Example Problems with Transformers

Transformer Example

Kevin Starkey, Nick Christman, Aric Vyhmeister

Problem:

A step down transformer has a winding of $N_{1}=10{\text{ turns and }}N_{2}=2$ turns. (a) If the input voltage is 1200V, what is the resulting output voltage? (b) If the input and output currents are $5A{\mbox{ and }}20A$, respectively, what is the current loss due to the leakage inductance, $i_{m}(t)$.

Figure 1: Model for an ideal transformer.

Figure 2: Magnetic circuit of an ideal transformer.

Solution:

a) By modifying equation 5-39 (Mohan 5-22) we can obtain an equation for the output voltage. That is,

$e_{2}=\left({\frac {N_{2}}{N_{1}}}\right)e_{1}$.

With the information provided we can now determine the output voltage:

$e_{2}=\left({\frac {N_{2}}{N_{1}}}\right)e_{1}=\left({\frac {2}{10}}\right)1200V=240V$

b) From the figure above, we need to find the true value of current (i.e. the current that is not lost by leakage inductance). To accomplish this we will use equation 5-40 (Mohan 5-23) to obtain the value of $i_{1}^{'}$. That is,

$i_{1}^{'}=\left({\frac {N_{2}}{N_{1}}}\right)i_{2}=\left({\frac {2}{10}}\right)20A=4A$.

Now from equation 5-42 (Mohan 5-23) we can obtain the current loss due to the leakage inductance, which is

$i_{m}=i_{1}-i_{1}^{'}=5A-4A=1A$.

Therefore, the current loss in our transformer is $1A$, which means this specific transformer is very "leaky."