Inductive Coupling Coefficient, Mutual Inductance, and Self Inductance

How a signed mutual inductance and two self inductances convert to a dimensionless inductive coupling coefficient, with orientation, matrix, and passivity boundaries explicit.

Node role: Concept.

Reader task: Convert consistently between self inductance, signed mutual inductance, and dimensionless inductive coupling, then test whether the resulting matrix can represent passive stored magnetic energy.

Why it matters

Field solvers, coupled-inductor models, and transmission-line matrices may describe the same magnetic coupling with different quantities. The conversion is simple only after all three inductances use the same current basis, orientation, units, frequency, and physical boundary.

SCQ_Design uses lowercase \(k_{ij}\) for the dimensionless inductive coupling coefficient. A bare uppercase \(K\) is avoided because \(\mathbf K_\Phi\) already denotes the flux-stiffness matrix. If an external artifact calls the coefficient K, its schema must explicitly map that field to \(k_{ij}\).

Read the conversion first, then follow magnetic energy to its passive bound and orientation rule. The multi-coordinate and per-length sections show why pairwise plausibility and matching units alone are insufficient.

The result to remember

Internal definition. The signed dimensionless coefficient below is defined from one declared reciprocal inductance matrix. It is not a Hamiltonian exchange rate. The energy model is the starting assumption; subsequent bounds and transformations are this-page algebra.

The books listed at the end are further-reading identities, not verified equation locators for this internal derivation. No extraction accuracy or lossy frequency-dependent conversion is established by that list.

For two reciprocal inductive coordinates with positive self inductances \(L_1\) and \(L_2\) and signed mutual inductance \(M_{12}\),

\[ \boxed{ k_{12} =\frac{M_{12}}{\sqrt{L_1L_2}}, \qquad M_{12}=k_{12}\sqrt{L_1L_2}. } \]

LaTeX source

\boxed{
k_{12}
=\frac{M_{12}}{\sqrt{L_1L_2}},
\qquad
M_{12}=k_{12}\sqrt{L_1L_2}.
}

Quantity	Meaning	SI unit
\(L_1,L_2\)	Self inductances in the declared current/flux-linkage basis	H
\(M_{12}\)	Signed mutual inductance in that same basis	H
\(k_{12}\)	Signed dimensionless inductive coupling coefficient	dimensionless

If a source publishes only a nonnegative coupling magnitude, its mapping is

\[ K:=|k_{12}|. \]

LaTeX source

K:=|k_{12}|.

The missing sign must come from its dot convention or current orientation; it cannot be reconstructed from the magnitude alone.

Start from the flux-linkage matrix

Give both currents declared positive directions and use the power-conjugate orientation for their flux linkages. The reciprocal model is

\[ \boxed{ \begin{pmatrix} \lambda_1\\ \lambda_2 \end{pmatrix} = \underbrace{ \begin{pmatrix} L_1&M_{12}\\ M_{12}&L_2 \end{pmatrix} }_{\mathbf L_{\mathrm{ind}}} \begin{pmatrix} I_1\\ I_2 \end{pmatrix}. } \]

LaTeX source

\boxed{
\begin{pmatrix}
\lambda_1\\
\lambda_2
\end{pmatrix}
=
\underbrace{
\begin{pmatrix}
L_1&M_{12}\\
M_{12}&L_2
\end{pmatrix}
}_{\mathbf L_{\mathrm{ind}}}
\begin{pmatrix}
I_1\\
I_2
\end{pmatrix}.
}

The stored magnetic energy is

\[ U_M =\frac12\vec I^{\,T}\mathbf L_{\mathrm{ind}}\vec I =\frac12L_1I_1^2+\frac12L_2I_2^2+M_{12}I_1I_2. \]

LaTeX source

U_M
=\frac12\vec I^{\,T}\mathbf L_{\mathrm{ind}}\vec I
=\frac12L_1I_1^2+\frac12L_2I_2^2+M_{12}I_1I_2.

Therefore, for a verified flux-linkage matrix in this basis,

\[ L_1=[\mathbf L_{\mathrm{ind}}]_{11}, \qquad L_2=[\mathbf L_{\mathrm{ind}}]_{22}, \qquad M_{12}=[\mathbf L_{\mathrm{ind}}]_{12}. \]

LaTeX source

L_1=[\mathbf L_{\mathrm{ind}}]_{11},
\qquad
L_2=[\mathbf L_{\mathrm{ind}}]_{22},
\qquad
M_{12}=[\mathbf L_{\mathrm{ind}}]_{12}.

The field-extracted matrix contract explains when a magnetostatic source-current matrix is eligible for this interpretation. A derived “mutual-form” table is not automatically the signed off-diagonal \(M_{12}\) used here.

