CR · Magnetic coupling

The Dot Convention and Polarity

When two coils are coupled, the sign of the mutual term M is set by the direction in which each winding is wound. This unit covers the convention that records that direction with a single dot, and how to read from the dots whether the mutual term is +M or −M.

Same side or opposite?

The arrow running through each coil is that coil's flux, Φ₁ and Φ₂. Switching between the Aiding and Opposing tabs changes whether the current in the right coil enters at its dot or at its undotted end, and the right-hand flux arrow flips accordingly. When both arrows point the same way the fluxes add and reinforce; when they point opposite ways they eat into each other and cancel, shown in red.

Total of the present connection

Lₜ = L₁ + L₂ + 2M

Series total inductance Lₜ

Reinforce · +2M ·

The dot marks the same-polarity end

Drawing out which way every coil is wound is tedious. So by convention a dot is placed on each coil at the end that has the same polarity at the same instant. When current enters one coil at its dot, the direction of that coil's flux is fixed, and while that current is increasing, the voltage induced in the other coil is positive (+) at that coil's dot.

Same side adds, opposite subtracts

If both currents enter at the dots (or both leave from them), the two fluxes overlap in the same direction and reinforce, and the mutual term is +M, the aiding case. If one current enters at a dot and the other at an undotted end, the fluxes point in opposite directions and cancel, and the mutual term is −M, the opposing case. Connecting the two coils in series gives a total inductance of Lₜ = L₁ + L₂ ± 2M, and it is the dots that decide the sign.

ObserveLₜ = L₁ + L₂ + 2M

Dots on the same side add +2M.

ChooseLₜ = L₁ + L₂ ? 2M

Dots on opposite sides subtract 2M.

Fill inM = (L_a − L_b) / ?

The two totals differ by 4M.

On your ownV₁ = jωL₁I₁ ? jωM I₂

In the aiding case, the M term in the voltage equation adds too.

The dots even measure M

Measure the same two coils in series once connected aiding and once opposing, and the two totals differ by exactly 4M. The aiding connection gives L₁ + L₂ + 2M and the opposing one gives L₁ + L₂ − 2M, so subtracting leaves only 4M. That is how the mutual inductance is actually measured: M = (L_aid − L_opp)/4.

Back to the first screen

With the dots on the same side, the two flux arrows pointed the same way and reinforced, and the total grew to L₁ + L₂ + 2M. Moving one dot to the opposite side flipped the right arrow, the fluxes cancelled in red, and the total shrank to L₁ + L₂ − 2M. The dot turned out to be a device that records the sign of the mutual term without drawing the windings at all. In any coupled-circuit problem, this dot is the first symbol to read.

The dot convention marks, on two coupled coils, the ends that have the same polarity at the same instant. If both currents enter at the dots, the fluxes reinforce and the mutual term is +M (aiding); if only one enters at a dot, they cancel and it is −M (opposing). The series total is Lₜ = L₁ + L₂ ± 2M, and by measurement M = (L_aid − L_opp)/4.