## Electromagnetics and Transmission Lines - Jun 2015

### Electronics & Telecom Engineering (Semester 5)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.

### Answer any one question from Q1 and Q2

**1 (a)** Derive the expression for electric field intensity E at a point 'P' due to infinite line charge with uniform line charge density 'Ρ_{L}'.(6 marks)
**1 (b)** Derive Laplace and Poisson equations for electronics & hence state physical significance of Laplace & Poisson equations.(6 marks)
**1 (c)** A current sheet k = 9a_{y} A/m is locate at z=0. The region 1 which is at z<0 has μ_{r1}=4 and region 2 which is at z>0 has μ_{r2}=3.

Given H_{2} =14.5a_{x} + 8a_{z} A/m Find H_{1}.(8 marks)
**2 (a)** Derive the expression for the capacitance of spherical plate capacitor.(6 marks)
**2 (b)** Derive expression for Biot & Savart law using magnetic vector potential.(6 marks)
**2 (c)** $$ \overline D = \dfrac {5x^3} {2} \widehat a x \ c/m^2 . $$ Prove divergence theorem for a volume of cube of side 1m. Centered at origin & edges parallel to the axis.(8 marks)

### Answer any one question from Q3 and Q4

**3 (a)** Define displacement current and displacement current density & hence show that $$ \nabla \times H = J_c + J_d \\
\begin {align*} where & J_c \rightarrow \ conduction \ current \ density \\ &J_d \rightarrow \ Displacement \ current \ density \end{align*} $$(8 marks)
**3 (b)** Select values of K such that each of the following pairs of fields satisfies Maxwell's equation. $$
i) \ \overline E = (Kx - 100t) \overline a_y \ V /m \\ \ \ \overline H = (x+20t)\overline a_z \ A/m \\ \ \ \mu=0.25 H/m \ \varepsilon=0.01F /m \\ \\ ii) \overline D = 5x \widehat a_x - 2 y \widehat a_y + Kz \widehat a _z \ \mu c/ m^2 \\ \ \ \overline B = 2 \overline a_y \ mT \\ \ \ \mu = \mu_0 \ \varepsilon= \varepsilon_0 $$(8 marks)
**4 (a)** What is mean by uniform plane wave, obtain the wave equation travelling in free space in terms of E.(8 marks)
**4 (b)** Derive Maxwell's equations in differential and integral form for time varying and free space.(8 marks)

### Answer any one question from Q5 and Q6

**5 (a)** Derive the expression for characteristic impedance (Z_{0} ) and propagation constant (r) in terms of primary constants of transmission line.(8 marks)
**5 (b)** A cable has an attenuation of 3.5dB/Km and a phase constant of 0.28 rad/km. If 3V is applied to the sending end then what will be the voltage at point 10 km down the line when line is terminated with Z_{0}.(8 marks)
**6 (a)** Explain the phenomenon of reflection of transmission line and hence define reflection coefficient.(6 marks)
**6 (b)** A transmission line cable has following primary constants.

R=11 Ω / km, G=0.8 μ℧ / km

L=0.00367 H/Km C=8.35 nF/km

At a signal of 1 kHz calculate

i) Characteristic impedance Z_{0}

ii) Attenuation constant (α) in Np/Km

iii) Phase constant (β) in radians / Km

iv) Wavelength (λ) in Km

v) Velocity of signal in Km/sec.(10 marks)

### Answer any one question from Q7 and Q8

**7 (a)** What is the impedance matching? Explain necessity of it, what is stub matching? Explain the single stub matching with its merits and demerits.(9 marks)
**7 (b)** Explain standing wave and why they generate? Derive the relation between the SWR and magnitude of reflection coefficient?(9 marks)
**8 (a)** What do you mean by distortionless line. Derive expression for characteristic impedance and propagation constant for distortionless line.(8 marks)
**8 (b)** The VSWR on a lossless line is found to be '5' and successive voltage minima are 40 cm a part. The first voltage minima is observed to be 15 cm from load. The length of a line is 160cm and characteristic impedance is 300 Ω Using Smith chart find load impedance, sending end impedance.(10 marks)