The passive bound comes from stored energy

This-page derivation. Nonnegative quadratic magnetic energy implies the two-coordinate determinant condition. It is a mathematical property of the stated passive model, not an added extraction tolerance.

For a real passive reciprocal two-inductor matrix,

\[ L_1>0, \qquad L_2>0, \qquad \det\mathbf L_{\mathrm{ind}} =L_1L_2-M_{12}^2 =L_1L_2(1-k_{12}^2). \]

LaTeX source

L_1>0,
\qquad
L_2>0,
\qquad
\det\mathbf L_{\mathrm{ind}}
=L_1L_2-M_{12}^2
=L_1L_2(1-k_{12}^2).

Positive semidefinite magnetic energy requires

\[ \boxed{|k_{12}|\le1.} \]

LaTeX source

\boxed{|k_{12}|\le1.}

A finite model with two independent inductive coordinates normally requires positive definiteness and hence \(|k_{12}|<1\). The limit \(|k_{12}|=1\) makes the matrix singular: it represents an ideal perfectly coupled constraint, not two independent finite leakage inductances. A reported \(|k_{12}|>1\) means that the self and mutual values do not form one passive matrix in the claimed basis, or that their definitions, units, orientations, or extraction boundaries were mixed.

The sign belongs to the orientation

\(M_{12}\) and \(k_{12}\) are signed quantities. Reversing the second current coordinate uses

\[ \mathbf S=\operatorname{diag}(1,-1), \qquad \vec I=\mathbf S\vec I', \qquad \mathbf L_{\mathrm{ind}}' =\mathbf S^T\mathbf L_{\mathrm{ind}}\mathbf S. \]

LaTeX source

\mathbf S=\operatorname{diag}(1,-1),
\qquad
\vec I=\mathbf S\vec I',
\qquad
\mathbf L_{\mathrm{ind}}'
=\mathbf S^T\mathbf L_{\mathrm{ind}}\mathbf S.

The self inductances stay unchanged while

\[ M_{12}'=-M_{12}, \qquad k_{12}'=-k_{12}. \]

LaTeX source

M_{12}'=-M_{12},
\qquad
k_{12}'=-k_{12}.

This is a coordinate change, not a change in the hardware. The energy remains the same because currents, flux linkages, voltages, and the inductance matrix are transformed together. A dot convention is a compact drawing of this orientation information.

More than two inductive coordinates

This-page derivation. Congruence-normalize the stated inductance matrix by its positive diagonal. Pairwise bounds do not establish positivity of the complete quadratic form.

For a reciprocal inductance matrix \(\mathbf L_{\mathrm{ind}}\) with positive diagonal entries, the self-inductance matrix is

\[ \mathbf D_L =\operatorname{diag}(L_{11},\ldots,L_{NN}), \]

LaTeX source

\mathbf D_L
=\operatorname{diag}(L_{11},\ldots,L_{NN}),

and the dimensionless coupling matrix

\[ \boxed{ \mathbf K_{\mathrm{cpl}} =\mathbf D_L^{-1/2} \mathbf L_{\mathrm{ind}} \mathbf D_L^{-1/2}. } \]

LaTeX source

\boxed{
\mathbf K_{\mathrm{cpl}}
=\mathbf D_L^{-1/2}
\mathbf L_{\mathrm{ind}}
\mathbf D_L^{-1/2}.
}

Its entries are

\[ [\mathbf K_{\mathrm{cpl}}]_{ii}=1, \qquad [\mathbf K_{\mathrm{cpl}}]_{ij} =k_{ij} =\frac{L_{ij}}{\sqrt{L_{ii}L_{jj}}}. \]

LaTeX source

[\mathbf K_{\mathrm{cpl}}]_{ii}=1,
\qquad
[\mathbf K_{\mathrm{cpl}}]_{ij}
=k_{ij}
=\frac{L_{ij}}{\sqrt{L_{ii}L_{jj}}}.

Conversely,

\[ \boxed{ \mathbf L_{\mathrm{ind}} =\mathbf D_L^{1/2} \mathbf K_{\mathrm{cpl}} \mathbf D_L^{1/2}. } \]

LaTeX source

\boxed{
\mathbf L_{\mathrm{ind}}
=\mathbf D_L^{1/2}
\mathbf K_{\mathrm{cpl}}
\mathbf D_L^{1/2}.
}

Checking every pair against \(|k_{ij}|\le1\) is necessary but not sufficient when \(N>2\). The complete \(\mathbf L_{\mathrm{ind}}\), equivalently \(\mathbf K_{\mathrm{cpl}}\), must still be positive semidefinite. This matrix property distinguishes incompatible collections of individually plausible pairwise coefficients.

Per-unit-length and finite-section values

This-page derivation. Multiply every entry of one uniform per-length matrix by the same section length; the common factor cancels in the definition of \(k\).

