Compatibility Analysis Among Vector Magnetic Circuit Theory, Electrical Circuit Theory, and Electromagnetic Field Theory

Due to the presence of the new magnetic component of magductance (or magnetic-inductace), the traditional scalar magnetic circuit theory is evolved into the vector magnetic circuit theory. Further research is still required to determine whether the vector magnetic circuit theory and the current electromagnetic theory are compatible with one another. In this paper, the reluctance and magductance parameters of the magnetic circuit are derived using electromagnetic field theory, vector magnetic circuit theory, and electrical circuit theory by taking a laminated magnetic core as an example. Additionally, based on the derived general expressions for the magnetic circuit parameters, the law of magnetic circuit parameters changing with magnetic circuit frequency is plotted and summarized, and the compatibility of electromagnetic field theory, vector magnetic circuit theory, and electrical circuit theory is analyzed. Finally, combined with the magnetoelectric power law, the derived general expressions for the magnetic circuit parameters have been verified by the experiments with the Epstein frame.


I. INTRODUCTION
Electrical machine as we are all aware, is a device that utilizes the principles of electromagnetism to perform energy conversion [1].In general, there exist various forms of three-dimensional alternating electromagnetic fields within electrical machines [2].It is difficult to accurately understand their spatial distribution and temporal variation of magnetic field for determining the dynamic and steady-state performance of electrical machines [3].Therefore, in order to simplify the analysis and calculation, the method of ''fieldto-circuit'' is still commonly used in many engineering problems, including the widely used ''electrical circuit'' [4] and ''magnetic circuit'' [5].As a result, electromagnetic field The associate editor coordinating the review of this manuscript and approving it for publication was Montserrat Rivas.
theory can be used to derive electrical circuit theory and magnetic circuit theory [6].
Magnetic circuit theory and electrical circuit theory are both important branches of electrical engineering.Although both involve electromagnetics and engineering, there are some differences between them.Compared with magnetic circuit theory, electrical circuit theory has many advantages including more circuit components, a superior theoretical framework, and a more straightforward calculation procedure, etc.For example, equivalent electrical circuits are still the favored method for analyzing the dynamic response and steady-state characteristics of electrical machines [4], [7].Beside, Zhu et al. and Hui et al. developed the transmission line modeling method from electrical circuit theory to examine the loss and nonlinear behavior of the magnetic core [8], [9].However, compared with electrical circuit theory, the behavior of magnetic fields in magnetic circuits, including the magnetic field distribution, magnetic flux etc., can be more precisely reflected by magnetic circuit theory [10].Nevertheless, as only the reluctance component is included in the traditional magnetic circuit theory, the extent of its applicability is rather limited, as is its use in engineering practice [11].To change this status quo, Cheng et al. defined the second component in the magnetic circuit, namely magductance, by means of vector magnetic circuit derivation [12].Moreover, Qin et al. derived the expressions for reluctance and magductance of silicon steel sheets at different frequencies, and completed the preliminary experimental validation using the Epstein frame in the case of the magnetic density of 0.01T [13].The proposed magductance component rewrites the history of scalar magnetic circuit theory, elevates the status of magnetic circuit theory in the existing physical system, and holds the promise of breaking the limitations of the application of magnetic circuit theory.
Although vector magnetic circuit theory has been proven to be feasible in [12] and [13], there is still much follow-up work to be done, with the most critical being the compatibility with electrical circuit theory and electromagnetic field theory.For a long time, people often choose one of the three theories to analyze electrical machines.Few scholars have explored and studied the relationship between the three theories, let alone the vector magnetic circuit theory involving magductance components.This research may be dated back to 1941, when Roters discovered the resemblance between magnetic circuits and electrical circuits when designing the equivalent magnetic circuit of a transformer [14].Later in 1948, Cherry conducted a study of the duality of the electrical circuit and magnetic circuit, and concluded that the equivalent electrical circuit of a transformer was derived from the magnetic circuit using the idea of dual topology [15].Subsequently, Carpenter first applied the duality of magnetic circuits and electrical circuits in the analysis of rotating electromagnetic devices in 1968 [16].Based on this, Fiennes employed equivalent magnetic circuits in 1973 to theoretically analyze threephase synchronous motors, three-phase induction motors, and DC motors under steady-state and unexpected shortcircuit situations [17].Due to the limitations of computer technology and experimental conditions, no experiment has ever been able to prove the validity of duality.In 2000, Brown et al. renewed the duality between the electrical circuit and the magnetic circuit and made related applications in the following years [18], [19], [20].In 2015, Lambert et al. evaluated and reviewed the duality of electrical and magnetic circuits that appeared in the past and proposed a solution for nonplanar magnetic circuits [11].Despite the fact that the relationship between magnetic circuits and electrical circuits has been extensively investigated and utilized in recent decades [21], [22], [23], the compatibility of electromagnetic field theory, magnetic circuit theory, and electrical circuit theory has always been a challenge.
To overcome the aforementioned existing predicament, this paper employs the laminated magnetic core as a breakthrough to conduct the analysis.Firstly, a magnetic circuit model and an equivalent magnetic circuit of the laminated magnetic core are established based on Thévenin's theorem and Norton's theorem of vector magnetic circuit theory.Secondly, the magnetic field distribution of the laminated magnetic core at different frequencies (uniform magnetic flux distribution and non-uniform magnetic flux distribution) is analyzed using electrical circuit theory and electromagnetic field theory.Further, the expressions of reluctance and magductance are obtained by combining vector magnetic circuit theory, and the regularity of the magnetic circuit parameters changing with frequency is analyzed.Thirdly, the analysis and discussion of the three theoretical compatibilities are carried out with the magnetic circuit parameters as the connection point, proving that the three theories are mutually compatible and applicable to each other.Finally, the expressions of the magnetic circuit parameters of the laminated magnetic core are verified by the experiments, and the feasibility and validity of the vector magnetic circuit theory are proved from the experimental results.

