Editing Inductance (section)

==Mutual inductance==
{{Further|Inductive coupling}}

=== Definition of Mutual induction or Coefficient of mutual induction ===
The mutual inductance or the coefficient of mutual induction of two magnetically linked coils is equal to the flux linkage of one coil per unit current in the neighboring coil.                OR

The mutual inductance or the coefficient of mutual induction of two magnetically linked coils is numerically equal  to the emf induced in one coil (secondary) per unit time rate of change of current in the neighboring coil (primary).

===Mutual inductance of two parallel straight wires===
There are two cases to consider:
# Current travels in the same direction in each wire, and
# current travels in opposing directions in the wires.
Currents in the wires need not be equal, though they often are, as in the case of a complete circuit, where one wire is the source and the other the return.

===Mutual inductance of two wire loops===
This is the generalized case of the paradigmatic two-loop cylindrical coil carrying a uniform low frequency current; the loops are independent closed circuits that can have different lengths, any orientation in space, and carry different currents. Nonetheless, the error terms, which are not included in the integral are only small if the geometries of the loops are mostly smooth and convex: They must not have too many kinks, sharp corners, coils, crossovers, parallel segments, concave cavities, or other topologically "close" deformations. A necessary predicate for the reduction of the 3-dimensional manifold integration formula to a double curve integral is that the current paths be filamentary circuits, i.e. thin wires where the radius of the wire is negligible compared to its length.

The mutual inductance by a filamentary circuit <math>m</math> on a filamentary circuit <math>n</math> is given by the double integral ''[[Franz Ernst Neumann|Neumann]] formula''<ref>
{{cite journal
 | last=Neumann | first=F.E. |author-link=Franz Ernst Neumann
 | year=1846
 | title=Allgemeine Gesetze der inducirten elektrischen Ströme  | language=de
 | trans-title=General rules for induced electric currents
 | journal=Annalen der Physik und Chemie
 | volume=143  | issue=1  | pages=31–44
 | issn=0003-3804  | doi=10.1002/andp.18461430103
 | bibcode = 1846AnP...143...31N
 | url = https://zenodo.org/record/1423608
 | publisher=Wiley
}}
</ref>
: <math display=block> L_{m,n} = \frac{\mu_0}{4\pi} \oint_{C_m} \oint_{C_n} \frac{\mathrm{d}\mathbf{x}_m\cdot \mathrm{d}\mathbf{x}_n}{\ \left| \mathbf{x}_m - \mathbf{x}_n \right|\ }\ ,</math>

where
: <math>C_m</math> and <math>C_n</math> are the curves followed by the wires.
: <math>\mu_0</math> is the [[permeability of free space]] ({{nowrap|4{{pi}}×{{10^|−7}} H/m}})
: <math>\mathrm{d}\mathbf{x}_m</math> is a small increment of the wire in circuit {{mvar|C}}{{sub|m}}
: <math>\mathbf{x}_m</math> is the position of <math>\mathrm{d}\mathbf{x}_m</math> in space
: <math>\mathrm{d}\mathbf{x}_n</math> is a small increment of the wire in circuit {{mvar|C}}{{sub|n}}
: <math>\mathbf{x}_n</math> is the position of <math>\mathrm{d}\mathbf{x}_n</math> in space.

===Derivation===
<math display=block> M_{ij} \mathrel\stackrel{\mathrm{def}}{=} \frac{\Phi_{ij}}{I_j} </math>

where
* <math>I_j</math> is the current through the <math>j</math>th wire, this current creates the magnetic flux <math>\Phi_{ij}\ \,</math>through the <math>i</math>th surface
* <math>\Phi_{ij}</math> is the [[magnetic flux]] through the ''i''th surface due to the [[electrical circuit]] outlined by {{nowrap|<math>C_j</math>:}}<ref>{{cite book |last=Jackson |first=J. D. |title=Classical Electrodynamics |url=https://archive.org/details/classicalelectro00jack_0|url-access=registration |date=1975 |publisher=Wiley |pages=[https://archive.org/details/classicalelectro00jack_0/page/176 176], 263|isbn=9780471431329 }}</ref> 
<math display=block>\Phi_{ij} = \int_{S_i} \mathbf{B}_j\cdot\mathrm{d}\mathbf{a} = \int_{S_i} (\nabla\times\mathbf{A_j})\cdot\mathrm{d}\mathbf{a}
  = \oint_{C_i} \mathbf{A}_j\cdot\mathrm{d}\mathbf{s}_i = \oint_{C_i} \left(\frac{\mu_0 I_j}{4\pi} \oint_{C_j} \frac{\mathrm{d}\mathbf{s}_j}{\left|\mathbf{s}_i-\mathbf{s}_j\right|}\right) \cdot \mathrm{d}\mathbf{s}_i </math>

