CR · Magnetic coupling

The Dot Convention and Polarity

When two coils couple, the sign of the mutual term M is set by how the windings are wound. Learn the convention that records that direction with a single dot, and how to read whether the dot means +M or −M.

Same side or opposite?

Switch how the currents enter the dots of the two coils. Do the two fluxes add up in the same direction, or cancel?

Total of the present connection

Lₜ = L₁ + L₂ + 2M

Series total inductance Lₜ

Reinforce · +2M ·

The dot marks the same-polarity end

Drawing the exact winding direction every time is tedious. So we agree to place one dot on the end of each coil that reaches the same polarity at the same instant. When current enters one dotted end, the flux direction of that coil is fixed, and the dot on the other coil tells you the polarity of the induced voltage to match.

Same side adds, opposite subtracts

When both currents enter the dots (or both leave them), the two fluxes overlap in the same direction and reinforce. The mutual term is then +M — aiding. If one enters the dot and the other enters the undotted end, the fluxes oppose and cancel, and the mutual term becomes −M — opposing. In series, Lₜ = L₁ + L₂ ± 2M, and the dots set the sign.

ObserveLₜ = L₁ + L₂ + 2M

Dots same side give +2M.

ChooseLₜ = L₁ + L₂ ? 2M

Dots opposite give −2M.

Fill inM = (L_a − L_b) / ?

The two totals differ by 4M.

On your ownV₁ = jωL₁I₁ ? jωM I₂

Aiding also adds the M term.

The dots even measure M

Measure the same two coils in series once aiding and once opposing, and the difference of the two totals is exactly 4M. The aiding connection gives L₁ + L₂ + 2M, the opposing one L₁ + L₂ − 2M, so subtracting leaves only 4M. Hence M = (L_aiding − L_opposing)/4 measures the mutual inductance directly.

Back to the first screen

With the dots on the same side, the two fluxes overlapped in the same direction and reinforced, and the total grew to L₁ + L₂ + 2M; on opposite sides they cancelled and it shrank to L₁ + L₂ − 2M. The dot is a device that records the sign of the mutual term without ever drawing the winding direction. In any coupled-circuit problem, the dot is the very first mark to read.

The dot convention marks, on two coupled coils, the ends that reach the same polarity at the same instant. When both currents enter the dots the flux reinforces and the mutual term is +M (aiding); when only one enters its dot it cancels and the term is −M (opposing). In series, Lₜ = L₁ + L₂ ± 2M, and by measurement M = (L_aiding − L_opposing)/4.