Characteristic eigen mode theory

This section provides a basic insight of the CMA based on the eigenmodes. It discusses the crucial role in the design of an antenna for the desired frequencies by extracting the fundamental modes and far-field behavior for an arbitrary structure (Perfect Electric Conductor) independent of the source. Harington and Mautz developed the CMA theory for antenna by considering the linear relationship between the surface current density and the tangential component of the incident electric field. The theory of characteristic modes is conceptually similar to that of the Fourier series. Any arbitrary periodic wave form can be written as a linear combination of the fundamental frequency and its harmonics weighted by suitable coefficients. The same concept can be used in electromagnetics as Maxwell’s equation is linear, the electric field can be written as the linear combination of their eigenmodes. The energy contained in the electromagnetic wave is bounded, this phenomenon enables Maxwell’s equation to form orthogonal basis functions and possible linear combinations of the physical quantity and also could find the coefficients of the harmonics. When no source is given, the computed current modes of the arbitrary antenna structure called is the characteristic modes or characteristic currents governs the resonance, gain, bandwidth and polarization which depends only on the size and shape of the design.

Mathematical formulations of the CMT

Based on the insights from [Reference Cabedo-Fabres, Antonino-Daviu, Valero-Nogueira and Ferrando Bataller41], the weighted eigenvalue ${\lambda _n}$ of the eigenfunctions is used to obtain the characteristic currents $\overrightarrow {{J_n}} $ of the patch by relating the real and the imaginary parts of the impedance Z = R + jX

(1)\begin{equation}\,X\left( {\overrightarrow {{J_n}} } \right) = \,{\lambda _n}R\,\left( {\overrightarrow {{J_n}} } \right)\end{equation}

From reciprocity theorem, the impedance operator Z has symmetricity and linearity, and hence the Hermitian parts R and X will also be linear and symmetrical. Therefore eigenvalue ${\lambda _n}$ given by equation (1) is all real and the eigenfunctions $\overrightarrow {{J_n}} $ over the defined surface is chosen to be a scalar value or equiphasal (a real function scaled by a complex constant). The orthogonality of the characteristic modes depends upon the R of the is a Z operator in (1), which is stated in terms of Kronecker delta function ${\delta _{mn}}$ as,

(2)\begin{equation}\left[ {\vec J_m^*,R{{\vec J}_n}} \right] = \,{\delta _{mn}}\end{equation}

(3)\begin{equation}\left[ {\vec J_m^*,X{{\vec J}_n}} \right] = \lambda_n {\delta _{mn}}\end{equation}

Depending in the size and the shape of the radiating patch, the surface current $\overrightarrow {{J_n}} $ define the characteristic modes. Equation (1) in the matrix form is,

(4)\begin{equation}\left[ X \right]\,\overrightarrow {{J_n}} \, = \,{\lambda _n}\,\left[ R \right]\,\overrightarrow {{J_n}} \,\end{equation}

The eigenvalues ( ${\lambda _n})$ and the eigenvectors ( $\overrightarrow {{J_n}} )\,$can be obtained by solving equation (4). The eigenvectors are calculated for the free space. The characteristic electric fields ( $\overrightarrow {{E_n}} )$ on the surface of the conducting body are produced by the characteristic currents ( $\overrightarrow {{J_n}} )$.

(5)\begin{align} {E_n}\left( {\overrightarrow {{J_n}} } \right) &= Z{\text{ }}(\overrightarrow {{J_n}} ) \nonumber\\ &=R \left( {\overrightarrow {{J_n}} } \right)\,\, + jX(\overrightarrow {{J_n}} ) \nonumber\\ {E_n}\left( {\overrightarrow {{J_n}} } \right)& = R\left( {\overrightarrow {{J_n}} } \right)\,\,\left( {1 + j{\lambda _n}} \right)\end{align}

The characteristic fields are R ( $\overrightarrow {{J_n}} )$ multiplied by a complex term $\left( {1 + \,j{\lambda _n}} \right)$ making it equiphasal. The orthogonality of these fields can be established by the complex Poynting theorem.

