EAS487: Radar Remote Sensing Spring 2014

HOMEWORK 1: Math review

Reference: Lectures 1-7 Due: 2014 February 5

This problem set is intended as a review of somemathematics useful for solving antenna problems.The problems should be fairly simple but may re-quire you to review some basic concepts in complexvector operations. They will also require you to be-come proficient with programs that perform numeri-cal integration such as Mathcad, Matlab, Mathemat-ica, Macsyma, or Maple. If you are not familiar withany of these programs, now is the time to become so!

Problem 1. Suppose

E1 = E(jx+ 2y)ej(tkz)

E2 = E(2x ejpi/4y)ej(tkz)

Find |E1|2, |E2|

2, E1 E2, and the average power

density associated with E2 if this is the electric fieldof an electromagnetic wave propagating in vacuum inthe z direction. Also, sketch or plot [E2(t, z = 0)].

Problem 2. Suppose a function defined in spheri-cal coordinates by f(, ) has a peak in the directionof the z axis where its value is fmax. Suppose fur-ther that we want to approximate this function witha sector pattern defined by:

g(, ) = fmax, 0 < <

= 0, >

where g is further constrained to have the same av-erage value as f . We can think of f as representingthe radiation pattern of an antenna and g as an ap-proximation of the pattern. Find an expression forthe angle in terms of the ratio of the peak to av-erage values of f . For small values of , you canshow that 2fmax/f = 4. Next, find the solid angle associated with the given . Finally, relate tofmax/f.

Hint: be sure you know how to evaluate the aver-age of a function in spherical coordinates. Making asketch of the problem might help you to understandit better.

Problem 3. We will soon learn that the radia-tion pattern of a half-wave dipole antenna has thefunctional form:

f() =

[cos(pi2 cos )

sin

]2

where is the usual polar angle measured from thez axis, which is aligned with the dipole.

Suppose that the dipole is now reoriented so thatit is aligned with the x axis. Express the radiationpattern for this configuration in polar coordinates(, ). Repeat the process for a dipole aligned withthe y axis. Note that you cannot just exchange and, for example. Your results will be helpful whenwe consider arrays of dipole antennas that are notaligned with the z axis.

Hint: note that cos() is another way to write z rand that sin2() = 1 cos2(). You can thereforereplace the trigonometric terms in f with factors ofz r. Next, replace z with x and y, and rewrite theseproducts in terms of polar coordinates to find thedesired answers.

Problem 4. Evaluate pi0f() sin d numerically

where f() is the function supplied in problem 3.Hint: the answer is somewhere between 1 and 2. Thisproblem pertains to finding the total power radiatedby a half-wavelength dipole antenna.

Problem 5. Using Cartesian coordinates, inte-grate numerically the function exp(x2 y2) in twodimensions over the circular region in the first quad-rant defined by x2+y2 1, x 0, and y 0. Checkyour answer by performing the integral analytically.Hint: do the second part in polar rather than rect-angular coordinates.

CORNELL UNIVERSITY c DAVE HYSELL (14/1/14)HW 11