View source for Gyrator–capacitor model

{{Short description|Model for magnetic circuits}}
{{About|modeling the magnetic field in magnetic components such as transformers and chokes|the circuit element|Gyrator}}
{{Electromagnetism|Magnetic circuit}}
[[File:Gyrator-Capacitor model of a simple transformer.png|thumb|upright=1.5|A simple transformer and its gyrator-capacitor model. R is the reluctance of the physical magnetic circuit.]]
The '''gyrator–capacitor model'''<ref name="Hamill">{{cite journal|title = Lumped equivalent circuits of magnetic components: the gyrator-capacitor approach|first=D.C.|last=Hamill|journal=IEEE Transactions on Power Electronics|volume=8|issue=2|date=1993| pages= 97–103|doi= 10.1109/63.223957|bibcode= 1993ITPE....8...97H}}</ref> - sometimes also the capacitor-permeance model<ref name="Lambert">{{Cite journal|last1=Lambert|first1=M.|last2=Mahseredjian|first2=J.|last3=Martı´nez-Duró|first3=M. |last4=Sirois| first4=F.| date=2015|title=Magnetic Circuits Within Electric Circuits: Critical Review of Existing Methods and New Mutator Implementations |journal=IEEE Transactions on Power Delivery|volume=30|issue=6|pages= 2427–2434|doi= 10.1109/TPWRD.2015.2391231|s2cid=38890643 }}</ref> - is a [[lumped element model|lumped-element model]] for [[magnetic circuit]]s, that can be used in place of the more common [[resistance–reluctance model]]. The model makes [[permeance]] elements analogous to electrical [[capacitance]] (''see [[magnetic capacitance]] section'') rather than [[electrical resistance]] (''see [[magnetic reluctance]]''). Windings are represented as [[gyrator]]s, interfacing between the electrical circuit and the magnetic model.

The primary advantage of the gyrator–capacitor model compared to the magnetic reluctance model is that the model preserves the correct values of energy flow, storage and dissipation.<ref name="González">{{Cite journal|last1=González|first1=Guadalupe G.| last2=Ehsani |first2=Mehrdad |date=2018-03-12|title=Power-Invariant Magnetic System Modeling|journal=International Journal of Magnetics and Electromagnetism|volume=4|issue=1|doi= 10.35840/2631-5068/6512 |pages=1–9|issn=2631-5068|doi-access=free|hdl=1969.1/ETD-TAMU-2011-08-9730|hdl-access=free}}</ref><ref name="Mohammad">{{Cite thesis|title=An Investigation of Multi-Domain Energy Dynamics| first=Muneer|last=Mohammad| url=https://oaktrust.library.tamu.edu/handle/1969.1/152720| date=2014-04-22|degree=PhD}}</ref> The gyrator–capacitor model is an example of a [[Mechanical–electrical analogies#Other energy domains|group of analogies]] that preserve energy flow across energy domains by making power conjugate pairs of variables in the various domains analogous. It fills the same role as the [[impedance analogy]] for the mechanical domain.

== Nomenclature ==
<!-- The magnetic flux in an inductor in the model magnetic circuit would have the awkward name "magnetic magnetic flux". Fortunately, inductors in the model circuit are rare in the literature. -->

<!-- Notes on symbols: for example a magnetic capacitor, that is, a capacitor that appears in the model magnetic circuit. Hamill, Lambert, Yan, and Kaushal use P because the magnetic capacitor represents a physical permeance. Mohammad and Gonzalez use C_M. -->

''Magnetic circuit'' may refer to either the physical magnetic circuit or the model magnetic circuit. [[Lumped-element model|Elements]] and [[Dynamical systems theory|dynamical variables]] that are part of the model magnetic circuit have names that start with the adjective ''magnetic'', although this convention is not strictly followed. Elements or dynamical variables in the model magnetic circuit may not have a one to one correspondence with components in the physical magnetic circuit. Symbols for elements and variables that are part of the model magnetic circuit may be written with a subscript of M. For example, <math>C_M</math> would be a magnetic capacitor in the model circuit.

