**4. Uncertainty evaluation of antenna factor** *F***<sup>a</sup> for biconical antennas with standard site method**

The setup of the standard site method (SSM) is shown in **Figure 13**. The Rx is swept from 1 m to 4 m in height to get the minimum SIL (and this is defined as site attenuation-SA). The horizontal separation *R* between the center (reference locations) of both antennas is 10 m.

After measuring three SAs for three pairs of antennas (#1 vs. #2, #1 vs. #3, and #2 vs. #3), respectively, the free-space antenna factor *Fa*<sup>1</sup> of antenna #1 can be calculated with Eq. (2) [5].

$$F\_{a\_1} = 10 \lg f\_{\text{M}} - 24.46 + \frac{1}{2} \left[ E\_{\text{D}}^{\text{max}} + A\_{12} + A\_{13} - A\_{23} \right] + \Delta F\_{\text{DevFree}-\text{space}} \tag{2}$$

Where *f* <sup>M</sup> is the frequency in MHz; *Ai*,*<sup>j</sup>* is the site attenuation between antenna # *<sup>i</sup>* and *<sup>j</sup>*, ð Þ *<sup>i</sup>* <sup>¼</sup> 1, 2, 3; *<sup>j</sup>* <sup>¼</sup> 1, 2, 3; *<sup>i</sup>* 6¼ *<sup>j</sup>* . *<sup>E</sup>*max <sup>D</sup> can be calculated according to Eq. (3),

**Figure 13.** *The setup of the SSM.*

$$E\_{\rm D}^{\rm max} = 20 \lg \left\{ \frac{\sqrt{49.2} \left[ d\_1^2 + d\_2^2 + 2d\_1 d\_2 \cos \left[ \beta (d\_2 - d\_1) \right] \right]^{1/2}}{d\_1 d\_2} \Bigg|\_{1 \le h\_1 \le 4} \right\}. \tag{3}$$

Where

$$d\_1 = \left[\mathbf{R}^2 + \left(h\_1 - h\_2\right)^2\right]^{1/2},\tag{4}$$

$$d\_2 = \left[ R^2 + \left( h\_1 + h\_2 \right)^2 \right]^{1/2},\tag{5}$$

$$
\beta = \frac{2\pi f\_{\text{M}}}{300}.\tag{6}
$$

where *ΔFDevFree*�*space* is the correction factor, as shown in Table G.1 in [5]. The meanings for other symbols are shown in **Figure 13**.

It is very hard to deduce an analysis equation for measuring antenna factor. Usually, a hybrid model (both analytical and dark box model) is adopted, as shown in Eq. (7) as an example,

$$\begin{aligned} F\_{\text{dJSSM}} &= 10 \text{lgf}\_{\text{M}} - 24.46 + \frac{1}{2} E\_{\text{D}}^{\text{max}} + \left[ \frac{1}{2} A\_{12} + \frac{1}{2} A\_{13} - \frac{1}{2} A\_{23} \right] + \frac{\sqrt{3}}{2} u(\delta A\_{\text{VMA}}) \\ &+ \frac{\sqrt{3}}{2} u\left( \delta A\_{\text{Impndance}} \right) + \frac{\sqrt{3}}{2} u(\delta A\_{\text{Cable}}) + \frac{\sqrt{3}}{2} u(\delta A\_{\text{Turb}}) + \frac{\sqrt{3}}{2} u(\delta A\_{\text{AN}}) \\ &+ \frac{\sqrt{3}}{2} u(\delta A\_{\text{AntPositi}}) + \frac{\sqrt{3}}{2} u(\delta A\_{\text{Site}} \delta \text{Mact}) + \frac{\sqrt{3}}{2} u\left( \delta A\_{\text{Repart}} \right) + \frac{\sqrt{3}}{2} u\left( \delta A\_{\text{Symmetry}} \right) \\ &+ \frac{\sqrt{3}}{2} u\left( \delta A\_{\text{X-pol}} \right) + \Delta F\_{\text{DerFrac}-\text{pacc}} \end{aligned} \tag{7}$$

The meaning of the other symbols is shown in **Table 2**. The evaluated standard uncertainty is shown in **Table 2**, too. The combined standard uncertainty can be calculated with Eq. (8), assuming the above uncertainty sources are independent.

*Design, Construction and Validation of a High-Performance OATS DOI: http://dx.doi.org/10.5772/intechopen.99727*

$$\begin{split} u\_{\mathrm{c}}^{2}(F\_{\mathrm{i1}}) &= \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{VMA}})\right]^{2} + \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{Impendit}})\right]^{2} + \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{Cable}})\right]^{2} \\ &+ \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{Tur}})\right]^{2} + \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{AN}})\right]^{2} + \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{AutPortion}})\right]^{2} \\ &+ \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{Site}}\delta \Delta \mathrm{at})\right]^{2} + \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{Repet}})\right]^{2} + \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{Symmetry}})\right]^{2} \\ &+ \left[\frac{\sqrt{3}}{2}u(\delta A\_{\mathrm{X-pol}})\right]^{2} + u^{2}\left(\Delta F\_{\mathrm{DerFre-space}}\right) \end{split} \tag{8}$$

Relacing the value in **Table 2**, there will be,

$$
u\_{\varepsilon} = \mathbf{0}.68 \text{ dB} \tag{9}$$

The expanded uncertainty can be calculated with Eq. (4) by assuming a normal distribution since there are many numbers, and their values are similar.


#### **Table 2.**

*Measurement uncertainty budget for* F*<sup>a</sup> of a biconical antenna with standard site method.*

$$U = k u\_c \tag{10}$$

Taking *k*≈2, there will be

$$U = \mathbf{1}.4 \text{ dB } (k=2) \tag{11}$$

This is the expanded uncertainty in measuring the free-space antenna factor of a biconical antenna with the standard site method, as shown in **Table 2**.
