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Review Paper: Analytical Calculation of Magnetic Flux Density within a Rotor
Slot of an Electric Motor Using Separation of Variables
Research · December 2015
DOI: 10.13140/RG.2.1.4655.4968
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Analytical Calculation of Magnetic Flux
Density within a Rotor Slot of an Electric
Motor Using Separation of Variables
Vempati Sai Ram Anand
Abstract— This paper demonstrates the application of an
analytical electromagnetic solving technique known as separation
of variables in the calculation of the magnetic flux density within
the rotor slot of a cylindrical electric motor. A Laplace equation
based on magnetic vector potential is formulated for the single
rotor slot along with the respective boundary conditions. The
separation of variables technique is used to find a solution for the
magnetic vector potential in this region. The magnetic flux density
is then calculated from this magnetic vector potential solution.
The other application use of this analytical method in the
calculation of the magnetic flux density within the air gap region
between the stator and rotor regions of the electric motor will be
introduced and discussed in brief.
Index Terms—Laplace equation, separation of variables,
electric motor
ordinary differential equations (ODE). The final solution of the
problem is a linear combination of the solution of each of these
ODEs. The procedure of the separation of variables technique
will be demonstrated with the example of magnetic flux density
calculation within a single slot of the cylindrical electric motor.
The organization of the paper is as follows. The problem for
which this technique will be used is introduced in Section II.
The analytical solution of the problem using the separation of
variables technique is described in Section III. Finally, the
possible use of this method for magnetic flux density
calculation in the air gap domain of the motor is briefly
introduced and discussed in section IV.
II. DESCRIPTION OF THE PROBLEM
A. Geometrical Configuration of the Problem
The first step that is needed for solving any electromagnetic
problem is to define its geometry and any possible assumptions
with respect to the specified geometry. The cross-section of the
electric motor is shown in Fig. 1.
I. INTRODUCTION
AN electromagnetic field problem can be solved using either
experimental, analytical or computational techniques. With the
advent of the 20th century, computational techniques were
introduced which were capable of solving extensive field
problems having increased geometrical complexity.
However, the analytical solution methods still hold a great
importance in solving electromagnetic problems of any type
due to the exact and closed-form nature of the solution that they
provide. These solutions can be further examined in detail to
obtain a physical understanding of the electromagnetic
phenomena behind the problem [1]. The analytical method also
provides a useful way to validate the solutions obtained from
numerical methods [2].
A large number of elementary and advanced problems in
electromagnetics are formulated in terms of differential
equations involving functions of more than one variable. These
are referred to as partial differential equations (PDE). The
Laplace equation is one such PDE. The general form of a
Laplace equation is defined in (1).
∇2 𝜑 = 0
(1)
From (1), 𝜑 represents the solution of an electromagnetic
problem in a predefined coordinate system such as Cartesian,
polar, spherical, or cylindrical and ∇2 represents the Laplacian
operator in the respective co-ordinate system.
The method of separation of the variables is one of the most
powerful and simple analytical methods to solve a Laplace
equation. The initial step in this method involves expressing a
multi-variable solution as a product of individual single
variable functions. One of the major steps in the procedure of
separation of variables is to convert the PDE to multiple
Fig. 1. Cross section of the electric motor [3].
The structure of the electric motor is divided into an annular
shaped air-gap region between the stator and rotor and the rotor
slot region consisting of 𝑄 slots [3].
The parameters which describe the geometry of the electric
motor are the inner radius of the rotor yoke (R1), the radius of
the outer surface (R2) and the bore radius of the stator (R3). The
opening angle for each slot is denoted as 𝛽 . The angular
position of the rotor is defined as 𝜃0 .
The expression for the angular position of the ith rotor slot
(𝜃𝑖 ) is shown in (2).
𝜃𝑖 = −
𝛽 2𝑖𝜋
+
+ 𝜃0
2
𝑄
(2)
In order to simplify the analytical solution process, the
following assumptions are made:
-- All end effects are neglected.
-- Stator and rotor cores have extremely high
permeability.
-- No current is flowing in the rotor slots.
-- The lateral shape of the rotor slots is radial.
The source generating the magnetic field inside the slots of
the rotor is represented by a current sheet 𝐾(𝜃) that is
distributed over the stator inner radius r = R3 which is only a
function of the angular position 𝜃 of any point in the surface
defined by this radius.
