The Ideal Transformer and the Turns Ratio
The turns ratio steps voltage up as much as current down
The primary voltage V1 = 10 V is fixed. Slide the turns ratio n = N2/N1 to change the number of turns on the secondary. The two coils share the same flux, so the secondary voltage V2 = n·V1 rises in direct proportion to the ratio, while to keep the power the same the secondary current falls by 1/n. Set the ratio so the secondary voltage reaches the target of 25 V.
Perfect coupling is the ideal transformer
As seen in mutual inductance, when the coupling coefficient k is 1 every flux line of one coil threads the other. With no leakage and no core loss, this is the ideal transformer. The two coils share the same flux Φ, so the voltage induced in each is proportional to its number of turns. The primary has N1 turns and the secondary N2, both seeing the same changing Φ, so V1 is proportional to N1 and V2 to N2. Take their ratio and V2/V1 = N2/N1 — this is the turns ratio n.
Step the voltage up and the current goes down
An ideal transformer neither stores nor loses energy, so the power in equals the power out: V1·I1 = V2·I2. If the secondary voltage becomes n times larger, then to keep this equality the secondary current must become 1/n times as large. I2/I1 = N1/N2 = 1/n. So the transformer steps the current down by exactly as much as it steps the voltage up. This is why transmission lines raise the voltage to hundreds of thousands of volts: carrying the same power at a smaller current cuts the I²R loss leaking from the wires by that much.
Impedance transforms by the square
Put a load Z_L on the secondary and what impedance does the primary see? The primary impedance is Z_in = V1/I1. Substituting V1 = V2/n and I1 = n·I2 gives Z_in = (V2/n)/(n·I2) = (V2/I2)/n² = Z_L/n². Impedance transforms by the square of the turns ratio. This makes the transformer an impedance-matching tool. To match a load Z_L to a source of internal impedance Z_s, insert a transformer with n = √(Z_L/Z_s) so that Z_L looks like Z_s. It is a common trick for drawing maximum power in audio output stages and RF.
Back to the first screen
As you slid the turns ratio n, the secondary gained turns and its voltage V2 rose as n·V1, reaching the target of 25 V at n = 2.5 (V1 = 10). To keep the power the same the secondary current fell by 1/n, so the transformer swapped voltage for current exactly. The load impedance was seen transformed by the square, 1/n². Two perfectly coupled coils, with a single turns ratio, govern voltage in proportion, current in inverse proportion, and impedance by the square all at once. The k = 1 ceiling of mutual inductance was precisely this device.