Power and current contradictions in transformer secondary

Question

Two contradictions that I couldn't find a solution for regarding a transformer:

For an ideal transformer, from conservation of energy we can say that: V1 * I1 = V2 * I2 Where V1, I1 are the primary voltage and current, and the same for V2, I2. Now, let's assume that we connect an AC voltage source, V, in the primary, and the winding ratio is N. The secondary is connected to a resistor.

The power dissipation on the load is (N * V)^2/R. So, it appears that I can change however I want the output power by changing N and R.

(a) How come? Will increasing N will increase the power? But the power in the primary side stays the same, no?

(b) Furthermore, V is given by the voltage source. the secondary sees N * V. Therefore from Ohm's law the current in the secondary is I2 = N * V/R. But from energy conservation we also know that I2 = V1 * I1 / V2 Is changing the resistor in the secondary changes the current in the primary?

The second contradiction is what happens when the secondary is open circuit. We know that the power on the primary is V1 * I1. But the secondary is V2 * 0 = 0. How does the energy conversation holds true? V1 * I1 = V2 * 0 = 0?

Answer 1

Will increasing N will increase the power?

Increase N on the secondary will increase the voltage on the secondary. If the load is purely resistive, increasing the voltage on the secondary will increase the current and power.

but the power in the primary side stays the same, no?

No. The power drawn by the primary side is equal to the losses plus the power consumed by the secondary load.

Is changing the resistor in the secondary changes the current in the primary?

Changing the resistor on the secondary side will change the current on the secondary. This will change the current on the primary.

How does the energy conversation holds true? V1 * I1 = V2 * 0 = 0 ?The second contradiction is what happens when the secondary is open circuit. We know that the power on the primary is V1 * I1 But the secondary is V2 * 0 = 0 How does the energy conversation holds true? V1 * I1 = V2 * 0 = 0 ?

If the transformer were lossless, when the secondary is open circuit, the primary would draw no power. In reality, transformers have losses, and even when the secondary is open circuit, the transformer will waste some power.