ANTENNA AND WAVE PROPAGATION – KEC-603
Section-wise Important Questions & Ready Answers
SECTION A
(Attempt all questions in brief – 2 marks each)
(a) Conversion of Point (0, −4, 3) into Cylindrical and Spherical Coordinates
For cylindrical coordinates:
ρ=x2+y2=4,ϕ=270∘,z=3\rho = \sqrt{x^2 + y^2} = 4,\quad \phi = 270^\circ,\quad z = 3ρ=x2+y2=4,ϕ=270∘,z=3
So the cylindrical coordinates are (4, 270°, 3).
For spherical coordinates:
r=x2+y2+z2=5,θ=cos−1(3/5),ϕ=270∘r = \sqrt{x^2+y^2+z^2} = 5,\quad \theta = \cos^{-1}(3/5),\quad \phi = 270^\circr=x2+y2+z2=5,θ=cos−1(3/5),ϕ=270∘
(b) Gradient of a Vector in Cylindrical Coordinate System
The gradient in cylindrical coordinates considers variations along radial, angular, and axial directions. It is expressed using scale factors and unit vectors aρ,aϕ,aza_\rho, a_\phi, a_zaρ,aϕ,az, enabling field analysis in circularly symmetric systems.
(c) Maxwell’s Equation for Electric Field
Gauss’s law for electricity states that the divergence of electric flux density equals the volume charge density:
∇⋅D=ρv\nabla \cdot \mathbf{D} = \rho_v∇⋅D=ρv
(d) Equation of Continuity and Its Application
The continuity equation expresses conservation of electric charge:
∇⋅J=−∂ρ∂t\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}∇⋅J=−∂t∂ρ
It is applied in antenna theory and wave propagation to ensure charge balance.
(e) Beam Area of an Antenna
Beam area is the solid angle subtended by the antenna radiation pattern:
ΩA=∫ ∫U(θ,ϕ)Umax dΩ\Omega_A = \int\!\!\int \frac{U(\theta,\phi)}{U_{max}} \, d\OmegaΩA=∫∫UmaxU(θ,ϕ)dΩ
(f) Gain and Directivity of an Antenna
Directivity measures how focused radiation is compared to an isotropic source. Gain includes antenna efficiency along with directivity.
(g) Radiation Resistance of Antenna
Radiation resistance is the equivalent resistance that accounts for power radiated by an antenna:
Rr=PradiatedI2R_r = \frac{P_{radiated}}{I^2}Rr=I2Pradiated
(h) Long Wire Antenna
A long wire antenna has a length much greater than wavelength and produces multiple lobes with high directivity.
(i) Maximum Range of Tropospheric Transmission
d=2ht+2hrd = \sqrt{2h_t} + \sqrt{2h_r}d=2ht+2hr
For ht=100h_t = 100ht=100 ft and hr=50h_r = 50hr=50 ft, the maximum range is obtained by substituting values in standard radio horizon formula.
(j) Maximum Electron Concentration of D and E Layers
Using:
fc=9Nmaxf_c = 9\sqrt{N_{max}}fc=9Nmax
Electron densities are calculated from given critical frequencies (2.5 MHz and 8.4 MHz).
SECTION B
(Attempt any three – 10 marks each)
2(a) Conversion of Vector from Cylindrical to Cartesian Coordinates
The given vector is transformed using:
aρ=cosϕ ax+sinϕ ay,aϕ=−sinϕ ax+cosϕ aya_\rho = \cos\phi\,a_x + \sin\phi\,a_y,\quad a_\phi = -\sin\phi\,a_x + \cos\phi\,a_yaρ=cosϕax+sinϕay,aϕ=−sinϕax+cosϕay
Final expression is written entirely in ax,ay,aza_x, a_y, a_zax,ay,az terms.
2(b) Boundary Conditions for Electric Field and Flux Density
At dielectric boundaries, tangential electric field remains continuous, while normal electric flux density changes based on surface charge density. These conditions are essential in wave transmission and reflection analysis.
2(c) Antenna with Field Pattern E(θ)=cos2θE(\theta)=\cos^2\thetaE(θ)=cos2θ
HPBW is found where power falls to half value.
FNBW is measured between first nulls.
Beam area is computed by integrating normalized power pattern.
A neat polar radiation pattern is drawn showing HPBW and FNBW.
2(d) Vertical Antenna and Folded Dipole Antenna
Vertical antennas radiate omnidirectionally in the horizontal plane and are widely used in broadcasting. Folded dipole antennas offer higher input impedance and broader bandwidth.
2(e) Critical Frequency, Multihop Propagation & Skip Distance
Critical frequency is the maximum frequency reflected by the ionosphere. Multihop propagation allows long-distance communication. Skip distance is the minimum distance from transmitter where sky wave returns to Earth.
SECTION C
3(a) Differential Length, Area and Volume Elements
In Cartesian coordinates, differential elements are simple products of dx, dy, dz.
In cylindrical coordinates, scale factors modify elements to ρdϕ\rho d\phiρdϕ, enabling integration over circular regions.
(Neat sketch expected in exam)
3(b) Divergence and Curl Evaluation
Divergence measures flux expansion, while curl measures field rotation. Calculations at point (1, −2, 3) are performed using standard vector calculus formulas and results are compared.
