# Chapter 7:Antennas¶

## Example 1,Page No:332¶

In :
import math

# Variable Declaration
Ldipole    = 50;         # Length of dipole in cm
c          = 3*10**10;   # velocity of EM wave in cm/s
BW         = 10*10**6;   # bandwidth in Hz

# Calculations
lamda      = 2*Ldipole;     # wavelength in cm
fo         = c/lamda;       # operating frequency in Hz
Q          = fo/BW;          # quality factor

# Result
print'Q = %d'%Q;

Q = 30


## Example 2,Page No:332¶

In :
import math

# Variable Declaration
Rr      = 72;       # Radiation resistance in ohms
Rl      = 8;        # Loss resistance in ohms
Ap      = 27;       # power gain

# Calculations
n       = Rr/float((Rr + Rl));     # radiation efficiency
D       = Ap/float(n);             # Directivity
D_dB    = 10*math.log10(D);      # directivity in dB

# Result
print'Directivity = %3.2f dB'%D_dB;

Directivity = 14.77 dB


## Example 3,Page No:333¶

In :
import math

# Variable Declaration
AZ_BW      = 0.5;      # beamwidth in degrees
E_BW       = 0.5;      # beamwidth in degrees
lamda      = 3*10**-2; # radar emission wavelength

# Calculations

AZ_BW_r    = AZ_BW*math.pi/float(180);     # azimuth beamwidth in radians
E_BW_r     = E_BW*math.pi/float(180);      # elevation beamwidth in radians
G          = (4*math.pi)/float((AZ_BW_r *E_BW_r ))     # antenna gain
G_db       = 10*math.log10(G);        # gain in dB
A          = (G*lamda*lamda)/float((4*math.pi));   # antenna aperture

# Output
print'Gain of Antenna = %3.2f dB\n'%G_db,'Antenna Aperture = %3.3f m'%A;

Gain of Antenna = 52.18 dB
Antenna Aperture = 11.818 m


## Example 4,Page No:333¶

In :
import math

# Variable Declaration
n_az      = 0.5;        #length efficiency in azimuth direction
n_el      = 0.7;        #length efficiency in elevation direction
A         = 10;         # area in square mts

# Calculations
n         = n_az * n_el;    # aperture efficiency
Ae        = n*A;            # Effective aperture

# Output
print'Effective aperture of the antenna = %3.1f sq.m'%Ae;

Effective aperture of the antenna = 3.5 sq.m


## Example 5,Page No:333¶

In :
import math

# Variable Declaration
Ptot        = 100;      # certain antenna radiating power
Ptot_iso    = 10*10**3;  # isotropic antenna radiating power

# Calculations
D           = 10*math.log10(Ptot_iso/Ptot);  # Directivity of antenna

# Output
print'Directivity of antenna = %d dB'%D;

Directivity of antenna = 20 dB


## Example 6,Page No:334¶

In :
import math

# Variable Declaration
D       = 3;        # diameter of the antenna in m
nl      = 0.7;      # length efficiency
nr      = 0.9;      # radiation efficiency
f       = 10*10**9;  # antenna operating freq.
c       = 3*10**8;   # vel of EM waves in m/s

# calculations
de     = D*(nl)  # Effective diameter
lamda   = c/float(f);      # wavelength in m
Beam_w  = lamda/de # beamwidth in radian
Beam_w_d= Beam_w*180/math.pi;       # beam width in degree;
n_a     = nl * nl;    # Aperture efficiency
AA      = (math.pi*D*D)/4;  # actual area in sq m
Ae      = AA*n_a;       # Effective aperture
G       = (4*math.pi*Ae)/float((lamda**2)); # Gain
G_db    = 10*math.log10(G);

# Output
print'Beam Width = %3.2f degrees\n '%Beam_w_d;
print'Effective Aperture = %3.2fsq m\n'%Ae;
print'Gain = %3.1f dB'%G_db;

Beam Width = 0.82 degrees

Effective Aperture = 3.46sq m

Gain = 46.8 dB


## Example 7,Page No:334¶

In :
import math
# given data
# given (lamda/10) wire dipole
# Radiation resistance of short dipoles is Rr  = 790*(1/lamda)**2;
# Rr   = 790*(lamda/(10*lamda))**2;
# Rr   = 7.9;

Radiation resistance = 7.9 ohms


## Example 8,Page No:334¶

In :
import math

# Variable Declaration
a_l     = 6;        # Azimuth length in m
n_a     = 0.7;      # Azimuth length efficiency
n_e     = 0.5;      # elevation length efficiency
e_l     = 4;        # elevation length in m
w       = 6;        # width of antenna
h       = 4;        # height of antenna
lamda   = 3*10**-2;  # wavelength

