**7. Dipole directivity**

**Directivity** is defined as the ratio between the amounts of energy propagating in a certain direction compared to the average energy radiated to all directions over a sphere as written in Eqs. (10) and (11).

$$ID = \frac{P(.)\,\text{maximal}}{P(.)\,\text{average}} = 4\frac{P(.)\,\text{maximal}}{\,\text{Prad}}\tag{10}$$

**8. Antenna impedance**

**Figure 8.**

**Figure 9.**

**11**

*Photo of loop antennas.*

*Duality relationship between dipole and loop antennas.*

Antenna impedance determines the efficiency of transmitting and receiving

*For a Dipole*: *Rrad* <sup>¼</sup> <sup>80</sup>*π*<sup>2</sup>*<sup>l</sup>*

Loop antennas are compact, low-profile, and low-cost antennas. Loop antennas are employed in wearable wireless communication systems. The loop antenna is referred to as the dual of the dipole antenna, see **Figure 8**. A small dipole has magnetic current flowing (as opposed to electric current as in a regular dipole), the

loop antenna fields are similar to that of a small loop. The short dipole has a

2

(16)

*λ*2

energy in antennas. The dipole impedance is given in Eq. (16).

*Introductory Chapter: Novel Radio Frequency Antennas DOI: http://dx.doi.org/10.5772/intechopen.93142*

**8.1 Loop antennas for wireless communication systems**

*Rrad* <sup>¼</sup> <sup>2</sup>*WT I* 2 0

$$P(,) 
average = \frac{1}{4}P(,) \sinh d = \frac{Prad}{4} \tag{11}$$

The radiated power from a dipole is calculated by computing the pointing vector P as given in Eq. (12).

$$\begin{aligned} P &= 0.5(E \times H) = \frac{15\pi I\_0^2 l^2 \sin^2 \theta}{r^2 \lambda^2} \\\ W\_T &= \int\_s P \cdot \mathbf{ds} = \frac{15\pi I\_0^2 l^2}{\lambda^2} \int\_0^\pi \sin^3 \theta d\theta \int\_0^{2\pi} d\rho = \frac{40\pi^2 I\_0^2 l^2}{\lambda^2} \end{aligned} \tag{12}$$

The overall radiated energy is WT. WT is computed as written in Eq. (12), by integration of P over an imaginary sphere surrounding the dipole. The power flow of an isotropic element equal to the overall radiated energy divided by the area of the sphere, 4*πr*2, see Eq. (13). The dipole directivity at θ = 90° is 1.5 as given in Eq. (14). For small antennas or for antennas without losses, D = G, losses are negligible. For a given θ and φ for small antennas, the approximate directivity is given by Eq. (15).

$$\oint\_{\gamma} \mathbf{ds} = r^2 \int\_0^{\pi} \sin \theta d\theta \Big|\_0^{2\pi} d\phi = 4\pi r^2$$

$$P\_{\text{iso}} = \frac{W\_T}{4\pi r^2} = \frac{10\pi I\_0^2 \mu^2}{r^2 \lambda^2} \tag{13}$$

$$D = \frac{P}{P\_{\text{iso}}} = 1.5 \sin^2 \theta$$

$$G\_{\rm dB} = 10 \log\_{10} G = 10 \log\_{10} 1.5 = 1.76 \text{dB} \tag{14}$$

$$D = \frac{\text{41253}}{\theta\_{\text{3dB}} \phi\_{\text{3dB}}} \tag{15}$$

$$G = \xi D \quad \xi = \mathbf{E} \\ \text{ffficiency}$$

Antenna losses degrade the antenna efficiency. Antenna losses consist of conductor loss, dielectric loss, radiation loss, and mismatch losses. For resonant small antennas, *ξ* ¼1. For reflector and horn antennas, the efficiency varies between *ξ* ¼ 0.5 and.

*ξ* ¼ 0.7. The beam width of a small dipole, 0.1 λ long, is around 90°. The 0.1 λ dipole impedance is around 2 Ω. The beam width of a dipole, 0.5 λ long, is around 80°. The impedance of 0.5 λ dipole is around 73 Ω.

*Introductory Chapter: Novel Radio Frequency Antennas DOI: http://dx.doi.org/10.5772/intechopen.93142*
