Download Electric Field around a Metal Disk within a Microwave Resonator

Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010
387
Electric Field around a Metal Disk within a Microwave Resonator:
Electrostatic Approximation
Gholamreza Shayeganrad1 and Leila Mashhadi2
1
2
Islamic Azad University, Karaj Branch, Karaj, Iran
Physics Department, Amirkabir University of Technology, Tehran, Iran
Abstract— In this work, a rectangular resonator cavity with non-ionization electromagnetic
radiation at frequency of 2.45 GHz or wavelength of 12.24 cm involving a metal disk has been
considered. We have shown electrostatic field is a suitable approximation to study a metal disk
within a cavity with alternative field at frequency of 2.45 GHz and it leads to solve Laplace’s
equation. We have solved this equation in elliptical coordinates in detail considering boundary
conditions and determined electric field around the disk. Solving Laplace’s equation in elliptical
coordinates is the appropriate method for determining electric and magnetic fields in the vicinity
of the metals with striped or circular types.
1. INTRODUCTION
A cavity can be considered as a volume enclosed by a conducting surface and within which an
electromagnetic field can be excited. The electric and magnetic energies are stored in the volume of
the cavity. The power loss effects of the finite conducting walls have been accurately investigated
in the previous work [1].
Here, we have briefly shown that the electrostatic approximation is the appropriate method
for determining electric field and electrostatic potential function around the conductors subjected
at microwave frequency. Laplace’s equation is derived for static electric field within free-source
rectangular and cylindrical cavities considering this approximation and therefore the potential
everywhere in the region can be found by solving it for specified Dirichlet, Neumann or mixed
boundary conditions. The method of separation of variables is straightforward. This method yields
solutions involving circular cosine and sine functions, Bessel and Legendre polynomials to Laplace’s
equation in cartesian, cylindrical and elliptical coordinates, respectively. However, the analytical
expressions are obtained only in rather idealized cases that certain symmetries must exist and the
boundaries must not be irregular which their mathematical analysis is not very difficult.
In other cases, field calculations often become very complicated and the ultimate objective, however, is obtaining a numerical result. Some numerical methods that have used to solve Laplace’s
equation in a given region are the finite-element method, finite-difference method, or the method
of moments [2–4]. Among them, the Finite Difference Time Domain (FD-TD) method is a numerical technique widely used in electromagnetics for solution of the Maxwell’s Curl equations in
those cases where exact analytical methods cannot be applied and approximated solutions are not
satisfactory [5].
An advantage of having analytical expressions is that the general behavior of the solution for
different values of the problem parameter may be observed. The curvilinear coordinates to solve
the Laplace’s equation offers many advantages in solving practical engineering problems, which
satisfies some special boundary conditions. We have proposed in this work a simple and exact
theoretical method for determining the explicit electric field and electrostatic potential function
in the vicinity of the metal disk subjected within a rectangular microwave cavity, considering
electrostatic approximation. We have generally derived an analytical expression for electric field
around the metal disk by solving Laplace’s equation in elliptical coordinate system. Substituting
numerical values for the problem parameters into these expressions yields the solution to a particular
problem.
2. THEORY
According to the Drude’s model, the electrical conductivity of a metal is given by [6]:
σ=
f0 Ne2
m(γ0 − iω)
(1)
PIERS Proceedings, Xi’an, China, March 22–26, 2010
388
in which f0 N is number of free electrons per unit volume in the metal and γ0 /f0 damping constant
that can be determined empirically from experimental, typically of the order of 1017 sec−1 . This
means that, for frequencies beyond the microwave region (ω ≤ 1011 s−1 ), ω in the dominator of the
Eq. (1) is negligible and σ is approximated by:
σ=
f0 Ne2
mγ0
(2)
It indicates that conductivity of metals in microwave region is essentially real and independent to
frequency. Therefore, electrostatic approximation is sufficient to investigate conductors located in
the microwave resonator. Figure 1 shows a thin metal disk lies in the xy-plane, with its center
at the origin and radius, a, introduced into a uniform electric field with magnitude E0 directed
~ = E0 x̂. The resonant modes inside the microwave cavity has the magnitude of
along the x-axis, E
5
E0 ≈ 10 V/m [1].
z
→
E
a
x
y
~ = E0 x̂.