\begin{equation*}P\,({J_m},{J_n}) = \left[ {\overrightarrow {J_m^*} ,Z\overrightarrow {{J_n}} } \right] = \left[ {\overrightarrow {J_m^*} ,R\overrightarrow {{J_n}} } \right] + \left[ {\overrightarrow {J_m^*} ,X\overrightarrow {{J_n}} } \right]\end{equation*}

\begin{equation*} = \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} {\left( {\overrightarrow {{E_m}} x\,\vec H_n{^*}} \right)} \,ds\, + \,j\omega \,\mathop {\iiint {\left( {\mu {{\vec H}_m} \bullet \vec H_n{^*}\, - \,\varepsilon {{\vec E}_m}\, \bullet \vec E_n{^*}} \right)}}\,d\tau \,\end{equation*}

(6)\begin{equation} = \,\,\left( {1 + \,j{\lambda _n}} \right)\,{\delta _{mn}}\end{equation}

This orthogonality is observed on the surface of the radiating patch and at the infinity sphere enclosing the patch. These modes transmit power without depending on other modes forming the basis set. The total current on the surface is given as J,

(7)\begin{equation}J\, = {\sum _n}\frac{{V_n^i\,{J_n}}}{{1 + j{\lambda _n}}}\end{equation}

(8)\begin{equation}V_n^i\, = \,\left[ {{J_{n,}}{{\textrm E}^i}} \right] = \oint {{J_n}} \bullet \,{{\textrm E}^i}ds\end{equation}

The product $V_n^i\,{J_n}$ denotes the coupling between the corresponding mode and the excitation and $V_n^i$ is the modal excitation coefficient. At resonance ${X_L} = {X_C}$, hence X = 0, from equation (1), | ${\lambda _n}$| = 0. When ${\lambda _n} \gt 0,$ the mode stores magnetic energy and when ${\lambda _n}$< 0, the mode stores electric energy. From equation (7), the eigenvalues and the modal expansion of the current is inversely proportional, it is more consistent to analyze a term called Modal Significance (MS), given by equation (9), as it gives the normalized amplitude of the modes of the current depending on the shape and the size of the structure.

(9)\begin{equation}MS\, = \,\left[ {\frac{1}{{1 + \,j{\lambda _n}\,}}} \right]\end{equation}

Another representation of the eigenvalues is the Characteristic angles defined by equation (10) giving an insight about the phase difference between the Modal currents, $\overrightarrow {{J_n}} $ and the corresponding electric field ${E_n}$.

(10)\begin{equation}{\alpha _n}\, = \,180^\circ \, - \,ta{n^{ - 1}}\left( {{\lambda _n}} \right)\end{equation}

Resonance occurs when $\lambda_n = 0, \alpha_n=180^\circ$.

Modal analysis of the structure of the antenna

The Characteristic modes of the dual L shaped antenna were analyzed to obtain a dominant mode at 5.9 GHz. The structure of the antenna was obtained through the following stages shown in Figure 1. Initially, the analysis of the design starts from a simple square patch of size 32.5 × 35 $m{m^2}$ had evolved to a novel structure. The modal significance and the characteristic angle graph of the stages A, B, C is given in Figure 1.

Figure 1.

Modal significance and characteristic angle and S parameters of stages A, B, and C.

Modal analysis of the patch at stage A, modes 2 and 3 are dominant produce resonance at 5.49 and 4.76 GHz, respectively. However, no resonance at the desired frequency. The modal study of the stage B, modes 2 and 4 dominates resonating at 7.54 and 3.84 GHz. Analyzing the characteristic modes of the patch in stage C, mode 3 is a dominant mode producing a resonance at 5.9 GHz. Suitably exciting the dominant modes produces resonance at 5.9 and 2.45 GHz. The corresponding return loss characteristics show the resonance at the dominant modes at each stage.

