An Ideal Transformer Example

Consider a simple, transformer with two windings. Find the current provided by the voltage source.

• Winding 1 has a sinusoidal voltage of $120{\sqrt {2}}\angle {0}$° applied to it at a frequency of 60Hz.
• ${\frac {N_{1}}{N_{2}}}=3$
• The combined load on winding 2 is $\ {Z_{L}}=(5+j3)\Omega$

Solution

$\ {e_{1}}(t)={V_{1}}\cos(\omega t)$

$\ \omega =2\pi f$, so $\omega =120\pi$

Therefore, $\ {e_{1}}(t)={V_{1}}\cos(120\pi t)$

Now the Thevenin equivalent impedance, ${Z_{th}}$, is found through the following steps:

${Z_{th}}={\frac {e_{1}}{i_{1}}}$

$={\frac {{\frac {N_{1}}{N_{2}}}{e_{2}}}{{\frac {N_{2}}{N_{1}}}{i_{2}}}}$

$=({\frac {N_{1}}{N_{2}}})^{2}{R_{L}}$

Now, substituting:

${Z_{th}}=3^{2}(5+j3)$

$=(45+j27)\Omega$

Since ${i_{1}}={\frac {e_{1}}{R_{th}}}$,

${i_{1}}={\frac {120{\sqrt {2}}}{45+j27}}$