# Calculations
Eff_A_l = a_l*n_a;  # effective azimuth length
Eff_E_l = e_l*n_e;  # effective elevation length
A       = w*h       # actual area
n       = n_a*n_e;  # aperture efficiency
Ae      = A*n;      # effective aperture
Az_BW   = lamda/float(Eff_A_l) # Azimuth beam width
E_BW    = lamda/float(Eff_E_l)  # elevation beam width
Az_BW_d = Az_BW*180/float(math.pi)  # rad to deg conv
E_BW_d  = E_BW*180/float(math.pi);  # rad to deg conv
G       = (4*math.pi*Ae)/float((lamda**2)); #Gain
G_dB    = 10*math.log10(G);  # gain in dB

# Result
print'Azimuth Beamwidth = %3.2f degrees'%Az_BW_d;
print'Elevation Beamwidth = %3.2f degrees'%E_BW_d;
print'Gain = %3.1f dB'%G_dB;

Azimuth Beamwidth = 0.41 degrees
Elevation Beamwidth = 0.86 degrees
Gain = 50.7 dB


## Example 9,Page No:335¶

In :
import math
# given data
Beam_w_3db  = 0.4;

# Calculations
N2N_Beam_w  = 2*Beam_w_3db; # Null to Null beamwidth

# output
print'Null to Null Beam width = degrees',N2N_Beam_w;

Null to Null Beam width = degrees 0.8


## Example 10,Page No:335¶

In :
import math

# given data
RSSR        = 20;   # Rx signal strength in horizontal polarised antenna when rx RHCP

# Calculations
# When incident polarisation is circularly polarised and the antenna is linearly polarised,there is a ploarisation loss of 3dB
# a
# when the Rx polarisation is same as the antenna polarisation , the polarisation loss is zero
RSS_HP      = ISS;      # rx signal strength for incident wave horizontally polarised
# b
# when the incident wave is vertically polarised ,the angle between the incident polarisation and the antenna polarisation is 90
# polarisation loss = 20log(1/cos( φ))
#                   = 20log(1/cos90) = ∞
RSS_VP      = 0;        # rx signal strength for incident wave vertically polarised
# c
# When the incident wave is LHCP and the antenna polarisation is linear ,there will be a 3dB polarisation loss and the
# Rx signal strength therefore will be 20 dB only
RSS_LHCP    = RSSR;      # rx signal strength for incident wave Left hand circularly polarised
# d
# The angle between the incident wave polarisation and the antenna polarisation is 60 degrees
phi         = 60;                               # rx wave polarisation angle with horizontal
PL          = 20*math.log10(1/float(math.cos(60*math.pi/float(180))));      # polarisation loss in dB
# Result
print'Received signal strength if incident wave Left hand circularly polarised is %d dB'%RSS_LHCP;

Received signal strength if incident wave horizontally polarised = 23 dB
Received signal strength if incident wave vertically polarised = 0 dB
Received signal strength if incident wave Left hand circularly polarised is 20 dB
Received signal strength if Received wave polarisation making 60deg angle with horizontal is  17 dB


## Example 11,Page No:337¶

In :
import math

# Variable Declaration
f       = 300*10**6;     # operating frequency in Hz
c       = 3*10**10;      # velocity of EM wave in cm/s

# Calculations
lamda   = c/float(f);          # wavelength in cm
# Physical length of antenna is made 5% shorter than desired length as per rule of thumb
l       = lamda/float(2);      # length of halfwave dipole
lphy    = l-(5/float(100))*l;  # as per rule of thumb

# Output
print'Length of a half wave dipole to be cut = %3.1f cm'%lphy;

Length of a half wave dipole to be cut = 47.5 cm


## Example 12,Page No:342¶

In :
import math

# Variable Declaration
Zi      = 72;       # input impedance in ohms
# A    = 1.5a      # area of cross section in sq.cm
# Zif  = Zi*[(sum of areas of cross section of various components)/(Area of cross section of the driven element )]**2;
# Zif  = 72*((a + 1.5a)/a)**2;
# Zif  = 72*(2.5*a/a)**2;
Zif     = 72*(2.5)**2;

print'Input impedance for a folded dipole = %d Ω'%Zif;

Input impedance for a folded dipole = 450 Ω


## Example 13,Page No:342¶

In :
import math
# given data
f       = 60*10**6;      # frequency in Hz
c       = 3*10**8;      # velocity of EM wave in m/s

# Calculations
lamda   = c/float(f);          # wavelength in m
l_dipole= lamda/float(2);      # length of diplole
# Physical length of antenna is made 5% shorter than desired length as per rule of thumb
L       = l_dipole - (5/float(100))*l_dipole;  # actual physical length
L_D     = L - (4/float(100))*L;                # length of director
L_R     = L + (4/float(100))*L;                # length of reflector
DDS     = 0.12*lamda;                   # director dipole spacing
RDS     = 0.2*lamda;                    # Reflector dipole spacing

# Output
print'Length of dipole = %3.3f m'%L;
print'length of Director = %3.2f m'%L_D;
print'length of Reflector = %3.2f m'%L_R;
print'director dipole spacing = %3.1f m'%DDS;
print'Reflector dipole spacing = %3.1f m'%RDS;