For a uniform coupled-line section with per-unit-length inductance matrix \(\mathbf L'\) in \(\mathrm{H/m}\),

\[ k_{ij}' =\frac{L_{ij}'}{\sqrt{L_{ii}'L_{jj}'}}. \]

LaTeX source

k_{ij}'
=\frac{L_{ij}'}{\sqrt{L_{ii}'L_{jj}'}}.

Multiplying every entry by the same section length \(\Delta z\) gives

\[ M_{ij}^{\mathrm{section}}=L_{ij}'\Delta z, \qquad L_i^{\mathrm{section}}=L_{ii}'\Delta z, \]

LaTeX source

M_{ij}^{\mathrm{section}}=L_{ij}'\Delta z,
\qquad
L_i^{\mathrm{section}}=L_{ii}'\Delta z,

so the common length cancels and

\[ k_{ij}^{\mathrm{section}}=k_{ij}'. \]

LaTeX source

k_{ij}^{\mathrm{section}}=k_{ij}'.

This cancellation requires one uniform common section. Unequal physical lengths, different current bases, or independently reduced subblocks must be assembled into one inductance matrix before the coefficient is calculated.

Numerical sanity check

Use

\[ L_1=10\,\mathrm{nH}, \qquad L_2=40\,\mathrm{nH}, \qquad k_{12}=-0.25. \]

LaTeX source

L_1=10\,\mathrm{nH},
\qquad
L_2=40\,\mathrm{nH},
\qquad
k_{12}=-0.25.

Then

\[ M_{12} =-0.25\sqrt{(10\,\mathrm{nH})(40\,\mathrm{nH})} =-5\,\mathrm{nH}, \]

LaTeX source

M_{12}
=-0.25\sqrt{(10\,\mathrm{nH})(40\,\mathrm{nH})}
=-5\,\mathrm{nH},

and

\[ \mathbf L_{\mathrm{ind}} = \begin{pmatrix} 10&-5\\ -5&40 \end{pmatrix}\mathrm{nH}. \]

LaTeX source

\mathbf L_{\mathrm{ind}}
=
\begin{pmatrix}
10&-5\\
-5&40
\end{pmatrix}\mathrm{nH}.

Its determinant is \(375\,\mathrm{nH}^2>0\), so the matrix is positive definite. Reversing the second current orientation changes the off-diagonal entries and \(k_{12}\) to positive values while describing the same physical coupled pair.

What this coefficient does not mean

\(k_{ij}\) is a dimensionless property of one declared inductance matrix. It is not the Hamiltonian coupling rate \(g\), \(J\), or \(G\) in rad/s.
\(k_{ij}\) alone does not determine resonance splitting. Capacitances, detuning, normalization, retained coordinates, and loads still matter.
An arbitrary complex impedance entry \(Z_{ij}(\omega)\) is not automatically a real mutual inductance. A lossy or dispersive model needs a declared frequency-dependent matrix and passivity/causality treatment.
This inductive definition does not define a capacitive coupling coefficient; a capacitance matrix has different sign and branch-lowering semantics.

Reporting checklist

Engineering choice. These reporting fields preserve interpretability; they do not define a universal promotion or numerical acceptance rule.

Before publishing \(k_{ij}\) or reconstructing \(M_{ij}\), record:

the complete inductance-matrix identity and basis;
current and flux-linkage orientations or dot convention;
self and mutual inductance units;
extraction frequency and loss/dispersion assumptions;
whether values are per unit length or finite section;
reciprocity, symmetry, and positive-semidefinite evidence; and
whether the reported coefficient is signed \(k_{ij}\) or magnitude only.

Connections

Field-Extracted Capacitance and Inductance Matrices owns the field-solver matrix and branch-map boundary.
Multiconductor RLGC Matrices applies the same conversion to a terminal-basis per-unit-length matrix.
Circuit Models to Bare Coordinates, Open EOM, and Normal Modes shows how the resulting inductive energy enters the circuit Lagrangian and Hamiltonian.
Parameter Layers separates matrix coefficients from Hamiltonian couplings and hybridized observables.

References

C. R. Paul, Inductance: Loop and Partial, Wiley-IEEE Press, 2010.
F. W. Grover, Inductance Calculations: Working Formulas and Tables, Dover, 2004 reprint.
Multiconductor Transmission-Line Characterization records the conductor/modal-basis and normalization source trail used by the per-unit-length application.

Review Links

Field	Value
Status	Seed
Used by	Field-Extracted Capacitance and Inductance Matrices and Multiconductor RLGC Matrices.
Open review question	Can a reader convert both directions, explain the sign change under one reversed current orientation, and distinguish \(k_{ij}\) from a Hamiltonian coupling rate?