A simple rectangular patch is considered as a parallel combination of resistance *R*_{1}, inductance *L*_{1}, and capacitance *C*_{1} circuit and its input impedance is represented as *Z*_{p}. The equivalent circuit of the rectangular patch is shown in Fig. 3(a), where *R*_{1}, *C*_{1} and *L*_{1} can be defined as [20,21],
$$\begin{array}{}{\displaystyle {C}_{1}=\frac{LW{\epsilon}_{0}{\epsilon}_{e}}{2H}\text{c}\text{o}{\text{s}}^{2}(\frac{\pi {X}_{0}}{L})}\end{array}$$(1)
$$\begin{array}{}{\displaystyle {R}_{1}=\frac{Q}{{\omega}_{r}^{2}{C}_{1}}}\end{array}$$(2)
$$\begin{array}{}{\displaystyle {L}_{1}=\frac{1}{{C}_{1}{\omega}_{r}^{2}}}\end{array}$$(3)

Quality factor,
$$\begin{array}{}{\displaystyle Q=\frac{c\sqrt{{\epsilon}_{e}}}{4fH}}\end{array}$$

*Where*

*L*- Length of rectangular patch,

*W*- Width of rectangular patch,

*H*- Thickness of the substrate material,

*X*_{0}-x coordinates of feed point, i.e.,*X*_{0} = *L*_{s},

*ε*_{e} - effective permittivity of the medium.

Figure 3 Equivalent Circuits diagram of (a) Simple patch (*Z*_{p}) (b) parasitic patch (*Z*_{pp} )(c) Gap between two patch (*Z*_{cc}) (d) Microstrip line patch (*Z*_{L})

The equivalent circuit of the parasitic patch is shown in Fig. 3(b), its input impedance is represented as *Z*_{pp}, where *R*_{2}, *C*_{2} and *L*_{2}can be calculated as *R*_{1}, *C*_{1} and *L*_{1} with same equations, here *X*_{0} is considered as 0. The equivalent circuit diagram of the gap between the fed patch and the parasitic patch *Z*_{cc} is shown in Fig. 3(c), which is represented as the combination of capacitances *C*_{g} and *C*_{p1}. The expression of gap capacitance *C*_{g} and plate capacitance *C*_{p1} of the microstrip line can be calculated as [22,23,24]
$$\begin{array}{}{\displaystyle {C}_{g}=\mathrm{0.5.}H.{Q}_{1}\mathrm{e}\mathrm{x}\mathrm{p}(-1.86(\frac{G}{H}))}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}[1+4.09\{1-\mathrm{e}\mathrm{x}\mathrm{p}(0.75\sqrt{\frac{H}{W}})\}]}\end{array}$$(4)
$$\begin{array}{}{\displaystyle {C}_{p1}={C}_{L}(\frac{{Q}_{2}+{Q}_{3}}{{Q}_{2}+1})}\end{array}$$(5)

Where,
$$\begin{array}{}{\displaystyle {Q}_{1}=0.04598\{0.03+(\frac{W}{H}{)}^{{Q}_{4}}\}(0.272+{\epsilon}_{r}0.07),}\end{array}$$

$$\begin{array}{}{\displaystyle {Q}_{2}=0.107[\frac{W}{H}+9](\frac{G}{H}{)}^{3.23}+2.09(\frac{G}{H}{)}^{1.05}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+[\frac{1.5+0.3(\frac{W}{H})}{1+0.6(\frac{W}{H})}]}\end{array}$$
$$\begin{array}{}{\displaystyle {Q}_{3}=\mathrm{e}\mathrm{x}\mathrm{p}(-0.5978)-0.55}\end{array}$$
$$\begin{array}{}{\displaystyle {Q}_{4}=1.23}\end{array}$$

*C*_{L} is the terminal capacitance of the open circuited conductor is given as,
$$\begin{array}{}{\displaystyle {C}_{L}={C}_{ll}\frac{\sqrt{{\epsilon}_{eff}}}{{z}_{0}c}}\end{array}$$

where *C*_{ll} is the conductor extension length, *ε*_{eff} is effective dielectric constant.

$$\begin{array}{}{\displaystyle {C}_{ll}=0.412[\frac{({\epsilon}_{e}+0.3)(\frac{W}{H}+0.264)}{({\epsilon}_{e}-0.258)(\frac{W}{H}+0.8)}]}\end{array}$$

*Z*_{0}is characteristic impedance of the patch, *c* is the velocity of light.

The microstrip line of the rectangular patch is considered as combination of *L* and *C*. The equivalent circuit of the microstrip line rectangular patch is shown in Fig. 3(d), where *L*_{L} and *C*_{L} are inductance and capacitance of strip [22, 23].

$$\begin{array}{}{\displaystyle {L}_{L}=100.H(4\sqrt{{W}_{s}/H}-4.21)nH}\end{array}$$(6)
$$\begin{array}{}{\displaystyle {C}_{L}={W}_{s}\{(9.5{\epsilon}_{r}+1.25){W}_{s}/H+5.2{\epsilon}_{r}+7.0\}pF}\end{array}$$(7)

Resonance frequency of the microstrip line antenna is given as,
$$\begin{array}{}{\displaystyle f=c/2{L}_{es}\sqrt{{\epsilon}_{re}}}\end{array}$$(8)

where
$$\begin{array}{}{\displaystyle {\epsilon}_{re}=1/2[({\epsilon}_{r}+1)+({\epsilon}_{r}-1)(1-12.H/{W}_{s}{)}^{-1/2}],}\end{array}$$

*L*_{es} = *L*_{S} +*ΔL*_{S},

*ε*_{re}- Effective dielectric constant,

L_{es} -Effective increase in length of strip,
$$\begin{array}{}{\displaystyle \mathrm{\Delta}{L}_{s}=H.0.412\frac{({\epsilon}_{re}+0.3)({W}_{s}/H+0.264)}{({\epsilon}_{re}-0.258)({W}_{s}/H+0.8)},}\end{array}$$

*ε*_{r} -Dielectric constant,

The characteristic impedance of microstrip line [20,21,22],
$$\begin{array}{}{\displaystyle {Z}_{L}=j\omega {L}_{L}+\frac{1}{j\omega {C}_{L}+\frac{1}{j\omega {L}_{L}}}}\end{array}$$(9)

Therefore, the total input impedance (*Z*_{in}) of antenna can be calculated by equivalent circuit diagram Fig. 4 as
$$\begin{array}{}{\displaystyle {Z}_{in}={Z}_{L}+\frac{1}{\frac{1}{\frac{1}{j\omega {C}_{c}+\frac{1}{j\omega {L}_{c}}}+\frac{1}{{Z}_{p}}+\frac{1}{{Z}_{pp}}+\frac{1}{{Z}_{cc}}}}}\end{array}$$(10)

where *L*_{c} and *C*_{c}is the inductance and capacitance of SRR’s can calculated as [24].

Figure 4 Equivalent circuit diagram of the antenna.

Now using equation (10) the total input impedance of the proposed antenna is calculated. Their various antenna parameters such as reflection coefficient, VSWR and return loss are calculated as:

Reflection Coefficient
$$\begin{array}{}{\displaystyle \Gamma =\frac{Z-{Z}_{in}}{Z+{Z}_{in}},}\end{array}$$

where

*Z* is the input impedance of the microstrip fed (50 Ω).

$$\begin{array}{}{\displaystyle VSWR=\frac{1+\Gamma}{1-\Gamma},}\end{array}$$

and RL=20 log |r|

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