where
{{plainlist|indent=1|1=
* <math>C_i</math> is the curve enclosing surface {{nowrap|<math>S_i</math>;}} and <math>S_i</math> is any arbitrary orientable area with edge <math>C_i</math>
* <math>\mathbf{B}_j</math> is the [[magnetic field]] vector due to the {{nowrap|<math>j</math>-th}} current (of circuit {{nowrap|<math>C_j</math>).}}
* <math>\mathbf{A}_j</math> is the [[vector potential]] due to the {{nowrap|<math>j</math>-th}} current.
}}

[[Stokes' theorem]] has been used for the 3rd equality step. For the last equality step, we used the [[retarded potential]] expression for <math>A_j</math> and we ignore the effect of the retarded time (assuming the geometry of the circuits is small enough compared to the wavelength of the current they carry). It is actually an approximation step, and is valid only for local circuits made of thin wires.

Mutual inductance is defined as the ratio between the EMF induced in one loop or coil by the rate of change of current in another loop or coil.  Mutual inductance is given the symbol {{mvar|M}}.

=== Derivation of mutual inductance ===
The inductance equations above are a consequence of [[Maxwell's equations]]. For the important case of electrical circuits consisting of thin wires, the derivation is straightforward.

In a system of <math>K</math> wire loops, each with one or several wire turns, the [[flux linkage]] of loop {{nowrap|<math>m</math>,}} {{nowrap|<math>\lambda_m</math>,}} is given by
<math display=block>\displaystyle \lambda_m = N_m \Phi_m = \sum\limits_{n=1}^K L_{m,n}\ i_n\,.</math>

Here <math>N_m</math> denotes the number of turns in loop {{nowrap|<math>m</math>;}} <math>\Phi_m</math> is the [[magnetic flux]] through loop {{nowrap|<math>m</math>;}} and <math>L_{m,n}</math> are some constants described below. This equation follows from [[Ampere's law]]: ''magnetic fields and fluxes are linear functions of the currents''. By [[Faraday's law of induction]], we have

<math display=block>\displaystyle v_m = \frac{\text{d}\lambda_m}{\text{d}t} = N_m \frac{\text{d}\Phi_m}{\text{d}t} = \sum\limits_{n=1}^K L_{m,n}\frac{\text{d}i_n}{\text{d}t},</math>

where <math>v_m</math> denotes the voltage induced in circuit {{nowrap|<math>m</math>.}} This agrees with the definition of inductance above if the coefficients <math>L_{m,n}</math> are identified with the coefficients of inductance. Because the total currents <math>N_n\ i_n</math> contribute to <math>\Phi_m</math> it also follows that <math>L_{m,n}</math> is proportional to the product of turns {{nowrap|<math>N_m\ N_n</math>.}}

=== Mutual inductance and magnetic field energy ===
Multiplying the equation for ''v<sub>m</sub>'' above with ''i<sub>m</sub>dt'' and summing over ''m'' gives the energy transferred to the system in the time interval ''dt'',
<math display=block>
  \sum \limits_m^K i_m v_m \text{d}t = \sum\limits_{m,n=1}^K i_m L_{m,n} \text{d}i_n
    \mathrel\overset{!}{=} \sum\limits_{n=1}^K \frac{\partial W \left(i\right)}{\partial i_n} \text{d}i_n.
</math>

This must agree with the change of the magnetic field energy, ''W'', caused by the currents.<ref>The kinetic energy of the drifting electrons is many orders of magnitude smaller than W, except for nanowires.</ref> The [[symmetry of second derivatives|integrability condition]]