Length of dipole = 2.375 m
length of Director = 2.28 m
length of Reflector = 2.47 m
director dipole spacing = 0.6 m
Reflector dipole spacing = 1.0 m


## Example 14,Page No:352¶

In :
import math
# given data
D       = 2;        # Mouth diameter in m
f       = 2;        # focal length in m
bw3db   = 90/float(100);   # beamwidth of antenna chosen to be 90% of angle subtended by feed

# Calculations
theta   = 4*math.atan(1/float((4*f/float(D))));    # angle subtended by the focal point feed at edges of reflector
theta_d = theta*180/float(math.pi);
Beam_w_3dB = bw3db*theta_d;       # 3 dB beam width
NNBW    = 2*(Beam_w_3dB );

# Output
print '3 dB Beamwidth = %3.1f°'%Beam_w_3dB,' Null-to-Null beam width = %3.2f°'%(NNBW);

3 dB Beamwidth = 50.5°  Null-to-Null beam width = 101.06°


## Example 15,Page No:352¶

In :
import math

# Variable DECLARATION
f       = 3;        # focal length in m
fpos    = 1.5;      # feed is placed 1.5m from pt of intersection os sec.reflector and antenna axis

# Calculation
f_hyp   = f-fpos;   # focal length of hyperboloid from figure;

# Result
print'focal length of hyperboloid = %3.1f m'%f_hyp;

focal length of hyperboloid = 1.5 m


## Example 16,Page No:353¶

In :
import math

# Variable Declaration
D       = 3;        # Mouth diameter in m
#f       = 2;      # focal length in m
bw3db   = 63;       # 3dB beam width
k       = 0.9;      # beam width is k times subtended angle

# Calculations
theta   = bw3db/k;  # subtended angle
theta_r = theta
#theta   = 4*atan(1/(4*f/D));
f       = D/(4*math.tan((theta_r/4)*(math.pi/180)));

# Result
print'Distance of feed from the point of intersection of antenna axis and the reflector surface = %3.2f m'%f;

Distance of feed from the point of intersection of antenna axis and the reflector surface = 2.38 m


## Example 17,Page No:365¶

In :
import math

# Variable Declaration
c       = 3*10**8;       # velocity of EM waves in m/s
f       = 2.5*10**9;     # operating frequency in Ghz
S       = 10*10**-2;     # inter element spacing
theta   = 10;            # steering angle

# Calculations
lamda   = c/f          # Wavelength in m
phi     = (360*(S/lamda))*math.sin(theta*(math.pi/180))
phi1    = 0*phi        # phase angle for element 1
phi2    = 1*phi        # phase angle for element 2
phi3    = 2*phi        # phase angle for element 3
phi4    = 3*phi        # phase angle for element 4
phi5    = 4*phi        # phase angle for element 5

# Result
print'Phase angles for elements 1,2,3,4,5 are %d°'%phi1, '%d°'%phi2,'%d°' %phi3,'%d°' %phi4,'%d°' %phi5

Phase angles for elements 1,2,3,4,5 are 0° 52° 104° 156° 208°


## Example 18,Page No:365¶

In :
import math
# Data is taken from Example 17. The beam steers towards left of the axis with all parameters remaining in Ex 17 are same
c       = 3*10**8;       # velocity of EM waves in m/s
f       = 2.5*10**9;     # operating frequency in Ghz
S       = 10*10**-2;     # inter element spacing
theta   = -10;          # steering angle

# Calculations
lamda   = c/f          # Wavelength in m
phi     = (360*S/lamda)*math.sin(theta*math.pi/180)
phi1    = 0*phi        # phase angle for element 1
phi2    = 1*phi        # phase angle for element 2
phi3    = 2*phi        # phase angle for element 3
phi4    = 3*phi        # phase angle for element 4
phi5    = 4*phi        # phase angle for element 5

# Output
print'Phase angles for elements 1,2,3,4,5 are %d°'%phi1, '%d°'%phi2,'%d°' %phi3,'%d°' %phi4,'%d°' %phi5

Phase angles for elements 1,2,3,4,5 are 0° -52° -104° -156° -208°


## Example 19,Page No:365¶

In :
import math
# given data
S       = 5*10**-2;      # inter spacing distance
lamda   = 6*10**-2;      # operating wavelength in cms
phi_Az   = 25            # angle in azimuth direction
phi_E    = 35            # angle in Elevation direction

# Calculations
theta_Az  = math.asin((lamda*phi_Az)/(360*S))
theta_E   = math.asin((lamda*phi_E)/(360*S))
Theta_Az  = theta_Az*(180/math.pi)
Theta_E   = theta_E*(180/math.pi)

# Output
print'Steering angle in Azimuth = %3.1f°'%Theta_Az
print 'Steering angle in Elevation = %3.1f°'%Theta_E;

Steering angle in Azimuth = 4.8°
Steering angle in Elevation = 6.7°