Electrical elements in an associated electrical circuit may be brought into the magnetic model for ease of analysis. Model elements in the magnetic circuit that represent electrical elements are typically the [[Duality (electrical circuits)|electrical dual]] of the electrical elements. This is because transducers between the electrical and magnetic domains in this model are usually represented by gyrators. A gyrator will transform an element into its dual. For example, a magnetic inductance may represent an electrical capacitance.

==Summary of analogy between magnetic circuits and electrical circuits==

The following table summarizes the mathematical analogy between electrical circuit theory and magnetic circuit theory.

{| class="wikitable"
|+ Analogy between magnetic circuits and electrical circuits used in the gyrator-capacitor approach
|-
! colspan=3 | Magnetic !! !! colspan=3 | Electric
|-
! Name !! Symbol !! Units !! !! Name !! Symbol !! Units
|-
|[[Magnetomotive force]] (MMF) || <math>\mathcal{F}= \int \mathbf{H}\cdot d\mathbf{l}</math> || [[ampere-turn]] || || [[Electromotive force]] (EMF) || <math>\mathcal{E}= \int \mathbf{E}\cdot d\mathbf{l}</math> || [[volt]]
|-
| [[Magnetic field]] || '''''H''''' || ampere/[[meter]] = 
[[Newton (unit)|newton]]/[[Weber (unit)|weber]]
| || [[Electric field]] || '''''E''''' || volt/[[meter]] = 
[[Newton (unit)|newton]]/[[coulomb]]
|-
|[[Magnetic flux]]||<math> \Phi </math>||[[weber (unit)|weber]]{{efn| Hamill parenthetically includes "(per turn)" on page 97. <ref name="Hamill"></ref>}}|| || Electric charge || Q || [[coulomb]]
|-
|Flux rate of change
|<math>\dot \Phi</math>
|weber/second = [[volt]]
|
|[[Electric current]]
|<math>I</math>
|coulomb/second = [[ampere]]
|-
|Magnetic admittance
|<math>Y_M(\omega)=\frac{\dot \Phi(\omega)}{\mathcal{F}(\omega)}</math>
|[[ohm]] = 1/siemens
|
|[[Admittance|Electric admittance]]
|<math>Y_E(\omega)=\frac{I(\omega)}{V(\omega)} = \frac{1}{\operatorname{Z_E(\omega)}} </math>
|[[siemens (unit)|siemens]] = 1/ohm
|-
|Magnetic conductance
|<math>G_M = \operatorname{Re}(Y_M(\omega))</math>
|ohm = 1/siemens
|
|[[Electric conductance]]
|<math>G_E = \operatorname{Re}(Y_E(\omega))</math>
|siemens = 1/ohm
|-
|Magnetic capacitance ([[Permeance]])||<math>\mathcal{P} = \frac{\operatorname{Im}(Y_M(\omega))}{\omega}</math>||[[Henry (unit)|henry]]|| || Electric [[capacitance]]||<math>C = \frac{\operatorname{Im}(Y_E(\omega))}{\omega}</math>
|[[farad]]
|}

==Gyrator==
[[File:Gyrator-Capacitor Model Gyrator Element.png|thumb|Definition of Gyrator as used by Hamill in the gyrator-capacitor approach paper.]]
{{main|Gyrator}}
A '''[[gyrator]]''' is a [[two-port network|two-port element]] used in network analysis. The gyrator is the complement of the [[transformer]]; whereas in a transformer, a voltage on one port will transform to a proportional voltage on the other port, in a gyrator, a voltage on one port will transform to a current on the other port, and vice versa.