<math display=block>\displaystyle\frac{\partial^2 W}{\partial i_m \partial i_n} = \frac{\partial^2 W}{\partial i_n \partial i_m}</math>

requires ''L<sub>m,n</sub>&nbsp;=&nbsp;L<sub>n,m</sub>''. The inductance matrix, ''L<sub>m,n</sub>'', thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents,
<math display=block>\displaystyle W\left(i\right) = \frac{1}{2} \sum \limits_{m,n=1}^K i_m L_{m,n} i_n.</math>

This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a build-up or decrease of magnetic field energy. The corresponding energy transfer requires or generates a voltage. A [[impedance analogy|mechanical analogy]] in the ''K''&nbsp;=&nbsp;1 case with magnetic field energy (1/2)''Li''<sup>2</sup> is a body with mass ''M'', velocity ''u'' and kinetic energy (1/2)''Mu''<sup>2</sup>. The rate of change of velocity (current) multiplied with mass (inductance) requires or generates a force (an electrical voltage).

[[File:Mutually inducting inductors.PNG|thumb|300px|right|Circuit diagram of two mutually coupled inductors.  The two vertical lines between the windings indicate that the transformer has a [[magnetic core|ferromagnetic core]] . "n:m" shows the ratio between the number of windings of the left inductor to windings of the right inductor. This picture also shows the [[dot convention]].]]

Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by which [[transformer]]s work, but it can also cause unwanted coupling between conductors in a circuit.

The mutual inductance, {{nowrap|<math>M_{ij}</math>,}} is also a measure of the coupling between two inductors. The mutual inductance by circuit <math>i</math> on circuit <math>j</math> is given by the double integral ''[[Franz Ernst Neumann|Neumann]] formula'', see [[#Calculating inductance|calculation techniques]]

The mutual inductance also has the relationship:
<math display=block>M_{21} = N_1\ N_2\ P_{21} \!</math>
where

{{plainlist|1=
* <math>M_{21}</math> is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 due to the current in coil 1.
* <math>N_1</math> is the number of turns in coil 1,
* <math>N_2</math> is the number of turns in coil 2,
* <math>P_{21}</math> is the [[permeance]] of the space occupied by the flux.
|indent=1}}

Once the mutual inductance <math>M</math> is determined, it can be used to predict the behavior of a circuit:
<math display=block> v_1 = L_1\ \frac{\text{d}i_1}{\text{d}t} - M\ \frac{\text{d}i_2}{\text{d}t} </math>
where

{{plainlist|1=
* <math>v_1</math> is the voltage across the inductor of interest;
* <math>L_1</math> is the inductance of the inductor of interest;
* <math>\text{d}i_1\,/\,\text{d}t</math> is the derivative, with respect to time, of the current through the inductor of interest, labeled 1;
* <math>\text{d}i_2\,/\,\text{d}t</math> is the derivative, with respect to time, of the current through the inductor, labeled 2, that is coupled to the first inductor; and
* <math>M</math> is the mutual inductance.
|indent=1}}

The minus sign arises because of the sense the current <math>i_2</math> has been defined in the diagram. With both currents defined going into the [[dot convention|dot]]s the sign of <math>M</math> will be positive (the equation would read with a plus sign instead).<ref>{{cite book|first1=Mahmood|last1=Nahvi |first2=Joseph|last2=Edminister |url=https://books.google.com/books?id=nrxT9Qjguk8C&pg=PA338|title= Schaum's outline of theory and problems of electric circuits|page=338|publisher=McGraw-Hill Professional|year=2002|isbn=0-07-139307-2}}</ref>

===Coupling coefficient===

The coupling coefficient is the ratio of the open-circuit actual voltage ratio to the ratio that would be obtained if all the flux coupled from one [[magnetic circuit]] to the other.  The coupling coefficient is related to mutual inductance and self inductances in the following way. From the two simultaneous equations expressed in the two-port matrix the open-circuit voltage ratio is found to be:

<math display=block>{V_2 \over V_1}_\text{open circuit} = {M \over L_1}</math>
where

{{plainlist|indent=1|1=
* <math>M^{2} = M_1 M_2</math>
}}

while the ratio if all the flux is coupled is the ratio of the turns, hence the ratio of the square root of the inductances

<math display=block>{V_2 \over V_1}_\text{max coupling} = \sqrt{ L_2 \over L_1\ }</math>

thus,

<math display=block>M = k \sqrt{L_1\ L_2\ } </math>
where

{{plainlist|indent=1|1=
* <math>k</math> is the ''coupling coefficient'',
* <math>L_1</math> is the inductance of the first coil, and
* <math>L_2</math> is the inductance of the second coil.
}}

