Ideal Transformer Principles

A transformer is an electrical device that transfers alternating current (AC) electrical energy from one circuit to another through electromagnetic induction. Transformers are critical in modern power grids because they allow power to be transmitted efficiently at very high voltages and then safely stepped down for consumer use.

The ideal transformer is a theoretical model assuming no power losses (no core losses, no copper losses), perfect magnetic coupling (no leakage flux), and infinite core permeability.

Turns Ratio and Voltage/Current Relationships

A transformer consists of two coils wound around a common magnetic core:

• The primary winding has $N_1$ turns and is connected to the source.
• The secondary winding has $N_2$ turns and is connected to the load.

The ratio of secondary turns to primary turns is defined as the turns ratio ($n$): $n = \frac{N_2}{N_1}$

For an ideal transformer, the voltage across the windings is directly proportional to the number of turns: $\frac{V_2}{V_1} = \frac{N_2}{N_1} = n$

Since an ideal transformer is 100% efficient, power into the primary equals power out of the secondary ($V_1 I_1 = V_2 I_2$). Therefore, currents are inversely proportional to the turns ratio: $\frac{I_1}{I_2} = \frac{N_2}{N_1} = n$

If $n > 1$, the transformer acts as a step-up transformer. If $n < 1$, it acts as a step-down transformer.

Impedance Reflection

A major engineering application of transformers is impedance matching. A load impedance ($Z_L$) connected to the secondary winding will "appear" differently when viewed from the source connected to the primary winding.

The equivalent impedance seen at the primary ($Z_{in}$) is: $Z_{in} = \frac{V_1}{I_1}$

Substituting the voltage and current relationships: $Z_{in} = \frac{V_2 / n}{n I_2} = \frac{1}{n^2} \frac{V_2}{I_2}$

Since $V_2 / I_2 = Z_L$: $Z_{in} = \frac{Z_L}{n^2} = \left(\frac{N_1}{N_2}\right)^2 Z_L$

This means an ideal transformer alters the magnitude of impedance by the square of the reciprocal of the turns ratio.

Engineering Check

While the ideal model is excellent for basic circuit analysis, real transformers deviate from this behavior. Real transformers have winding resistance (copper losses), eddy currents and hysteresis in the core (core losses), and leakage inductance. These non-ideal factors cause voltage drops under load and generate heat, necessitating cooling systems in large power transformers.