II. VECTOR MAGNETIC CIRCUIT MODEL OF THE LAMINATED CORE
The magnetic core is the constituent part of the magnetic circuit and acts as a carrier of the magnetic flux.Its purpose is to provide a predictable and distinct path for the magnetic flux, and to concentrate the magnetic field energy in the magnetic core [6], [24].By using magnetic cores, the size and weight of electrical machines can be reduced, increasing efficiency and power density.Silicon steel is one of the most commonly used magnetic materials due to its high resistance and low hysteresis loss, making it suitable for electrical machines and other electromagnetic devices [25].As seen in Fig 1(a), silicon steel is typically formed into sheets to create a laminated magnetic core along the direction of sinusoidal alternating magnetic flux flow, which helps to further reduce eddy current losses.In Fig 1(a), denotes the total magnetic flux existing in the laminated magnetic core.I denotes the equivalent eddy current formed in the laminated magnetic core, and n denotes the number of silicon steel sheets.
Assuming that the silicon steel sheets in the laminated magnetic core are isolated from one another, the magnetic flux of each silicon steel sheet does not interfere with one another, and the effect of magnetic flux leakage and hysteresis loss are not taken into consideration.When an alternating magnetic flux circulates in the laminated magnetic core, an induced magnetomotive force (eddy current) is formed in each silicon steel sheet to oppose the change in magnetic flux, following Lenz's law.Assuming that the source magnetomotive force(MMF) F applied at both ends of each silicon steel sheet is the same, the magnetic flux in each silicon steel sheet is determined by both reluctance and magductance.According to the vector magnetic circuit theory [12], the equivalent magnetic circuit of the laminated magnetic core is depicted in Fig. 1(b), where 1 , 2 . . .n denote the magnetic flux flowing through each silicon steel sheet; R 1 , R 2 . . .R n denote the reluctance of each silicon steel sheet; L 1 , L 2 . . .L n denote the magductance of each silicon steel sheet; ω denotes the angular frequency of the alternating magnetic flux.
For ideal laminated core magnetic circuits, Thévenin's theorem and Norton's theorem convert a complex magnetic circuit to a simpler series or parallel equivalent magnetic circuit for easier analysis [26], which is shown in Fig. 1(c) and Fig. 1(d), respectively, where R eq and L eq are the equivalent reluctance and equivalent magductance of the laminated magnetic core, respectively.Ideally, when the magnetic circuit parameters of each silicon steel sheet are exactly the same, yielding From the laws for magnetic impedances in parallel [12], Hence, the expression for the equivalent reluctance R eq is And the expression for the equivalent magductance L eq is From ( 4), it is clear that the basis of solving the magnetic circuit parameters of the laminated core is to solve the magnetic circuit parameters of each silicon steel sheet, which necessitates more extensive analysis and derivation.