From Fig. 1, the region labeled slot i denotes ith rotor slot.
The circular region present between the rotor slots and the
stator current sheet is defined as the air gap region.
Section III describes the procedure for calculation of the
magnetic flux density within the ith slot of the rotor.
III. ANALYTICAL SOLUTION FOR MAGNETIC FIELD WITHIN A
SINGLE ROTOR SLOT
A. Geometry of the ith Rotor Slot
The geometrical representation of the ith slot of the rotor is
shown in Fig. 2.
The magnetic flux density is then calculated from the resulting
magnetic vector potential solution.
B. Laplace Equation for the Vector Magnetic Potential
Since the magnetic vector potential depends only on the r
and 𝜃 co-ordinates, it is appropriate that a Laplace equation for
the magnetic vector potential is defined in the polar co-ordinate
system. The Laplace equation in the polar co-ordinate system
can be derived from the two-dimensional Laplace’s equation in
the Cartesian co-ordinate system [4].
The polar form of the Laplace equation is given in (5), where
𝑢 denotes the solution of the given problem.
∇2 𝑢 =
𝜕 2 𝑢 1 𝜕𝑢
1 𝜕2𝑢
+
+ 2 2
2
𝜕𝑟
𝑟 𝜕𝑟
𝑟 𝜕𝜃
(5)
The slot domain shown in Fig. 2 is the region bounded by the
inner radius R1, the outer radius R2, and the angles 𝜃𝑖 and
𝜃𝑖 + 𝛽 . The Laplace equation in polar coordinates for the
magnetic vector potential defined in (6) needs to be solved
within this region corresponding to the slot domain.
𝜕 2 𝐴𝑖 1 𝜕𝐴𝑖
1 𝜕 2 𝐴𝑖
𝑅1 ≤ 𝑟 ≤ 𝑅2
+
+ 2
= 0 for {
2
𝜃𝑖 ≤ 𝜃 ≤ 𝜃𝑖 + 𝛽
𝜕𝑟
𝑟 𝜕𝑟
𝑟 𝜕𝜃 2
(6)
After defining the Laplace’s equation for the magnetic
vector potential, the boundary conditions need to be defined for
the single slot structure. Laplace’s equation along with the
boundary conditions are needed to be solved for obtaining
unique and exact solutions to the problem.
C. Defining Boundary Conditions
The boundary conditions for the single slot of the motor
shown in Fig. 2 can be divided into individual conditions for
tangential magnetic fields at the sides of the slot, the bottom of
the slot, and the top of the slot [3].
Fig. 2. Geometry of a single rotor slot [3]
Based on the assumptions made in section II, the magnetic
vector potential for a single slot has only one component along
the z-direction which depends on the r and 𝜃 co-ordinates. The
expression for the magnetic vector potential for the ith slot is
given in (3).
𝑨𝒊 = 𝐴𝑖 (𝑟, 𝜃) ∙ 𝒆𝒛
(3)
Similarly the magnetic vector potential in the air gap region
is defined in (4).
𝑨𝑰 = 𝐴𝐼 (𝑟, 𝜃) ∙ 𝒆𝒛
(4)
The first step towards solving the magnetic flux density for
the ith rotor slot involves formulating a Laplace equation based
on the magnetic vector potential and defining its boundary
conditions. The Laplace equation along with its boundary
conditions is then solved using the separation of variables
technique to find a solution for the magnetic vector potential.
Boundary conditions on sides of the slot
The tangential magnetic field component at the sides of the
slot is equal to zero. This is mathematically expressed in the
Neumann boundary conditions for the magnetic vector
potential. A Neumann boundary condition [5] specifies that the
normal derivative of the solution to the given problem must be
equal to zero on the boundary of the solution region. The
boundary conditions at for the magnetic vector potential are
stated in (7) and (8).
𝜕𝐴𝑖
|
=0
𝜕𝜃 𝜃=𝜃𝑖
𝜕𝐴𝑖
|
=0
𝜕𝜃 𝜃=𝜃𝑖 +𝛽
(7)
(8)
After stating the boundary conditions for the sides of the slot,
the boundary conditions at the bottom of the slot will be defined
next.
Boundary conditions at bottom of the slot
The tangential component of the magnetic field at the bottom
of the slot, bounded by r = R1, is also equal to zero.
