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Let $V_1$, $I_1$, $R_1$ and $V_2$, $I_2$, $R_2$ be voltages, currents, resistance in first, second circuit.

And we assume $R_1=R_2$, $I_1\neq I_2$.

The conservation of electrical power $P=V_1I_1=V_2I_2$ holds in the transformer circuits.

Hence $I_1^2R_1=I_2^2R_2$, $R_2=(I_1^2/I_2^2)R_1$ which leads to a contradiction.

What am I missing?

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What you're missing is a source that's driving the whole thing. As you wrote it, there's no contradiction: $I_1=I_2=0$. With a source included, it's no longer true that $I_1^2R_1 = I_2^2 R_2$.

You also need to watch the polarities on your transformer V's and I's: remember power in = power out.

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You're not missing anything. If you assume that the resistances are the same but currents are different, then it is indeed a contradiction because the power in isn't equal to the power out but they should be equal.

If $R_1=R_2$, then also $I_1=I_2$ (assuming no losses, and so on). Your simple argument is a "proof by contradiction" of this simple assertion.

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  • $\begingroup$ In an effort to reconcile our apparently contradictory answers: If the transformer has resistors across the primary and secondary windings, and each resistor current equals its transformer winding current (so you can write the I1^2 R1=I2^2 R2 equation from Pin=Pout), I think there is no room left for a source to drive the whole thing (since either a resistor voltage or current would no longer be equal to its transformer v or i). With no source, all signals are 0. $\endgroup$ – Art Brown Aug 29 '13 at 8:29

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