Impedance Matching With Transformers

I like using specific numerical examples to introduce the concept of impedance transformation, because I find abstract mathematical presentations tend to “lose” a lot of students.

The setup of this problem may confuse some students, with reference to the amount of impedance that a source “sees”. Hopefully, the anthropomorphic language will not be a barrier to understanding. The point is, for students to realize that just as a load can have a voltage or a current “impressed” upon it, a source can have a load “impressed” upon it as well. In this particular question, the issue is how the 1:2 step-down transformer ratio affects the amount of loading impressed upon the 240 VAC source by the 30 ohm resistor. That the resistor “sees” the same source voltage should be obvious. That the sources see very different impedance loadings (due to the transformer) is the purpose of this question.

Determining the voltage and current ratios should be trivial. Calculating the impedance ratio will likely require the set-up of an example problem, based on known values of voltage and current.

The most important part of this question is the identification of mathematical patterns and trends relating the turns ratio to the requested ratios. Of particular note are the inductance and impedance ratios. Why are they 9:1 and not 3:1? Ask your students what mathematical operation relates the number 3 to the number 9? If necessary, have them work through another example problem (with a different turns ratio) to see the impedance transformation ratio there, and the resulting relationship between that ratio and the turns ratio.