This is expressed mathematically in the form of a Neumann
boundary condition for the magnetic vector potential in (9).
𝜕𝐴𝑖
|
=0
𝜕𝑟 𝑟=𝑅1
(9)
substituting (14) in (13). This ODE is stated in (16).
𝑟 2 𝜌𝑖′′ (𝑟) + 𝑟𝜌𝑖′ (𝑟) + 𝜆𝜌𝑖 (𝑟) = 0
(16)
By substituting (11) in (7) and (8) respectively, the boundary
conditions for the two sides of the slot can be rewritten in terms
of the function Θ(𝜃) as
Θ𝑖 ′(𝜃𝑖 ) = 0 and Θ𝑖 ′(𝜃𝑖 + 𝛽) = 0 .
(17)
th
Continuity condition between the i slot and air gap region
In the top region of the slot, defined by R2, a continuity
condition exists between the slot and the air gap region. In this
region, the magnetic vector potential of the slot (𝐴𝑖 (𝑟, 𝜃)) is
equal to vector potential of the air gap (𝐴𝐼 (𝑟, 𝜃)) shown in (10).
𝐴𝑖 (𝑅2 , 𝜃) = 𝐴𝐼 (𝑅2 , 𝜃)
(10)
D. Solving the Laplace Equation
In accordance with the initial step of the separation of
variables method, the solution of the Laplace equation for
vector magnetic potential of the ith slot stated in (3) is written
as a product of two individual solutions, one being only a
function of variable r and the other being a function of variable
θ [3]. This is shown in (11).
𝐴𝑖 (𝑟, 𝜃) = 𝜌𝑖 (𝑟)Θ𝑖 (𝜃)
(11)
By substituting (12) in (7), a single equation is obtained in
terms of 𝜌𝑖 (𝑟) and Θ𝑖 (𝜃). This is shown in (12).
𝜌𝑖′′ (𝑟)Θ𝑖 (𝜃) +
1 ′
1
𝜌𝑖 (𝑟)Θ𝑖 (𝜃) + 2 𝜌𝑖 (𝑟)Θ𝑖 ′′(𝜃) = 0
𝑟
𝑟
𝜆0 = 0
𝑘𝜋 2
𝜆𝑘 = − ( ) where 𝑘 = 1,2,3 …
𝛽
(18)
(19)
The eigenfunctions Θ𝑖0 (𝜃) and Θ𝑖𝑘 (𝜃) that correspond to
the eigenvalues 𝜆0 and 𝜆𝑘 are obtained by substituting (18) and
(19) in (15) and then solving (15) for each eigenvalue. The
corresponding eigenfunctions are stated in (20) and (21).
Θ𝑖0 (𝜃) = 1
𝑘𝜋
Θ𝑖𝑘 (𝜃) = cos ( (𝜃 − 𝜃𝑖 ))
𝛽
(20)
(21)
(12)
𝜌𝑖′′ (r)
Where
and Θ𝑖 ′′(𝜃) are the second order partial
differential operations in r and 𝜃 respectively. Now, the second
order PDE in terms of 𝜌𝑖 (𝑟) and Θ𝑖 (𝜃) needs to be separated
into two single variable ODEs. Dividing (12) throughout by
Θ𝑖 (𝜃) and rearranging the terms, we get
𝜌𝑖 (𝑟)Θ′′
𝑖 (𝜃)
𝑟 2 𝜌𝑖′′ (𝑟) + 𝑟𝜌𝑖′ (𝑟) = −
.
Θ𝑖 (𝜃)
The problem of finding 𝜆 for which there exist non-zero
solutions of (15) that satisfy the boundary conditions in (17) is
classified as a Sturm-Liouville problem [6]. Taking the
boundary conditions in (17) into account, 𝜆 corresponds to the
eigenvalues of this problem and the resulting solutions of
Θ𝑖 (𝜃) are referred to as eigenfunctions of the problem. The
calculated eigenvalues 𝜆 are stated in (18) and (19).
The solutions for 𝜌𝑖 (𝑟) corresponding to the eigenvalues 𝜆0
and 𝜆𝑘 are obtained by substituting (18) and (19) in (16). These
solutions are stated in (22) and (23), where 𝐴𝑖0 , 𝐵0𝑖 , 𝐴𝑖𝑘 and 𝐵𝑘𝑖
are arbitrary constants.
