IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 6, JUNE 2010
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Extension of the Concept of Windings in Magnetic Field—Electric Circuit Coupled Finite—Element Method W. N. Fu and S. L. Ho The Hong Kong Polytechnic University, Kowloon, Hong Kong A more general formulation of two-dimensional (2-D) transient finite element method (FEM) for the computation of magnetic field and electric circuit coupled problems is presented. The concept of windings is extended to include groups of solid conductors. All excitations, including stranded windings, solid conductors, are regarded as special cases of the basic winding. The formulation also includes displacement current in the direction of the model depth. The coefficient matrix of the last system algebraic matrix equation is symmetrical. The developed method is applied to model electric devices. Index Terms—Coupling, electric circuit, finite element method, magnetic field, modeling, solid conductor, winding.
I. INTRODUCTION
T
HE MAGNETIC field-electric circuit coupled transient finite element method (FEM) has been widely used to simulate the dynamic operations of electric motors [1]. The electric circuit is established by either loop or nodal methods [2]. If there is a solid coil, in which several solid conductors are connected together, such as the shaded ring in shaded-pole motor, it is incorrect to deal with this conductor group as a normal winding, because usually the eddy-current densities in solid conductors are distributed unevenly. As the current density distributions on each conductor are different, the conductor on each side of the coil should be set up as separate solid conductors and the two conductors in the two sides should be connected by an external circuit. Such methodology is not convenient to users. In this paper the winding concept is extended to solid conductors, which generalizes the definition of windings and facilitates the users to set up excitations. A general method for coupling magnetic field and electric circuit in electric devices using 2-D FEM is presented. Similar to stranded windings, each solid winding can also have several series-connected solid conductors. Therefore, the stranded windings and solid conductors can all be set up as general windings. The merit is that it is convenient for users to set up a group of solid conductors and no external circuit connection is required. The deduced formulation is more general and can include the displacement current in the direction of the model depth. The coefficient matrix of the last system equation is symmetrical. II. DEFINITION OF WINDINGS In magnetic field regions, stranded windings and solid conductors connected to the electric circuits are the coupling ports between the magnetic field and electric circuit. To model a coupled system, the definition of windings proposed in this paper is that each winding consists of a group of coils (Fig. 1). These coils are, depending on the number of parallel branches, connected in series or in parallel. The polarity of the coils ( 1 or 1) represents, respectively, the forward or return paths. Each
Fig. 1. Structure of a winding.
coil also consists of a group of conductors connected in parallel. The conductor can be stranded conductor (consisting of very thin conductors coated with insulation and has negligible eddy current), or solid conductor (which is a lump of conductor). Usually solid conductors cannot be grouped together as the current density distributions in each solid conductor are different. However, with the novel formulation presented in this paper, this problem can be overcome. III. BASIC FORMULATIONS OF ONE SOLID CONDUCTOR A. Basic Field Equations In the 2-D FEM, the following hypotheses are introduced: the magnetic flux density has a component in the plane only; therefore the magnetic vector potential has a component in the direction only. For the component of electric field strength in the direction (1) where is the electric scalar potential of the conductor in the domain of magnetic field. If is the voltage drop between the two terminals of the conductor with a length in the direction, one has (2)
Manuscript received October 31, 2009; revised January 18, 2010; accepted January 19, 2010. Current version published May 19, 2010. Corresponding author: W. N. Fu (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2041433
where is the polarity ( 1 or 1) to represent either the forward or return path. Substituting (2) into (1), one has
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(3)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 6, JUNE 2010
According to the Maxwell-Ampere’s law
According to (8), one has (4)
where is the reluctivity of the material, is the conductivity, and is the permittivity. In 2-D problems (5) where is the unit vector along the z direction. Therefore, substituting (3) into (4), the magnetic field equation in the region of one conductor can be written as (6) where
is simplified as
.
(13)
The basic system equations in the region of one solid conductor are (11), (12) and (13). IV. BASIC FORMULATIONS OF ONE COIL Each coil can have several geometric objects. These objects are connected in parallel. For the stranded coil, one coil can have conductors. All conductors in the same coil are connected is always equal to one. in series. For the solid conductor, A. One Stranded Coil
B. Basic Circuit Branch Equations The total current density flowing in the conductor along the axis is
Because the stranded coil is usually connected with a relatively large voltage source or current source, the displacement current can be neglected. The magnetic field equation in the region of the stranded coil can be expressed as
(7) The total current in the conductor in the
(14)
axis is where is the total cross-sectional area of the coil and is the number of total conductors of this coil. The branch equation of the stranded coils is
where is the region of the conductor and of the conductor
(8)
(15)
is the resistance
is the voltage on the two terminals of the coil; and where is the total DC resistance of the coil
(9)
(16)
where is the cross-sectional area of the conductor. The back e.m.f. of the conductor is
B. One Solid Coil
(10)
Even though each solid coil has only one conductor , for the purpose of deducing a unified formulation with stranded coils, is still included in the formulation. According to (11), one has
To make the coefficient matrix of the last system equation symmetrical and also allow the model depth (which is the dimension of the device through the axis) of different objects to be different [3], the model depth is multiplied to the two sides of (6)
(17) According to (12), one has
(11)
According to (13), one has
C. Summary of the Equations of One Conductor
Multiplying
(18)
to the two sides of (3), one has (12)
(19)
FU AND HO: EXTENSION OF THE CONCEPT OF WINDINGS IN MAGNETIC FIELD—ELECTRIC CIRCUIT COUPLED FEM
The loop method is used to establish the circuit equations. An is introduced to keep the coefficient additional unknown matrix of the last system equations symmetrical
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is the winding current; is the polarity of the is the total cross-sectional area of the coil; is the total conductor number of the coil. The branch equation of the stranded windings is
where; coil;
(20) (26) According to the circuit branch equation (19), the branch voltage can be expressed as
where the winding has coils; is the voltage difference between the two terminals of the winding; and is the total DC resistance of the winding
(21) Substituting the branch equation (21) into (17), the magnetic field equation can be written as (27) B. One Solid Winding (22)
According to (22), the field equation in the solid winding is
Substituting (21) into (18), the electric field equation can be written as (28)
(23) Multiply
to the additional (20), it can be written
According to (23), the electric field equation can be written as
as (29) (24) The system equations of the coil comprising of solid conductors are (22), (23), (24) and (19).
