5. Integral form of coupling formula

### 5.1. Introduction

the aperture antenna, which is well-known to radiate toward the direction of front-side of the aperture. Thus, in order to include the low-gain antennas, the modified antenna gain can be defined using total radiated power over a half sphere. The modified gain, namely the frontside gains GFS, can be used instead of the far-field gain G presented in (10). The front-side gain

, γFSð Þ¼ Δ 1 � α<sup>U</sup>

where Um: the maximum radiation intensity, PF: the total radiated power over a half sphere

Figure 4 shows the gain reduction factors using the front-side gain, which can achieve sufficient convergence of all curves. The trend of all the curves suggests that an empirical coefficient α can be α = 0.06. The major weakness of this method is that we need to calculate the

Instead of using front-side gain, the simpler adjusted gain would be more desirable with respect to the instant calculation of the mutual coupling. The proposed antenna gain can be obtained depending on its classification: low-gain or high-gain antennas defined as below:

, FGð Þ¼ <sup>2</sup>:<sup>5</sup> � <sup>a</sup>

GA ¼ G Gð ≥ 10 dB : high GÞ ¼ G þ 3 dB ð Þ G < 10 dB : low G , γAð Þ¼ Δ 1 � α<sup>U</sup>

2λGA=π<sup>2</sup> �<sup>2</sup>

Otherwise, more sophisticated switching model can be used such as:

R 2λGFS=π<sup>2</sup> �<sup>2</sup>

(12)

/λ and (b) using normalized

(13)

R 2λGA=π<sup>2</sup> �<sup>2</sup>

<sup>π</sup> � arctan c½ � � ð Þ <sup>G</sup> � <sup>b</sup> (14)

and associated gain reduction factor can be expressed as:

Figure 4. The gain reduction factor (a) using normalized distance R with respect to 2D<sup>2</sup>

62 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

GFS <sup>¼</sup> <sup>4</sup>πUm PF

front-side gain, based on complex 3D radiation pattern.

where GA is the proposed adjusted gain.

<sup>γ</sup>Að Þ¼ <sup>Δ</sup> <sup>1</sup> � <sup>α</sup>UF Gð Þ <sup>R</sup>

where the constant is found to be a = 3,b= 10, and c = 1.

where maximum occurs.

distance R with respect to 2λGFS/π<sup>2</sup> (right).

Another method to evaluate mutual coupling is the use of integral form of coupling formula, which is advantageous in its flexibility. The power transmission for various antenna geometries such as rotation and offset in a closer distance maybe required for a comprehensive coupling evaluation. The formula is advantageous since it estimates the coupling in the very close distance, which covers an entire radiating near-field region. The integral form of coupling formula takes a form of a scalar integral of two vector far-field patterns. Transmission integral was studied, which is based on plane-wave scattering matrix (PWSM) theory, and its validity is evaluated through the power transmission between two identical apertures [20]. Two different computer programs were developed to calculate the coupling between two antennas located in a longitudinal and transverse displacement [21]. The coupling in both near-field and far-field is evaluated. The measured far-fields of array and reflector antennas are used for the calculation of near-field pattern [22]. An advancement in computer program increases freedom of possible antenna orientation and displacement [23, 24].

#### 5.2. Integral form of coupling formula

The integral form of coupling formula is practical and flexible, which enables to calculate power transmission in diverse scenarios such as rotation and off-axis cases. The complexity of required information slightly increases, compared to the Friis formula with a correction term. While simple bore-sight gain and antenna geometry are employed in Friis formula, the integral form of coupling formula utilizes 3D vector far-field patterns of two antennas and antenna displacement. Several cases are studied using the integral form of coupling formula and validity of the result is evaluated comparing with the full-wave simulation FEldberechnung für Körper mit beliebiger Oberfläche (FEKO).

