Power-Invariant Magnetic System Modeling

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Research Article: Open Access Received: University, CollegeStation, TX77840,USA,E-mail: Texas A&M Computer Engineering, and Department ofElectrical G González, *Corresponding author:Guadalupe Introduction Department ofElectricalandComputerEngineering,TexasA&MUniversity,USA Guadalupe GGonzález Power-Invariant MagneticSystemModeling that iseitherstoredinamediaordissipatedasheat. three passiveelementsthatestablishtheamountofenergy these quantitiesastheconservedquantity,flow,effort and havior ofanythesemedia(seeTable2).Wegeneralized six recurrentquantitiesthatdescribethemacroscopic be- electrical andmechanicalmodelswerealizethatthere are model ofmagneticssystems.Furthermore,bystudying the true inthereluctancemodel,whichisconventional lic. However,thisgeneralpowerrelationshipdoesnothold in atleastthreemedia,electrical,mechanicalandhydrau- Table 1wherethegeneralizedpowerequationholdstrue source arecredited. author and the original in anymedium,provided Attribution License,whichpermits unrestricteduse,distribution,andreproduction Copyright: power equationcanbegeneralizedas, a floworrateofchangeresults(γ).Consequently,the or force(χ)isneededtocreateaflowinmediumand transmit poweraresimilarinfunctionality,aneffort 4:012 Citation: The previous statement can be supported by analyzing The previousstatementcanbesupportedbyanalyzing In allenergysystemstheparametersnecessaryto P ⋅ = χγ VIBGYOR January10,2018: González GG, Ehsani M (2018) Power-Invariant MagneticSystemModeling. Int J Magnetics Electromagnetism © 2018 González GG, et al. This is an open-access article distributed under thetermsofCreativeCommons articledistributed GG, © 2018González et al.Thisisanopen-access

losses Magnetic analysis,Magneticcircuits,hysteresis, modeling, Electromagnetic Keywords study systemformagneticnetworkidentificationtovalidateourmodel. case a in performed are verifications theoretical Finally, presented. are model magnetic proposed of the that aretheunderpinnings materials andmagnetic ofelectromagnetism the fundamentals basedonlumped models ofother review the differencesbetweenthoseandourmodel.Then, parameters ispresentedinordertohighlight and aliterature new approach our for justification the First, of magneticsystemswhichuptonowhasnotbeenavailable. and aclearerunderstanding tofrom the engineer draw analogiesotherenergysystems models andallows to gather insights system with otherenergy is consistent introduce we Themodel parameters. to lumped approach consistent a more systems using magnetic to modeling alternative an we present In thispaper, Abstract

Accepted:March10,2018: * andMehrdadEhsani

(1) Published: [email protected] our modelbasedonelectromagnetismandmagneticma - Then, wewillintroducethefundamentaljustification of which wecalledthepower-invariantmagneticmodel. es betweenthesemodelsandtheoneweareproposing, on lumpedparametersinordertoestablishthedifferenc - ent aliteraturereviewofthethreemagneticmodelsbased netic mediausinglumpedparameters.First,wewillpres - the otherenergyregimemodels. that modificationscouldbemadetoconsistentwith in thepowerequationatmagneticregimesuggests transmission pattern.Therefore,theinconsistencyseen ples, weassumedthatallmediafollowasimilarpower tance/resistance analogymisconception[1]. being definedandfinally,thereisawell-knownreluc- of thesixquantities that standardizethemodels havenot generalized powerequationisnotsatisfied,second,three not follow the same pattern as the other media. First, the González andEhsani.IntJMagneticsElectromagnetism2018,4:012 Electromagnis Magnetics d Inter In thispaper,wepresentaconsistentmodelformag- Based onconservationofenergyandrelatedprinci- It isnotedthattheconventionalmagneticmodeldoes March12,2018 natiol Jur nal of ISSN: 2631-5068 González and Ehsani. Int J Magnetics Electromagnetism 2018, 4:012 ISSN: 2631-5068 | • Page 2 of 9 •

Table 1: Calculation of power in different media. Electrical Mechanical Hydraulic Magnetic Voltage × Current Force × Velocity Pressure × Flow MMF × Flux V × A N × m/s N/m2 × m3/s A × Wb Watts Watts Watts Joules!

