Question

In an ideal transformer, which of the following is correct?

• A. \( \left(\frac{\text{V}_1}{\text{V}_2}\right) = \left(\frac{\text{I}_1}{\text{I}_2}\right) = \left(\frac{\text{N}_1}{\text{N}_2}\right) \)
• B. \( \left(\frac{\text{V}_1}{\text{V}_2}\right) = \left(\frac{\text{I}_2}{\text{I}_1}\right) = \left(\frac{\text{N}_2}{\text{N}_1}\right) \)
• C. \( \left(\frac{\text{V}_1}{\text{V}_2}\right) = \left(\frac{\text{I}_2}{\text{I}_1}\right) = \left(\frac{\text{N}_1}{\text{N}_2}\right) \)
• D. \( \left(\frac{\text{V}_2}{\text{V}_1}\right) = \left(\frac{\text{I}_1}{\text{I}_2}\right) = \left(\frac{\text{N}_1}{\text{N}_2}\right) \)

Hint:

In any ideal transformer equation, remember that voltage ($V$) is always directly proportional to the number of turns ($N$), while current ($I$) is always inversely proportional to the turns. Therefore, the indices for $V$ and $N$ will match, while the index for $I$ will be flipped: $$V \propto N \quad \text{and} \quad I \propto \frac{1}{N}$$

Answer 1

The Correct Option is C

Concept: An ideal transformer is a theoretical model of a transformer that has no energy losses. This means it has perfectly zero winding resistances, no magnetic core losses (hysteresis and eddy current losses), and perfect magnetic coupling between the primary and secondary windings (zero leakage flux). The basic equations governing an ideal transformer are derived from Faraday's Law of Electromagnetic Induction and the principle of conservation of energy:
• Faraday's Law & Voltage Ratio: The electromotive force (emf) induced per turn is identical in both the primary and secondary windings because they share the same mutual magnetic flux ($\phi$). Thus, the ratio of induced voltages is directly proportional to the ratio of their respective number of turns: $$\frac{V_1}{V_2} = \frac{N_1}{N_2}$$
• Conservation of Power & Current Ratio: Since there are no losses in an ideal transformer, the apparent input power delivered to the primary winding must exactly equal the apparent output power delivered by the secondary winding to the load: $$P_{\text{in}} = P_{\text{out}} \quad \Rightarrow \quad V_1 \cdot I_1 = V_2 \cdot I_2$$

Step 1: Relate voltage and turns ratio.
According to Faraday's law of induction, the voltage induced in a winding is proportional to the rate of change of magnetic flux flux ($d\phi/dt$) multiplied by the number of turns in that winding ($N$). Let $\phi$ be the common mutual flux in the core. The primary induced voltage $V_1$ and secondary induced voltage $V_2$ can be stated as: $$V_1 = N_1 \frac{d\phi}{dt}$$ $$V_2 = N_2 \frac{d\phi}{dt}$$ Taking the ratio of these two equations yields the standard transformer turn ratio rule: $$\frac{V_1}{V_2} = \frac{N_1}{N_2} \quad \cdots (1)$$

Step 2: Relate voltage and current ratio using the conservation of energy.
In an ideal transformer, the efficiency ($\eta$) is $100%$, meaning there is absolutely no active or reactive power dissipation within the device. Therefore, the input volt-amperes must be equal to the output volt-amperes: $$V_1 \cdot I_1 = V_2 \cdot I_2$$ By rearranging the variables to group the voltages on one side and currents on the other side, we obtain the inverse relationship between voltage and current: $$\frac{V_1}{V_2} = \frac{I_2}{I_1} \quad \cdots (2)$$

Step 3: Combine both relationships into a single unified equation.
By equating expression (1) and expression (2), we establish the complete proportional relationship governing an ideal transformer: $$\frac{V_1}{V_2} = \frac{I_2}{I_1} = \frac{N_1}{N_2}$$ This demonstrates that while voltage scales directly with the number of turns, the current scales inversely with the number of turns. Evaluating the given choices, option (3) perfectly matches this derived truth.