When the changing flux of one coil links a second coil, a voltage appears in the second coil without any wire between them. That shared flux is the secret behind transformers, current sensors, wireless chargers and even unwanted crosstalk on a printed circuit board.
Two coils placed close together share part of their magnetic flux, and a change of current in one induces a voltage in the other. Mutual inductance measures how strong that link is, and five simple formulas cover almost every practical calculation.
What Is Mutual Inductance?
Mutual inductance is the property of two coils by which a changing current in one coil induces a voltage in the other coil through their shared magnetic flux. It is the two coil partner of self inductance, which is explained in inductor working principle.
The induced voltage follows Faraday’s law of electromagnetic induction, and its direction follows Lenz's law. The symbol is M and the unit is the henry, the same unit used for self inductance L.
The dots in the figure mark the ends of the two windings that have the same polarity at any instant. You will see these dots on transformer symbols and datasheets, and they decide whether the mutual term is added or subtracted.
How Shared Flux Creates a Voltage
When current i1 flows in coil 1, it produces a magnetic field. Part of that flux, called the mutual flux, also passes through the turns of coil 2, while the rest, called leakage flux, closes its path without reaching coil 2.
If i1 changes, the mutual flux changes and a voltage appears across coil 2 even with no electrical connection. The process works in both directions, and for linear materials M12 equals M21, so one value M describes the pair.
Formula 1b: M = N2 × Φ21 ÷ i1
Example: M = 3 mH, current in coil 1 rises by 2 A in 1 ms
di1 ÷ dt = 2 ÷ 0.001 = 2000 A per second
v2 = 0.003 × 2000 = 6 V
Because only a changing current induces voltage, steady DC through coil 1 gives no output from coil 2. This is why a transformer does not work on pure DC, a point covered in transformer turns ratio and secondary voltage.
Michael Faraday showed mutual induction in 1831 with two coils wound on one iron ring. The galvanometer on the second coil kicked only when he closed or opened the switch on the first coil, never while current flowed steadily.
5 Essential Mutual Inductance Formulas
| No. | Formula | Use |
|---|---|---|
| 1 | v2 = M × di1 ÷ dt | Induced voltage in coil 2 |
| 2 | M = k × √(L1 × L2) | M from self inductances and coupling |
| 3 | L aiding = L1 + L2 + 2M | Series, fluxes add |
| 4 | L opposing = L1 + L2 minus 2M | Series, fluxes oppose |
| 5 | M = (L aiding minus L opposing) ÷ 4 | Measuring M on the bench |
These five relations solve most textbook and field problems on coupled coils. The lecture notes from the University of Ruse also give the stored energy as W = ½ L1 i1² + ½ L2 i2² ± M i1 i2, where the sign depends on whether the fluxes aid or oppose.
Coupling Coefficient k Explained
In mutual inductance work, the coupling coefficient k tells what fraction of the flux of one coil links the other. It ranges from 0, with no shared flux, to 1, where every line of flux links both coils.
k = M ÷ √(L1 × L2), with 0 ≤ k ≤ 1
Example: L1 = 4 mH, L2 = 9 mH, k = 0.5
√(4 × 9) = √36 = 6 mH
M = 0.5 × 6 = 3 mH
Mutual inductance can never exceed the geometric mean of the two self inductances, which is why k cannot be greater than 1. Coils on a closed iron or ferrite core have k close to 1, while air core coils spaced apart may have k well below 0.5.
Common closed core, windings interleaved, k close to 1.
Air gap or distance between coils, k roughly 0.05 to 0.5.
Coils far apart or at right angles, k near 0.
Wireless phone chargers and RFID tags work with loose coupling, so they use resonance to transfer energy efficiently. On a PCB, by contrast, designers try to keep k near zero between neighbouring traces, as shown in PCB crosstalk reduction.
To reduce unwanted mutual inductance between two inductors on a board, place them at right angles and keep them apart. Shielded inductors also confine their flux and cut coupling sharply.
Dot Convention and the Sign of Mutual Inductance
The dot rule says that a current entering the dotted end of one coil induces a voltage that is positive at the dotted end of the other coil. In other words, the dotted ends rise and fall together.
With these signs in place, each coil loop is solved with ordinary Kirchhoff’s voltage law. In AC analysis the mutual term becomes j ω M times the other current, a reactance just like those in impedance and reactance in AC circuits.
Transformer makers mark polarity with dots, with H1 and X1 labels, or with letters such as 1U and 2U. Reversing one winding by mistake when paralleling two transformers can cause a large circulating current.
Mutual Inductance in Series Aiding and Opposing Coils
When two coupled coils are connected in series so that current enters both dotted ends, their fluxes add and the total inductance rises. This is called series aiding, and Circuit Digest gives L = L1 + L2 + 2M for it.
If one coil is reversed, current enters one dot and leaves the other, the fluxes oppose and the total becomes L = L1 + L2 minus 2M. Circuit Digest notes that k can reach a maximum of 1, which means 100 percent coupling.
Formula 4: L opposing = L1 + L2 minus 2M
Example: L1 = 4 mH, L2 = 9 mH, M = 3 mH
L aiding = 4 + 9 + 6 = 19 mH
L opposing = 4 + 9 minus 6 = 7 mH
For coupled coils in parallel the result is L = (L1 L2 minus M²) ÷ (L1 + L2 minus 2M) when aiding and L = (L1 L2 minus M²) ÷ (L1 + L2 + 2M) when opposing. Compare these with the simple rules in series vs parallel circuits, which hold only when M is zero.
