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 :
Radiation resistance for a small square loop (single turn of side length 'l') of the size λ/10 is approximately :
3.12 ohms
A small loop's radiation resistance depends on the square of its area measured in wavelengths:
\(R_{r}=31171\left(\dfrac{A}{\lambda^{2}}\right)^{2}\ \Omega\)
Step 1 — the area in wavelengths. For a square of side \(l=\lambda/10\):
\(A=l^{2}=\dfrac{\lambda^{2}}{100}\quad\Rightarrow\quad \dfrac{A}{\lambda^{2}}=0.01\)
Step 2 — substitute.
\(R_{r}=31171\times\left(0.01\right)^{2}=31171\times10^{-4}=3.12\ \Omega\)
— option 2.
Option 3 is the deliberate trap. 73 Ω is the radiation resistance of a half-wave dipole, a figure so familiar that it is easy to reach for. But it belongs to a resonant, half-wavelength structure, whereas this loop is a tenth of a wavelength on a side — electrically tiny. Option 4's 273 Ω is another standard figure, that of a folded dipole with its impedance stepped up, and is equally irrelevant here.
Why such a small resistance is a serious problem. The loop's conductor also has an ohmic loss resistance \(R_{L}\), and the radiation efficiency is
\(\eta=\dfrac{R_{r}}{R_{r}+R_{L}}\)
With \(R_{r}\) only a few ohms, a loss resistance of the same order halves the efficiency — and for a loop of \(\lambda/100\) the radiation resistance falls to milliohms and almost all the input power is dissipated as heat. This is the fundamental difficulty of every electrically small antenna.
Note the fourth-power dependence on size. Since \(R_{r}\propto A^{2}\propto l^{4}\), halving the side reduces the radiation resistance by sixteen times. Two remedies follow directly: use N turns, which multiplies \(R_{r}\) by \(N^{2}\), or fill the loop with ferrite to raise the effective permeability — which is exactly what the ferrite rod aerial in a portable radio does.
Hence, the radiation resistance is approximately 3.12 Ω.
