# Step-by-Step Solution ## Step 1: Understanding Transformer Design Basics Transformers are designed to transfer electrical energy between two or more circuits through electromagnetic induction. The key design parameters include: - **Voltage rating** (primary and secondary) - **Frequency of operation** - **Core material and geometry** - **Number of turns of wire** - **Cross-sectional area of the core** For a **given voltage**, the design must ensure that the **core does not saturate** and that losses are minimized. ## Step 2: The Transformer Equation The induced emf (voltage) per turn in a transformer is given by: \[ E = 4.44\, f\, N\, \Phi_{max} \] Where: - \( E \) = rms voltage (V) - \( f \) = frequency (Hz) - \( N \) = number of turns - \( \Phi_{max} \) = maximum flux in the core (Wb) The maximum flux in the core is: \[ \Phi_{max} = \frac{E}{4.44\, f\, N} \] For a given voltage \( E \), as frequency \( f \) increases, \( \Phi_{max} \) decreases if \( N \) is kept the same. ## Step 3: Effect of Frequency on Size - **At higher frequencies**, the same voltage can be achieved with fewer turns and/or a smaller core cross-sectional area, because \( \Phi_{max} \) can be smaller. - The **core size** depends on \( \Phi_{max} \) and the **cross-sectional area** required to prevent core saturation. ## Step 4: Evaluating the Choices Let's analyze each statement: 1. **Transformer of a higher frequency is smaller in size because the increase of frequency allows to reduce cross sectional area and/or number of turns.** - This is correct. Higher frequency means higher rate of change of flux, so for the same voltage, you can use less core material (smaller area) and/or fewer turns. 2. **Transformer of a higher frequency is smaller in size because magnetic core used in high frequency transformers have much higher flux densities, resulting in smaller cross sectional area and/or number of turns.** - This is partially correct, but the main reason is **frequency**, not just higher flux density of the core material. 3. **Transformer of a higher frequency is larger in size to accommodate higher losses associated with higher frequency of operation.** - Incorrect. While eddy current and hysteresis losses increase with frequency, transformers are made smaller at high frequencies because the core can be made smaller. 4. **Transformer of a higher frequency is larger in size because a higher number of turns (proportional to frequency) should be used with it.** - Incorrect. Actually, higher frequency **allows for fewer turns**, not more. ## Step 5: Final Answer ### Correct Explanation > **Transformer of a higher frequency is smaller in size because the increase of frequency allows to reduce cross sectional area and/or number of turns.** --- ## Summary Table | Frequency | Core Size | Number of Turns | |----------------|--------------------|-----------------| | Low | Larger | More | | High | Smaller | Fewer | --- ## Final Answer **The correct choice is:** **"Transformer of a higher frequency is smaller in size because the increase of frequency allows to reduce cross sectional area and/or number of turns."** --- **Reason:** At higher frequencies, the same voltage requires less magnetic flux (and hence a smaller core) and/or fewer turns, which reduces the overall size of the transformer.