Impedance-transfer ratio:

One of the most significant applications of transformers is in the audio frequency (AF) and radio frequency (RF) electronic circuits. In these applications, the transformers are generally employed to match impedances. Hence, you might hear of an impedance step up transformer, or an impedance step down transformer.

The impedance transfer ratio of the transformer varies according to square of the turns ratio, and according to the square of the voltage-transfer ratio as well. The formula for the voltage transfer ratio:

E_pri/E_sec = T_pri/T_sec

If the source and load, impedances are purely resistive, and can be denoted by Zpri (at primary winding) and Zsec (at secondary), then

Z_pri/Z_sec = (T_pri/T_sec)²

and

Z_pri/Z_sec =(E_pri/E_sec)²

The inverses of the above formulas, in which the turns ratio or voltage transfer ratio are expressed in the terms of impedance transfer ratio, are

Tpri/Tsec   (Zpri/Zsec)1/2

and

E_pri/E_sec =(Z_pri/Z_sec)^1/2

The half power is same thing as the square root.

Problem : 

A transformer is required to match the input impedance of 50.0 Ω, purely resistive, to the output impedance of 300 Ω, also purely resistive. What will be the ratio of T_pri/T_sec be?

The  desired  transformer  will  have  a  step up  impedance  ratio  of  Z_pri/Z_sec 50.0/300=1:6.00. From formulas given above,

T_pri/T_sec =(Z_pri/Z_sec)^1/2 = (1/6.00)^1/2 = 0.16667^1/2 =0.40829

A couple of extra digits are included (as they show up on the calculator) to prevent the sort of error introduction you recall from earlier chapters. The decimal value 0.40829 can be changed into ratio notation by taking its reciprocal,  and then writing "1:" followed by that reciprocal value

0.40829=1:(1/0.40829)=1:2.4492

This can be rounded to 3 significant figures, or 1:2.45. This is primary to secondary turns ratio for transformer. The secondary winding has 2.45 times turns as the primary winding.