Download ECE 6341 HW 4 Solutions: Leaky-Wave Antennas and Magnetic Currents and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!
ECE 6341
Spring 2009
HW 4
- Consider an infinite line source of current having the form
I ( z ) = exp ( − jk z 0 z),
which is flowing on the z axis in an infinite medium with wavenumber k. By matching the magnetic vector potential between cylindrical coordinates and rectangular coordinates (the latter solution was done in ECE 6340), derive the following mathematical identity:
' (^) (2) ' 0 0 z z
jkR e (^) e jk z (^) dz j H k e jk z R^ ρ
∞ (^) − − − −∞
where ( )
2 2
R = ρ + z − z' and ( )
2 2 1/ 2 k (^) ρ 0 = k − kz 0.
- Consider a slotted rectangular waveguide leaky-wave antenna, as shown below, which has an infinite narrow slot. Assume that the structure is fed with a source at z = 0 and a bi- directional leaky wave propagates in both directions from this point. Therefore, the slot has a voltage given by
jk LWz z V z = A e− ,
where k zLW= β − jα. Use the equivalence principle and image theory to show that the field
above the infinite conducting baffle is that of a magnetic current flowing on the z axis in free space,
jkz LWz K z = A e−.
V(z) + -
a
b
baffle
x
y
source
- Consider a leaky-wave magnetic current flowing on the z axis in free space that has the form
jkz LWz K z = A e− ,
where k zLW= β − jα. Using duality along with Fourier transform theory, as was done in class
for an arbitrary line current I (z), derive an expression for the electric vector potential
Fz ( ρ ,z)at any point in space (your answer will be in the form of an integral).
- Derive expressions for all of the field components for the previous leaky-wave antenna. This includes Ez, Hz, E ρ, H ρ, E φ, H φ. (Note: Some of these components may be zero.)
5. Find the far field components E θ (r, θ, φ) and E φ (r, θ, φ) of the leaky-wave current in the
previous problem. Do this by first finding the electric vector potential Fz in the far field by using the “far-field identity” that was discussed in class, namely
( ) (^2 )^ ( ) z ~ 2 1 ( cos )
jkr jk z (^) n z n z
e f k H k e dk j f k ρ (^) r ρ θ
+∞ − − (^) + −∞
This is valid as r → ∞ in spherical coordinates, with θ ≠ 0, 180D^.
- A circular loop of uniform current I 0 is in free space as shown below. The loop has a radius b and it lies in the xy plane. Because the current is uniform, the field produced will be purely TEz. The field inside and outside the loop may be written as
( ) (^ )^ ( )
1 0
2 2 0
z
z
jk z z z z
jk z z z z
F A k J k e dk b
F B k H k e dk b
ρ
ρ
ρ ρ π
ρ ρ π
∞ − −∞ ∞ − −∞
where
2 2 1/ k (^) ρ = k 0 − kz.
Solve for the unknown coefficient functions A(kz) and B(kz) by applying boundary conditions
at ρ = a. These include the condition that the field E φ should be continuous at this boundary,
and the condition that the field Hz should be discontinuous. Note that the ring of current can
be thought of as a surface current at ρ = a, having the form
J s= φˆ I 0 δ ( z).
( ) ( ) ( )^ ( ) 1
V 2 = jω M I 0 = 2 π b E φ h b,
where V 2 is the open-circuit voltage induced at the terminals of coil 2 (assuming that we introduce a terminal pair at some point on loop 2), I 0 is the current on the transmitter coil 1,
and E φ ( )^1 ( h b, ) is the electric field produced by coil 1 (radiating in the presence of the pipe)
at z = h and ρ = b.
Derive a formula for the mutual inductance M between the two coils, using your solution
form the previous problem to determine the field E φ (^1 )^ ( h b, ).
I 0
a
b
x
y
z