The role gyrators play in the gyrator–capacitor model is as [[transducer]]s between the electrical energy domain and the magnetic energy domain. An emf in the electrical domain is analogous to an mmf in the magnetic domain, and a transducer doing such a conversion would be represented as a transformer. However, real electro-magnetic transducers usually behave as gyrators. A transducer from the magnetic domain to the electrical domain will obey [[Faraday's law of induction]], that is, a rate of change of magnetic flux (a magnetic current in this analogy) produces a proportional emf in the electrical domain. Similarly, a transducer from the electrical domain to the magnetic domain will obey [[Ampère's circuital law]], that is, an electric current will produce a mmf.

A winding of N turns is modeled by a gyrator with a gyration resistance of N ohms.<ref name="Hamill"/>{{rp|100}}

Transducers that are not based on magnetic induction may not be represented by a gyrator. For instance, a [[Hall effect sensor]] is modelled by a transformer.

== Magnetic voltage ==
'''Magnetic voltage''', <math> v_m </math>, is an alternate name for ''[[magnetomotive force]]'' (mmf), <math>\mathcal{F} </math> ([[SI unit]]: [[ampere|A]] or [[amp-turn]]), which is analogous to electrical [[voltage]] in an electric circuit.<ref name="Mohammad"/>{{rp|42}}<ref name="González" />{{rp|5}} Not all authors use the term ''magnetic voltage''. The magnetomotive force applied to an element between point A and point B is equal to the line integral through the component of the magnetic field strength, <math> \mathbf{H} </math>.
<math display="block">v_m = \mathcal{F}= - \int_A^B \mathbf{H}\cdot d\mathbf{l}</math>
The [[resistance–reluctance model]] uses the same equivalence between magnetic voltage and magnetomotive force.

== Magnetic current ==
{{distinguish | text = [[Magnetic current]], an electromagnetic field quantity}}
'''Magnetic current''', <math>i_m</math>, is an alternate name for the ''time rate of change of flux'', <math>\dot \Phi</math> ([[SI unit]]: [[Weber (unit)|Wb]]/sec or [[volts]]), which is analogous to electrical current in an electric circuit.<ref name="Lambert"/>{{rp|2429}}<ref name="Mohammad"/>{{rp|37}} In the physical circuit, <math>\dot \Phi</math>, is [[Magnetic current#Magnetic displacement current|magnetic displacement current]].<ref name="Mohammad"/>{{rp|37}} The magnetic current flowing through an element of cross section, <math>S</math>, is the area integral of the magnetic flux density <math> \mathbf{B} </math>.

<math display="block"> i_m = \dot \Phi = \frac {d} {dt} \int_S \mathbf{B} \cdot d\mathbf{S}</math>
The resistance–reluctance model uses a different equivalence, taking magnetic current to be an alternate name for flux, <math> \Phi</math>. This difference in the definition of magnetic current is the fundamental difference between the gyrator-capacitor model and the resistance–reluctance model. The definition of magnetic current and magnetic voltage imply the definitions of the other magnetic elements.<ref name="Mohammad"/>{{rp|35}}

==Magnetic capacitance==
{{anchor|Magnetic capacitance}}
[[File:Gyrator-Capacitor Model Permeance Element.png|thumb|upright=1.2|Permeance of a rectangular prism element]]
'''Magnetic capacitance''' is an alternate name for [[permeance]], ([[SI unit]]: [[Henry (unit)|H]]). It is represented by a capacitance in the model magnetic circuit. Some authors use <math>C_\mathrm{M}</math> to denote magnetic capacitance while others use <math>P</math> and refer to the capacitance as a permeance. Permeance of an element is an [[extensive property]] defined as the magnetic flux, <math>\Phi</math>, through the cross sectional surface of the element divided by the [[magnetomotive force]], <math>\mathcal{F} </math>, across the element'<ref name="González" />{{rp|6}}
<math display="block">C_\mathrm{M} = P = \frac{\int \mathbf{B}\cdot d\mathbf{S}}{\int \mathbf{H}\cdot d\mathbf{l}}= \frac{\Phi}{\mathcal{F}}</math>

For a bar of uniform cross-section, magnetic capacitance is given by,
<math display="block">C_\mathrm{M} = P=\mu_\mathrm{r} \mu_0\frac{S}{l}</math>
where:
*<math>\mu_\mathrm{r} \mu_0 = \mu</math> is the [[Permeability (electromagnetism)|magnetic permeability]], 
*<math>S</math> is the element cross-section, and
*<math>l</math> is the element length.