Table 2: Different media analogs. Generalization Electrical Mechanical Magnetic Conserved Quantity Charge [C] Distance [m] ? Flow Current [A] Speed [m/s] Flux [Wb] Effort Voltage [V] Force [N] MMF [t·A] Dissipative Element Resistance [Ω] Friction Reluctance [H-1] Storage Element Capacitance [F] Compliance ? Storage Element Inductance [H] Mass [kg] ?

terials. Finally, we will conceptually illustrate our theory (Pm) cannot be calculated using an analogue of Joules law by modeling a simple but relevant magnetic system: an as done for electric losses (Pe) in the electric regime. That iron core coil. is, = 2 Literature Review Pe IR (4) 2 Several lumped parameter models have been devel- Pm ≠ ϕ R (5) oped for magnetic systems. The reluctance model is the One of the reasons why the reluctance model has oldest and most widely used in magnetic analysis there- been successfully used for so long is that it is a quasi-stat- fore we consider it as the conventional magnetic model. ic model. At low frequencies, the magnetic material In this section, we present a brief description of the reluc- shows almost zero heat losses and the reluctance model tance model as well as other lumped parameter models represents quite accurately the material’s behavior under that have been developed in order to provide a solution this condition. However, when alternating currents at to the deficiencies of the reluctance model. medium or high frequency excite the magnetic material, The reluctance model heat losses are more relevant and the reluctance model becomes inconsistent. Consequently, other methods are This model is based on Hopkinson’s law, which is applied in order to calculate the magnetic losses. For in- analogous to Ohm’s law for electric circuits. stance, hysteresis losses which are heat losses resulting In electrical circuits, Ohm’s law is an empirical re- from the tendency of the material to oppose a change in lation between the electromotive force (emf), potential magnetism can be calculated using the area of the hys-

(U) or voltage (ve) needed in order to produce a flow of teresis loops. These hysteresis power losses (Ph) are given current (I) through a conductor. It can be expressed as, by, = emf IR [V] (2) Ph =⋅⋅ VfAloop (6) Where R is the resistance [Ω]. Where V is the volume of the specimen [m3], f is the

In magnetic circuits, Hopkinson’s law is an empiri- frequency of operation [Hz] and Aloop is the area of the cal relation between the magnetomotive force (mmf) and hysteresis loop. the magnetic flux (φ) given by, The permeance-capacitor model mmf = ϕR [t·A] (3) In 1969, R.W. Buntenbach from the University of Where R is the reluctance [H-1]. California stated that there are several stumbling blocks that inhibit the progress of magnetic circuit theory [1]. When the reluctance model was defined, it was considered equivalent to the electric model in both form and In an effort to correct the conventional model, Bun- functionality, but through the years it has been well es- tenbach proposed a different one also based on an anal- tablished that this is not the case. The main difference ogy between electrical and magnetic quantities. In his between these expressions is that in electrical circuits, the model, the flux is analogous to electric charge, contrary resistance is a measure of how much energy can be dis- to the reluctance model where flux was analogous to elec- sipated as heat while current flows in the material. How- tric current. Also, the terms flux rate, magnetic resistance, ever, in magnetic circuits the reluctance is a measure of magnetic inductance and magnetic impedance were in- magnetic energy storage rather than a measure of mag- troduced. However, the main contribution of this model netic energy dissipation. In other words, magnetic losses is that it presents the Permeance-Capacitor analogy.

Buntenbach’s model suggests that the magnetic cir- that either the effort or flow quantity is not properly de- cuit is better represented as an analogue to an electric fined. We will use gyrator theory in order to identify the capacitance rather than an electric resistance, which is ill-defined quantity and redefine it. true in functionality and can be validated by examining energy relations. Gyrator theory The gyrator is a power-invariant, multiport network Assuming energy in the magnetic regime (E ) is cal- m that converts the effort (χ) or flow (γ) quantity of one culated as the energy an electric capacitor (E ) the di- e of the networks into its dual with respect to its gyration mensions are consistent, 1 constant (g) [5], this is E= CV ⋅ 2 [J] (7) e χγ=−⋅ 2 12g (9)