Design of the antenna

After the CMA, the proposed design is optimized over the Rogers 4534 substrate ( ${\varepsilon _r}$=3.5 and tan δ = 0.0022). The thickness of the substrate is 1.524 mm. Figure 2 prototypes the structure of the presented antenna. The electrical dimension of the antenna is 0.286 $\,{{{\lambda }}_0}$ x 0.265 $\,{{{\lambda }}_0}$ x 0.012 ${ }{{{\lambda }}_0}\,m{m^3}$, where ${{{\lambda }}_0}$ is the free space wavelength at 2.45 GHz. The antenna excites the fundamental mode at 2.45 GHz and the higher-order mode at 5.9 GHz. Table 1 provides the summary of Figure 3 showing the return loss of −37 dB, 190 MHz bandwidth and 3.49 dB gain for 2.45 GHz and −21.4 dB return loss, 720 MHz bandwidth and 3.2 dB gain in simulation and nearly similar measured results.

Figure 2.

The prototyped antenna and dimensions.

Figure 3.

Simulated and measured (a) return loss plot of the antenna and (b) gain plot of the antenna.

Table 1.

Summary and interpretation of Figure 3

Radiation mechanism of the antenna

More the surface current density at the specific sites of the radiating patch at resonance leads to concentrating the energy. This concentrated energy can enhance the antenna’s coupling ability with the electromagnetic field and hence more efficiently radiate or receive the energy. Observing Figure 4 explains the surface current density of the proposed antenna when the dominant mode 3 is excited. It has higher current density at the resonating length of the structure. Figure 5(a) and (b) shows radiation pattern of the antenna at 2.45 and 5.9 GHz. On observing, this kind of radiation pattern is not suitable for WBAN application as the radiation pattern has a shape of “8.” The lower lobe of the radiation pattern gets absorbed by the user’s tissue. This back-lobe must be mitigated.

Figure 4.

Current distribution of the dominant eigen mode 3 of the proposed antenna.

Figure 5.

Antenna’s radiation pattern at (a) 2.45 GHz and (b) 5.9 GHz.

Equivalent circuit model of the dominant modes of the antenna structure

The equivalent circuit model of the designed antenna which resonates at dual band is presented. The proposed equivalent circuit model is shown in Figure 6(a) and its response is shown in Figure 6(b). The resistor $\,{R_1}$and ${C_0}$ are for impedance matching purpose. The inductor ${L_1}$ and the capacitor ${C_1}$ in parallel is responsible for the resonance at the lower band. The inductor ${L_2}$ and the capacitors ${C_{21}}$ and ${C_{22}}$ in parallel is responsible for the resonance at the upper band. The impedance Z = R + jX, where R is the resistance and X is the reactance. Z can be determined for 2.45 GHz ( ${f_1})$ and 5.9 GHz ( ${f_2})$ from the Smith chart. Depending on the Z value, either L or C can be calculated for individual frequency bands.

(11)\begin{equation}{X_L}\, = \,\omega L\,\,{\text{and}}\,{X_C} = 1/\omega C\end{equation}

Determining one of the unknown for the required frequency, the other value can be calculated.

(12)\begin{equation}{f_1} = \frac {1} {2\pi \sqrt {{L_1}{C_1}} }\end{equation}

(13)\begin{equation}{f_2} = \frac{1}{{2\pi \sqrt {{L_2}({C_{21}} + {C_{22}})} }}\end{equation}

Figure 6.

(a) Equivalent circuit of the proposed antenna ${R_1}\,50{{\Omega }}$, ${C_0},\,{C_1}$ = 1.8 pF, ${L_1}$=1 nH, ${C_{21}},\,{C_{22}}$=0.6 pF, ${L_2}$ = 0.4 nH. (b) S parameters of the antenna’s equivalent circuit.

Design of the artificial magnetic conductor

The proposed antenna is efficient, but still more electric field gets absorbed by the human tissues. In order to satisfy the criteria for Wearable Body Area Network, an AMC was incorporated along, backing the antenna. The AMC plays the role of a reflecting surface for the antenna, due to its behavior as a perfect magnetic conductor (PMC) at the desired frequency which is characterized by a zero-degree reflection phase, with a reflection phase bandwidth of range ± 90°. A unit cell of AMC having two concentric square rings as shown in Figure 7(a), is excited with TE10 mode to produced 0° phase angle of the reflection at 2.45 and 5.9 GHz hence acting a PMC. The in-phase reflection interferes constructively with the incident wave and hence enhancing the performance of the antenna.