The coupling coefficient is a convenient way to specify the relationship between a certain orientation of inductors with arbitrary inductance.  Most authors define the range as {{nowrap|<math> 0 \le k < 1</math>,}} but some<ref>{{cite book |first=Stephen C. |last=Thierauf |title=High-speed Circuit Board Signal Integrity |url=https://archive.org/details/highspeedcircuit00thie_269 |url-access=limited |page=[https://archive.org/details/highspeedcircuit00thie_269/page/n70 56] |publisher=Artech House |year=2004 |isbn=1580538460}}</ref> define it as {{nowrap|<math> -1 < k < 1\,</math>.}} Allowing negative values of <math>k</math> captures phase inversions of the coil connections and the direction of the windings.<ref>{{cite journal|doi=10.5573/JSTS.2009.9.4.198|title=Design of a Reliable Broadband I/O Employing T-coil |year=2009 |last1=Kim |first1=Seok |last2=Kim |first2=Shin-Ae |last3=Jung |first3=Goeun |last4=Kwon |first4=Kee-Won |last5=Chun |first5=Jung-Hoon |journal=Journal of Semiconductor Technology and Science |volume=9 |issue=4 |pages=198–204 |s2cid=56413251 |via=ocean.kisti.re.kr |s2cid-access=free |url=http://ocean.kisti.re.kr/downfile/volume/ieek/E1STAN/2009/v9n4/E1STAN_2009_v9n4_198.pdf |url-status=live |archive-url=https://web.archive.org/web/20180724115136/http://ocean.kisti.re.kr/downfile/volume/ieek/E1STAN/2009/v9n4/E1STAN_2009_v9n4_198.pdf |archive-date= Jul 24, 2018 }}</ref>

===Matrix representation===
Mutually coupled inductors can be described by any of the [[two-port network]] parameter matrix representations. The most direct are the [[z parameters]], which are given by<ref>{{Cite book |last=Aatre |first=Vasudev K. |url=https://archive.org/details/networktheoryfil0000aatr/page/n1/mode/2up |title=Network Theory and Filter Design |date=1981 |publisher=John Wiley & Sons |isbn=0-470-26934-0 |location=US, Canada, Latin America, and Middle East |publication-date=1981 |pages=71, 72 |language=EN}}</ref>

<math display=block> [\mathbf z] = s \begin{bmatrix} L_1 \ M \\ M \ L_2 \end{bmatrix} .</math>

The [[Admittance parameters|y parameters]] are given by

<math display="block"> [\mathbf y] = \frac{1}{s} \begin{bmatrix} L_1 \ M \\ M \ L_2 \end{bmatrix}^{-1} .</math>  Where <math>s</math> is the [[complex frequency]] variable, <math>L_1</math> and <math>L_2</math> are the inductances of the primary and secondary coil, respectively, and <math>M</math> is the mutual inductance between the coils.

==== Multiple Coupled Inductors ====
Mutual inductance may be applied to multiple inductors simultaneously.  The matrix representations for multiple mutually coupled inductors are given by<ref>{{Cite book |last1=Chua |first1=Leon O. |url=https://archive.org/details/linearnonlinearc0000leon |title=Linear and Nonlinear Circuits |last2=Desoer |first2=Charles A. |last3=Kuh |first3=Ernest S. |date=1987 |publisher=McGraw-Hill, Inc. |isbn=0-07-100685-0 |publication-date=1987 |pages=459 |language=EN}}</ref><math display="block"> \begin{align}
&[\mathbf z] = s 
\begin{bmatrix} 
L_1 & M_{12} & M_{13} & \dots & M_{1N} \\ 
M_{12} & L_2 & M_{23} & \dots & M_{2N} \\
M_{13} & M_{23} & L_3  & \dots & M_{3N} \\
\vdots & \vdots &\vdots & \ddots \\
M_{1N} & M_{2N} & M_{3N} & \dots & L_N \\
\end{bmatrix}  \\