1 2 χγ21=g ⋅ (10) Em =P ⋅ mmf [J] (8) 2 A symbolic representation of a conventional two- Where C represents the electric capacitance [F], V the port gyrator is presented in Figure 1. It can be shown −1 electric voltage [V], the permeance [H] which is R that an iron core coil is a gyrator where the primary net- and mmf is the magnetomotive force [t·A]. work is the electric side and the secondary network is the magnetic side and the two networks are coupled through The gyrator-capacitor model the gyrator constant which is the number of turns (N) In 1993, Dr. Hamill applied gyrator theory in order as shown in Figure 2. We can identify the effort and to redefine the quantities of the reluctance model. In the flow quantities of each regime by comparing the gyrator Gyrator-Capacitor model, the permeance is also used equations (9, 10) with the system’s performance equa- analogous to the electric capacitance, however, it is the tions, Faraday’s and Ampere’s laws, given by (11) and only circuit quantity of the model, contrary to the Per- (12) respectively. meance-Capacitor model which includes expressions dϕ vN=−⋅ (11) for magnetic resistance and inductance. However, in e dt later publications appear extensions to the original gy- mmf= N ⋅ i (12) rator-capacitor model where a magnetic resistance and e magnetic inductance are defined but there is neither a It is evident that these set of equations are similar in comprehensive description nor an application of them form and functionality. Equations (9) and (11) suggest [2]. that the electric voltage (ve) is the effort quantity of the electric network while the rate of change of flux (dφ/dt) Hamill applied the gyrator-capacitor approach to is the flow quantity of the magnetic network. Similar- model: A gapped-core inductor, a two-winding trans- ly, (10) and (12) suggest that the magnetomotive force former and a Ćuk converter in quasi-static conditions (mmf) and the electric current (ie) are the effort and flow with satisfactory results [3,4]. quantities of the magnetic and electric regimes, respec- Power-Invariant Magnetic Model tively. As mentioned earlier, in the conventional magnetic In order to verify that these quantities satisfy the model the power equation is not satisfied. This means power equation we perform a dimensional analysis in

g γ1 γ2

χ2 χ1 Port 1 Port 2

Figure 1: Symbolic representation of a two-port gyrator.

γm ie ie γ N N m

νe mmf νe mmf

a) Physical model b) gyrator model Figure 2: Representation of an iron core coil as a) Physical; b) Gyrator models.

(13) and (14) obtaining watts units in both cases. in a volume (V) is equivalent to the electric flux density (D) through a surface (S) which encloses a volume (V). p= vi ⋅ (13) e ee Therefore, we will assume the electric flux as a represen- dϕ p= mmf ⋅ (14) tation of the conserved quantity in dielectric materials. m dt Using the rate of change of magnetic flux (dφ/dt) as Fundamentally, the magnetic phenomenon is the flow is the fundamental modification to the magnetic result of the magnetization of the magnetic dipole mo- model. It is the key to defining the rest of the parameters ments, similar to the electric phenomenon being the re- (magnetic resistance, inductance and capacitance) need- sult of the polarization of the electric-bound charges in ed to standardize the magnetic model. dielectric materials. We will use this analogy to define the magnetic conserved quantity. Magnetic fundamental quantities The scalar quantity that represents the magnetization In this section, we will define the six scalar quantities (M) of the magnetic material due to a magnetic field in- we consider standard to fully model energy transfer pro- tensity (H) is the magnetic flux φ( ) and it is expressed as, cess specifically for the magnetic regime. We will use an ϕµ= = + electric analogy as the basis to derive the magnetic model ∫∫BdS0 ( M H) dS (16) because there is an intrinsic coupling between the two SS systems. Where µ0 represents the permeability of air. Conserved quantity: In the electric regime, the con- Similar to the electric flux, we will use the magnetic flux (φ), which is the magnetic flux density (B) through a served quantity is the free charge (qe). Its iteration with an electromagnetic field or with other charged particles surface (S), as the magnetic conserved quantity. In order is the main cause of many electric phenomena. For in- to be consistent with the electric analogy, we will refer to stance, when charge carriers move in different media, the magnetic conserved quantity as the flux (φ) or as a such as metallic conductors and free space, an electric magnetic- bound charge (qm). current is produced. Flow: The flow can be defined as the movement of a In dielectric materials, on the other hand, due to the conserved quantity through certain cross-sectional area tight atom-electron bonds there are no free charges. The per unit of time. In the electric regime, the flow or cur- electric phenomenon is the result of the interaction of rents are divided into two classes, conduction currents electric fields and the polarization charges of the dielec- and displacement currents. Conduction currents ( ie, cond ) tric material. These charges, which are basically electric result from the movement of free charged particles in a dq dipoles, are known as bound charges to distinguish them conductive medium ( e ). The motion of these particles from the free charges. dt generates heat and therefore, it is usually associated with The scalar quantity that represents the polarization of energy losses. the bound charges (P) due to an electric field intensity (E) is the electric flux (ψ) and it is expressed as, Mathematically it is given by, dq i = e ψε=DdS =( P + E) dS (15) e, cond [A] (17) ∫∫0 dt ε Where 0 represents the permittivity of air. Displacement currents ( ie, cond ) were introduced by According to Gauss’s law, the free charge contained James Clerk Maxwell in 1861 in order to satisfy Ampère’s