The ratio between input wave and output wave presented in [21, 22] can be expressed as normalized vector far-field pattern of transmit antenna f TX k � � and receive antenna <sup>f</sup> RX �<sup>k</sup> � �.

$$\frac{b\_0'}{a\_0}(R) = -\frac{\mathcal{C}}{4\pi k} \iint\_{\sqrt{k\_x^2 + k\_y^2} < k} \frac{\overline{f}\_{\text{Tx}}(\overline{k}) \cdot \overline{f}\_{\text{RX}}(-\overline{k})}{k\_z} e^{-\overline{\beta}\cdot\overline{\tau}} \,d\mathbf{k}\_x d\mathbf{k}\_y \tag{15}$$

a0 is the amplitude of input wave of transmitting antenna, b0 is the amplitude of output wave of receiving antenna, and k is the wave vector. The Eq. (15) has e jwt time dependence in free space, and the constant C can defined as:

$$\mathcal{L} = -\frac{Z\_{\text{wg.Rx}}}{Z\_0} \frac{1}{1 - \Gamma\_{L,Rx}\Gamma\_{0,Rx}} \tag{16}$$

Zwg, Rx, Z0 are the impedance of feed waveguide in receive antenna and free space impedance. ГL,Rx, Г0,Rx are reflection coefficient of the load impedance and the receive antenna impedance, respectively. The discretization of the sampling space kx-ky and discrete Fourier transform is required in order to obtain the coupling quotient, and the sampling frequency is defined as

$$f\_s = 2\kappa \times (D\_{\rm Tx} + D\_{\rm Rx}) \tag{17}$$

DTx and DRx are the diameters of a sphere, which encloses the geometry of transmit and receive antenna, respectively. The constant κ is the oversampling ratio, which limits the movement of antennas in the transverse direction. The separation distance between two antennas can also be defined as:

$$\frac{D\_{\rm Tx} + D\_{\rm Rx}}{2} \le R \le \frac{\left(D\_{\rm Tx} + D\_{\rm Rx}\right)^2}{2} \tag{18}$$

and simulated results is made as shown in Figure 5. The Friis formula with correction term agrees well with the simulated coupling level. In addition, it does provide a significant enhancement after the nearest distance of Fresnel region while the standard Friis formula results in huge deviation. However, it is also shown that the calculated result drops drastically before the nearest distance of Fresnel region, which indicates that its effectiveness is limited to the Fresnel region and far-field region. The formula is made up of quadratic form of the gain reduction factor, which turns out to be negative at a very close distance. In order to predict the coupling at a closer distance, higher order term of interaction should be considered in the quadratic form of gain reduction factor. The nearest prediction ranges for the high-gain

Figure 5. (a) Mutual coupling between two dipole antennas operated at 1.4GHz and (b) mutual coupling between two

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The first step is to obtain the 3D vector far-field pattern with phase and amplitude using FEKO. Next step is to take the integral of scalar product between two far-field patterns so that the coupling with respect to the antenna geometries can be calculated. For the purpose of evaluation, 2 m and 1.9 m prime-focus reflector operated at 5.5GHz are used as an example. The far-field pattern is simulated using FEKO, and the maximum directivity of 42.5 dB and 42.1 dB is obtained for each reflector antenna, respectively. Integral coupling formula is calculated for the bore-sight scenario using the far-field patterns of both antennas. The calculated result is compared to the Friis formula and FEKO simulation as shown in Figure 7. The integral coupling agrees well with the FEKO simulation within 0.5 dB deviation. It is worthy noting that the formula is effective within the radiating near-field region, and no sudden decrease of the coupling level in the nearest distance is observed. The second example is selected as the ground station antennas pointing to the multiple satellites. The interference between two antennas is investigated using the integral form of a formula. In order to communicate with satellites, antennas are both tilted at θ = 45 toward the sky, and the separation

/2λ). The nearest range for dipole antenna is approximately 0.35λ.

antenna is around 1.44(D<sup>2</sup>

Ku-band horn antennas operated at 12.7GHz.

6.2. Evaluation using integral form of coupling formula

At the minimum separation distance, the accuracy deteriorates due to the growing reactive field component. When R approaches the maximum separation distance, the only near-axis plane wave is taken into consideration, which leads to the inaccurate result of calculation [22].