For [[phasor analysis]], the magnetic permeability<ref name=Arkadiew>Arkadiew W. ''Eine Theorie des elektromagnetischen Feldes in den ferromagnetischen Metallen''. – Phys. Zs., H. 14, No 19, 1913, S. 928-934.</ref> and the permeance are complex values.<ref name=Arkadiew/><ref name=Popov/>

Permeance is the reciprocal of [[reluctance]].

==Magnetic inductance==
{{distinguish||magnetic induction (disambiguation){{!}}magnetic induction}}
[[File:Magnetic Inductance.png|thumb|upright=1.5|Circuit equivalence between a magnetic inductance and an electric capacitance.]]
In the context of the gyrator-capacitor model of a magnetic circuit, '''magnetic inductance''' <math>L_\mathrm{M}</math>([[SI unit]]: [[Farad|F]]) is the analogy to inductance in an electrical circuit.

For phasor analysis the magnetic inductive reactance is:
<math display="block">x_\mathrm{L} = \omega L_\mathrm{M}</math>
where:
*<math>L_\mathrm{M}</math> is the magnetic inductance
*<math>\omega</math> is the [[angular frequency]] of the magnetic circuit

In the complex form it is a positive imaginary number:
<math display="block">j x_\mathrm{L} = j\omega L_\mathrm{M}</math>

The magnetic potential energy sustained by magnetic inductance varies with the frequency of oscillations in electric fields. The average power in a given period is equal to zero. Due to its dependence on frequency, magnetic inductance is mainly observable in magnetic circuits which operate at [[Very high frequency|VHF]] and/or [[Ultra high frequency|UHF]] frequencies.{{cn|date=April 2020|reason=reference Mohammad says magnetic inductance will not considered a magnetic element on page 43}}

The notion of magnetic inductance is employed in analysis and computation of circuit behavior in the gyrator–capacitor model in a way analogous to [[inductance]] in electrical circuits.

A magnetic inductor can represent an electrical capacitor.<ref name="Mohammad"/>{{rp|43}} A shunt capacitance in the electrical circuit, such as intra-winding capacitance can be represented as a series inductance in the magnetic circuit.

==Examples==
===Three phase transformer===
[[File:Gyrator-Capacitor Model Example Three Phase Transformer.png|thumb|upright=1.5|left|Three phase transformer with windings and permeance elements.]]
[[File:Gyrator-Capacitor Model Example Three Phase Transformer Schematic.png|thumb|upright=1.5|Schematic using gyrator-capacitor model for transformer windings and capacitors for permeance elements]]
This example shows a [[three-phase]] [[transformer]] modeled by the gyrator-capacitor approach. The transformer in this example has three primary windings and three secondary windings. The magnetic circuit is split into seven reluctance or permeance elements. Each winding is modeled by a gyrator. The gyration resistance of each gyrator is equal to the number of turns on the associated winding. Each permeance element is modeled by a capacitor. The value of each capacitor in [[farads]] is the same as the inductance of the associated permeance in [[Henry (unit)|henrys]].

N<sub>1</sub>, N<sub>2</sub>, and N<sub>3</sub> are the number of turns in the three primary windings. N<sub>4</sub>, N<sub>5</sub>, and N<sub>6</sub> are the number of turns in the three secondary windings. Φ<sub>1</sub>, Φ<sub>2</sub>, and Φ<sub>3</sub> are the fluxes in the three vertical elements. [[Magnetic flux]] in each permeance element in [[Weber (unit)|webers]] is numerically equal to the charge in the associated capacitance in [[Coulomb|coulombs]]. The energy in each permeance element is the same as the energy in the associated capacitor.

