction | Copper loss underestimated | Correct to reference |
| Comparing only rated efficiency | Actual losses misjudged | Use the load profile |
| Ignoring leading loads | Overvoltage | Check voltage rise |
I'd like to learn related topics.
Try the voltage regulation tool and transformer efficiency tool. Related pages include core loss, leakage inductance, winding loss and three-phase transformers.
Related Topics
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