This method was used to calculate the characteristic impedance of Bow-Tie antenna [11] . The theoretical quasi-static impedance [3] of an infinite Bow-Tie printed on a dielectric half-space, obtained using conformal mapping is given by:

where is the free space wave impedance (=),is the substrate dielectric constant and represents a complete elliptic integral function with argument k,

and: complementary function given by

Based on Equation (1) a matlab code is built and the results are shown in Figure 2. Referring to Figure 2, it can be noticed that a good agreement is found with the curves the curve of that was published in [10] , in which it has been calculated as a function of bow-tie angle for different values of.

In a multi-section quarter-wave transformer used to match two transmission lines with different characteristic impedances, the change in impedance level is obtained in a number of discrete steps. An alternative is to use a tapered transition which has characteristic impedance that varies continuously in a smooth fashion from the impedance of one line to that of the other line. A transition or matching section, of this type is referred to as a tapered transmission line [12] , Figure 3.

Wide line:

Narrow line:

Taking the average value of Z(z):

where is the half width at impedance, is the half width at impedance [1] [13] ,

Figure 3, , are the effective dielectric constants at and, h is the height of dielectric

substrate, is the characteristic impedance function of distance z along the taper and is the length of the taper.

The problem of finding the input terminal resistance and also reactance taking into account the effect of conductor thickness is most simply approach by Schelkunoff’s treatment of the biconical antenna. This method was used to determine the characteristic impedance of finite biconical antenna [2] . This is analogues to finite or terminated transmission line. A TEM mode wave can exist along the biconical conductor, but in the space beyond the cones transmission can be only in higher order modes. Schelkunoff’s has defined the sphere coinciding with the ends of the cones as a boundary sphere and the sphere radius is equal to radius of cone (r = L₁). Inside the sphere TEM wave can exist and also higher order modes may be presented, but outside only the higher order modes can exist. At the cones most of the outgoing TEM wave is reflected, but near the equator most of the energy escapes, Figure 4(a). But a step of imagine from the impedance view point, the magnetic shell acting as a load impedance Z_L connected across the open end of the cones. Neglecting the end caps of the cones, the finite biconical antenna can now be treated as a transmission line of characteristic impedance Z_k terminated in the load impedance Z_L, Figure 4(b). If the load impedance Z_L can be found, the impedance Z_in1 at the input terminal of biconical antenna can be calculable.

where is the characteristic impedance of finite biconical antenna and is the half angle of the cone.

where is the phase constant for bow-tie width, Figure 1, is the free space wavelength, is the guide wave length, and was derived by substituting instead of in Equation (4).

Schelkunoff’s method was used to determine Z_L. This method consists first of calculating Z_m at current maximum on a very thin biconical antenna, a sinusoidal current distribution being assumed, where Z_m is the impedance which appears between the current maximum on one cone and corresponding point on other cone. Since this impedance occurs at from the open end of the antenna, Z_L is then equal to Z_m transformed a line, Figure 5(a). Finally, the input impedance Z_in1 is Z_L transformed over a line of characteristic impedance Z_k and length L₁, Figure 5(b).

and since, so

where is the phase constant, and x is equal to.

is the radiation resistance at a current maximum of a very thin linear antenna, is the radiation reactance, is the length of one cone [2] , Figure 5. Schelkunoff’s has extended his analysis for thin biconical antennas as outlined above, to thin antennas of other shapes by considering the characteristic impedance of the antenna. So, on the same manner based on biconical approximation method, substitute in Equation (11) and Equation (13) to get the input impedance of Bow-Tie antenna and load impedance using characteristic impedance based on quasi static method Equation (1), tapered transmission line method Equation (9).

The input impedance of Bow-Tie antenna and load impedance, Figure 1 are given by:

where is the characteristic impedance for different methods quasi static, Equation (1), tapered transmission line, Equation (9) and biconical methods, Equation (10).

A Bow-Tie antenna, Figure 6 was designed [14] . This antenna is fed with a characteristic impedance of value Z_o = 300 Ω at f = 900 MHz. The corresponding parameters of this antenna are shown in Table 1.

The general input impedance, Figure 1 is given by

where is the phase constant for feeding line width, Figure 1, was derived by substituting instead of in Equation (6).

The aforementioned methods were used to calculate the Bow-Tie antenna input impedance, Figure 7 and Figure 8. These figures indicate that the aforementioned conventional analytical methods for obtaining the Bow-Tie input impedance gave accurate results as indicated in Table 2. Also, proposed methodsare used to form alternative equation for solving the input impedance of Bow-Tie antenna presented in Section 2.4.