The schematic shows a three phase generator and a three phase load in addition to the schematic of the transformer model.
{{clear}}

===Transformer with gap and leakage flux===
[[File:Gyrator-Capacitor Model Example Transformer with Gap and Leakage Flux.png|thumb|upright=1.5|left|Transformer with gap and leakage flux.]]
[[File:Gyrator-Capacitor Model Example Transformer with Gap and Leakage Flux Schematic.png|thumb|upright=1.5|Gyrator-capacitor model of a transformer with a gap and leakage flux.]]
The gyrator-capacitor approach can accommodate [[leakage inductance]] and air gaps in the magnetic circuit. Gaps and leakage flux have a permeance which can be added to the equivalent circuit as capacitors. The permeance of the gap is computed in the same way as the substantive elements, except a relative permeability of unity is used. The permeance of the leakage flux may be difficult to compute due to complex geometry. It may be computed from other considerations such as measurements or specifications.

C<sub>PL</sub> and C<sub>SL</sub> represent the primary and secondary leakage inductance respectively. C<sub>GAP</sub> represents the air gap permeance.
{{clear}}
==Magnetic impedance==

=== Magnetic complex impedance ===
[[File:Magnetic impedance.png|upright=1.5|thumb|Circuit equivalence between a magnetic impedance and an electric admittance.]]
'''Magnetic complex impedance''', also called full magnetic resistance, is the [[quotient]] of a complex sinusoidal magnetic tension ([[magnetomotive force]], <math>\mathcal{F}</math>) on a passive [[magnetic circuit]] and the resulting complex sinusoidal magnetic current (<math>\dot \Phi</math>) in the circuit. Magnetic impedance is analogous to [[electrical impedance]].

Magnetic complex impedance ([[SI unit]]: [[Siemens (unit)|S]]) is determined by:
<math display="block">Z_M = \frac{\mathcal{F}}{\dot \Phi} = z_M e^{j\phi}</math>
where <math>z_M</math> is the modulus of <math>Z_M</math> and <math>\phi</math> is its phase. The [[argument]] of a complex magnetic impedance is equal to the difference of the phases of the magnetic tension and magnetic current.
Complex magnetic impedance can be presented in following form:
<math display="block">Z_M = z_M e^{j\phi} = z_M \cos \phi + j z_M \sin \phi = r_M + j x_M </math>
where <math>r_M = z_M \cos \phi</math> is the real part of the complex magnetic impedance, called the effective magnetic resistance, and <math>x_M = z_M \sin \phi</math> is the imaginary part of the complex magnetic impedance, called the reactive magnetic resistance. 
The magnetic impedance is equal to
<math display="block">z_M = \sqrt{r_{M}^2 + x_{M}^2},</math> <math display="block">\phi = \arctan {\frac{x_M}{r_M}}</math>

====Magnetic effective resistance====
'''Magnetic effective resistance''' is the [[Real analysis|real]] component of complex magnetic impedance. This causes a magnetic circuit to lose magnetic potential energy.<ref name="Pohl" /><ref name="Küpfmüller">[[Karl Küpfmüller|Küpfmüller K.]] Einführung in die theoretische Elektrotechnik, Springer-Verlag, 1959.</ref> Active power in a magnetic circuit equals the product of magnetic effective resistance <math>r_\mathrm{M}</math> and magnetic current squared <math>I_\mathrm{M}^2</math>.

<math display="block">P = r_\mathrm{M} I_\mathrm{M}^2</math>

The magnetic effective resistance on a complex plane appears as the side of the resistance triangle for magnetic circuit of an alternating current. The effective magnetic resistance is bounding with the effective magnetic conductance <math>g_\mathrm{M}</math> by the expression
<math display="block">g_\mathrm{M} = \frac{r_\mathrm{M}}{z_\mathrm{M}^2}</math>
where <math>z_\mathrm{M}</math> is the full magnetic impedance of a magnetic circuit.