𝜌𝑖0 (𝑟) = 𝐴𝑖0 + 𝐵0𝑖 ln𝑟
𝜌𝑖𝑘 (𝑟) = 𝐴𝑖𝑘 𝑟
(13)
The following step is one of the characteristic steps of the
method of separation of variables. An arbitrary separation
constant 𝜆, where 𝜆 ∈ ℜ, is chosen for separating (13) into two
individual ODEs. The separation constant is defined as
𝑘𝜋
−
𝛽
+ 𝐵𝑘𝑖 𝑟
(22)
(23)
𝑘𝜋
𝛽
The solution for the magnetic vector potential is now given
as a linear combination of the ODE solutions for 𝜌𝑖 (𝑟) and
Θ(𝜃) from (20)-(23). The solution is stated in (24).
∞
𝐴𝑖 (𝑟, 𝜃) = Θ𝑖0 (𝜃)𝜌𝑖0 (𝑟) + ∑ Θ𝑖𝑘 (𝜃)𝜌𝑖𝑘 (𝑟).
(24)
𝑘=1
𝜆=
Θ′′
𝑖 (𝜃)
.
Θ𝑖 (𝜃)
(14)
Rearranging the terms of (14), we get one ODE in terms of
the function Θ𝑖 (𝜃) stated as
Θ′′ (𝜃) − 𝜆Θ(𝜃) = 0 .
The solution for 𝐴𝑖 (𝑟, 𝜃) which is obtained by substituting
(20)-(23) in (24) is given in (25).
∞
𝐴𝑖 (𝑟, 𝜃) = 𝐴𝑖0 + 𝐵0𝑖 ln𝑟 + ∑ (𝐴𝑖𝑘 𝑟
𝑘=1
(15)
The second ODE in terms of the variable r is derived by
−
𝑘𝜋
𝛽
𝑘𝜋
+ 𝐵𝑘𝑖 𝑟 𝛽 )
𝑘𝜋
. cos ( (𝜃 − 𝜃𝑖 ))
𝛽
(25)
By using the boundary conditions at the bottom of the ith slot
from (10) and the continuity condition at the top of the slot
from (11), the solution for 𝐴𝑖 (𝑟, 𝜃) in (25) can be simplified to
(26).
𝑃𝑘𝜋 (𝑟, 𝑅1 )
∞
𝐴𝑖 (𝑟, 𝜃) =
𝐴𝑖0
+
∑ 𝐴𝑖𝑘 .
𝑘=1
𝛽
𝑃𝑘𝜋 (𝑅1 , 𝑅2 )
. cos (
𝑘𝜋
(𝜃 − 𝜃𝑖 ))
𝛽
𝛽
(26)
From (26), 𝑃𝑘𝜋 (𝑟, 𝑅1 ) and 𝑃𝑘𝜋 (𝑅1 , 𝑅2 ) are defined as,
𝛽
𝛽
𝑘𝜋
𝑘𝜋
𝑟 𝛽
𝑅1 𝛽
𝑃𝑘𝜋 (𝑟, 𝑅1 ) = ( ) + ( )
𝑅1
𝑟
𝛽
𝑘𝜋
𝑘𝜋
𝑅1 𝛽
𝑅2 𝛽
𝑃𝑘𝜋 (𝑅1 , 𝑅2 ) = ( ) + ( ) .
𝑅2
𝑅1
𝛽
(27)
V. CONCLUSION
𝐴𝑖0 =
1 𝜃𝑖 +𝛽
∫
𝐴𝐼 (𝑅2 , 𝜃) . 𝑑𝜃
𝛽 𝜃𝑖
(29)
𝐴𝑖𝑘 =
2 𝜃𝑖+𝛽
𝑘𝜋
∫
𝐴𝐼 (𝑅2 , 𝜃). cos ( (𝜃 − 𝜃𝑖 )) . 𝑑𝜃
𝛽 𝜃𝑖
𝛽
(30)
REFERENCES
[2]
[3]
[4]
(31)
(32)
The total magnetic flux density (𝐵) of the ith rotor slot can be
calculated by finding the resultant of its radial (𝐵𝑖𝑟 ) and
tangential components (𝐵𝑖𝜃 ) stated in (31) and (32).