Citation: González GG, Ehsani M (2018) Power-Invariant Magnetic System Modeling. Int J Magnetics Electromagnetism 4:012 González and Ehsani. Int J Magnetics Electromagnetism 2018, 4:012 ISSN: 2631-5068 | • Page 5 of 9 • circuital law which establishes that currents always flow Where E is the electric field intensity and dl is a differ- in a closed path. It satisfies the current through capaci- ential element of a path. tors as there are no free charges moving inside the dielec- In magnetic systems, the effort quantity has been tric medium. Mathematically it is given by specified as the magnetic potential (U ) or magnetomo- dψ m i = [A] (18) tive force (mmf), analogous to the electromotive force. e, disp dt In this paper, we will also refer to it as magnetic voltage Where ψ is used to denominate the electric flux [C]. analogous to electric voltage. Mathematically it can be expressed as, Displacement currents play little role in metallic con- mmf= v = Hdl [t·A] (23) duction. In metals, they are dwarf into insignificance m ∫ by the magnitude of the conductive current; but in in- Where H is the magnetic field intensity and dl is a sulators where the conduction currents are weak, the differential element of a closed path. displacement current may well play a role even at low frequencies and most certainly when the frequencies are In order to satisfy the analogy, the mmf must also be elevated [6]. defined as the force needed to generate a magnetic current. This statement contradicts the popular notion that In magnetic systems, we propose to define the mag- there is no flow of anything in a magnetic circuit cor- i netic current as a displacement current ( m, disp ) similar to responding to the flow of charge in an electric circuit. a dielectric material. This is based in the fact that both However, we have already established that the magnetic materials need to be polarized in order to react. Further- current is a displacement current and can be considered more, the expression of flow obtained from the iron core as the result of magnetization of the magnetic dipoles in coil as a gyrator suggests that magnetic materials, similar to the displacement current dϕ im, disp = [Wb/s] (19) related to the bound or polarized charge of an electric dt insulator. In both cases, an effort is needed to create the Which is similar in form, functionality and origin to polarization of the bound charges. (18) as both are based on flux. In this case φ is used to Resistance: As a standard model element, resistance specify the magnetic flux. can be generalized as the parameter that causes energy Similar to electric displacement currents, the mag- to be released as heat due to the flow of current in a ma- netic displacement currents are not based on the move- terial. Its generalized definition in terms of flow (γ) and ment of magnetic free charged particles as the existence effort (χ) is as follows, of such charges have not been discovered. Instead, they χ R = (24) are based in the rate of change of magnetic fields. How- γ ever, the displacement currents can be recognized as a moving-charge current related to the movement of the In electric circuits, the resistance relates the electric magnetic dipoles during the magnetization of the mag- voltage (ve) and current (ie) flowing through a conductive netic material. It is given by (19) or in terms of its current material, vRe density as, Re = [Ω] (25) dB iRe J = [T/s] (20) m, disp dt Also, its value can be expressed in terms of the intrinsic property of the material (resistivity, ρ [Ω·m] or Since B=µ ( MH + ) , the magnetic displacement e 0 conductivity, σ [Ω-1/m]) and the geometry (length, l [m] current can also be expressed in terms of the magneti- e and area, S [m2]) of the specimen. This is given as zation as, dM( + H) Edl Edl = µ ∫∫ l Jm,0 disp [T/s] (21) R = = = ρ [Ω] (26) dt eeσ S ∫∫Jee dS EdS Effort: In general terms, the effort can be defined as Similarly, in magnetic circuits we can define a mag- the force needed to generate a flow or as the force needed netic equivalent resistance based on the general defini- to produce work. In electrical systems, the effort quantity tion. It is the quantity that measures the amount of mag- is most commonly known as voltage (V), electric poten- netic current the material allows to flow when certain tial (U ) or electromotive force (emf). It is the force need- e voltage is applied. It also causes energy to be lost as heat. ed to move a charged particle from one point to another in the presence of an electric field. Mathematically it can We can relate the magnetic current (im) and the mag- be expressed as, netic voltage (vm) using, v emf= v = Edl [V] (22) Rm -1 e ∫ Rm = [Ω ] (27) iRm

