Editing Inductance (section)

===Equivalent circuits===
====T-circuit====
[[File:Mutual inductance equivalent circuit.svg|thumb|''T'' equivalent circuit of mutually coupled inductors]]
Mutually coupled inductors can equivalently be represented by a T-circuit of inductors as shown. If the coupling is strong and the inductors are of unequal values then the series inductor on the step-down side may take on a negative value.<ref>{{Cite book |last=Eslami |first=Mansour |url=https://archive.org/details/circuitanalysisf0000esla/mode/2up |title=Circuit Analysis Fundamentals |date=May 24, 2005 |publisher=Agile Press |isbn=0-9718239-5-2 |location=Chicago, IL, US |publication-date=May 24, 2005 |pages=194 |language=EN}}</ref>

This can be analyzed as a two port network. With the output terminated with some arbitrary impedance {{nowrap|<math>Z</math>,}} the voltage gain {{nowrap|<math>A_v</math>,}} is given by:

<big><math display=block> A_\mathrm{v}
  = \frac{s M Z}{\, s^2 L_1 L_2 - s^2 M^2 + s L_1 Z \,}
  = \frac{k}{\, s \left(1 - k^2\right) \frac{ \sqrt{L_1 L_2} }{Z} + \sqrt{\frac{L_1}{L_2}} \,}
</math></big>

where <math>k</math> is the coupling constant and <math>s</math> is the [[complex frequency]] variable, as above.
For tightly coupled inductors where <math>k = 1</math> this reduces to

<math display=block> A_\mathrm v = \sqrt {L_2 \over L_1} </math>

which is independent of the load impedance. If the inductors are wound on the same core and with the same geometry, then this expression is equal to the turns ratio of the two inductors because inductance is proportional to the square of turns ratio.

The input impedance of the network is given by:

<big><math display=block>Z_\text{in}
  = \frac {s^2 L_1 L_2 - s^2 M^2 + s L_1 Z}{sL_2 + Z}
  = \frac{L_1}{L_2}\, Z\, \left( \frac{ 1 }{ 1 + \frac{Z}{\, s L_2 \,} } \right) \left( 1  + \frac{1 - k^2}{ \frac{Z}{\, s L_2 \,} } \right)
</math></big>

For <math>k = 1</math> this reduces to

<math display=block> Z_\text{in} = \frac{s L_1 Z}{sL_2 + Z} = \frac{L_1}{L_2}\, Z\, \left( \frac{ 1 }{ 1 + \frac{Z}{\, s L_2 \,} } \right)</math>

Thus, current gain <math>A_i</math> is {{em|not}} independent of load unless the further condition

<math display=block>|sL_2| \gg |Z|</math>

is met, in which case,

<math display=block> Z_\text{in} \approx {L_1 \over L_2} Z </math>

and

<math display=block> A_\text{i} \approx \sqrt {L_1 \over L_2} = {1 \over A_\text{v}} </math>

====π-circuit====
[[File:Mutual inductance pi equivalent circuit.svg|thumb|''π'' equivalent circuit of coupled inductors]]
Alternatively, two coupled inductors can be modelled using a ''π'' equivalent circuit with optional ideal transformers at each port. While the circuit is more complicated than a T-circuit, it can be generalized<ref>{{Cite journal |doi = 10.1109/JSSC.2012.2204545|title = Simultaneous 6-Gb/s Data and 10-mW Power Transmission Using Nested Clover Coils for Noncontact Memory Card|journal = IEEE Journal of Solid-State Circuits|volume = 47|issue = 10|pages = 2484–2495|year = 2012|last1 = Radecki|first1 = Andrzej|last2 = Yuan|first2 = Yuxiang|last3 = Miura|first3 = Noriyuki|last4 = Aikawa|first4 = Iori|last5 = Take|first5 = Yasuhiro|last6 = Ishikuro|first6 = Hiroki|last7 = Kuroda|first7 = Tadahiro|bibcode = 2012IJSSC..47.2484R|s2cid = 29266328}}</ref> to circuits consisting of more than two coupled inductors. Equivalent circuit elements {{nowrap|<math>L_\text{s}</math>,}} <math>L_\text{p}</math> have physical meaning, modelling respectively [[magnetic reluctance]]s of coupling paths and [[magnetic reluctance]]s of [[leakage inductance|leakage paths]]. For example, electric currents flowing through these elements correspond to coupling and leakage [[magnetic flux]]es. Ideal transformers normalize all self-inductances to 1&nbsp;Henry to simplify mathematical formulas.

Equivalent circuit element values can be calculated from coupling coefficients with
<math display=block>\begin{align}
L_{S_{ij}} &= \frac{\det(\mathbf{K})}{-\mathbf{C}_{ij}} \\[3pt]
   L_{P_i} &= \frac{\det(\mathbf{K})}{\sum_{j=1}^N\mathbf{C}_{ij}}
\end{align}</math>

where coupling coefficient matrix and its cofactors are defined as
: <math>\mathbf{K} =
  \begin{bmatrix}
         1 & k_{12} & \cdots & k_{1N} \\
    k_{12} &      1 & \cdots & k_{2N} \\
    \vdots & \vdots & \ddots & \vdots \\
    k_{1N} & k_{2N} & \cdots &      1
  \end{bmatrix}\quad
</math> and <math>\quad \mathbf{C}_{ij} = (-1)^{i+j}\,\mathbf{M}_{ij}.</math>

For two coupled inductors, these formulas simplify to
: <math>L_{S_{12}} = \frac{-k_{12}^2 + 1}{k_{12}}\quad</math> and <math>\quad L_{P_1} = L_{P_2} \!=\! k_{12} + 1,</math>

and for three coupled inductors (for brevity shown only for <math>L_\text{s12}</math> and <math>L_\text{p1}</math>)

: <math>
  L_{S_{12}} = \frac{2\,k_{12}\,k_{13}\,k_{23} - k_{12}^2 - k_{13}^2 - k_{23}^2 + 1} {k_{13}\,k_{23} - k_{12}}
\quad</math> and <math>\quad
  L_{P_1} = \frac{2\,k_{12}\,k_{13}\,k_{23} - k_{12}^2 - k_{13}^2 - k_{23}^2 + 1} {k_{12}\,k_{23} + k_{13}\,k_{23} - k_{23}^2 - k_{12}-k_{13} + 1}.
</math>