Additional equation in each coil is
V. BASIC FORMULATIONS OF ONE WINDING
(30)
Each winding can have several paralleled branches, which are reflected by the number of parallel branches . For a solid conductor, is always equal to one. Periodic boundary conditions are used to reduce the solution domain whenever possible. Similar to the stranded windings, the periodic boundary conditions can also be used for the solid conductors as their electric and magnetic field quantities satisfy either periodic condition or antiperiodic condition. Here is the symmetry multiplier which is defined as the ratio of the original full cross-sectional area to the solution area. When , the coils on the outside of the domain can be connected in series or in parallel. For simplicity, here it is supposed that all coils on the outside are connected in series with the coil in the domain. A. One Stranded Winding
According to (23), the voltage balance equation of one winding is
(31)
C. Summary of One Winding For voltage sources and external circuit coupling, the basic equations of the field and branch circuit are summarized as
All conductors in the same stranded winding are grouped together. The magnetic field equation in the region of the stranded windings can be expressed as (25)
(32)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 6, JUNE 2010
additional equation and the branch equation, one obtains the recurrence formulas (33)
(37)
(34)
where the superscript corresponds to the quantities at the step. The branch equation of the arbitrary connected circuits (not in the magnetic field region) can be expressed as
(35)
(38)
where the number of additional equations and the number of additional unknown are , respectively. The second and the third term on the left side of (32), as well as (33) and (34) only exist in solid winding region. If this formulation is applied to the stranded windings, (33) and (34) and the unknown in each coil are not required. If the displacement current are not in the z direction is neglected, (33) and the unknown required.
is the matrix of the resistance and is the column where to the two matrix associated with the sources. Multiplying sides of (38), one has (39) Adding the circuit equations (39) into (37), one obtains
VI. FORMULATIONS FOR ARBITRARY COUPLING A. Coupled System Equations The coupled field and circuit branch equations in the magnetic field region can be written in the matrix format
(40)
Using the loop method, the relationship between the branch current and the loop current is (41) where is the loop-to-branch incidence matrix. The Kirchhoff’s voltage law can be expressed as, (42)
(36)
Substituting these relationships into the system equations, one obtains the final global equations, shown in (43) at the top of the next page, where the coefficient matrix is symmetrical. VII. EXAMPLE
where , , , , , , , , and are sub matrixes which can be deduced using a similar method presented by the authors in [2]. Using the backward Euler’s to the method to discretize the time variable and multiply
A simple example is used to verify the proposed method. Fig. 2 shows a transformer, where all dimensions are in mm. The model depth is 1 m. The primary stranded winding has 10 turns, a resistance of 1 ; a voltage at is applied on it. The secondary winding has
FU AND HO: EXTENSION OF THE CONCEPT OF WINDINGS IN MAGNETIC FIELD—ELECTRIC CIRCUIT COUPLED FEM
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(43)
Fig. 4. Currents in the stranded winding and the solid winding. Fig. 2. A test example with a stranded winding and a solid winding.
is verified. The proposed method has been widely used to simulate electric devices. VIII. CONCLUSION
Fig. 3. Field plots when t density distribution.
= 2 ms. (a) Flux line distribution. (b) Eddy current
six solid conductors and they are connected in series. The iron core’s relative permeability is 4000. The flux line distribution is shown in Fig. 3(a) and the distribution of eddy-current density vectors is as shown in Fig. 3(b). It can be observed that the flux cannot totally penetrate into the secondary winding due to strong skin effect in the conductors of the secondary winding. The currents in the windings are shown in Fig. 4. The current in Fig. 4 is in correspondence with 10 turns in the primary stranded winding and six solid conductors in the secondary solid winding. Because of the strong eddy current in solid conductors, this example needs correct modeling of solid conductors. The case study shows that the solutions are exactly the same irrespective of whether the solid conductors are set up as one solid winding or as six individual solid conductors in external circuits. Therefore, the correctness of the proposed formulations
A formulation of 2-D magnetic field—electric circuit coupled transient FEM is deduced. It can include the displacement current in the direction of the model depth. The concept of stranded windings is extended; a group of solid conductors can also be taken as a winding which has many series-connected coils. The coefficient matrix of the last algebraic equations is symmetrical. ACKNOWLEDGMENT This work was supported in part by The Hong Kong Polytechnic University under Grant 87RX and Grant B-Q18X. REFERENCES [1] R. Escarela-Perez, E. Melgoza, and J. Alvarez-Ramirez, “Coupling circuit systems and finite element models: A 2-D time-harmonic modified nodal analysis framework,” IEEE Trans. Magn., vol. 45, no. 2, pt. 1, pp. 707–715, Feb. 2009. [2] W. N. Fu, P. Zhou, D. Lin, S. Stanton, and Z. J. Cendes, “Modeling of solid conductors in two-dimensional transient finite-element analysis and its application to electric machines,” IEEE Trans. Magn., vol. 40, no. 2, pp. 426–434, Mar. 2004. [3] W. N. Fu, S. L. Ho, H. L. Li, and H. C. Wong, “A multislice coupled finite-element method with uneven slice length division for the simulation study of electric machines,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1566–1569, May 2003.