Figure 1: A metal disk with radius, a, to a uniform electric field E
Of particular interest is the determination of the distribution of electric field around the disk.
First, we obtain the electrostatic potential function, φ, by solving Laplace’s equation in elliptical
~ from E
~ = −∇φ. At
coordinate. Knowing the potential function, we can find the electric field, E,
this point, two assumptions have made: i) at points far away from the disk compared to its radius,
the field becomes to E0 x̂, and ii) the electric field becomes zero on the surface of disk.
We specify the location of a point in this coordinate system by giving its coordinates in the
form (u, ϕ, v). The conversion between the cylindrical coordinate (ρ, ϕ, z) and elliptical (u, ϕ, v)
coordinate systems are given by:
ρ = a cosh u cos v
z = a sinh u sin v
ϕ = ϕ
− π/2 < v < π/2
0<u<∞
0 < ϕ < 2π
(3a)
(3b)
(3c)
The boundary conditions in elliptical coordinate for this problem are:
φ = 0
at v = 0
φ = −aE0 cosh u cos v cos ϕ at u → ∞
Expanding ∇2 φ in elliptical coordinate, we find that Laplace’s equation given by:
µ
¶
1
1
∂
∂
1 ∂
∂
1
∂2φ
2
∇ φ= 2
cosh u
+
cos v
φ+ 2
=0
D (u, v) cosh u ∂u
∂u cos v ∂v
∂v
a cosh2 u cos2 v ∂ 2 ϕ
(4a)
(4b)
(5)
where D2 (u, v) = a2 (cosh2 u − cos2 v).
Equation (5) can be solved by the method of separation of variables. The result would have
been:
φ = ψm (u, v)(Am cos mϕ + Bm sin mϕ)
(6)
where m is an integer. Finally, we obtain for most general form:
φ=
∞
X
m=0
ψm (u, v)(Am cos mϕ + Bm sin mϕ)
(7)
Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010
389
From the boundary conditions (4a) and (4b) for all values of (0 < ϕ < 2π) we can obtain:
φ = ψ(u, v) cos ϕ
Considering ψ(u, v) =
P
µ
(8)
Xµ (u) Yµ (v) combination with (8) and substituting in (5), we obtain
·
¸
∂
∂
1
1
cosh u
+
X = µX
cosh u ∂u
∂u cosh2 u
·
¸
1 ∂
∂
1
cos v
−
Y = −µY
cos v ∂v
∂v cos2 v
The Eq. (9b) can be rewritten as Legendre’s equation:
·
¸
¡
¢ 2
∂
1
2 ∂
1−ξ
Y = −µY
− 2ξ
−
∂2ξ
∂ξ 1 − ξ 2
(9a)
(9b)
(10)
in which ξ = sin v. Considering µ = l(l + 1), the solution is the first order of Legendre’s polynomial.
Therefore:
∞
X
ψ=
Xl (u)Pl1 (sin v)
(11)
l=1
Substituting (11) in (8) and comparing by (4b), we keep only first term, and consequently it becomes
ψ (u, v) = X1 (u) cos v.
