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Mutual inductance between two coils

How strongly are these two coils coupled?

The formula
M=k⁢L1·L2
M
Mutual inductance, H
k
Coupling coefficient
L1
First inductance, H
L2
Second inductance, H
LaTeX 
M = k \sqrt{L1 \cdot L2}

Work it out

mH

The self-inductance of the drive coil.

mH

The self-inductance of the pickup coil.

1 if every field line from one coil passes through the other; 0 if none do. Falls off quickly with separation and misalignment.

Compare with a variation

Result

M Mutual inductance 45 mH

How much EMF one coil induces in the other per unit rate of change of its own current.

With your numbers
45 mH=0.9⁢50 mH·50 mH
LaTeX 
45\,\mathrm{mH} = 0.9 \sqrt{50\,\mathrm{mH} \cdot 50\,\mathrm{mH}}

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

What this looks like

Mutual inductance against coupling coefficient Coupling coefficient swept from 0.45 to 1 with everything else held at your numbers. The dashed lines cross where you are.
Mutual inductance against coupling coefficientMutual inductance rises from 22.5 mH to 50 mH as coupling coefficient rises from 0.45 to 1. At your coupling coefficient of 0.9 it is 45 mH.203040500.40.60.810.945 mHCoupling coefficientMutual inductance (mH)
The formula behind the curve
M=k⁢L1·L2
M
Mutual inductance, H
k
Coupling coefficient
L1
First inductance, H
L2
Second inductance, H
LaTeX 
M = k \sqrt{L1 \cdot L2}
What moves the answer Each input moved 10% either way, with the others held still, and the effect on mutual inductance.
What moves the answerMutual inductance is most sensitive to Coupling coefficient, which moves it by about 10% for a 10% change. It is least sensitive to Second coil's inductance, at about 5.13%.Change in the answer when each input moves by 10%-10%-5%0%5%10%Coupling coefficient±10First coil's inductance±5.13Second coil's inductance±5.13
The formula behind the curve
M=k⁢L1·L2
M
Mutual inductance, H
k
Coupling coefficient
L1
First inductance, H
L2
Second inductance, H
LaTeX 
M = k \sqrt{L1 \cdot L2}

Method

  1. Convert both inductances to henries.
  2. Take the geometric mean of the two — √(L₁ × L₂). This is the mutual inductance two perfectly coupled coils would have.
  3. Multiply by the coupling coefficient k to scale that ideal figure down to what these two coils, at this separation and alignment, actually achieve.

Assumptions

  • k is supplied, not derived from geometry. Estimating it from coil spacing, core material and alignment is a separate, harder problem this calculation does not attempt — it takes k as a measured or assumed figure.
  • k cannot exceed 1, which is the theoretical limit of perfect coupling; a value above that is not physically possible for two real coils and the result should not be trusted.
  • Both coils are linear — neither core is near saturation, where its effective permeability, and so its inductance and coupling, would fall.