The Ideal Transformer and the Turns Ratio
The turns ratio steps voltage up as much as current down
The primary voltage V1 = 10V is fixed, and sliding the turns ratio n = N2/N1 changes the number of turns on the secondary coil on the right. Because both coils share the same flux, the secondary voltage V2 = n·V1 rises in direct proportion to the turns ratio, and to keep the power the same the secondary current must drop to 1/n. The lower bar is that voltage; set the turns ratio so the secondary voltage reaches the target of 25V on the orange dashed line.
Perfect coupling is the ideal transformer
As in mutual inductance, when the coupling coefficient k is 1, all of one coil's flux passes through the other. Add the assumption of no leakage and no core loss and you have an ideal transformer. Since the two coils share the same flux Φ, the voltage induced in each coil is proportional to its number of turns. The primary's N1 turns and the secondary's N2 turns see the same change in Φ, so V1 is proportional to N1 and V2 to N2. Taking the ratio gives V2/V1 = N2/N1, which is called the turns ratio n. In the plot, the number of turns drawn on the right coil also grows and shrinks with this ratio.
Step the voltage up and the current goes down
An ideal transformer neither stores nor loses energy, so the power going in equals the power coming out: V1·I1 = V2·I2. If the secondary voltage becomes n times as large, the secondary current must become 1/n times as large to keep this equality: I2/I1 = N1/N2 = 1/n. So a transformer lowers the current by exactly as much as it raises the voltage. This is why power lines step the voltage up to hundreds of thousands of volts: carrying the same power with a smaller current shrinks the I²R loss that leaks away as heat in the wires, in proportion to the square of the current.
Impedance transforms by the square
Attach a load Z_L to the secondary: what impedance does the primary see? The primary-side impedance is Z_in = V1/I1, and substituting V1 = V2/n and I1 = n·I2 gives Z_in = (V2/n)/(n·I2) = (V2/I2)/n² = Z_L/n². The impedance appears changed by the square of the turns ratio, which makes the transformer a tool for impedance matching. To match a load Z_L to a source with 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 method for drawing maximum power from audio output stages and high-frequency circuits.
Back to the first screen
As you slid the turns ratio n, the secondary coil on the right gained turns and the secondary voltage V2 rose as n·V1, reaching the target of 25V at n = 2.5 (V1 = 10). To keep the power the same the secondary current fell to 1/n, so the transformer traded voltage for current exactly, and the load impedance appeared changed by the square, 1/n². With two perfectly coupled coils, choosing just the turns ratio sets voltage in proportion, current in inverse proportion and impedance by the square, all at once. The k = 1 ceiling you met in mutual inductance was exactly this device.