Figure 7.

(a) Unit-cell with dimension. (b) AMC array. (c) Current distribution when EM wave incidents.

At the reflection phase range of −90° to −180° and + 90° to + 180°, the unit cell behaves as a conductor. The single unit cell is designed to an array of 3 × 3 elements to make it act as an AMC. The AMC is designed on Rogers 4534 substrate ( ${\varepsilon _r}$=3.5 and tan δ = 0.0022) of thickness 1.524 mm. The length of each concentric square ring was calculated in accordance with the following equations.

(14)\begin{equation}\Theta = \beta l\,\quad or\quad \,l = \Theta / \times \beta \end{equation}

Where β (= ${k_o},{\text{ wave number}}$) is the phase constant and l is the physical length and Θ is the reflection phase

(15)\begin{equation}l\, = \,\frac{{\phi \,( {{\frac{\pi }{{180}}}} )}}{{\sqrt {{\varepsilon_{eff}}} {k_0}}}\;{\text{where}},\,\;{k_o} = {\frac{{2\pi f}} {c}}\end{equation}

The unit-cell is extended as a 3 × 3 array to behave as an AMC layer as shown in Figure 7(b). Figure 7(c) shows the current density at the surface of the ring structures showing the outer rings of the AMC array having higher current density is responsible for 0° reflection at 2.45 GHz and in other case, the inner rings of the AMC array having higher current density is responsible for 0° reflection at 5.9 GHz. Figure 8 shows the zero-degree reflection phase at 2.45 and 5.9 GHz produced.

Figure 8.

Reflection phase of the AMC array.

The AMC reflector behaves like a mirror for the EM waves at the design frequencies, thus, altering the field distribution in the antenna vicinity. The interference of the EM waves between the antenna and AMC is constructive and modifies the surface current distribution of the antenna.

Design of the artificial magnetic conductor

The proposed antenna is placed over the AMC layer. A thin layer of styrofoam is placed in between the antenna and the AMC layer to serve the purpose of isolation. The thickness of the foam layer is about 4–5 mm. The overall dimensions of the antenna are 0.613 ${{{\lambda }}_0}$ x0.613 ${{{\lambda }}_0}$ x0.049 ${{{\lambda }}_0}$ $m{m^3}$. Figure 9 shows the antenna incorporated with the AMC layer and the cross-sectional assembly of the AMC backed antenna.

Figure 9.

Top-view and side view of the antenna + AMC.

Figure 10(a) and (b) shows the return loss and gain of the antenna along with the AMC. The antenna has a return loss of −21.9 dB, a bandwidth of 230 MHz and a gain of 6.18 dB at 2.45 GHz. Also, a return loss of −36 dB, a bandwidth of 770 MHz and a gain of 10.02 dB at 5.9 GHz and is tabulated in Table 2. The effects of number of unit cells impact the performance of the antenna is tabulated in Table 3. The 3 × 3 is chosen based on optimized performance and size of the magnetic in phase reflector.

Figure 10.

Simulated and measured (a) return loss plot of the antenna and (b) gain plot of the antenna with AMC.

Table 2.

Summary and interpretation of Figure 10

Table 3.

Effect of number of unit-cell of AMC on the antenna performance

Radiation mechanism of the antenna with AMC

Figure 11 shows the surface current distribution of the antenna and AMC while placed together while at (a) lower and (b) upper band. On comparing the current distribution of the antenna alone as given by Figure 4, the surface current value has increased after the incorporation of the AMC surface. The effect of the number of unit-cell in the AMC array is studied. Higher current distribution on the surface of the antenna is achieved after the AMC. This contributes to better impedance matching and better performance of the antenna.

Figure 11.

Radiation mechanism of the antenna with AMC while resonance at (a) 2.45 GHz and (b) 5.9 GHz.

