MC-A3 · Ideal transformer

The Ideal Transformer and the Turns Ratio

When two coils share one flux, the ratio of their turns rescales the voltage directly. Turn the ratio, pin down what changes and what is conserved, and learn to read a transformer in one line.

What is conserved as you turn the ratio?

A single flux threads both coils. Change the turns ratio and the secondary voltage and current shift with it. Pick what stays put no matter how the others move.

Tap to switch the ratio (primary:secondary). The shared flux stays.

Secondary voltage, current and power

V₂=200V · I₂=1.0A · P=200W

However you turn the ratio, P = V₁I₁ = V₂I₂ is pinned at 200 W. The current drops by the same factor the voltage rises.

Raise the voltage by a — current?

Misaligned

Shared flux makes the voltage ratio

When the same flux Φ threads both coils, each coil’s induced voltage is proportional to its turns. From V = N·dΦ/dt with a shared dΦ/dt, V₁/V₂ = N₁/N₂. So the turns ratio a = N₁/N₂ is the voltage ratio itself.

ObserveV₁V₂ = N₁N₂

With shared flux, voltage is proportional to turns.

Power conservation flips the current ratio

An ideal transformer is lossless, so the power in comes out unchanged: V₁I₁ = V₂I₂. If the voltage becomes a times larger, the current must become 1/a times to keep the product. That is why a step-up transformer lowers the current, and why power lines ship at high voltage and low current to cut losses.

ChooseV₁ I₁ = ?

An ideal transformer is lossless — power is conserved.

Fill inI₁I₂ = ?

From power conservation, the current ratio inverts the turns ratio.

Impedance transforms by the square

A load Z₂ on the secondary looks like a²Z₂ from the primary. Since voltage is a times and current is 1/a times, the impedance seen at the primary is V₁/I₁ = (aV₂)/(I₂/a) = a²·(V₂/I₂). A transformer is thus a tool for impedance matching, not only voltage and current.

On your ownZ₁ = ?

Voltage ×a and current ×1/a give impedance ×a².

Back to the first screen

However you turned the ratio, what stayed conserved was power. The shared flux rescales voltage by the ratio of turns, and the lossless transformer cancels that voltage change with an exactly opposite change in current. Raise the voltage by a and the current falls by 1/a, pinning the product VI = 200 W. A transformer leaves power untouched and only swaps the pairing of voltage and current.

The ideal transformer: two coils share one flux and move energy without loss. For a turns ratio a = N₁/N₂, voltage is V₁/V₂ = a, current is I₁/I₂ = 1/a, and power is conserved as V₁I₁ = V₂I₂. A secondary load appears at the primary scaled by a².

Once you hold this transformation

The turns-ratio transformation is the backbone of power systems. Real transformers depart from this ideal through leakage flux, magnetising current and core and copper losses. The next unit captures that departure in an equivalent circuit to compute efficiency and voltage regulation (MC-A4). The same shared-flux principle carries into the stator–rotor coupling of induction and synchronous machines.