There are two proposed methods to obtain input impedance of Bow-Tie antenna, analytical and graphical.

In this section the real and imaginary parts of the input impedance of the Bow-Tie antenna are calculated using IE3D simulator, CST simulator, and the adopted analytical and graphical methods for the same geometric dimensions stated before, Figure 11 and Figure 12.

Referring to Figure 11 and Figure 12, it can be noticed that the proposed analytical input impedance calculation gives good result, while the proposed graphical method gives more accurate result exactly similar to Zeland IE3D, Table 6, because in this method we depend on Zeland software package in obtaining the graphical input impedance results. Also, only the frequency bandwidth from 700 MHz to 1000 MHz is considered that is why the Bow-Tie antenna real part input impedance is quite similar with that using graphical method except at the low frequency (0.5 - 0.7GHz) and at the high frequency (over 1 GHz).

The dimensions of the modified bow-tie antenna are given in Table 7. The T-matched height y-parameter is varied starting from 6 mm to 8 mm, other dimension, Table 7 are kept at the same values, and the simulations were carried out, Figure 21(a). It is quite clear at y = 7 mm the antenna can achieves acceptable matching and

bandwidth, below and above this value both matching and bandwidth are degraded. On the same manner W_T-parameter is varied starting from 0.6 mm to 1.2 mm. It can be noticed from Figure 21(b) that as W_T value increases, both antenna matching and bandwidth are enhanced increase until W_T reaches 1 mm after that both matching and bandwidth are degraded. Also, for b-parameter which is varied from 17.54 mm to 19.54 mm, Figure 21(c), good results occurs at b = 18.54 mm.

By changing the value of slot width W_slot from 0.6 mm to 0.8 mm, Figure 21(d). It can be noticed that slot width is inversely proportional with antenna matching performance and bandwidth, as it decreases they increase, so good result achieved at W_slot = 0.6mm. The results are compared with conventional one, Table 8. The obtained bandwidth is extended from 859 MHz to 1.02 GHz, which indicates that the antenna achieves a wide bandwidth that covers the chip operating frequency, Figure 22. The antenna input impedance is which has a good matching to chip impedance at frequency 900 MHz.

It can be noticed from Figure 22 that due to the presence of the slot, a second resonant frequency of 995 MHz can be properly excited close to the fundamental mode. The ratio of the frequency of the two modes is made low enough to realize a wideband operation thus able to cover the entire UHF band [16] . The reading range of conventional and modified Bow-Tie tag antennas [17] with readers able to transmit at maximum power P_EIRP = 4 watt for most of that bandwidth, P_chip = −20 dBm [18] , Z_chip and Z_A is shown in Figure 23.

The maximum directivity for modified Bow-Tie antenna increased to 3.3 dBi, as the electrical path has been increased. It can be indicated from Figure 19 that the existence of slots in the modified Bow-Tie antenna increases the bandwidth. For RFID applications, the antenna efficiency and antenna gain defined by IE3D may not be useful as stated in Table 8 because RFID devices behave more like constant voltage sources. For constant voltage sources, it is required to achieve conjugate matching for best efficiency [15] [19] . The current distribution for both conventional and modified bow-tie antennas are shown in Figure 24. The arrows in Figure 24(a) and Figure 24(b) indicates the average current density path through the antennas where their lengths indicate the amplitude which is of same magnitude for both antenna sides but with 180˚ out of phase, as well known for the dipoles shapes.

The simulated coplanar and cross polarization radiation pattern of both conventional and modified bow-tie antennas are shown in Figure 25(a) and Figure 25(b). In this figure, the E_theta-field is the co-polar component,

and the E_phi-field is the cross-polar component. These patterns show that the designed antenna provides linear polarization with a cross polarization level about −30.3 dBi lower than the co-polarization level −8.9 dBi for conventional bow-tie antenna and about −27.4 dBi lower than the co-polarization level −10.29 dBi for modified bow-tie antenna.

Figure 26 shows the radiation pattern of both conventional and modified bow-tie antennas which is almost omnidirectional at the H-plane and figure of eight at the E-plane that is suitable for the RFID surveillance application

[15] . The design procedure is given by the flow chart, Figure 27, where the proposed graphical method is used to find the initial geometry of the Bow-Tie antenna. Analytical method is used to calculate the antenna input impedance. IE3D simulator is used to confirm the result. If the required antenna bandwidth and required antenna matching to chip impedance are achieved, the antenna design is finished else a modification is needed and the result again has to be confirmed with IE3D simulator.