Or in terms of the intrinsic property of the magnet- Where l [m] and S [m2] represent the depth and area -1 ic material (magnetic resistivity, ρm [Ω ·m] or magnetic of the dielectric, respectively and ε [F/m] represents the conductivity, σm [Ω/m]) and the geometry (length, l [m] permittivity of the dielectric material, which determines and area, S [m2]) of the specimen as, the ability of a material to polarize in response to an elec- Hdl Hdl tric field. ∫∫ l -1 Rmm= = = ρ [Ω ] (28) J dSσ HdS S The energy stored by a capacitor can be expressed as ∫∫mm W= v dq [J] (34) It is important to point out that there is a fundamental Ce∫ CC ee difference between the electric and magnetic resistance In magnetic media, the capacitance determines the and this has its origins in the mechanism that produc- ability of the media to store energy due to a given mag- es the flow of currents. The electric resistance describes netic potential or effort (vCm). The amount of energy the opposition that the conductive media presents to the stored in the magnetic medium can be expressed as, flow of free charges. Therefore, it reflects the heat losses q due to the kinetic energy released by the movement of C = Cm [H] (35) m v these free charges. However, as we established earlier, the Cm magnetic currents result from the polarization or mag- Similar to the resistance, the capacitance depends on netization of the magnetic dipoles in the magnetic ma- a property of the material medium. Its numerical value terial, similar to the displacement electric currents. The depends on the intrinsic property of the magnetic mate- power dissipated as heat is the result of the resistance of rial (permeability, µ [H/m]) and the geometry of the ma- the magnetic material to being magnetized. terial (length, l [m] and area, S [m2]). This is expressed as, The power dissipated by the magnetic resistance can ∫∫BdSµ HdS S also be expressed in terms of an analogy to the Joule’s law Cm = = = µ [H] (36) when the resistance is a constant value. That is, ∫∫Hdl Hdl l

221 At this point, it is evident that our magnetic capaci- PRm=⋅=⋅=⋅ V m,, rms I m rms RI m m , rms Vm , rms (29) Rm tance plays the same role as the permeance in the con- Which is the average of the instantaneous power [W]. ventional reluctance model. Therefore, we have verified that the permeance and its inverse, the reluctance, are in If the resistance is nonlinear, the power dissipated by fact energy storage elements and not the perceived dissi- the resistance can be calculated directly by calculating pative elements. the average of the instantaneous power, 1 n Energy stored by a capacitance can be generally ex- = ⋅ PRm ∑( viRm,, k Rm k ) [W] (30) pressed as n k =1 Capacitance: The capacitance can be generalized as W= v dq [J] (37) Cm∫ CC mm the parameter that determines the amount of energy stored in the medium due to an applied effort (χ) in that If the capacitance is a constant value, then the energy medium. Mathematically it is expressed as, stored can be expressed using a more familiar expression,

qrw 1 2 C = (31) WC= CV mC ⋅ [J] (38) χ mm2

Where qrw stands for conserved quantity in the me- Inductance: In order to explain the inductance as a dium. generalized element we need to remember the first law In the electrical regime, the capacitance measures the of thermodynamics, energy is neither created nor de- ability of the medium to hold a charge (q ) due to a giv- stroyed only transformed. This is important because en- e ergy can either be kept in the same medium of work or en potential (ve). For example, in an air capacitor it describes the amount of energy stored in the electric field. transformed into another medium in order to be stored. Electrical capacitance is expressed as, The most common example of this process occurs in q the electric regime when energy is stored in the magnetic C = Ce e [F] (32) medium due to the flow of electric current. The element vCe that describes this process is known as electric induc- For a dielectric capacitor with its intrinsic dielectric tance (L ) and is expressed as, property and the geometry, capacitance can be expressed e Nϕ as, L = DdSε EdS e (39) ∫∫ S iLe Ce = = = ε [F] (33) ∫∫Edl Edl l

