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The dual band equilateral triangular microstrip antennas are realized by cutting the slots of either quarter wave or half wave in length, inside the patch. In this design, however these simpler approximations of slot length against the frequency do not give closer results for different slot lengths and there positions inside the patch. In this paper, the modal variations of slot cut patch antennas over wide frequency range are studied. It is observed that the slot does not introduce any mode but reduces the higher order mode resonance frequency of the patch and along with the fundamental mode realizes dual band response. The formulations of the resonant length for the mode introduce by the slots in these antennas are proposed. The resonance frequencies calculated using proposed formulations agree well with the simulated results with an error of less than 5%.

The dual band microstrip antenna (MSA) is realized by cutting a slot at an appropriate position inside the patch [1-3]. Since the slots are cut inside the patch, they neither increase the patch size nor largely affect the radiation pattern of the patch. These slots can take different shapes like, rectangular or square slot, step slot, tooth-brush shaped slot, V-slot, U-slot, etc [4-8]. The slot adds another resonant mode near the fundamental mode of the patch and realizes dual frequency response. The dual band equilateral triangular MSA (ETMSA) have also been realized by cutting the pair of rectangular or U-slot inside the patch [4-9]. While designing these slotted MSAs for the required frequency, depending upon where they are cut in the patch, the slot length is taken equal to nearly half wave or quarter wave in length. Towards the shorted end of the slot, the surface currents encircles over a finite length, which stores magnetic energy towards the shorted slot end [

In this paper, the slot frequencies for U-slot or rectangular slots cut dual band ETMSAs, calculated by using their half wave or quarter wave length approximations are compared with the simulated values obtained using IE3D software [_{11} mode of ETMSA and along with its fundamental TM_{10} mode realizes dual band response. The radiation pattern at the TM_{10} mode for ETMSA is in the broadside direction whereas at TM_{11} mode it is in the conical direction. Since the slot affects the current distribution on the patch at TM_{11} mode, with the increasing slot length the radiation pattern at the modified TM_{11} frequency becomes in the broadside direction. The cross-polarization levels decreases with the increase in slot length. The surface current distributions for slot cut ETMSAs were studied for different slot lengths and the resonant length formulation for the slot (modified higher order) mode and patch mode is proposed. The resonant frequencies calculated using the proposed formulations agrees well with the simulated results with an error of less than 5% over the entire slot length range. The formulations are first proposed on glass epoxy substrate (e_{r} = 4.3, h = 0.16 cm, and tan d = 0.02) and further they are validated on RT-Duroid substrate (e_{r} = 2.33, h = 0.16 cm, and tan d = 0.001).

In most of the reported slotted antennas the slot length is taken to be equal to half or quarter wave in length. In dual band pair of rectangular slots cut ETMSA as shown lengths (l) from 0.5 to 4.0 cm, the equivalent slot resonant length is calculated by adding the correction length as given in Equation (1). The slot frequency is calculated by using Equation (2). Since the surface currents circulate around the slot, the effective dielectric constant (e_{re}) is calculated by using the average of the widths on the two sides of the slot [

Using this approach for some values of Y and only for larger l, a smaller E is obtained. The dual band U-slot cut ETMSA is shown in _{re} is calculated by using the inner U-slot width (L_{v} – 2w). The resonant length and frequency is calculated by using Equations (4) to (6). For different values of L_{v} and L_{h}, the slot frequency calculated by using the following equations and that obtained using IE3D are shown in Figures 2(b-d). For all the values of U-slot dimension, a larger E as calculated by using Equation (3), is observed.

Thus the quarter wave or half wave length approximations of the slot length against the frequency do not give closer results. Hence in-depth analysis of slot cut MSAs is required to understand the slot mode. The ETMSA in above configurations, on glass epoxy substrate has, TM_{10 }and TM_{11} mode resonance frequencies of 958 and 1690 MHz as shown in its resonance curve plot in _{10} mode the currents show half wavelength variation along side length (S) whereas at TM_{11} mode the currents show half wave length variations along S as well as along base of the ETMSA [_{1} = 940 MHz and f_{2} = 1176 MHz, as shown in the resonance curve plot in _{11} mode resonance frequency from 1690 to 1176 MHz as shown in _{2} is governed by this modified (due to the slots) TM_{11} resonance frequency. The f_{1 }is governed by TM_{10} mode of ETMSA, as half wavelength variation in surface currents is present along S. This frequency is also marginally affected by slots as shown in _{11} mode and for slot cut ETMSA for two different slot lengths are shown in Figures 4(a-c). At TM_{11} mode, the pattern is in the conical direction. However since the slots affects the current distributions on the patch, with the increasing slot length the radiation pattern becomes in the broadside direction. Also as observed from the patterns, the cross polarization levels decreases with the slot length.