Figure 12 represents the simulated and measured E plane and H plane radiation pattern at lower cutoff and upper cut off frequencies, respectively. The radiation pattern explains that the unwanted back-lobes are mitigated due to the Magnetic Conductor causing an in-phase reflection. This effect reduces the undesirable lobes of the antenna. Figure 13 visually explains the role of AMC layer in mitigating the back radiation at (a) 2.45 GHz and (b) 5.9 GHz.

Figure 12.

Radiation pattern of the antenna with AMC at (a) 2.45 GHz and (b) 5.9 GHz.

Figure 13.

Influence of the AMC layer on the radiation pattern of the antenna at (a) 2.45 GHz and (b) 5.9 GHz.

Design of the artificial magnetic conductor

The functioning of the antenna drastically changes when kept over a human. The human body acts as an electrical conductor due to presence of conductive elements such as blood, sweat, electrolytes and hence couples with the antenna and changes the resonant frequency as its electrical length changes. Also, the antenna strongly interacts with the human body leading to loading effect by altering the impedance of the antenna results in frequency detuning, a shift in the resonant frequency. Due to these factors the antenna performs less efficiently while placed over the body. Figure 14(a) shows the image of antenna placed on human 3 layered model consisting of the skin, fat and the muscle. The electrical nature of the tissues is tabulated in Table 4.

Figure 14.

(a) Antenna. (b) Antenna and AMC on the 3 layered human model.

Table 4.

Electrical nature of human tissues

The plots in Figure 15 show the return loss characteristics of the antenna placed over human tissue model. From the graph, frequency detuning is observed. The resonance had shifted from 2.45 GHz to 1.6 GHz with ${S_{11\,dB}}$ of −19.9 dB and no sharp resonance at the upper band. When placed over hand and measured using a Vector Network Analyzer (VNA), the resonance occurs at 1.78 GHz of ${S_{11\,dB}}$ −35 dB and at 6.04 GHz of ${S_{11\,dB}}$ −12.3 dB and when measured at chest, the resonance occurs at 2.2 GHz with ${S_{11\,dB}}\,$of −26.3 dB and −20.26 dB ${S_{11\,dB}}$ at 5.8 GHz. Measurement is done by placing the antenna over the pork (as it has similar electrical nature to that of human) in anechoic chamber and the gain and the pattern were deteriorated.

Figure 15.

Simulated and measured return loss plot of the antenna.

Antenna with AMC: human proximity effect

To mitigate the body effects, AMC is incorporated to enhance the functionality of the antenna. The AMC provides isolation between the antenna and human body and therefore preventing the frequency detuning. By reflecting the EM waves, it enhances the radiation efficiency and improves the gain of the antenna. Figure 14(b) represents antenna along with AMC when placed over the 3 layered human tissue model consisting of the skin, fat and the muscle.

Figure 16 shows the simulated and measured (a) return loss characteristics and (b) gain of the antenna with AMC placed over the body (and pork when kept in anechoic chamber). The results are summarized in Table 5. The frequency detuning effect is very insignificant while placed over the human body with AMC.

Figure 16.

Simulated and measured (a) return loss plot of the antenna and (b) gain plot of the antenna & AMC placed on human body.

Table 5.

Summary and interpretation of Figure 16

Figure 17 shows the on-body pattern of antenna + AMC at (a) 2.45 GHz and (b) 5.9 GHz. The back radiation is mitigated in the presence of the AMC layer. Figure 18 shows the images taken from the Keysight-N9926A hand-held VNA when antenna with AMC is placed over the a) hand b) Chest c) when placed over pork in anechoic chamber.

Figure 17.

On-body antenna + AMC radiation pattern at (a) 2.45 GHz and (b) 5.9 GHz.

Figure 18.

Measurement of S parameters when placed over (a) chest, (b) hand, and (c) anechoic chamber setup placed over pork.