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The procedure for the analytical method of separation of
variables was demonstrated with the help of the application of
finding the magnetic flux density within the single rotor slot
domain of an electric motor. The Laplace equation in terms of
the magnetic vector potential in the slot was described followed
by the stating the boundary conditions for all four sides of this
slot. The separation of variables technique was then used to
convert the Laplace equation into two single variable ODEs
which were solved using the concept of Sturm-Louville
problem. Finally the solution for the magnetic vector potential
is given as a solutions of the individual ODEs. The main
advantage of the separation of variables method is that it allows
the simplification of the partial differential equation into
multiple single variable ODEs which greatly reduces the
complexity of the analytical solving procedure.
[1]
E. Calculating Magnetic Flux Density for ith Slot
The radial and tangential components of the magnetic flux
density are calculated [4] from the vector magnetic potential
solution 𝐴𝑖 (𝑟, 𝜃) in (27) by using the relations
1 𝜕𝐴𝑖
𝑟 𝜕𝜃
𝜕𝐴𝑖
𝐵𝑖𝜃 = −
.
𝜕𝑟
A. Calculating Flux Density in Air Gap Domain of Motor
The method of separation of variables can also be applied in
the calculation of vector magnetic potential in the air gap
domain of the electric motor described in [3].The Laplace
equation for the air gap domain is given in terms of the air gap
vector magnetic potential 𝐴𝐼 (𝑟, 𝜃). The air gap domain for the
electric motor is the annular region between the rotor radius 𝑅2
and the stator inner radius 𝑅3 as shown Fig. 1.
The boundary condition for the air gap domain at r = 𝑅3
takes the infinite permeability of the stator back iron and the
current sheet 𝐾(𝜃) described in section II into consideration.
The other boundary condition at r = 𝑅2 takes the continuity
condition stated in (11) into account. This application will not
be discussed in detail in the scope of this review paper.
(28)
The vector magnetic potential solution 𝐴𝑖 (𝑟, 𝜃) for the ith
slot in (26) is analogous to a Fourier series [2] expansion of
𝐴𝑖 (𝑟, 𝜃) with 𝐴𝑖0 and 𝐴𝑖𝑘 as the Fourier series coefficients. The
coefficient 𝐴𝑖0 can be found by integrating the vector magnetic
potential 𝐴𝑖 (𝑟, 𝜃) for the ith slot with over the slot interval
[𝜃𝑖 , 𝜃𝑖 + 𝛽] with r = 𝑅2 .This is equivalent to integrating the
vector magnetic potential 𝐴𝐼 (𝑟, 𝜃) for the air gap at r = 𝑅2 as
per the continuity condition in (11). The other coefficient 𝐴𝑖𝑘
can be calculated by utilizing the orthogonal condition of
cosine function [3] in (26) and then integrating 𝐴𝐼 (𝑅2 , 𝜃) over
the interval [𝜃𝑖 , 𝜃𝑖 + 𝛽]. The expressions for finding 𝐴𝑖0 and 𝐴𝑖𝑘
are given in (29) and (30).
𝐵𝑖𝑟 =
IV. OTHER APPLICATIONS OF THIS ANALYTICAL METHOD
[5]
[6]
G. S. Smith, An introduction to classical electromagnetic radiation, 1st
ed. New York, NY: Cambridge University Press, 1997, p. 71.
R. Garg, Analytical and computational methods in electromagnetics.
Boston, MA: Artech House, 2008, pp. 29–35.
T. Lubin, S. Mezani and A. Rezzoug, "Exact Analytical Method for
Magnetic Field Computation in the Air Gap of Cylindrical Electrical
Machines Considering Slotting Effects," IEEE Trans. Magn., vol. 46, no.
4, pp. 1092-1099, Apr. 2010.
Tolosa and M. Vajiac, “An Introduction to Partial Differential Equations
in the Undergraduate Curriculum,” PCMI Undergraduate Faculty
Program[Online], 2003, pp.2-3 Available:
http://www.math.hmc.edu/~ajb/PCMI/lecture11.pdf
M. N. O. Sadiku, Numerical techniques in electromagnetics, 2nd ed.
Boca Raton, FL: CRC Press, 2001, pp. 18–19.
W. F. Trench, Elementary differential equations with boundary value
problems, 1.01 ed [Online], 2001, p. 582 Available:
http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_FREE_DIFF
EQ_II.PDF