#### 6. Evaluation using numerical methods

#### 6.1. Evaluation using Friis formula with a correction term

The on-axis power transmission between two identical antennas is evaluated at various separation distances. Both far-field antenna gain of the antenna and separation distance is required information. In addition, the point of phase center is important to measure the accurate separation distance. The first example is selected as two half-wavelength dipole antennas, which operate at 1.4GHz with the far-field gain of 2.14 dB. The next example uses two standard gain horns at 12.7GHz with the far-field gain of 15.2 dB, and the phase center of the horn antenna is located at 5 mm below the antenna aperture.

$$\begin{split} \frac{P\_r}{P\_t} &= \frac{\lambda^2 G\_l G\_r}{16\pi^2 R^2} \left( 1 - a\_\mathbb{E} \left( \frac{R}{2\lambda G/\pi^2} \right)^{-2} \right) \left( 1 - a\_\mathbb{E} \left( \frac{R}{2\lambda G/\pi^2} \right)^{-2} \right) \\ &\times \left( 1 - |\Gamma\_l|^2 \right) \left( 1 - |\Gamma\_r|^2 \right) |\widehat{\rho}\_t \cdot \widehat{\rho}\_r| \end{split} \tag{19}$$

The coupling level is obtained using the Friis formula with correction term, and the validity of the formula is evaluated by comparing it with the full-wave simulation FEKO. The comparison among standard Friis formula, Friis formula with correction term (asymptotic Friis formula), Electromagnetic Computation of the Short-range Wireless Linkbuget for Biomedical Communication http://dx.doi.org/10.5772/intechopen.76141 65

Figure 5. (a) Mutual coupling between two dipole antennas operated at 1.4GHz and (b) mutual coupling between two Ku-band horn antennas operated at 12.7GHz.

and simulated results is made as shown in Figure 5. The Friis formula with correction term agrees well with the simulated coupling level. In addition, it does provide a significant enhancement after the nearest distance of Fresnel region while the standard Friis formula results in huge deviation. However, it is also shown that the calculated result drops drastically before the nearest distance of Fresnel region, which indicates that its effectiveness is limited to the Fresnel region and far-field region. The formula is made up of quadratic form of the gain reduction factor, which turns out to be negative at a very close distance. In order to predict the coupling at a closer distance, higher order term of interaction should be considered in the quadratic form of gain reduction factor. The nearest prediction ranges for the high-gain antenna is around 1.44(D<sup>2</sup> /2λ). The nearest range for dipole antenna is approximately 0.35λ.

#### 6.2. Evaluation using integral form of coupling formula

<sup>C</sup> ¼ � Zwg,Rx Z0

64 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

DTx þ DRx

6. Evaluation using numerical methods

6.1. Evaluation using Friis formula with a correction term

horn antenna is located at 5 mm below the antenna aperture.

� <sup>1</sup> � <sup>Γ</sup><sup>t</sup> j j<sup>2</sup> � �

Pr Pt ¼ λ2 GtGr <sup>16</sup>π<sup>2</sup>R<sup>2</sup> <sup>1</sup> � <sup>α</sup><sup>E</sup>

can also be defined as:

1 1 � ΓL,RxΓ0,Rx

f <sup>s</sup> ¼ 2κ � ð Þ DTx þ DRx (17)

2

<sup>2</sup> (18)

Zwg, Rx, Z0 are the impedance of feed waveguide in receive antenna and free space impedance. ГL,Rx, Г0,Rx are reflection coefficient of the load impedance and the receive antenna impedance, respectively. The discretization of the sampling space kx-ky and discrete Fourier transform is required in order to obtain the coupling quotient, and the sampling frequency is defined as

DTx and DRx are the diameters of a sphere, which encloses the geometry of transmit and receive antenna, respectively. The constant κ is the oversampling ratio, which limits the movement of antennas in the transverse direction. The separation distance between two antennas

<sup>2</sup> <sup>≤</sup><sup>R</sup> <sup>≤</sup> ð Þ DTx <sup>þ</sup> DRx

At the minimum separation distance, the accuracy deteriorates due to the growing reactive field component. When R approaches the maximum separation distance, the only near-axis plane wave is taken into consideration, which leads to the inaccurate result of calculation [22].

The on-axis power transmission between two identical antennas is evaluated at various separation distances. Both far-field antenna gain of the antenna and separation distance is required information. In addition, the point of phase center is important to measure the accurate separation distance. The first example is selected as two half-wavelength dipole antennas, which operate at 1.4GHz with the far-field gain of 2.14 dB. The next example uses two standard gain horns at 12.7GHz with the far-field gain of 15.2 dB, and the phase center of the

> R 2λG=π<sup>2</sup> � ��<sup>2</sup> !