Specific absorption rate (SAR) analysis

The SAR must be calculated and must be kept within the safe limit. Commissions like FCC of united states has imposed a limited of 1.6 W/Kg averaged for 1 g tissue. The International Commission on Non-Ionizing Radiation Protection (ICNIRP) of Europe has imposed a limit of 2 W/Kg averaged for 10 g tissue. The SAR analysis is simulated averaging the values over 1 and 10 g of tissues. The values of the SAR of the designed antenna and that of with the AMC is evaluated and compared. The simulation is carried out with an input power of 0.5 W for 1 and 10 g of average masses of the human tissues. Mathematically, SAR can be analyzed by equation 17,

(17)\begin{equation}SAR\, = \int\frac{{\sigma \,[\,\overrightarrow {\left| {{E_x}} \right|} \, + \left| {\overrightarrow {{E_y}|} \, + } \right|\overrightarrow {{E_z}} |}}{{2\,\rho }}\,dV\,W/Kg\end{equation}

Where $\sigma $ is the $\,\overrightarrow {\left| {{E_{}}} \right|} \,,\left| {\overrightarrow {{E_y}|} \,,} \right|\overrightarrow {{E_z}|} $ is the Electric field along the x, y, z directions, respectively, and $\rho $ is the mass density of the human tissue.

When the antenna is placed over the human body, the side lobes and the back lobes of the antenna get absorbed by the skin, fat and the muscle as they are directed into the body gets dissipated contributing to the high SAR value. The analyzed SAR value when antenna alone placed over the human body is 59.16 W/Kg for 1 g and 85.42 W/Kg for 10 g at 2.45 GHz and 15.13 W/Kg for 1 g and 26.47 W/Kg for 10 g at 5.9 GHz for 0.5 W input power. The higher SAR values at 2.45 GHz shows higher conductivity of the body at that frequency. The SAR value of the antenna alone placed over the human body is far more greater than the values limited by FCC and the ICNIRP. To enable it to be used for a longer time in the nearness of a human body, AMC is incorporated along the antenna to reduce the SAR value to its limits. The simulation results of the SAR analysis along with AMC is 1.105 W/Kg for 1 g and 0.169 W/Kg for 10 g at 2.45 GHz and 0.447 W/Kg for 1 g and 0.086 W/Kg for 10 g at 5.9 GHz for 0.5 W input power. Figure 20 shows the results of SAR analysis for antenna alone and for the antenna with AMC and their peak values are summarized.

Figure 19 compares the return loss characteristics of the antenna in free space, with AMC, and with AMC on the body, showing ability of AMC to preserve impedance matching and minimize frequency detuning in human proximity.

Figure 19.

Comparison of the S parameters of antenna, antenna with AMC, on-body antenna with AMC.

Figure 20.

SAR analysis of the antenna alone (top) and antenna with AMC (bottom) when placed near human at 2.45 and 5.9 GHz and their maximum SAR values for 0.5 W input power is summarized.

Effect of bending on antenna performance

The WBAN slightly suffers deformation and the extent to which it bends when placed over the body depends on the material used. In this paper, Rogers 4534 substrate is used which is slightly flexible. Hence the performance of the antenna when slightly bent is analyzed using simulation tool. The designed antenna is bent along xy and xz planes. The antenna is bent around various radius of r = 90 mm, 100 mm, 110 mm and their scattering parameters are analyzed. Figure 21 renders the pictorial representation of the bending at various radius of curvature alone both the xy and xz planes. Figure 22 shows the functioning of the antenna when bent along various axis with abovementioned radius. The results are summarized in Table 6.

Figure 21.

Antenna bent along (a) xy plane and (b) xz plane.

Figure 22.

Performance of the antenna under bent condition along (a) xy and (b) xz plane with radius of curvature r = 90 mm, 100 mm, 110 mm and at flat condition.

Table 6.

Summary of Figure 22

The lower band of the antenna remained stable with very negligible frequency shift. The upper band shows slight variations when observed in both xy and xz planes. This confirms the antenna’s resilience. The performance of the proposed antenna is studied with various research works and is summarized in Table 7. Comparing to the state of art works, the proposed work is having better performance in terms of gain, bandwidth, return loss and SAR.

Table 7.

Comparison of this work with those of the work presented in the literature