====Magnetic reactance====
{{see also|Magnetic complex reluctance}}

'''Magnetic reactance''' is the parameter of a passive magnetic circuit, or an element of the circuit, which is equal to the square root of the difference of squares of the magnetic complex impedance and magnetic effective resistance to a magnetic current, taken with the sign plus, if the magnetic current lags behind the magnetic tension in phase, and with the sign minus, if the magnetic current leads the magnetic tension in phase.

Magnetic reactance <ref name=Pohl>{{Cite book| last = Pohl | first = R. W.| title=Elektrizitätslehre| location=Berlin-Gottingen-Heidelberg | publisher=Springer-Verlag| year=1960| language=German}} </ref><ref name=Popov>{{Cite book| author=Popov, V. P.| title=The Principles of Theory of Circuits| publisher=M.: Higher School| year=1985| language=Russian}}</ref><ref name=Küpfmüller/> is the component of magnetic complex impedance of the [[alternating current]] circuit, which produces the phase shift between a magnetic current and magnetic tension in the circuit. It is measured in units of <math>\tfrac{1}{\Omega}</math> and is denoted by <math>x</math> (or <math>X</math>). It may be inductive <math>x_L = \omega L_M</math> or capacitive <math>x_C = \tfrac{1}{\omega C_M}</math>, where <math>\omega</math> is the [[angular frequency]] of a magnetic current, <math>L_M</math> is the [[magnetic inductivity|magnetic inductiance]] of a circuit, <math>C_M</math> is the magnetic capacitance of a circuit. The magnetic reactance of an undeveloped circuit with the inductance and the capacitance which are connected in series, is equal: <math display="inline">x = x_L - x_C = \omega L_M - \frac{1}{\omega C_M}</math>. If <math>x_L = x_C</math>, then the net reactance <math>x = 0</math> and [[resonance]] takes place in the circuit. In the general case <math display="inline">x = \sqrt{z^2 - r^2}</math>. When an energy loss is absent (<math>r = 0</math>), <math>x = z</math>. The angle of the phase shift in a magnetic circuit <math display="inline">\phi = \arctan{\frac{x}{r}}</math>. On a complex plane, the magnetic reactance appears as the side of the resistance triangle for circuit of an alternating current.

== Limitations of the analogy ==
The limitations of this analogy between magnetic circuits and electric circuits include the following;

* The current in typical electric circuits is confined to the circuit, with very little "leakage". In typical magnetic circuits not all of the magnetic field is confined to the magnetic circuit because magnetic permeability also exists outside materials (see [[vacuum permeability]]). Thus, there may be significant "[[leakage flux]]" in the space outside the magnetic cores. If the leakage flux is small compared to the main circuit, it can often be represented as additional elements. In extreme cases, a lumped-element model may not be appropriate at all, and [[Field theory (physics)|field theory]] is used instead.
* Magnetic circuits are [[Nonlinear element|nonlinear]]; the permeance in a magnetic circuit is not constant, unlike capacitance in an electrical circuit, but varies depending on the magnetic field. At high magnetic fluxes the [[ferromagnetic materials]] used for the cores of magnetic circuits [[Saturation (magnetic)|saturate]], limiting further increase of the magnetic flux, so above this level the permeance decreases rapidly. In addition, the flux in ferromagnetic materials is subject to [[hysteresis]]; it depends not just on the instantaneous MMF but also on the history of MMF. After the source of the magnetic flux is turned off, [[remanent magnetism]] is left in ferromagnetic materials, creating flux with no MMF.

==References==
{{reflist|group=lower-alpha}}

{{DEFAULTSORT:Gyrator-capacitor model}}
[[Category:Electronic engineering]]
[[Category:Electrical analogies]]
[[Category:Magnetic circuits]]