Example: Ideal Transformer Exercise

Author

John Hawkins

Problem Statement

An ideal transformer has a primary winding with 500 turns and a secondary winding with 2000 turns. Given that $\ e_{1}=120\angle {0^{\circ }}{\text{ V, RMS}}$ and $\ i_{1}=(2+3j){\text{ A}}$, find the load impedance, $\ Z_{L}$ and the Thevenin equivalent, $\ Z_{th}$.

Solution

We could find the Thevenin impedance directly, but we will save that until the end as a checking mechanism. First, we will find the actual load impedance by finding the current and voltage in the secondary winding and finding their ratio. The equations used are those derived in class by Professor Frohne.

$e_{2}={\frac {N_{2}}{N_{2}}}e_{1}={\frac {2000}{500}}(120)=480{\text{ V}}$

$i_{2}={\frac {N_{1}}{N_{2}}}i_{1}={\frac {500}{2000}}(2+3j)=\left({\frac {1}{2}}+{\frac {3}{4}}j\right){\text{ A}}$

$Z_{L}={\frac {e_{2}}{i_{2}}}={\frac {480}{{\frac {1}{2}}+{\frac {3}{4}}j}}=\mathbf {(295.4-443.1j)\ \Omega \ =(532.5\angle {-56.3^{\circ }})\ \Omega }$

$Z_{th}=\left({\frac {N_{1}}{N_{2}}}\right)^{2}Z_{L}=\left({\frac {500}{2000}}\right)^{2}(295.4-443.1j)=\mathbf {(18.5-27.7j)\ \Omega \ =(33.3\angle {-56.3^{\circ }})\ \Omega }$

As mentioned at the beginning, this should be the impedance found using the ratio of the primary voltage and current. Using this method, we find that

$Z_{th}={\frac {e_{1}}{i_{1}}}={\frac {120}{2+3j}}=\mathbf {(18.5-27.7j)\ \Omega \ =(33.3\angle {-56.3^{\circ }})\ \Omega }$

This is the same answer as above, which verifies the solutions.

Reviewed By

Tyler Anderson

Comments

Tyler Anderson: it may be helpful to the readers if you referenced what equations you are using. For example: $e_{2}={\frac {N_{2}}{N_{2}}}e_{1}$ $EQ(5-39)$ Otherwise it looks sound to me.

John Hawkins: I didn't use the textbook, so such a reference is not required. I agree that it would be useful for those in the class, but I don't have the same textbook as everyone else, and I doubt anyone would care to know my book's equation numbers. Thanks for reminding me about references, though. I mentioned the class derivation above in the text.

Tyler Anderson: haha fair enough then. props for that. perhaps I could barrow your book sometime? cause ours is absolute crap.

John Hawkins: I hate the book I'm using as well, but if you want to use it sometime that would be fine.

J. Apablaza: Everything looks sound to me. Perhaps, you should include an image so that you can earn some extra points.