> > <sup>b</sup>r<sup>t</sup> � <sup>b</sup>r<sup>r</sup> j j

The coupling level is obtained using the Friis formula with correction term, and the validity of the formula is evaluated by comparing it with the full-wave simulation FEKO. The comparison among standard Friis formula, Friis formula with correction term (asymptotic Friis formula),

<sup>1</sup> � <sup>Γ</sup><sup>r</sup> j j<sup>2</sup> � �

1 � α<sup>E</sup>

R 2λG=π<sup>2</sup> � ��<sup>2</sup> ! (16)

(19)

The first step is to obtain the 3D vector far-field pattern with phase and amplitude using FEKO. Next step is to take the integral of scalar product between two far-field patterns so that the coupling with respect to the antenna geometries can be calculated. For the purpose of evaluation, 2 m and 1.9 m prime-focus reflector operated at 5.5GHz are used as an example. The far-field pattern is simulated using FEKO, and the maximum directivity of 42.5 dB and 42.1 dB is obtained for each reflector antenna, respectively. Integral coupling formula is calculated for the bore-sight scenario using the far-field patterns of both antennas. The calculated result is compared to the Friis formula and FEKO simulation as shown in Figure 7. The integral coupling agrees well with the FEKO simulation within 0.5 dB deviation. It is worthy noting that the formula is effective within the radiating near-field region, and no sudden decrease of the coupling level in the nearest distance is observed. The second example is selected as the ground station antennas pointing to the multiple satellites. The interference between two antennas is investigated using the integral form of a formula. In order to communicate with satellites, antennas are both tilted at θ = 45 toward the sky, and the separation of two antennas is around 3 m. The configuration of both antennas is also shown in Figure 6. In a similar way, coupling integral is calculated, and the comparison of the results is shown in Table 1. Considering that the cross-pol component becomes significant in off-axis radiation pattern, the comparison shows reasonable agreement within 3 dB difference.

7. Evaluation using the two numerical methods

The coupling level between two identical Ku-band standard gain horns is measured in order to evaluate the proposed formula. The standard gain horn operates at 12.7GHz, the center frequency of Ku-band, and the radius of the sphere, which encloses the horn antenna, is around 2.15λ. The radiation pattern of the Ku-band standard gain horn is measured inside the UCLA spherical near-field range as shown in Figure 7. The amplitude and phase of the radiated near-field are acquired by rotating the horn mounted on the positioner. The near-field information can be converted into far-field radiation pattern. The comparison between measured far-field and the simulated far-field pattern using FEKO is depicted in Figure 8. The max

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As shown in Figure 8, the two horn antennas are mounted on an optical table and the mutual coupling S21 is measured using a vector network analyzer. To properly measure S21, we need to

1. The polarization of both horn antennas must be perfectly matched in order to avoid any

2. The insertion loss proportional to the length of cable should be carefully measured and

3. Separation distance R needs to be measured from one phase center to the other phase center. Prior to measuring the distance, it is critical to determine the phase center where the phase response of far-field pattern is uniform. In general, the simulated model can be used

The separation distance R varies from 4 to 31λ in the on-axis direction. The mutual coupling levels using different methods are compared as shown in Figure 8. The simulated results show

Figure 8. (a) Indoor measurement of mutual coupling and (b) measured mutual coupling level in order to evaluate the

simulated and measured gain patterns are 15.28 dB and 15.47 dB, respectively.

7.1. Evaluation using mutual coupling measurement

carefully setup the measurement as below:

loss due to the mismatch of polarization.

counted in calculating the coupling level.

to find the location of the phase center.

simulated coupling level.

Figure 6. (a) On-axis mutual coupling between C-band 2 m and 1.9 m reflector antennas and (b) off-axis mutual coupling between the two reflector antennas.

Figure 7. (a) Setup of horn antenna inside near-field anechoic chamber and (b) far-field radiation pattern from the measurement.


Table 1. Off-axis mutual coupling between the two reflector antennas.
