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A transformer transfers energy by inductive coupling between its circuit windings. It is a static electrical device. There is varying current in the primary windings which creates a varying magnetic flux through transformer’s core and hence through the secondary windings. This varying magnetic flux induces a varying emf (electromotive force) in secondary windings. Transformers are used to vary the relative voltage of circuits or isolate them or both. Let us consider the ideal, lossless, perfectly coupled transformer shown in the diagram below. It has primary windings as NP and secondary windings as NS. The induced secondary voltage and primary voltage are related by the following relation: Vp/Vs = Ep/Es = Np/Ns = a, where, VP/VS = EP/ES = a is the voltage ratio and NP/NS = a is the winding turns ratio. The value of these ratios is higher than one for step-down transformer and less that one for step-up transformer. In the above equation, VP denotes source voltage, VS denotes output voltage and EP and ES denotes respective emf induced voltage.

ideal Transformers circuit diagram

Fig (1): Ideal Transformer Circuit Diagram.

If we connect any load impedance, Zl, to the secondary windings of the ideal transformers the current flows without losses from primary to secondary circuit, which results in input and output apparent power (which are equal): Ip*Vp = Is*Vs. When we combine the two equations we get the following identity: Vp/Vs = Ep/Es = Np/Ns = a; This formula is a reasonable approximation for the typical commercial transformer, with voltage ratio and winding turns ratio both being inversely proportional to the corresponding current ratio. The load impedance is defined as follows: Zl= Vl/Il =Vs/Is. The apparent impedance Zl’ of this secondary circuit load referred to the primary winding circuit is governed by following relation: Zl’ = Vp/Ip = a*Vs/Is/a = a*a*Vs/Is= a*a*Zl.

Referring to the below diagram, a practical transformer's physical behaviour may be represented by an equivalent circuit model. Primary winding: RP, XP , Secondary winding: RS, XS are the series loop impedances of the model considering winding joule losses and leakage reactances. Usually, for circuit equivalence transformation, RS and XS are in practice usually referred to the primary side by multiplying these impedances by the turns ratio squared, (NP/NS) 2 = a2. Core loss and reactance is represented by the following shunt leg impedances of the model: Core or iron losses: RC, Magnetizing reactance: XM. RC and XM are collectively termed the magnetizing branch of the model.

real Transformer equivalent circuit

Fig (2): Real Transformer equivalent circuit.

Transformers Example

For the circuit shown below, calculate the phasor currents and voltages and the power delivered to the load.

Transformers example Transformers example Transformers example

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