Ideal Transformers

A transformer consists of several coils wound on a common core that usually consists of laminated iron (to reduce eddy-current loss).

Transformers are usually used to adjust the values of ac voltages. A transformer can step up or step down voltage. Transformers are used in electric power distribution.

 To transport power over long distance, large voltages are used. Power delivered by an ac source is given by:

P = V_rmsI_rmsCosϴ

For a fixed power factor Cosϴ, many combinations of voltage and current can be used in transferring a given amount of power. The wires that carry the current have nonzero resistances hence; some power is lost in the transmission lines, given by:

P_loss = R_lineI²_rms

Where R_line is the resistance of the transmission line

By designing the power distribution system with a large voltage value and a small current value, the line loss can be made to be a small fraction of the power transported. Therefore larger voltage yields higher efficiency in power distribution.

For safety and other reasons, relatively small voltages are employed especially in residential homes, thus transformers are used in stepping voltage levels up or down where necessary in a power distribution system.

The Voltage Ratio

From the figure above, an ac voltage is connected to the primary coil, which consists of N₁ turns of wire. Current flows into the primary side and causes an ac magnetic flux Φ (t) to appear in the core. This flux induces a voltage in the N₂-turn secondary coil, which delivers power to the load. Depending on the turns’ ratio , the rms secondary voltage can be greater or less than the rms primary voltage.

Since we assume an ideal transformer, we neglect the resistances of the coils and the core loss. Furthermore we assume the reluctance of the core is very small and that all the flux links all of the turns of both coils.

The primary voltage is given by:

Assuming that all of the flux links all of the turns, secondary voltage is given by:

From equation 7, we can see the voltage across each coil is proportional to its number of turns.

Also we can deduce from lenz’s law, the induced voltages, have positive polarity at both dotted terminals when Φ is increasing and negative polarity at both dotted terminals when Φ is decreasing.

From the above, we have established the fact that voltage across each winding is proportional to the number of turns. The peak and rms values of the voltages are also related by the turns’ ratio:

Current Ratio

Considering the image above, you will notice that the current i₁ and i₂ produce opposing magnetic fields (because i₁ enters a dotted magnetic fields and i₂ leaves a dotted terminal).

Thus the total magnetomotive force (mmf) applied to the core is:

F_m = N₁i₁(t) – N₂i₂(t)  {equation 8}

Furthermore, the mmf is related to the flux (Φ) and the reluctance (R) of the core by:

F_m =RΦ    {equation 9}

In a well-designed transformer, the core reluctance is very small. Ideally, the reluctance is zero, and the mmf required to establish the flux in the core is zero. Then equation 8 becomes:

F_m = N₁i₁(t) – N₂i₂(t) = 0  {equation 10}

Therefore:

This relationship also applies for the rms values of the currents:

Power in an Ideal Transformer

Considering the image above, the power delivered to the load by the secondary winding is:

P₂(t) = V₂(t)i₂(t)   {equation 13}

Using equation 7 and 11 to substitute for V₂(t), respectively, we have:

However the power delivered to the primary winding by the source is:

 P₁(t) = V₁(t)i₁(t) , and we get P₂(t) = P₁(t)

Therefore, we have established that, the power delivered to the primary winding by the source is delivered in turn to the load by the secondary winding. Net power is neither generated nor consumed by an ideal transformer.