Flux linkage

Flux linkage $\Lambda$ is the total Magnetic flux coupled to a coil, summed over all of its turns. For a coil of $N$ turns each enclosing the same flux $\Phi$ per turn:

$\Lambda =N\Phi \text{(Wb, or "weber-turns")}.$

Single-turn loops have $\Lambda =\Phi$; multi-turn coils get the multiplicative $N$. This is the quantity that appears in the induced EMF equation:

$V_{\text{emf}}=-\frac{d\Lambda }{dt}=-N\frac{d\Phi }{dt},$

assuming all turns link the same flux (Faraday’s law with the $N$-turn factor).

Why “flux linkage” and not just “flux”

A single loop is “linked” by flux that pierces it once. A coil of $N$ tightly-wound turns is linked by the same flux $N$ times over: each turn separately encloses the flux pattern. The total “linkage” is $N\Phi$ — flux multiplied by the number of times it’s linked.

This is the right quantity for inductors and transformers because Faraday’s law applied turn-by-turn gives an EMF of $-d\Phi /dt$ on each turn. The turns are in series, so their EMFs add: total $V_{\text{emf}}=-Nd\Phi /dt=-d\Lambda /dt$. Flux linkage is the “summed flux as seen by the circuit.”

Versus flux

Quantity	Symbol	Units	What it means
Flux	$\Phi$	Wb	${\int }_S\mathbf{B}\cdot d\mathbf{s}$ through a single surface
Flux linkage	$\Lambda$	Wb (or Wb-turns)	$N\Phi$ for a coil; sum over linkages

In SI, both have the same unit (weber). The “turns” in “weber-turns” is a counting label, not a physical dimension. Some authors keep them separate to avoid confusing single-loop flux with $N$-turn linkage.

When the flux varies turn-to-turn

For a tightly-wound solenoid or torus, every turn sees the same field, so $\Lambda =N\Phi$ exactly. For a loosely-wound coil — or one with non-uniform field — different turns may link different amounts of flux. Then:

$\Lambda ={\sum }_{i=1}^N{\Phi }_i,$

a sum over per-turn fluxes. In practice this matters mainly for transformers with imperfect coupling and for irregular geometries.

In the definition of inductance

Inductance is defined as flux linkage per ampere:

$L=\frac{\Lambda }{I}\text{(H, henry)}.$

A 1 H inductor produces 1 Wb of flux linkage per amp. Equivalently, $\Lambda =LI$ — the flux linkage is directly proportional to the current that produced it (in linear, non-saturating cores).

The ”$N^2$ scaling” of self-inductance — $L=\mu N^{2}S/l$ for a solenoid — comes from this: $N$ appears once in the field magnitude $B=\mu NI/l$ and once in the linkage factor $\Lambda =N\Phi$. Doubling turns quadruples inductance.

For Mutual inductance $M=N_2{\Phi }_{12}/I_1$, the linkage on coil 2 is ${\Lambda }_{12}=N_2{\Phi }_{12}=M{I}_1$ — the cross-coil version of the same idea.

Why it’s a separate concept

Distinguishing flux from flux linkage matters because:

1. EMF formula uses linkage, not flux. $V_{\text{emf}}=-d\Lambda /dt$ correctly accounts for the $N$-turn multiplier. Writing $-d\Phi /dt$ would miss it.

2. Inductance is defined via linkage. $L=\Lambda /I$ packages “geometry plus number of turns” into a single coefficient. The combination $\Lambda$ — not $\Phi$ alone — is what appears in the circuit-level $v=Ldi/dt$ relation.

3. In transformer analysis, the primary and secondary windings have different $N$s; their flux linkages ${\Lambda }_1=N_1\Phi$ and ${\Lambda }_2=N_2\Phi$ relate by the turns ratio $N_2/N_1$, which determines the voltage transformation.

Worked example: $N$-turn solenoid

Solenoid of length $l$, $N$ turns, cross-section $S$, current $I$, vacuum core.

Interior $\mathbf{B}$ from Ampère’s law: $B={\mu }_{0}NI/l$.

Flux per turn: $\Phi =BS={\mu }_{0}NIS/l$.

Flux linkage:

$\Lambda =N\Phi =\frac{{\mu }_0{N}^{2}IS}{l}.$

Inductance:

$L=\frac{\Lambda }{I}=\frac{{\mu }_0{N}^{2}S}{l}.$

Note that the $N^2$ behavior of $L$ comes from two separate factors of $N$: one in the field generation (Ampère), one in the linkage summation. Each is essential.

In flux-linkage-based circuit analysis

In some modeling frameworks (especially for nonlinear magnetic materials), $\Lambda$ is used as a state variable of an inductor:

$v=\frac{d\Lambda }{dt}.$

This works even when $L$ isn’t constant — saturating iron cores have $\Lambda (I)$ nonlinear, but $v=d\Lambda /dt$ remains valid. Linearizing around a bias point recovers $v=Ldi/dt$ with $L=d\Lambda /dI$ as the incremental inductance.

Permanent magnets and superconductors similarly benefit from flux-linkage formulations, where the constraint is “trapped flux linkage” rather than “trapped current.”

Connecting back

The full chain from microscopic field to circuit equation:

$\mathbf{B}\stackrel{{\int }_S\cdot d\mathbf{s}}{\to }\Phi \stackrel{\times N}{\to }\Lambda \stackrel{/I}{\to }L\stackrel{V=LdI/dt}{\to }\text{circuit}.$

Flux linkage is the bridge from field-level $\Phi$ to circuit-level $L$ and $V$. Skipping it conflates single-turn and multi-turn coils.