III. SOLUTION OF LUMPED MAGNETIC CIRCUIT PARAMETERS FOR LAMINATED CORES
Based on the analysis above, assuming that the magnetic field distribution of each silicon steel sheet of the laminated magnetic core is the same, the parameters of the laminated magnetic core can be calculated by solving the magnetic circuit of the silicon steel sheet.Therefore, the threedimensional mathematical model of the silicon steel sheet is established, and the center of the silicon steel sheet crosssection is used to create a coordinate system [6], as shown in Fig. 2. The silicon steel sheet is a, b, and h in thickness, width, and height, respectively.
There are two possible distributions of the magnetic flux in the silicon steel sheet: a uniform distribution of magnetic flux (low-frequency condition) or an uneven distribution of the magnetic flux (high-frequency condition).Therefore, different theories combined with vector magnetic circuit theory are developed to analyze the above two conditions below.

A. UNIFORM MAGNETIC FLUX DISTRIBUTION
Assuming that a sinusoidal MMF is applied to the laminated magnetic core where F m denotes the amplitude of the MMF.Due to the existence of reluctance and magductance in the magnetic circuit, according to Kirchhoff's law of the magnetic circuit [12], the resulting magnetic flux is where m and θ denote the magnetic flux amplitude and phase, respectively.And B m denotes the amplitude of magnetic flux density.When the frequency of magnetic flux in the silicon steel sheet is relatively low, according to the definition of skin depth δ [6], we can obtain where σ and µ refer to the conductivity and permeability of the silicon steel sheet, respectively.In this case, the distribution of magnetic flux (t) flowing through the silicon steel sheet can be considered to be uniform.According to the definition of reluctance [5], [26], the expression of reluctance is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Taking the yellow loop with thickness dx and area A = b(dx), as shown in Fig. 2. By Faraday's law, the induced voltage e(t) is equal to the derivative of the magnetic flux passing through the area enclosed by the yellow loop where e m denotes the amplitude of e(t).Since h ≫ a, the resistance of the two short edges in the yellow loop is ignored, According to the definition of resistance [26], the resistance of the yellow loop is approximated as dr ≈ 2h σ bdx (10) According to the electrical circuit theory, the active power of the yellow loop is Resulting in the active power caused by eddy current in the silicon steel sheet at low frequencies Considering only the active loss caused by the magductance of the silicon steel sheet, according to the magnetoelectric power law [12], yielding Combining ( 6), ( 12) and ( 13), the expression of magductance of the silicon steel sheet is Substituting ( 8) and ( 14) into (3) and ( 4), respectively, the expressions for the reluctance and magductance of the laminated magnetic core with uniform magnetic flux distribution are

B. NON-UNIFORM MAGNETIC FLUX DISTRIBUTION
As the frequency of the magnetic flux becomes higher, the effect of magductance becomes stronger against the magnetic flux change, i.e., the skin depth δ becomes smaller than the thickness of the silicon steel sheet, and the magnetic flux in the silicon steel sheet becomes unevenly distributed.As a result, electromagnetic field theory and vector magnetic circuit theory are used to derive the magnetic circuit parameters.
Typically, the eddy currents of electromagnetic devices belong to the region of quasi-stationary effects.That is to say, the displacement current within conductors may always be neglected with respect to the conductive current [27].At any frequency, the magnetic field intensity of a silicon steel sheet is described by Maxwell's equations for quasistationary electromagnetic fields.
∇ × H = J (16) where H and J refer to the magnetic field intensity and current density in the magnetic core, respectively.Taking the curl of ( 16) Considering the material equations Substituting ( 18) into ( 17) yields the Helmholtz equation In Fig. 2, the only component of H is assumed to be in the y-axis and is constrained to be a spatial function of x alone and to vary sinusoidally with time.Thus the eddy currents are induced entirely in the z-axis [27].Sinusoidal time variation of the field quantities enables us to substitute jω for ∂/∂t [28], therefore, (19) becomes where A general solution of ( 20) is Considering the condition of even symmetry requires from which Substituting ( 24) into ( 22) yields The amplitude of the magnetic field intensity at x = a/2 is Substituting ( 26) into (25) yields the solution of the Helmholtz equation According to Ampere's law [6], the applied source MMF is The magnetic flux flowing through the cross-section of the silicon steel sheet is given by Using tanh (1 + j) x = sinh 2x − j sin 2x cosh 2x + cos 2x (30) we obtain According to Kirchhoff's magnetomotive force law (KML) of the vector magnetic circuit, the expression for the magnetic impedance of the silicon steel sheet is Thus far, we have effectively employed KML derived from vector magnetic circuit theory in the computation of highfrequency characteristics for laminated magnetic cores.The expression for the reluctance of the silicon steel sheet is The expression for the magnetic reactance is The expression for the magductance is Substituting ( 33) and ( 35) into ( 3) and ( 4), respectively, the expressions for the reluctance and magductance of the laminated magnetic core with non-uniform magnetic flux distribution are

IV. COMPATIBILITY ANALYSIS OF THREE THEORIES
The magnetic circuit parameters are derived for laminated magnetic cores with uniform and non-uniform magnetic flux distributions.In this section, an analysis of the compatibility of electromagnetic field theory, electrical circuit theory, and vector magnetic circuit theory based on the derived expressions is carried out below.