\end{align}
 </math>

===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>

===Resonant transformer===
{{main|Resonant inductive coupling}}
When a capacitor is connected across one winding of a transformer, making the winding a [[tuned circuit]] (resonant circuit) it is called a single-tuned transformer. When a capacitor is connected across each winding, it is called a [[double tuned|double tuned transformer]]. These ''[[Transformer types#Resonant transformer|resonant transformers]]'' can store oscillating electrical energy similar to a [[resonant circuit]] and thus function as a [[bandpass filter]], allowing frequencies near their [[resonant frequency]] to pass from the primary to secondary winding, but blocking other frequencies. The amount of mutual inductance between the two windings, together with the [[Q factor]] of the circuit, determine the shape of the frequency response curve. The advantage of the double tuned transformer is that it can have a wider bandwidth than a simple tuned circuit. The coupling of double-tuned circuits is described as loose-, critical-, or over-coupled depending on the value of the [[Coupling coefficient (inductors)|coupling coefficient]] {{nowrap|<math>k</math>.}} When two tuned circuits are loosely coupled through mutual inductance, the bandwidth is narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond the critical coupling, the peak in the frequency response curve splits into two peaks, and as the coupling is increased the two peaks move further apart. This is known as overcoupling.

Stongly-coupled self-resonant coils can be used for [[wireless power transfer]] between devices in the mid range distances (up to two metres).<ref name="Kurs">{{cite journal |last1=Kurs |first1=A. |last2=Karalis |first2=A. |last3=Moffatt |first3=R. |last4=Joannopoulos |first4=J. D. |last5=Fisher |first5=P. |last6=Soljacic |first6=M. |title=Wireless Power Transfer via Strongly Coupled Magnetic Resonances |journal=Science |date=6 July 2007 |volume=317 |issue=5834 |pages=83–86 |doi=10.1126/science.1143254 |pmid=17556549 |bibcode=2007Sci...317...83K |citeseerx=10.1.1.418.9645 |s2cid=17105396 }}</ref> Strong coupling is required for a high percentage of power transferred, which results in peak splitting of the frequency response.<ref>{{cite journal|doi=10.1109/TIE.2010.2046002|title=Analysis, Experimental Results, and Range Adaptation of Magnetically Coupled Resonators for Wireless Power Transfer |year=2011 |last1=Sample |first1=Alanson P. |last2=Meyer |first2=D. A. |last3=Smith |first3=J. R. |journal=IEEE Transactions on Industrial Electronics |volume=58 |issue=2 |pages=544–554 |s2cid=14721 }}</ref><ref>{{cite book|doi=10.1109/MEMS51670.2022.9699458|chapter=Magnetically Coupled Microelectromechanical Resonators for Low-Frequency Wireless Power Transfer |title=2022 IEEE 35th International Conference on Micro Electro Mechanical Systems Conference (MEMS) |year=2022 |last1=Rendon-Hernandez |first1=Adrian A. |last2=Halim |first2=Miah A. |last3=Smith |first3=Spencer E. |last4=Arnold |first4=David P. |pages=648–651 |isbn=978-1-6654-0911-7 |s2cid=246753151 }}</ref>

===Ideal transformers===
When {{nowrap|<math>k = 1</math>,}} the inductor is referred to as being closely coupled. If in addition, the self-inductances go to infinity, the inductor becomes an ideal [[transformer]]. In this case the voltages, currents, and number of turns can be related in the following way:

<math display=block>V_\text{s} = \frac{N_\text{s}}{N_\text{p}} V_\text{p} </math>
where

{{plainlist|1=
* <math>V_\text{s}</math> is the voltage across the secondary inductor,
* <math>V_\text{p}</math> is the voltage across the primary inductor (the one connected to a power source),
* <math>N_\text{s}</math> is the number of turns in the secondary inductor, and
* <math>N_\text{p}</math> is the number of turns in the primary inductor.
|indent=1}}

Conversely the current:

<math display=block>I_\text{s} = \frac{N_\text{p}}{N_\text{s}} I_\text{p} </math>
where

{{plainlist|1=
* <math>I_\text{s}</math> is the current through the secondary inductor,
* <math>I_\text{p}</math> is the current through the primary inductor (the one connected to a power source),
* <math>N_\text{s}</math> is the number of turns in the secondary inductor, and
* <math>N_\text{p}</math> is the number of turns in the primary inductor.
|indent=1}}

The power through one inductor is the same as the power through the other. These equations neglect any forcing by current sources or voltage sources.