Read the following passage and answer the questions that follow : Antennas have become increasingly importance to the society and at present, they are indispensable. They are being used every places. They are available in vast varieties. They are operating at various frequencies which are depending on different application. They operate on the principle of Maxwell's equation. They have different types of radiation patterns. There are several atmospheric losses in the way of propagation of waves. Due to which signal fades down, when it travels from transmitter to receiver antennas. Based on the above para, answer the following questions :
The propagation constant for uniform plane wave is given by the expression given below :
\(\sqrt{\left(R+j\omega L\right)\left(G+j\omega C\right)}\)
A propagation constant is always the square root of a product, never of a ratio — which identifies option 2 immediately.
\(\gamma=\sqrt{\left(R+j\omega L\right)\left(G+j\omega C\right)}=\alpha+j\beta\)
Product against ratio — the distinction that settles every option. The two fundamental quantities of any propagating medium are formed from the same two parameters in different ways:
| Quantity | Form | Describes |
|---|---|---|
| Propagation constant γ | √(series × shunt) | How the wave attenuates and phase-shifts per metre |
| Characteristic impedance Z0 | √(series ÷ shunt) | The ratio of voltage to current in the wave |
Option 1 is the ratio form, so it is \(Z_{0}\), not \(\gamma\). Option 3 is also a ratio — \(\sqrt{j\omega\mu/(\sigma+j\omega\varepsilon)}\) is the intrinsic impedance of a medium, the field-theory counterpart of \(Z_{0}\). The corresponding propagation constant would be the product \(\sqrt{j\omega\mu\left(\sigma+j\omega\varepsilon\right)}\), which is not among the options. Option 4 is the reflection coefficient at a boundary between two media, and is dimensionless.
What the two parts of γ mean. Separating real and imaginary parts,
\(e^{-\gamma z}=e^{-\alpha z}e^{-j\beta z}\)
The real part \(\alpha\) is the attenuation constant in nepers per metre, describing how the amplitude decays; the imaginary part \(\beta\) is the phase constant in radians per metre, from which the wavelength and phase velocity follow:
\(\lambda=\dfrac{2\pi}{\beta},\qquad v_{p}=\dfrac{\omega}{\beta}\)
The lossless case shows the structure plainly: with R = G = 0,
\(\gamma=j\omega\sqrt{LC}\)
so \(\alpha=0\) — no attenuation — and \(\beta=\omega\sqrt{LC}\), giving a velocity independent of frequency and therefore no dispersion. That is the ideal a real line only approximates.
A memory aid worth keeping : \(\gamma\) and \(Z_{0}\) are built from the same two ingredients, one by multiplying and one by dividing — and their product recovers the series impedance while their quotient recovers the shunt admittance.
Hence, the propagation constant is √[(R + jωL)(G + jωC)].
Similar Questions
Consider the following statements
A. Reflection coefficient is change in the magnitude of reflected wave with constant phase with respect to incident wave.
B. Transmission coefficient is change in the magnitude and phase of transmitted wave with respect to incident wave
C. On smith chart, \(\dfrac{\lambda}{2}\) distance is equivalent to 2π.
D. For distortion less line, RL = GC
E. Directivity of an antenna can be less than 1 (unity).
Choose the most appropriate answer from the options given below :
In a loss less transmission line of length 50 cm with L = 10 μH/m, C = 40 pF/m is operated at 25 MHz. Its electrical path length is
The normalized impedance of a transmission line is given by expression :
A lossless line has a characteristic impedance of 50 ohms. It is terminated in a load resistance of 75 ohms. The line is energised by a generator which has an output impedance of 50 ohms and an output voltage of 30 V (rms). The line is assumed to be 2.25 wavelength long. The input impedance and instantaneous load voltages are given :
(a) Zin = 50 ohms (b) VL (instantaneous) = 36 V (c) Zin = 33.33 ohms (d) VL (instantaneous) = 12 V
Which of the above are correct :
Arrange the below mentioned transmission lines in order of their increasing frequency handling capabilities :
(a) wave guide (b) Parallel wire (c) Rigid co-axial cables (d) Flexible co-axial cables
For a quarter wave ideal transmission line of characteristic impedance of 50 Ω and load impedance of 100 Ω, the input impedance of line will be :
The disadvantage of co-axial cable is/are :
(A) Support higher bandwidth than twisted pair cable
(B) Light weight
(C) Relatively expensive compared to twisted pair cable
(D) EMI resistant
(E) Doesn't at very high frequencyChoose the most appropriate answer from the options given below :
Important Questions from Transmission Lines
A characteristic impedance does NOT satisfy which of the following statements?
The dielectric constant of the material used in a transmission line is 2. What is the velocity factor of this line if its characteristic impedance is 300 Ω?
A transmission line of \(50{\rm{\;\Omega }}\) characteristic impedance is terminated with a \(\rm 100 \ Ω\) resistance. The minimum impedance measured on the line is equal to
Twisting of live and return lines in long signal lines is done to reduce the effect of
The input impedance of short circuited lossless transmission line quarter wavelength is
