Question
Accepting the relation $\frac{v_s}{v_p}=\frac{ N _s}{N_s}$ for ideal transformer, and using thé formula of power; derive $\frac{I_p}{I_s}=\frac{v_s}{v_p}=\frac{N_s}{ N _p}$ and from that explain the types of the transformers.
✓
Answer
$\rightarrow$ If the transformer is assumed to be $100 \%$ efficient $($no energy losses$),$ the input power is equal to the output power,
$\therefore I _p U _p= I _s U _s$
$\therefore \frac{ U _s}{ v _p}=\frac{ I _p}{ I _s}$
But $\frac{ U _s}{ v _p}=\frac{ N _s}{N_p}$
$\rightarrow $ But $\frac{v_s}{v_p}=\frac{ N _s}{N_p}$
we get $\frac{v_s}{v_p}=\frac{ N _s}{N_p}=\frac{ I _p}{ I _s}$.
$\rightarrow$ From this equation
$\frac{v_s}{v_p}=\frac{ N _s}{N_p} \frac{ N _s}{N_p}=\frac{ I _p}{ I _s}$
$\therefore v _s=\left(\frac{ N _s}{N_p}\right) \cdot v _p \ldots(2) $
$\therefore I _s=\frac{ N _p}{N_s} \cdot I _p \cdots$
$\rightarrow$ From eq. $(2)$ and $(3)$
$(i)$ If $N _s> N _p$ then $v_s>v_p$ which means the voltage is stepped up.
Such a transformer is called step up transformer.
$(ii) $ If $N _s< N _p$ then $v _s< v _p$ which means the voltage reduces $($or voltage is stepped down$)$.
Such a transformer is called step down transformer.
$\therefore I _p U _p= I _s U _s$
$\therefore \frac{ U _s}{ v _p}=\frac{ I _p}{ I _s}$
But $\frac{ U _s}{ v _p}=\frac{ N _s}{N_p}$
$\rightarrow $ But $\frac{v_s}{v_p}=\frac{ N _s}{N_p}$
we get $\frac{v_s}{v_p}=\frac{ N _s}{N_p}=\frac{ I _p}{ I _s}$.
$\rightarrow$ From this equation
$\frac{v_s}{v_p}=\frac{ N _s}{N_p} \frac{ N _s}{N_p}=\frac{ I _p}{ I _s}$
$\therefore v _s=\left(\frac{ N _s}{N_p}\right) \cdot v _p \ldots(2) $
$\therefore I _s=\frac{ N _p}{N_s} \cdot I _p \cdots$
$\rightarrow$ From eq. $(2)$ and $(3)$
$(i)$ If $N _s> N _p$ then $v_s>v_p$ which means the voltage is stepped up.
Such a transformer is called step up transformer.
$(ii) $ If $N _s< N _p$ then $v _s< v _p$ which means the voltage reduces $($or voltage is stepped down$)$.
Such a transformer is called step down transformer.
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