Download تکلیف سری اول درس آنتن1 موعد تحویل : 3/12/84

84/12/3 : ‫ موعد تحویل‬1‫تکلیف سری اول درس آنتن‬
1- Instead of using the exact expression for computing the directivity, it is often convenient to derive
simpler expressions, even if they are approximations. These are very important for Design purposes.
For antennas with one narrow major lobe and very negligible minor lobes, the beam solid angle is
approximately equal to the product of the half-power beam width in the two perpendicular planes
shown in figure 1.
(a) Nonsymmetrical pattern
(b) Symmetrical pattern
Figure 1 Beam solid angles for nonsymmetrical and symmetrical patterns
The directivity can then be approximated by
4
41253
D0 

(Kraus Equation)
1r  2 r 1d  2 d
The radiation intensity of the major lobe of many antennas can be adequately represented by
U  B0 cos 
Where B0 is the maximum radiation intensity. The intensity exists only in the
upper hemisphere 0     / 2, 0    2 
? Find the maximum directivity using the given approximations and compare it with its
exact value!
Another approximation has been published by Tai and Pereira in 1976:
1
22.181
72815
 2
(Tai and Pereira Equation)
2
2
1r   2 r 1d   22 d
Find the maximum directivity of the following power pattern
D0 
B0 cos n 
U  ,    
 0
0     / 2 , 0    2
elsewhere
for n = 5 and 20 in comparison to the Kraus equation and the exact solution!
2- Circular polarization is often generated by using two linearly polarized antennas. In this case, it is
important to know how accurate the two linear excitations need to be relative to each other. Consider
an x-polarized plane wave of unit amplitude and a y-polarized plane wave of amplitude jAe j , both
propagating in the positive z-direction.
? What sort of field will be generated with A = 1 and ΔΦ = 0°?
? Find the requirements to the maximum acceptable deviation of A (in dB) or ΔΦ (in deg) from this
values which give a crosspolarisation decoupling better than 25 dB.
? What is the axial ratio in this case?
3- A dipole antenna, with a total loss resistance of 1Ω, is connected to a generator whose internal
impedance is 50Ω+j25Ω. Assume that the peak voltage of the generator is 2V and the input
impedance of the dipole, excluding the loss resistance, is 73Ω+j42.5Ω.
? Find the power radiated by the antenna.
? Find the power dissipated by the antenna.
? What is the efficiency of the antenna?
4- Given is an idealized radiation pattern of an antenna:
100
U  ,    
 1
0     / 20 , 0    2
elsewhere
? Calculate the directivity of this antenna.
Given is a brightness temperature around the antenna with:
0   /2
5K
Tb    
 /2  
300 K
Calculate the antenna temperature of the antenna, when the antenna is pointing towards
  00
Give the G/T of the antenna for the loss less case.
2
Modify the antenna pattern in a form:
100

U  ,     1
0.01

0     / 20 , 0    2
 / 20     / 2 ,0    2
elsewhere
Recalculate the antenna temperature.
5- If F  Af r  where A is a constant vector and f is a function of r
(a) show:
df r 
  F  rˆ  A
dr
e  jkr
(b) In particular, if f r  
, show that for kr  1 (corresponding to far field)
r
  F   jkrˆ  F
(c) More generally, if A  A ,  and F  F r , ,    A
e  jkr
, show that in the far field
r
  F   jkrˆ  F
    F  k 2 rˆ  rˆ  F 
In other words, in far field the operator  can be simply replaced by  jkrˆ .
(d) show
 rˆ  rˆ  F   F  rˆF.rˆ
(e) From the results of part (c) and (d) interpret the      far field operator. What are the Eigen
vectors of this operator.
6- With the results of the previous exercise show that any finite (in space) electric current density , J(r)
e  jkr
, produces spherical TEM waves in the far field.( Spherical waves are proportional to
)
r
3