Table 3: Comparison of the Power-Invariant and reluctance models. Quantities Power-Invariant Reluctance Conserved Quantity Magnetic Flux [Wb] ? Flow Magnetic Current [Wb/s] Flux [Wb] Effort Magnetic Voltage [t·A] MMF [t·A] Dissipative Element Magnetic Resistance [Ω-1] Reluctance [H-1] Energy Storage Element Magnetic Capacitance [H] ? Energy Storage Element Magnetic Inductance [F] ?

medium. Where ϕ is the magnetic flux and iLe is the electric current. The number of turns (N) appears in (39) because To summarize, Table 3 presents a comparison of the a gyration from the magnetic to electric regime is taking power-invariant magnetic parameters proposed in this place. paper and those presented in the conventional reluctance model. Due to its completeness, the power-invari- Equation (39) can also be expressed in terms of the ant magnetic model will be helpful in more accurately intrinsic property of the magnetic material (permeabil- modeling the magnetic phenomena. ity, µ [H/m]) and the geometry of the material (length, l [m] and area, S [m2]) as, Case Study: Iron Core Coil BdSµ HdS To illustrate the elements described in the power-in- 22∫∫ 2S LNe = = N= Nµ [H] (40) Hdl Hdl l variant magnetic model we will use an iron core coil for a ∫∫ case study (see Figure 2). We use this case for several rea- Theenergy stored by the inductance can be expressed sons. First, it is a simple structure, second, we are famil- as, iar with its physical behavior, and third, we want to rep- W= idϕ [J] (41) resent the magnetic medium using the known material LLee∫ characteristics (B-H Loops, permeability curves, etc.). Similarly, in magnetic systems the inductance deter- According to [7], in quasi-static conditions the mag- mines the amount of energy stored in a dual medium due netic material can be characterized as a magnetic capaci- to the flow of magnetic current (im). As the electric and tor, as suggested by Hamill’s gyrator-capacitor modeling magnetic regimes are coupled, we consider the electric technique [4]. However, the capacitance element alone medium as the dual of the magnetic regime; therefore, does not represent the real behavior of an iron core coil. the magnetic inductance (Lm) can be expressed as, Hysteresis and magnetic losses, among other factors,

Nqe need to be taken into consideration when modeling the Lm = (42) iLm magnetic medium. According to our theory, in order to The magnetic inductance can also be expressed in represent the real behavior of the magnetic material we terms of the geometry of the material, length (l) and area should model the medium using the magnetic resistance, inductance and capacitance. We understand that the ef- (S), and its permittivity (ε) as, fect of some of those elements is more dominant than DdSε EdS 22∫∫ 2S others in the model. However, for completeness we will LNm = = N= Nε [F] (43) ∫∫Edl Edl l represent the medium based on all the three elements. To develop an equivalent circuit model of the mag- The energy stored by a magnetic inductance (WLm) can be expressed either by, netic medium from scratch, there are eight possible con- figurations by which the magnetic elements could be W= i dq [J] (44) Lm ∫ Lm e arranged. However, after studying the system’s response Or in the case that the inductance is a constant value to different magnitude, frequency and waveform exci- by, tations, it is realized that there is only one configuration 1 2 that satisfied the observed behavior of a magnetic mate- WL= LI mL ⋅ [J] (45) mm2 rial. Further, to produce the observed hysteresis behavior Finally, the inductance can be generalized as the ele- under ac excitation conditions, a nonlinear RLC series ment that stores energy in a dual regime or medium due configuration is the suitable model (Figure 3). to a flow (γ) and it can be expressed as, The equation that represents this circuits is

qdrw v= RLCi ++ ⋅ L = (46) m( m m mm) (47) γ Being the magnetic resistance, inductance and capac-

Where qrw stands for conserved quantity in the dual itance modeled as non-linear elements.

m m im R L a

νm Cm

Figure 3: Magnetic model of an iron core coil.

