TITLE: What is the ratio of the turns in the transformer when it is impedance-matching?
QUESTION [0 upvotes]: I study physics on my own. There is a question in my textbook:  

A transformer may be used to provide maximum power transfer between two AC circuits that have diffirent impedances $Z_{1}$ and $Z_{2}$. This process is called impedance matching. Show that the ratio of turns     $N_{1}/N_{2}$ for this transformer is
  $\frac{N_{1}}{N_{2}}=\frac{\sqrt{Z_{1}}}{\sqrt{Z_{2}}}$


I tried to solve the problem with two equations: 
$$P=V_{1}I_{1}\cos\phi_{1}=V_{2}I_{2}\cos\phi_{2}\tag{1}$$
$$\frac{V_{1}}{V_{2}}=\frac{N_{1}}{N_{2}}\tag{2}$$
And I got:
$$\frac{N_{1}^{2}}{N_{2}^{2}}\frac{R_{1}}{R_{2}}=\frac{Z_{1}^{2}}{Z_{2}^{2}}$$
From the result, only when $\cos{\phi_{1}}=\cos{\phi_{2}}$, I could obtain the answer: $\frac{N_{1}}{N_{2}}=\frac{\sqrt{Z_{1}}}{\sqrt{Z_{2}}}$. But it should not be assumed that $\cos{\phi_{1}}=\cos{\phi_{2}}$. And I have read another article. In the article, it replaced my equation one to
$$P=V_{1}I_{1}=V_{2}I_{2}$$
The power factors are ignored!

Why the power factors of two diffirent AC circuits can be considered as the same when matching the impedance of the transformer?

REPLY [0 votes]: But it should not be assumed that $\cos\phi_1=\cos\phi_2$.

For an ideal transformer, there is no other possibility than that $\phi_1 = \phi_2$
First, one can easily derive the impedance transformation of a transformer with just the voltage and current relationships:
$$V_1 = \frac{N_1}{N_2}V_2$$
$$I_1 = \frac{N_2}{N_1}I_2$$
Assuming the load has impedance $Z_L$, then
$$V_2 = I_2Z_L$$
and it immediately follows that
$$V_1 = \frac{N_1}{N_2}I_2Z_L$$
$$I_1 = \frac{N_2}{N_1}I_2$$
$$\Rightarrow Z_1 \equiv \frac{V_1}{I_1} = \left(\frac{N_1}{N_2}\right)^2Z_L$$
Now, the power delivered to the load impedance is (assuming rms phasors)
$$P_L = \mathrm{Re}\{V_2I_2^*\} = \mathrm{Re}\{|I_2|^2Z_L\} = |I_2|^2|Z_L|\cos\phi_L,\quad \phi_L = \tan^{-1}\left(\frac{X_L}{R_L}\right)$$
and the power delivered to the primary is
$$P_1 = \mathrm{Re}\{V_1I_1^*\} = \mathrm{Re}\left\{|I_2|^2Z_L\right\} = |I_2|^2|Z_L|\cos\phi_L = P_L$$
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