Mutual Inductance Calculator
Enter the two self inductances and the coupling coefficient. The calculator returns M and the total inductance for series aiding and series opposing connections.
The default values match the worked example above, so you can check the formula and the tool against each other. Try k = 1 and see that the opposing value falls to 1 mH, the smallest possible for these two coils.
Second Example: Measuring M With an LCR Meter
You rarely know k in advance, but you can find mutual inductance with an LCR meter. Connect the two coils in series and measure L, then reverse one coil and measure again, which gives L aiding and L opposing.
Example: L aiding = 30 mH, L opposing = 14 mH
M = (30 minus 14) ÷ 4 = 4 mH
L1 + L2 = (30 + 14) ÷ 2 = 22 mH, so L1 = L2 = 11 mH if identical
k = 4 ÷ √(11 × 11) = 0.364
The larger of the two readings is always the aiding connection, which also tells you where the dots are. This simple bench method is handy for checking transformer polarity and coil samples before a design is frozen.
Measure at a frequency close to the real operating frequency, because ferrite permeability and winding capacitance change the reading. Keep the meter leads short and twisted.
Where Mutual Inductance Is Used
In a current transformer, the primary conductor and secondary winding share flux on a toroidal core. In an LVDT, a moving core changes the coupling to two secondaries, so the output voltage tracks position.
A common mode choke uses two tightly coupled windings so that noise currents in the same direction see high inductance while the differential signal passes. A flyback converter uses a coupled inductor whose leakage, set by imperfect k, causes the voltage spike that the snubber must absorb.
Common Mistakes and Troubleshooting
- Confirm the dot or polarity marks before connecting windings.
- Use the correct sign of M in every KVL equation.
- Check that k is between 0 and 1 in all calculations.
- Measure inductance at the real operating frequency.
- Allow for leakage inductance in converter and transformer designs.
- Keep sensitive coils at right angles to noisy inductors.
A frequent design error is ignoring leakage, which is the part of each self inductance not shared with the other coil. In galvanic isolation transformers, higher isolation spacing usually lowers k and raises leakage, so the trade off must be checked.
- Transfers energy with no electrical contact.
- Provides galvanic isolation between circuits.
- Allows voltage and current scaling by turns ratio.
- Enables non contact sensing of position and current.
- Works only with changing current, not steady DC.
- Leakage flux causes losses and voltage spikes.
- Unwanted coupling creates crosstalk and noise.
- Core saturation limits the useful current range.
Lecture Notes on Mutually Coupled Inductors
Dot Convention Video Lesson
Mutual Inductance FAQ
It is the ability of one coil to induce a voltage in a nearby coil when its current changes. The two coils share part of their magnetic flux.
The unit is the henry, the same as self inductance. A value of 1 henry means a change of 1 ampere per second in one coil induces 1 volt in the other.
It is the fraction of flux from one coil that links the other coil. Its value lies between 0 for no coupling and 1 for perfect coupling.
It is calculated as k = M ÷ √(L1 × L2). Power transformers have k very close to 1, while air core coils spaced apart have much lower values.
The dots mark winding ends that have the same instantaneous polarity. A current entering a dot induces a positive voltage at the dot of the other coil.
This rule fixes the sign of the mutual inductance term in every circuit equation you write. Getting it wrong changes the predicted voltages and the total inductance of series coils.
In series aiding, current enters both dotted ends and the fluxes add, so the total is L1 plus L2 plus twice the mutual inductance. In series opposing, one coil is reversed and the total becomes L1 plus L2 minus twice that value.
The difference between the two totals is four times the mutual term. Measuring both with an LCR meter is the easiest way to find the mutual inductance of real coils.
Only a changing current produces a changing flux, so steady DC induces no voltage in the second coil. A short pulse appears only when the DC supply is switched on or off.
This is why transformers need AC or switched DC to operate, as mutual inductance acts only on change. Current transformers also give no output on a steady DC primary current.
Neighbouring traces and loops share flux, so a fast current change in one induces noise in another. This unwanted mutual inductance is a major cause of crosstalk on circuit boards.
Designers reduce it by shrinking loop areas, adding solid ground planes and spacing fast traces apart. Placing inductors at right angles also cuts unwanted magnetic coupling.
No, energy considerations limit M to the square root of L1 times L2. So the coupling coefficient can approach 1 but never exceed it in any real pair of coils.
Real transformers reach values very close to 1 with closed cores and interleaved windings. Any calculation that gives k above 1 points to a measurement or arithmetic error.
Related Articles
- Inductor Working Principle
- Faraday's Law of Electromagnetic Induction
- Transformer Turns Ratio and Secondary Voltage
- Common Mode Choke Working Principle
- LVDT Linear Variable Differential Transformer
External References
- Lecture Notes on Mutually Coupled Inductors, University of Ruse
- Inductor Coupling, Series and Parallel Combinations, Circuit Digest
- Inductance and Mutual Inductance, Wikipedia
What We Learn Today
- Mutual inductance M links two coils through shared flux, so a changing current in one coil induces a voltage v2 = M di1 ÷ dt in the other.
- The coupling coefficient k = M ÷ √(L1 L2) runs from 0 to 1, with transformers near 1 and spaced air core coils far lower.
- Series aiding gives L1 + L2 + 2M and series opposing gives L1 + L2 minus 2M, so M equals the difference of the two divided by 4.