MC-A4 · Real transformer equivalent circuit

The Real Transformer and Its Equivalent Circuit

A real transformer departs from the ideal. Sort that departure by where it sits in the circuit and the equivalent circuit appears. Start by pinning whether the exciting branch is in series or a shunt.

Where does the exciting branch belong?

Even at no load with the secondary open, a small exciting current still flows in the primary. For that current to flow it must close a loop with the source. Place the exciting branch in three positions and find the one where the no-load current flows as it really does.

Tap to move the exciting branch. No load = secondary open.

No-load primary current (secondary open)

I₁ = 0 A

Putting the exciting branch in series with the load leaves no closed loop when the secondary opens, so the primary current is zero. That flatly contradicts the real exciting current.

Misaligned

The flux is not shared 100%

The ideal model assumes both coils share exactly the same flux, but in reality some flux leaks through air and oil paths and links only one coil. This leakage flux is proportional to the load current, so it appears as a series leakage reactance X that the load current drops voltage across. The resistance R of the winding wire adds at the same series spot.

ObserveZ₁ = R₁ + jX₁

Winding resistance is in series — the load current passes it.

ChooseX₁ → ?

Leakage reactance scales with load current — in series.

Magnetising the core costs exciting current

Building the flux takes current, and the alternating flux also heats the core through hysteresis and eddy-current loss. Both flow whenever voltage is applied, even at no load, so they sit as a shunt across the primary terminals. The core loss is a conductance Gc and the magnetisation a susceptance Bm, bundled into the exciting admittance Y0 = Gc - jBm.

Fill inY₀ → ?

The exciting admittance hangs on the voltage — a shunt.

Refer the secondary to the primary as one circuit

Referring the secondary impedance by the turns ratio as a²Z2 merges the series branches into Req = R1 + a²R2 and Xeq = X1 + a²X2. The exciting admittance is usually small, so an approximate circuit with it moved to the input terminals simplifies the maths. The series branch accounts for copper loss and voltage drop; the shunt branch for core loss and no-load current.

On your ownReq = R?

Refer the secondary by a² and merge the series branch.

Back to the first screen

The only place the no-load current flowed as it really does was the shunt. Put the exciting branch in series and the loop breaks the moment the secondary opens, dropping the current to zero; put it as a shunt and it closes a loop with the source, leaving a small exciting current. Sorting the non-idealities by position is exactly the equivalent circuit. Winding resistance and leakage reactance, which the load current passes, are in series; core loss and magnetisation, which hang on the voltage and leak even at no load, are the shunt.

The real-transformer equivalent circuit: an ideal a:1 transformer plus a series branch (R1 + a²R2) + j(X1 + a²X2) that the load current passes, and a shunt exciting branch Y0 = Gc - jBm that hangs on the voltage. The series branch accounts for copper loss and voltage drop; the shunt for core loss and no-load current.

Once you hold this equivalent circuit

The equivalent circuit is what turns a transformer’s performance into numbers. Copper loss in the series branch and core loss in the shunt set the efficiency, while the voltage drop across the series impedance sets the voltage regulation. The next unit computes the maximum-efficiency condition and the regulation directly from these two losses (MC-A5). The same series-versus-shunt split repeats in the equivalent circuits of rotating machines.