Question

Draw a labelled diagram of a step-up transformer. Obtain the ratio of secondary voltage to primary voltage in terms of number of turns in the two coils.

Hint:

To easily remember transformer principles: Voltage simply follows the turns.
More turns inherently mean more voltage (Step-up).
Fewer turns inherently mean less voltage (Step-down).
Current, however, does the exact mathematical opposite to conserve energy.

Answer 1

Concept:
• A transformer operates strictly on the fundamental principle of mutual induction.
• It effectively consists of two separate coils (primary and secondary) wrapped tightly around a common laminated magnetic core.
• A step-up transformer specifically increases the alternating voltage, which necessitates having far more turns in the secondary coil than in the primary coil.

Step 1: Labelled Diagram of a Step-up Transformer
\includegraphics[width=0.5\linewidth]{31bi_sol.png}
The proper diagram must clearly display a laminated iron core forming a closed loop.
The primary coil is wound on one limb and contains a relatively small number of turns ($N_p$).
The secondary coil is deliberately wound on the opposite limb (or over the primary) and contains a much larger number of turns ($N_s$).
An AC voltage source is explicitly connected across the primary terminals, and the high voltage output is taken across the secondary terminals.
Crucial labels should prominently include: Laminated soft iron core, Primary coil ($N_p$), Secondary coil ($N_s$), Input AC voltage ($V_p$), and Output AC voltage ($V_s$).

Step 2: Derivation of the Voltage Ratio
When a continuously varying alternating voltage is applied directly to the primary coil, it aggressively drives an alternating current through it.
This alternating current generates a constantly changing magnetic flux within the highly permeable iron core.
Assuming an ideal scenario where there is absolute zero flux leakage, all the magnetic flux ($\Phi$) perfectly links with both the primary and the secondary coils simultaneously.
According to Faraday's fundamental law of electromagnetic induction, the induced electromotive force (emf) in the primary coil is given precisely by:
\[ e_p = -N_p \frac{d\Phi}{dt} \quad \text{--- (Equation 1)} \]
Similarly, the induced emf in the extensively wound secondary coil is:
\[ e_s = -N_s \frac{d\Phi}{dt} \quad \text{--- (Equation 2)} \]

Step 3: Formulate the transformation ratio
By directly dividing Equation 2 by Equation 1, the time derivative of flux cancels out completely:
\[ \frac{e_s}{e_p} = \frac{-N_s \frac{d\Phi}{dt}}{-N_p \frac{d\Phi}{dt}} = \frac{N_s}{N_p} \]
For an ideal transformer with negligible coil resistances, the applied primary voltage $V_p$ perfectly equals the back emf $e_p$ ($V_p = e_p$), and the secondary terminal voltage $V_s$ exactly equals the induced emf $e_s$ ($V_s = e_s$).
Substituting these voltage relations directly yields the final ratio:
\[ \frac{V_s}{V_p} = \frac{N_s}{N_p} \]
This highly important relation proves that the ratio of secondary voltage to primary voltage is strictly equal to the ratio of their respective number of turns, known as the transformation ratio ($k$).