# Design of Two Compact Wideband Monopoles Through Loading with Linear Approximated Lumped Components
## 2 Antenna Design  
### Parallel Lumped Load of Monopole  
Compared with ideal lumped components, practical lumped components behave differently across a wide bandwidth, especially coil inductors. Since the parasitic capacitance between each turn of a coil inductor is not negligible, this capacitance needs to be considered carefully in wideband applications. With the parasitic capacitance, a coil inductor would act as a parallel RC circuit and might resonate in a wide bandwidth, as shown in Figure 1. The impedance of a coil inductor increases more rapidly with respect to frequency. In contrast to a constant-value element, the effective inductance \(L_{effective}\) as a function of frequency can be deduced from the classical impedance function of an inductor. \[Z=j\omega L_{effective} \tag{1}\]  
A coil inductor with N = 3, \(p\) = 0, D = 1 cm, and d= 1.5 cm has an effective inductance of 30-800 MHz, as Figure 2 illustrates. Before resonating, the coil inductor still exhibits inductance and varies with frequency. As Figure 2 depicts, the inductance of the coil inductor could be further simplified to a linear frequency-dependent approximation as follows:  
\[L_{appro}(k)=b_{L}\cdot(A_{L}\cdot k+B_{L}) \tag{2}\]  
with  
\[A_{L}=\frac{M_{L}-1}{f_{2}-f_{1}}\cdot\frac{c_{0}}{2\pi} \tag{3}\]  
\[B_{L}=1-\frac{M_{L}-1}{f_{2}-f_{1}}\cdot f_{1} \tag{4}\]  
Figure 1: Coil inductor with parasitic capacitor behaves as parallel resonant circuit. Figure 2: Inductance of coil inductor varies versus frequency in wide bandwidth. where \(L_{approx}\) as a function of the wave number \(k\) is the approximate inductance of the concerned band, \(f_{1}\) and \(f_{2}\) are the lower and upper frequency of the concerned band, \(M_{L}\) is the ratio of the inductance at the upper frequency to the inductance at the lower frequency, \(c_{0}\) is the velocity of light, and \(b_{L}\) is the inductance at the lower frequency. In addition, a similar approximation could also apply to the practical patch resistance and capacitance as follows:  
\[R_{approx}(k)=b_{R}\cdot(A_{R}\cdot k+B_{R}) \tag{5}\]  
\[C_{approx}(k)=b_{C}\cdot(A_{C}\cdot k+B_{C}) \tag{6}\]  
where \(R_{approx}\) and \(C_{approx}\) are the approximated resistance and capacitance of the practical patch resistor and capacitor, respectively. \(b_{R}\), \(A_{R}\), \(B_{R}\), \(b_{C}\), \(A_{C}\), and \(B_{C}\) are the corresponding coefficients of linear frequency-dependent models, similar to the approximation of the inductor. Even though the resistance of the practical patch resistor and the capacitance of the practical capacitor would not vary intensively like the coil inductor in a wide bandwidth, alleviation could be made for the tuning process since there would be multiple loads arranged on the monopole. The impedance of the loaded parallel lumped RLC circuit as a function of wave number can be expressed as follows:  
\[Z(k)=\frac{1}{1/R+1/(jc_{0}kL)+jc_{0}kC} \tag{7}\]  
Substituting (2), (5), and (6) into (7) gives the following:  
\[Z=j\omega L_{effective} \tag{1}\]0  
with  
\[Z=j\omega L_{effective} \tag{1}\]1  
\[Z=j\omega L_{effective} \tag{1}\]2  
\[Z=j\omega L_{effective} \tag{1}\]3  
\[Z=j\omega L_{effective} \tag{1}\]4  
\[Z=j\omega L_{effective} \tag{1}\]5  
\[Z=j\omega L_{effective} \tag{1}\]6  
\[Z=j\omega L_{effective} \tag{1}\]7  
\[Z=j\omega L_{effective} \tag{1}\]8  
\[Z=j\omega L_{effective} \tag{1}\]9  
\[L_{appro}(k)=b_{L}\cdot(A_{L}\cdot k+B_{L}) \tag{2}\]0  
To obtain an accurate simulation result across a wide bandwidth for the loaded antenna, adequate sweep points should be set in the concerned wide band range. Since too many frequency points have to be covered, sweeping these points discretely by the method of moments (MoM) would be time-consuming. Employing the asymptotic waveform evaluation (AWE) procedure could accelerate the sweeping process in the simulation, since a bandwidth ratio (BWR) of over 2:1 could be covered by AWE [37].