Department of Electrical Engineering and Computer Science
EEL6482 Electromagnetic Theory
Homework/Project Set No. 2
Problem 7.
A line current I flows along an elliptical ring located in the xoy plane. If we put thumb in
the z direction, current will follow right-hand rule to flow. The elliptical equation is
𝑥2
𝑦2
given as 𝑎2 + 𝑏2 = 1 . Let a = 1, b = 0.75, 𝐼 = 107 A. Find magnetic flux density B on
yoz plane defined by [y,z] = meshgrid(-2:0.5:2, -2:0.5:2) and plot using quiver. Discuss
whatever physics you can find through investigation.
(Note: let 𝑥 ′ = acos(𝜙 ′ ) , 𝑦 ′ = 𝑏 sin(𝜙 ′ ) so that 𝐝𝐥′ = 𝑑𝑥 ′ 𝐮𝐱 + 𝑑𝑦′𝐮𝐲 )
Problem 8.
The following figure shows a simple linear actuator. By controlling the DC voltage
source, we can control the movement of the bar. Assuming the mass of the bar is M, the
DC magnetic field is B, the length of the bar is l, and the electrical loss of the system is
described using resistor R. The force from the load is 𝐅load.
(1) Before 𝑡 = 0,, the bar is stationary. There must be a current going through the bar to
get a force 𝐅ind = 𝐅load . Find the DC voltage 𝑉𝐵1 of this case.
(2) At 𝑡 = 0, apply 𝑉𝐵2 > 𝑉𝐵1 so that the bar will start to accelerate. Find the the
dynamical equation for transient velocity 𝑣(𝑡)and solve using MatLab function
dsolve.
(3) At 𝑡 = 𝑡′, apply 𝑉𝐵3 < 𝑉𝐵1 (can be negative) so that the bar will start to decelerate.
Find the the dynamical equation for transient velocity 𝑣(𝑡)and solve using MatLab
function dsolve.
(4) When the velocity decelerates to 0, suddenly apply 𝑉𝐵1so that the bar will be held at
a new position. Find the total actuation time from the old to new position. Also find
the stroke (distance from old to new position) by integrating the velocity.
(5) If M = 1 kg, B = 0.2 T, l = 10 m, 𝑉𝐵2 = 120V, 𝑉𝐵3 = −120𝑉, R = 2Ω, 𝐹load =
30 N, and stroke s = 1 m. Find 𝑉𝐵1, actuation time 𝑇𝑎𝑐𝑡 and 𝑡′. Plot velocity 𝑣(𝑡)
and current 𝑖(𝑡)versus time from 0 to 𝑇𝑎𝑐𝑡 .
(6) There is another way to drive the device. Instead of changing the DC voltage source,
we can change the magnetic field B. If this is your research project, discuss how you
are going to do.
Problem 9.
A two legged core with all the dimensions is shown in the following figure. The windings
on the left leg of the core has 600 turns, and the windings on the right has 200 turns. The
coils are wound in the directions shown in the figure. Neglect magnetic leakage.
Assume the relative permeability of the core is 1200 (constant).
(1) Find reluctance of each leg (left, right, top and bottom);
(2) Find the self inductances of these two windings and mutual inductance between them;
(3) Find the flux produced by currents i1  0.5A and i2  1A .
(4) If we do not assume a constant permeability for the core and the material will be M5
steel, find the flux produced by currents i1  0.5A and i2  1A . The BH relationship
of M5 steel (M5DC_B2H.m) was given in class. (Hint: You need to solve the
transcendental equation:




H(
)lleft  H (
)lright  H (
)ltop  H (
)lbottom  N1i1  N 2i2 to find  .
Aleft
Aright
Atop
Abottom
MatLab function fsolve can help to solve transcendental equation. )
Problem 10.
The time harmonic magnetic field inside a source-free perfect magnetic conducting
(PMC) rectangular waveguide is given by

H  H 0 sin( y )e  jkz z a x , 0  x  a, 0  y  b
b
Assume that the media inside the waveguide is free space, and four side walls at x  0, a
and y  0, b are perfect magnetic conductor. For a section of the waveguide of length d
along z axis from z  0 to z  d , find:
(1) time domain expression of electric fields;
(2) k z ;
(3) the exiting complex power;
(4) the supplied complex power;
(5) the dissipated real power;
(6) time average electric energy;
(7) time average magnetic energy;
(8) verify the conservation of energy equation for this set of fields.
Problem 11.
The phasor form expression for the electric field of a uniform plane wave propagating in
lossless isotropic medium is
E(x) = 60j)ay -(1-j)az]exp(j4x) V/m.
The magnetic field at point (0,0,0) when t = 0 is 2 A/m. The unit of t is second and the
unit of x is meter. The radian frequency is 3 10 8 rad/s. Find:
(1) the time domain expression of the electric field;
(2) the wave impedance of the media;
(3) the relative dielectric constant r and relative permeability r;
(4) the time domain expression for magnetic field;
(5) the time-average power density.
Problem 12.
The magnetic field intensity of a plane wave traveling in a lossy earth is given by
H  (a y  ja z ) H 0 e x e  jx
where H 0  1 μA/m. Assuming the lossy earth has a conductivity of 10 4 S/m, a
dielectric constant of 9, and the frequency of operation is 1 GHz, find inside the earth:
(1) the corresponding time domain electric field;
(2) the average power density;
(3) the phase constant (rad/m);
(4) the phase velocity (m/s);
(5) the wavelength;
(6) the attenuation constant (Np/m);
(7) the skin depth.
Problem 13.
For sea water, assuming  r  81,  r  1,   4S/m, and f  10 4 GHz, find:
(1) the complex propagation constant;
(2) the phase velocity (m/s);
(3) the wavelength;
(4) the attenuation constant (Np/m);
(5) the skin depth.