MC-A4 · Real transformer equivalent circuit

The Real Transformer and Its Equivalent Circuit

A real transformer departs a little from the ideal one. Sorting out where in the circuit each departure sits reveals the equivalent circuit, so start by pinning down whether the exciting branch is in series or in shunt.

Where does the exciting branch belong?

Even at no load, with the secondary open, a small exciting current flows in the primary. For that current to flow it must close a loop with the source, and if you place the exciting branch in each of the three positions with the tabs, only one of them lets the no-load current flow as it really does. Which position that is, and why, is the question of this lesson.

The tabs move the exciting branch. No load = secondary open.

Primary current at no load (secondary open)

I₁ = 0 A

Put the exciting branch in series with the load and, once the secondary is opened, there is no closed loop, so the primary current is 0. That flatly contradicts the exciting current that flows in a real transformer.

Misaligned

The flux is not shared 100%

The ideal transformer assumes both coils share the flux completely, but in reality some flux leaks along paths through air or oil and links only one coil. This leakage flux is proportional to the current in that coil, so it shows up as a series leakage reactance X across which the load current drops voltage. The resistance R of the winding wire is added in the same series position.

ObserveZ₁ = R₁ + jX₁

The winding wire's resistance sits in the series position the load current passes through.

ChooseX₁ → ?

Leakage reactance is proportional to the load current too, so it is in series.

Magnetising the core costs exciting current

Setting up flux takes current, and the alternating flux also heats the core through hysteresis loss and eddy-current loss. Both flow at no load as long as a voltage is applied, so they are placed as a shunt across the primary terminals. Grouping the core loss as a conductance Gc and the magnetization as a susceptance Bm gives the exciting admittance Y0 = Gc - jBm. The exciting current is only a few percent of the rated current, and smaller still in large transformers, but because the core saturates it is not sinusoidal: it is a distorted wave with odd harmonics mixed in.

Fill inY₀ → ?

The exciting admittance sits across the voltage, so it is in shunt.

Referring the secondary to the primary, one circuit

Referring the secondary impedance through the turns ratio a as a²Z2 merges the series branches into Req = R1 + a²R2 and Xeq = X1 + a²X2. The exciting admittance is usually small, so an approximate equivalent circuit with it moved to the input terminals simplifies the calculation. The series branch represents copper loss and voltage drop, and the shunt branch represents core loss and no-load current; the actual values come from measuring them separately, the shunt branch in a no-load test with the secondary open and the series branch in a short-circuit test with the secondary shorted.

On your ownReq = R?

Refer the secondary by a² and merge the series branches.

Back to the first screen

The only position where the no-load current flowed as it really does was the shunt. With the exciting branch in series, the loop breaks the moment the secondary opens and the current is 0, but in shunt it closes a loop with the source and a small exciting current remains. Sorting the non-ideal effects by position is exactly what the equivalent circuit is: the winding resistance and leakage reactance the load current passes through are in series, and the core loss and magnetizing current, which sit across the voltage and flow even at no load, are in shunt.

Real-transformer equivalent circuit: to the ideal transformer a:1, add the series branch (R1 + a²R2) + j(X1 + a²X2) that the load current passes through and the shunt exciting branch Y0 = Gc - jBm across the voltage. The series branch represents copper loss and voltage drop, and the shunt branch core loss and no-load current.

Once you hold this equivalent circuit

The equivalent circuit is a device for turning a transformer's performance into numbers. The copper loss of the series branch and the core loss of the shunt branch set the efficiency, and the voltage drop across the series impedance sets the voltage regulation. The next lesson uses these two losses to calculate the condition for maximum efficiency and the voltage regulation directly (MC-A5). The same split into series and shunt repeats in the equivalent circuits of rotating machines.