1.5

0.5

-0.5

Magnetic Field Density (B) [T] Hmax=5 Hmax=10 -1 Hmax=20 Hmax=40 Hmax=60 -1.5 Hmax=80 Hmax=100

-2 -100 -80 -60 -40 -20 0 20 40 60 80 100 Magnetic Field Intensity (H) [A/m]

Figure 4: Family of hysteresis loops at different amplitudes at 60 Hz.

1.5

0.5

-0.5 Magnetic Field Density (D) [T]

f=20 Hz -1 f=40 Hz f=60 Hz f=100 Hz -1.5 f=300 Hz f=500 Hz

-2 -80 -60 -40 -20 0 20 40 60 80 Magnetic Field Intensity (H) [A/m]

Figure 5: Family of hysteresis loops at different frequencies.

Figure 4 and Figure 5 show the B-H loops that result done in this section was based on a hypothetical magnet- from modeling the magnetic medium as a nonlinear RLC ic material and a simulation experiment was done to val- series configuration, excited at different amplitudes and idate that the general behavior of the magnetic iron-core frequencies. It is important to point out that the study model as we know it to the proposed power-invariant

Citation: González GG, Ehsani M (2018) Power-Invariant Magnetic System Modeling. Int J Magnetics Electromagnetism 4:012 González and Ehsani. Int J Magnetics Electromagnetism 2018, 4:012 ISSN: 2631-5068 | • Page 9 of 9 • model. The magnetic voltage is related to the magnetic The above theory can facilitate the development of field intensity and the magnetic current is related to the better computer models of complex magnetic systems. magnetic flux density, both considering the geometry of It can also be instrumental in developing better design the core. tools for magnetic devices and materials. Conclusion Acknowledgment In this paper, we have developed a unified magnet- This work was partially supported by the govern- ics model that is power-invariant under transforma- ment of the Republic of Panama through SENACyT-IF- tion of power from across electrical media. This model ARHU’s Professional Excellency Scholarship Program. is consistent with other standard models used for other Also, the generous gifts of the Industrial Affiliates of the energy and power transfer media, such as electrical, me- Power Electronics and Motor Drives Laboratory at Texas chanical, and hydraulic. This is useful for many reasons. A&M University are gratefully acknowledged. First, following the principle of conservation of energy, all energy systems are expected to behave in a similar References pattern during an energy transfer process (some ener- 1. RW Buntenbach (1969) Analogs between magnetic and electrical circuits. Electronic Products 12: 108-113. gy is dissipated and some energy is reactively stored). Therefore, new insights and a different prospective can 2. VP Popov (1845) The principles of theory of circuits. Higher School, Russia, Moscow, 496. be attained for understanding magnetic system behavior 3. DC Hamill (1994) Gyrator-capacitor modeling: A better way by symmetry with other regimes. Second, the same an- of understanding magnetic components. In: Proc 9th Annu alytical and numerical techniques can be applied across Applied Power Electronics and Conference and Exposition, coupled power media in order to analytically compose Orlando, Florida, 1: 326-332. and solve system differential equations. For instance, 4. DC Hamill (1993) Lumped equivalent circuits of magnetic when working in the linear region of the characteristics components: The gyrator-capacitor approach. IEEE Trans of the magnetic material we could use phasors or Laplace Power Electron 8: 97-103. transformations in order to solve the system’s equations. 5. RY Barazarte, GG González, M Ehsani (2010) Generalized This practice is not common in magnetic system analy- gyrator theory. IEEE Trans Power Electron 25: 1832-1837. sis. Finally, we have now created a model where we are 6. SS Attwood (1956) Electric and magnetic fields. In: Electric current. (3rd edn), John Wiley & Sons, New York, 116-118. able to calculate power and energy for magnetic systems in a simpler and more traditional way, using the power 7. GG González (2011) Power-invariant magnetic system modeling. Ph.D. dissertation, Texas A&M University, Col- equations. lege Station, TX.

Citation: González GG, Ehsani M (2018) Power-Invariant Magnetic System Modeling. Int J Magnetics Electromagnetism 4:012