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I am hoping for help with the concept of impedance matching with regard to transformers.

I understand that

$$\frac{V_1}{I_1} = \frac{N_1^2}{N_2^2}*R$$

I also understand that

$$\frac{V_1}{I_1} = Z_1 = \sqrt{r^2+(X_L-X_C)^2}, \text{where r is internal resistance of source}$$

The above equation, from my understanding, is identically true, meaning, if an ac source provides a certain voltage and current amplitude, the impedance is defined as the proportion of the two. In this case, the proportion gives the impedance of the circuit to which the primary coil belongs.

My interpretation of the two above equations tells me that, a resistor of $R$ on the secondary coil is equivalent to a resistor of $\frac{N_1^2}{N_2^2}*R$ on the primary coil if the primary coil had no other factors adding to its impedance. The problem here is that the coil provides an inductance, and unless there is a capacitor with reactance equal to the inductive reactance, then there is no way to match the effective resistance $\frac{N_1^2}{N_2^2}*R$ with the internal resistance.

To say it another way, with impedance matching, I know we would want to match the equivalent resistance $\frac{N_1^2}{N_2^2}*R$ to the given internal resistance of the ac source, thus maximizing the power delivered to the load on the secondary coil.

However, based on my understanding, $\frac{N_1^2}{N_2^2}*R$ is identically equal to the impedance of the circuit of the primary coil. Furthermore, according to the equations above, matching internal resistance to effective resistance $\frac{N_1^2}{N_2^2}*R$ would be impossible because of the $(X_L-X_C)^2$ term (unless $X_L=X_C$)

I've come to the conclusion that something in my reasoning must be wrong. Again, I understand how to match internal resistance to the effective resistance $\frac{N_1^2}{N_2^2}*R$. What I am struggling to reconcile this with is that $\frac{V_1}{V_2}$ is identically the impedance of the circuit to which the primary coil belongs.

Help please? Thank you in advance!

EDIT:

I believe I have found my own answer, but would like confirmation. The definition of impedance (for a series RLC circuit) given by:

$$\frac{V_1}{I_1} = Z_1 = \sqrt{r^2+(X_L-X_C)^2}, \text{where r is internal resistance of source}$$

is for the circuit of the primary coil unconnected to a transformer. Once we connect it to a transformer whose secondary coil has a load of R, the impedance of the circuit connected to the primary coil, in its entirety, becomes:

$$\frac{V_1}{I_1} = \frac{N_1^2}{N_2^2}*R$$

As the current must be adjusted in the circuit with the primary coil once it is connected so that the power input and output (i.e. $V_1*I_1$ and $V_2*I_2$ respectively) are equal, the impedance must change too, and so my assumption that

$$\frac{N_1^2}{N_2^2}*R = \sqrt{r^2+(X_L-X_C)^2}, \text{where r is internal resistance of source}$$

is patently false. This makes setting internal resistance equal to effective resistance (i.e. $r=\frac{N_1^2}{N_2^2}*R$) possible, and this is how we maximize the power pushed through the load attached to the secondary coil.

Is all this correct? Or am I still missing something?

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The main reason for the confusion is that the formula Z1=N12/N22∗R (and many other familiar relationships between voltages and currents of the transformer) are valid for an ideal transformer only.

In an ideal or close to an ideal transformer, the impedance seen by the source is dominated by the reflected load, i.e., N12/N22∗R and, therefore all other impedances associated with transformer windings and core, can be ignored.

You can check out this notes or many other sites to see how this formula is derived from basic transformer equations for the primary and secondary loops.

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