A. ELECTRICAL CIRCUIT THEORY WITH NON-UNIFORM MAGNETIC FLUX DISTRIBUTION
By Maxwell's equations, substituting ( 27) into ( 16) gives the current density of the silicon steel sheet Further, the amplitude of the current density is According to the differential form of Joule's law in electrical circuit theory [6], [26], the time-average active power loss dissipated in the silicon steel sheet is According to (31), the amplitude of the magnetic flux is Combined with (13), the expression for the magductance of the silicon steel sheet is Further, the expression for the magnetic-reactance is Next, we derive the expression for the reluctance.Based on (27), the amplitude of the magnetic field intensity is Assuming that µ is uniform for the entire silicon steel sheet, the magnetic energy stored in the magnetic field inside the silicon steel sheet is [1], [29]: 113012 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
On the other hand, based on electrical circuit theory [26], [30], the magnetic energy stored in the magnetic field inside the silicon steel sheet is given by Equating the two expressions for the magnetic energy W m stored inside the silicon steel sheet, we have Easily found that the expressions (33) and (47) for reluctance and ( 35) and (42) for magductance are completely consistent, thus, the expression of the magnetic circuit parameters of the laminated magnetic core is shown in ( 36) and (37), proving theoretically that the three theories are fully compatible and complementary to each other.

B. ELECTROMAGNETIC FIELD THEORY WITH UNIFORM MAGNETIC FLUX DISTRIBUTION
Considering that the expansions of the trigonometric and hyperbolic functions found in Dowell's equation into the Maclaurin series for −∞ < x < ∞ are typically as follows [6]: When the magnetic flux is evenly distributed over each silicon steel sheet of the laminated magnetic core, this indicates that the skin effect is minimal and the flux frequency is low.Based on (7), i.e. δ ≫ we obtain Therefore, simply the first two terms of (48) are taken into account, while the higher-order terms are ignored.Substituting (48) into (36) and (37) yields Upon comparing equations ( 15) and (50), it can be inferred that vector magnetic circuit theory exhibits applicability not only within the realm of electromagnetic field theory but also in the context of electrical circuit theory.The correspondence and mutual reinforcement between these theories are evident, indicating a harmonious synergy between them.