(12)
As a result, the (9a) becomes:
·
¸
∂
∂
1
1
cosh u
+
X1 = 2X1
cosh u ∂u
∂u cosh2 u
(13)
The solutions of this differential equation are:
and
X1 = cosh u
(14a)
X1 = tanh u + cosh u tan−1 (sinh u)
(14b)
According to the boundary conditions of (4a) and (4b), (14b) is the acceptable solution and then
we obtain:
£
¤
ψ (u, v) = tanh u + cosh u tan−1 (sinh u) cos v
(15)
Substituting (15) in (8) gives the final solution as:
φ=−
¤
2aE0 £
tanh u + cosh u tan−1 (sinh u) cos v cos ϕ
π
(16)
It is perhaps most common to find the components of electric field in cylindrical coordinates. The
electric field in the cylindrical coordinate system can be expressed as
µ
¶
µ
¶
∂u
∂
∂v
∂
∂u
∂
∂v
∂
∂φ
~ = −∇φ
~ = ρ̂
E
+
φ + ẑ
+
φ + ϕ̂
(17)
∂ρ ∂u ∂ρ ∂v
∂z ∂u ∂z ∂v
∂ϕ
Assuming the Jacobian J(ρ, z) defined by


∂u ∂v
µ
¶
 ∂ρ ∂ρ 
a sinh(u) cos(v) −a cosh(u) sin(v)
1


¡
¢
J(ρ, z) ≡ 
=
(18)
a cosh(u) sin(v) a sinh(u) sin(v)
∂u ∂v  a2 cosh2 (u) − cos(v)
∂z ∂z
PIERS Proceedings, Xi’an, China, March 22–26, 2010
390
and Eqs. (16) and (17), the electric components in cylindrical coordinate system are given by:
¸
·
2E0 (cosh2 (u) + cos2 (v)) sinh(u)
−1
Eρ =
+
tan
(sinh(u))
cos ϕ
(19a)
π (cosh2 (u) − cos2 (v)) cosh2 (u)
·
¸
−2E0 sinh(u)
−1
Eϕ =
+ tan (sinh(u)) sin ϕ
(19b)
π
cosh2 (u)
·
¸
2E0
sin(2v) cos ϕ
Ez =
(19c)
π cosh(u)(cosh2 (u) − cos2 (v))
where
p
(a2 + ρ2 + z 2 )2 − aρ2 a2
cos v =
2
p 2a
2
2
2
a + ρ + z + (a2 + ρ2 + z 2 )2 − aρ2 a2
cos2 hu =
2a2
2
a2 + ρ2 + z 2 −
Notice that at ϕ = 0 and ϕ = π the ϕ-component of electric field is zero, Eϕ = 0. The other
¡ a ¢1/2
¡ a ¢1/2
0
0
components are Eρ ≈ 2E
and Ez ≈ 2E
at ρ = a and for points that are sufficiently
π
z
π
z
x
0
close to the edge of the disk by allowing z → 0 and ρ = a + 10−x a, they simplified to Eρ = 2E
π 10
and Ez = 0.
3. CONCLUSIONS
The curvilinear coordinates to solve the Laplace’s equation analytically offers many advantages in
solving practical engineering problems, which satisfies some special boundary conditions. We have
shown the electrostatic approximation is suitable for determining electric field and electrostatic
potential functions around the conductors subjected at microwave frequency. Solving Laplace’s
equation in curvilinear elliptical coordinate is a simple method for determining the electric field
analytically around a metal disk subjected within a source-free rectangular microwave cavity. Finally, we have derived the electric field components in the cylindrical coordinates that are most
convenient.
REFERENCES
1. Shayeganrad, G. and L. Mashhadi, “Theoretically and experimentally investigation of sparking
of metal objects inside a microwave oven,” PIERS Proceedings, 632–641, Beijing, China, March
2009.
2. De Vries, P. L., A First Course in Computational Physics, 4th Edition, John Wiley & Sons,
Inc., New York, 1994.
3. Chapra, S. C. and R. P. Canale, Numerical Method for Engineers: With Software and Programming Applications, Tata McGrow-Hill, 2002.
4. Harrington, R. F., Field Computation by Moment Methods, Wiley-IEEE Press, 1994.
5. Bellanca, G., P. Bassi, and G. Erbacci, “Optimized microwave oven design via fd-td simulation
on parallel computer,” Science and Supercomputing at CINECA, 2001.
6. Jackson, J. D., Classical Electrodynamics, 3rd Edition, John Wiley & Sons, Inc., New York,
1999.