Relation between radiation resistance and efficiency formula
Answers
Radiation efficiency is another important parameter to describe how efficiently an antenna transmits and receives RF signals, which is defined as the ratio of the total power radiated by an antenna to the total input power received from the generator. An antenna with high radiation efficiency efficiently radiates the input power to free space. In the case of low radiation efficiency, the input power is mostly dissipated because of the internal losses such as metal conduction, dielectric and magnetic losses within the antenna. As presented in the last section, the power loss of the ferrite antenna is closely related to the loss factors of tan δɛ and tan δμ of the ferrite. Other contributions to antenna loss include input impedance mismatch and polarization loss between transmitting and receiving antennas. However, these degradation factors can be minimized by using impedance matching circuitry and proper positioning of the antennas.
The basic operation of a transmitting antenna is illustrated in Fig. 8.13. A generator with internal impedance Zg is connected to the transmitting antenna with impedance ZA
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Figure 8.13. Transmitting antenna and its equivalent circuit. The radiated waves are in the form of spherical wave in near-field region and become a plane wave in far-field region.
(8.129)Zg=Rg+jXg,
(8.130)ZA=RA+jXA=Rr+RL+jXA.
Rg and Xg are the resistance and reactance of generator impedance, respectively, RA is the antenna resistance being the sum of radiation resistance Rr and loss resistance RL of the antenna, and XA is the antenna reactance. The generator transmits the input power to the antenna for electromagnetic radiation. The radiated power Pr for the antenna can be written as
(8.131)Pr=Ig2Rr,
where
(8.132)Ig=VgZg+ZA.
Vg is the peak generator voltage and Ig is the generated current in the equivalent circuit. Since the antenna has internal losses, some of the input power delivered to the antenna is dissipated as heat. The power loss PL is given by
(8.133)PL=Ig2RL.
Therefore, the total input power of the antenna is then obtained as
(8.134)PT=Pr+PL=Ig2Rr+Ig2RL.
The radiation efficiency η is defined by the ratio of the power radiated over the input power
(8.135)η=PrPT=PrPr+PL.
Applying Eqs. (8.131) and (8.133) to Eq. (8.135) leads to the following equation of the radiation efficiency
(8.136)η=Ig2RrIg2Rr+Ig2RL=RrRr+RL=11+RL/Rr.
This demonstrates that high radiation efficiency can be achieved by decreasing the loss resistance and increasing the radiation resistance.
The radiation efficiency η can also be expressed in terms of quality factor Q by Eq. (8.137) [65]
(8.137)η=RrRr+RL=1/Qr1/QT=QTQr,
where QT is the total quality factor of antenna and Qr is the radiated quality factor (= 1/Rr). The total quality factor QT describes the total antenna losses including radiation loss, conduction loss, dielectric and magnetic losses and is given by [61,62]
(8.138)1QT=1Qr+1Qc+1Qμ+1Qɛ,1QT=1Qr+1Qc+tanδμ+tanδɛ,
where Qμ is
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