Great Deal! Get Instant $10 FREE in Account on First Order + 10% Cashback on Every Order Order Now

Please solve the problems provided. Make sure to include all work, steps and details leading to the answer

1 answer below »
ECE 407 – ELECTROMAGNETIC COMPATIBILITY
ECE
Advanced Electromagnetic Fields and Waves I
1. A dipole antenna lies on the z-axis between −?/2 and ?/2 in free space.
Let’s consider a triangular distribution for the cu
ent over the dipole:
?(?) = ?0 (1 − |
?
?/2
|)
a) Derive the far field radiated by the dipole.
) Derive the radiation resistance and plot the radiation resistance as a function
of the length for 0 < ?/2 ≤ 1. The involved integral for this question should
e calculated numerically.
c) Assume that ?/2 ≪ 1 and determine an approximate formula for the
adiation resistance. Compare this to the result for a linear dipole antenna
from Chapter 2 of the notes. Compare the numerical value to that obtained in
part (b) for ? = λ/10.
2. Consider a Hertzian dipole at the origin of a cartesian system and oriented along
a generic direction �̂�.
(a) Derive the radiated field.
(b) Derive the equivalent cu
ents over the surface of a sphere centered with the
dipole and with a radius of 5λ.
3. Consider a sphere with center at the origin of a cartesian system and with radius 5
λ. The following electric (?) and ( �⃗⃗⃗�) magnetic cu
ent distributions are provided
over the surface of the sphere:
? = ?0 sin ? ?
�⃗⃗⃗� = ?0 ζ sin ? �̂�
With ?0 a constant and ζ the characteristic impedance in free space.
(a) (15 pts) Derive the field radiated by such a distribution in a generic point in
free space within and outside the sphere. Hint: Link these cu
ents to the point
(b) of Ex. 2 for a dipole along the z-axis.
(b) The far-field distance associated to such distribution.
Answered 5 days After Oct 21, 2024

Solution

Bhaumik answered on Oct 27 2024
3 Votes
ECE
Advanced Electromagnetic Fields and Waves I
Question 1:
Part (a): Deriving the Far Field Radiated by the Dipole
1. Understanding the Vector Potential: In the far-field approximation, the vector potential for a cu
ent element in free space is:
where:
· is the distance to the observation point in the far field.
· is the wave number.
· in the far field, which leads to an approximation for as
2. Substituting the Cu
ent Distribution: Substitute into the expression:
3. Evaluating the Integral: Since changes based on the sign of we split the integral into two parts over :
4. Calculating Each Integral: Each of these integrals involves exponential and linear terms in , which require integration by parts or substitution techniques. After simplification, the resulting expressions yield the vector potential in terms of , L, k, .
5. Determining The far-field electric field is related to by:
where is the -component of the vector potential derived from the above expression. The resulting far-field expression shows the dependency on and decreases as which is characteristic of far-field radiation.
Part (b): Deriving the Radiation Resistance
To find the radiation resistance we need the total power radiated by the dipole and related it to the cu
ent.
1. Radiated Power using the Poynting Vector: The power radiated by the antenna is obtained by integrating the time-averaged Poynting vector over a spherical surface at distance
2. Expression for Substitute the expression for derived in part (a). After simplification, can be expressed in terms of , L, and k, showing dependence on .
3. Integrating over Solid Angle: Using , the integration over and is performed. The integration over simply contributes a factor of 2 due to symmetry. The integral over involves , which can be evaluated to yield:
4. Calculating Radiation Resistance Radiation resistance is defined as:
Substituting from above:
which depends on L and...
SOLUTION.PDF

Answer To This Question Is Available To Download

Related Questions & Answers

More Questions »

Submit New Assignment

Copy and Paste Your Assignment Here