MC-A3 · Ideal transformer

The Ideal Transformer and the Turns Ratio

When two coils share the same flux, the ratio of their turns becomes the ratio of their voltages. Change the turns ratio, pin down what changes and what stays put, and learn to read a transformer as a single relationship.

What is conserved as you turn the ratio?

One flux passes through both coils. When you change the turns ratio with the tabs, the height of the voltage bar and the thickness of the current bar on the secondary coil change with it, while the voltage and current of the primary coil stay as they are. What does not change however the turns ratio is set is the question of this lesson.

The tabs change the turns ratio (primary : secondary). The flux the two coils share stays the same.

Secondary voltage, current and power

V₂=200V · I₂=1.0A · P=200W

However the turns ratio is set, P = V₁I₁ = V₂I₂ stays at 200 W, because the current goes down by the same factor the voltage goes up.

Raise the voltage by a — current?

Misaligned

Shared flux makes the voltage ratio

When the same flux Φ passes through both coils, the voltage induced in each is proportional to its number of turns. In V = N·dΦ/dt, dΦ/dt is common to both coils, so V₁/V₂ = N₁/N₂, and the turns ratio a = N₁/N₂ becomes the voltage ratio as it is. It helps to think of the voltage per turn as the same in both coils. Because a voltage appears only when the flux changes, a transformer works only on AC; apply DC and no voltage comes out on the secondary, while only a large current flows in the primary coil.

ObserveV₁V₂ = N₁N₂

Because the two coils share the flux, voltage is proportional to the number of turns.

Power conservation flips the current ratio

An ideal transformer has no losses, so the power that goes in comes out unchanged: V₁I₁ = V₂I₂. If the voltage becomes a times larger, the current has to become 1/a times as large for the product to stay the same, so a transformer that raises voltage lowers current. This is also why power is transmitted at high voltage and low current: line loss goes with the square of the current, so raising the voltage to cut the current cuts the loss sharply. When a load on the secondary draws current, the primary current rises by the matching magnetomotive force so that the two magnetomotive forces oppose each other, and that is why the flux stays the same.

ChooseV₁ I₁ = ?

An ideal transformer has 0 loss, so the power is kept as it is.

Fill inI₁I₂ = ?

Since the power stays the same, the current ratio is the reciprocal of the turns ratio.

Impedance transforms by the square

A load Z₂ hung on the secondary looks like a²Z₂ from the primary. The voltage is a times larger and the current 1/a times as large, so the impedance seen from the primary is V₁/I₁ = (aV₂)/(I₂/a) = a²·(V₂/I₂). So a transformer is a device that changes voltage and current and also a tool for matching impedances; the output transformer of a vacuum-tube amplifier, matching the high output impedance to a speaker's low impedance, is one example.

On your ownZ₁ = ?

Voltage a times, current 1/a times, so the impedance becomes a² times.

Back to the first screen

What stayed the same however you set the turns ratio was the power. The shared flux scales the voltage by the turns ratio, and a lossless transformer makes up exactly for that voltage change with the opposite change in current. Raise the voltage a times and the current drops to 1/a times, so the product VI = 200 W is fixed. A transformer is a device that leaves the power alone 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, the voltage is V₁/V₂ = a, the current I₁/I₂ = 1/a, and the power stays the same as V₁I₁ = V₂I₂. A load on the secondary appears at the primary as an impedance a² times as large.

Once you hold this transformation

Turns-ratio conversion is the backbone of power systems. Real transformers, though, depart from this ideal because of leakage flux, magnetizing current, and core and copper losses, and the next lesson captures that departure in an equivalent circuit to calculate efficiency and voltage regulation (MC-A4). The same principle of shared flux carries on into the stator–rotor coupling of induction and synchronous machines.