V. EXPERIMENTAL VALIDATION
Based on the analysis presented in Section IV, it becomes evident that (36) and (37) serve as a pivotal link that establishes a coherent connection between electrical circuit theory, vector magnetic circuit theory, and electromagnetic field theory.Hence, it is crucial to verify the correctness and validity of the equations by experiment.
Typically, The Epstein frame is commonly employed in the investigation of magnetic circuit parameters due to its notable advantages such as strong repeatability, and ease of operation [31], [32].Therefore, in this paper, the Epstein frame is selected as the research subject to validate the derived magnetic circuit parameters of the laminated magnetic core.The Epstein frame is assembled from 35 pieces of 0.5 mm thick silicon steel sheet and the pieces are insulated from each other, the geometric dimensions and a physical photograph of the Epstein frame employed in this study are depicted in Fig. 3, and the key parameters of the Epstein frame are summarized in Table 1.Furthermore, an experimental platform was established to investigate the parameters of the magnetic circuit associated with the Epstein frame, as depicted in Fig. 4. The programmable AC power supply (ITECH IT7627) is employed to generate the alternating magnetomotive force applied to the primary-side winding of the Epstein frame, thereby inducing an alternating magnetic flux within the  laminated magnetic core constituting the magnetic circuit.Power analyzer (YOKOGAWA WT1806E), interconnected on the primary-side winding, enables real-time monitoring and acquisition of various parameters, including active power, reactive power, primary-side voltage, primary-side current, secondary-side voltage, and other pertinent variables associated with the Epstein frame.Leveraging these measured parameters, computational methods are employed to ascertain the magnetic circuit parameters of the Epstein frame.In addition, the voltage probe (Sapphire Instrument SI-9002) is utilized to capture the waveforms of both the primary-side and secondary-side voltages, while the current probe (Tektronix TCP305A) is employed to capture the primary-side current waveform.The waveforms generated during the experimental process are recorded using a scope corder (YOKOGAWA DL850E).The primary-side winding temperatures are monitored using a temperature tester (Anbai AT4716) to ensure the temperature of the Epstein frame remains stable at approximately 17 • C before the start of each test, as depicted in Fig. 5.
By following the magnetoelectric power law in [12] and [13], the expressions for the active and reactive power of the laminated magnetic core can be derived as follows: where L total is the total equivalent magductance of the laminated magnetic core.By combining the real-time power data obtained from the power analyser, the magnetic circuit parameters of the laminated magnetic core under different magnetic densities with frequency variation can be determined after removing the loss of the primary-side winding, as shown in Fig. 6.As shown in Fig. 6(a), with a constant magnetic flux density, the increasing frequency of the magnetic flux accentuates the skin effect, which in turn reduces the effective cross-sectional area of the magnetic flux.Consequently, this phenomenon leads to an elevated reluctance as the frequency rises.
We have conducted the frequency characteristics of the magductance of the laminated magnetic core for the first time.At lower-frequency, the equivalent magductance of the laminated magnetic core is notably influenced by the magnetic flux density, primarily due to the dominant influence of hysteresis losses on the active power.However, as the frequency increases, the magductance for various magnetic flux densities tends to converge.This convergence can be attributed to the increasing prominence of eddy current losses as the primary component of the active power.At higherfrequency, the magductance exhibit reduced sensitivity to changes in magnetic flux density, which aligns with the findings reported in [12].
Besides, according to (31) and (38), it is evident that as the frequency increases, the laminated magnetic core exhibit not only the skin effect of magnetic flux but also the accompanying skin effect of eddy currents.The skin effect of the magnetic flux reduces the equivalent cross-sectional area of the magnetic circuit, resulting in an increase in the equivalent reluctance of the magnetic circuit, and the skin effect of eddy currents reduces the equivalent magductance of the magnetic circuit, which aligns consistently with the outcomes illustrated in Fig. 6 and further reinforces the earlier discussions.
Based on the experimental results depicted in Fig. 6, we have selected the curve corresponding to the linear region (B = 0.6T) to validate (36) and (37).The results are compared in Fig. 7.It is evident that both the theoretically computed values and the experimentally measured values demonstrate a remarkable level of concordance, thereby validating the efficacy and precision of (36) and (37).

VI. CONCLUSION
This paper takes laminated magnetic cores as a starting point and conducts a comprehensive analysis and discussion on the compatibility issues among the newly developed vector magnetic circuit theory, and traditional electrical circuit theory, and electromagnetic field theory.Here are the three main points summarized: 1) Vector magnetic circuit theory, electrical circuit theory, and electromagnetic field theory can be derived from each other, compatible with each other, and combined with each other.2) Vector magnetic circuit theory is more direct than electrical circuit theory in describing the relationship of variables of a magnetic circuit, is easier to understand than electromagnetic field theory, and is more suitable for actual engineering applications.
3) Vector magnetic circuit theory incorporates magnetic circuit components such as reluctance and magductance, and also applies to Thévenin's theorem and Norton's theorem.4) The constructed experimental platform was utilized to validate magnetoelectric power law and the derived expressions for magnetic circuit parameters.It contributes to the development of the vector magnetic circuit theory system.

FIGURE 1 .
FIGURE 1. Magnetic circuit analysis of the laminated magnetic core.(a) Ideal laminated magnetic core model.(b) Equivalent magnetic circuit.(c) Thévenin equivalent magnetic circuit.(d) Norton equivalent magnetic circuit.

FIGURE 2 .
FIGURE 2. Mathematical model of the silicon steel sheet.

FIGURE 3 .
FIGURE 3. Sizes of core in the Epstein frame.(a) Geometric dimensions of the laminated magnetic core.(b) Epstein frame physical model.

FIGURE 4 .
FIGURE 4. The experimental platform for measuring magnetic circuit parameters of Epstein frame.

FIGURE 5 .
FIGURE 5. Variation curve of the magnetic core temperature during the experimental operation.

FIGURE 6 .
FIGURE 6. Frequency response of magnetic circuit parameters of the laminated core.(a) Reluctance.(b) Magductance.

FIGURE 7 .
FIGURE 7. Frequency response of magnetic circuit parameters of the laminated core.(a) Reluctance.(b) Magductance.

TABLE 1 .
Key parameters of epstein frame.