Electrical Machines - Ideal Transformer

Ideal Transformer

An ideal transformer is an imaginary model of the transformer which possesses the following characteristics −

• The primary and secondary windings have negligible (or zero) resistance.
• It has no leakage flux, i.e., whole of the flux flows through the magnetic core of the transformer.
• The magnetic core has infinite permeability, which means it requires negligible MMF to establish flux in the core.
• There are no losses due winding resistances, hysteresis and eddy currents. Hence, its efficiency is 100 %.

Working of an Ideal Transformer

We may analyze the operation of an ideal transformer either on no-load or on-load, which is discussed in the following sections.

Ideal Transformer on No-Load

Consider an ideal transformer on no-load, i.e., its secondary winding is open circuited, as shown in Figure-1. And, the primary winding is a coil of pure inductance.

When an alternating voltage $\mathit{V_{\mathrm{1}}}$ is applied to the primary winding, it draws a very small magnetizing current $\mathit{I_{\mathit{m}}}$ to establish flux in the core, which lags behind the applied voltage by 90. The magnetizing current Im produces an alternating flux $\mathit{\phi_{m}}$ in the core which is proportional to and in phase with it. This alternating flux ($\mathit{\phi_{m}}$) links the primary and secondary windings magnetically and induces an EMF $\mathit{E_{\mathrm{1}}}$ in the primary winding and an EMF $\mathit{E_{\mathrm{2}}}$ in the secondary winding.

The EMF induced in the primary winding $\mathit{E_{\mathrm{1}}}$ is equal to and opposite of the applied voltage $\mathit{V_{\mathrm{1}}}$ (according to Lenzs law). The EMFs $\mathit{E_{\mathrm{1}}}$ and $\mathit{E_{\mathrm{2}}}$ lag behind the flux ($\mathit{\phi_{m}}$) by 90, however their magnitudes depend upon the number of turns in the primary and secondary windings. Also, the EMFs $\mathit{E_{\mathrm{1}}}$ and $\mathit{E_{\mathrm{2}}}$ are in phase with each other, while $\mathit{E_{\mathrm{1}}}$ is equal to $\mathit{V_{\mathrm{1}}}$ and 180 out of phase with it.

Ideal Transformer on On-Load

When a load is connected across terminals of the secondary winding of the ideal transformer, the transformer is said to be loaded and a load current flows through the secondary winding and load.

Consider an inductive load of impedance connected across the secondary winding of the ideal transformer as shown in Figure-2. Then, the secondary winding EMF $\mathit{E_{\mathrm{2}}}$ will cause a current $\mathit{I_{\mathrm{2}}}$ to flow through the secondary winding and load, which is given by,

$$\mathrm{\mathit{I_{\mathrm{2}}}\:=\:\frac{\mathit{E_{\mathrm{2}}}}{\mathit{Z_{\mathit{L}}}}\:=\:\frac{\mathit{V_{\mathrm{2}}}}{\mathit{Z_{\mathit{L}}}}}$$

Where, for an ideal transformer, the secondary winding EMF $\mathit{E_{\mathrm{2}}}$ is equal to the secondary winding terminal voltage $\mathit{V_{\mathrm{2}}}$.

Since we considered an inductive load, therefore, the current $\mathit{I_{\mathrm{2}}}$ will lag behind $\mathit{E_{\mathrm{2}}}$ or $\mathit{V_{\mathrm{2}}}$ by an angle of $\mathit{\phi_{\mathrm{2}}}$. Also, the no-load current $\mathit{I_{\mathrm{0}}}$ being neglected because the transformer is ideal one.

The current flowing in the secondary winding ($\mathit{I_{\mathrm{2}}}$) sets up an MMF ($\mathit{I_{\mathrm{2}}}\mathit{N_{\mathrm{2}}}$) which produces a flux $\mathit{\phi_{\mathrm{2}}}$ in opposite direction to the main flux ($\mathit{\phi_{\mathit{m}}}$). As a result, the total flux in the core changes from its original value, however, the flux in the core should not change from its original value. Therefore, to maintain the flux in the core at its original value, the primary current must develop an MMF which can counter-balance the demagnetizing effect of the secondary MMF $\mathit{I_{\mathrm{2}}}\mathit{N_{\mathrm{2}}}$.

Hence, the primary current $\mathit{I_{\mathrm{1}}}$ must flow so that

$$\mathrm{\mathit{I_{\mathrm{1}}}\mathit{N_{\mathrm{1}}}\:=\:\mathit{I_{\mathrm{2}}}\mathit{N_{\mathrm{2}}}}$$

Therefore, the primary winding must draw enough current to neutralize the demagnetizing effect of the secondary current so that the main flux in the core remains constant. Hence, when the secondary current ($\mathit{I_{\mathrm{2}}}$) increases, the primary current ($\mathit{I_{\mathrm{1}}}$) also increases in the same manner and keeps the mutual flux ($\mathit{\phi_{\mathit{m}}}$) constant.

In an ideal transformer on-load, the secondary current $\mathit{I_{\mathrm{2}}}$ lags behind the secondary terminal voltage $\mathit{V_{\mathrm{2}}}$ by an angle of $\mathit{\phi _{\mathrm{2}}}$.