Similar study is carried out for U-slot cut ETMSA. The resonance curve plot for ETMSA and for U-slot cut ETMSA (U-slot dimension: L_{h} = 3.0, L_{v} = 1.0, w = 0.2 cm) are shown in _{11} mode resonance frequency and f_{2} is governed by this modified TM_{11} mode. This frequency has reduced from 1690 to 1324 MHz. The f_{1} is governed by TM_{10} mode. This frequency is also reduced since the vertical U-slot length is orthogonal to surface currents at TM_{10} mode and it increases the surface currents length. The radiation patterns at f_{2} for two different U-slot lengths are shown in Figures 5(d, e). Similar to the above dual band ETMSA, the U-slot affects the current distributions on patch at f_{2} and gives broadside radiation pattern. The cross polar level decreases with increasing U-slot length. In the following section by studying current distributions at dual frequencies the formulation of resonant length at dual frequencies for the above slot cut ETMSAs are proposed.

The modal variations for ETMSA in voltage and currents are with respect to the centroid of the patch [_{10} mode the current maxima is present at the centroid point whereas at TM_{11} mode the voltage maximum is present at the centroid point [

cut ETMSA as shown in _{2} is smaller whereas decrease in f_{1} is larger.

The resonance frequency of ETMSA depends upon the effective side length (S_{e}) [_{1 }(modified TM_{10} mode) for pair of slots cut ETMSA, S_{e} is modified with respect to the l and the resonance frequency is calculated by using Equations (7) to (12).

, m = 1, n = 0 for TM_{10} (patch mode) (11)

The width as seen by the surface currents varies from base of ETMSA towards the vertex point. Towards the base, it is equal to S and towards the opposite vertex point it is nearly equal zero. Hence the effective width (W_{e}) is taken equal to be the average of the width along ETMSA length as given in Equation (7). The e_{re} is calculated by using Equation (8). For smaller l, decrease in f_{1} is smaller and for larger l, decrease in f_{1} is larger. To account for this non-linear variation, a weighting function (A) in terms of l is used as given in Equation (10). The equation for A is derived based on the variation in frequency of TM_{10} mode against l. For a given l, with increase in Y, the perturbation in surface current length increases (or the frequency reduces) since the slots are placed towards the maximum current location. This variation is accounted by using the sinusoidal term, as given in Equation (9). The fringing field extension is accounted by the last term in the right hand side of Equation (9). The frequency is calculated by using Equation (11) and the % error (E) between the calculated and simulated values with respect to the simulated value, is calculated by using Equation (12) and they are plotted in Figures 6(a-f). For the entire slot length range, an E of less than 5% is obtained. The resonant length at f_{2} is formulated by modifying S_{e} with respect to l as given in Equations (13) to (16).

, m = 1, n = 1 for TM_{11} (slot mode) (15)

The factor ‘2’ in Equation (13) is to account for the circulation of surface currents around the slot length. The perturbation in surface current length is modeled by using the weighting function A and the equation for the same is derived based on the frequency variation with respect to l at TM_{11} mode. The variation in the frequency with Y, for given l, is modeled by using sinusoidal function in Equation (13). The Y is varied from 1 to 3 cm and the l is varied from 0.5 to 4.0 cm. The frequency f_{2} obtained

using simulation, calculated by using Equation (15), and E calculated by using Equation (16) are plotted in Figures 7(a-f). For all the values of Y, a close match is obtained between the simulated and calculated frequencies.

While designing the pair of rectangular slots cut ETMSA for the given two frequencies, the separation between the pair of slots is selected first as it affects the impedance matching at the two frequencies. Due to the ETMSA patch geometry, Y is taken to be less than S/4, so as to have larger variation in l and hence the frequency. Thus Y = 2 is selected. For the desired dual frequencies of 860 and 1060 MHz, the slot length obtained from Figures 6 and 7 is = 2.5 cm. The slot width of 0.2 cm is selected. The slotted ETMSA is simulated and dual frequencies and BW’s are 888, 1073 MHz and 15, 15 MHz, respectively as shown in ^{0} and 90^{0}, respectively. The pattern is in the broadside direction at both the frequencies with the cross polar levels less than 12 dB as compared to the co-polar levels. Although the radiation pattern at TM_{11} mode of ETMSA is conical, but since the pair of slots affects the current distributions on the patch at f_{2}, the radiation pattern is in the broadside direction.

As seen from the current distributions for U-slot cut ETMSA, the U-slot affects the surface current distributions at TM_{10} as well as TM_{11} mode and realizes dual band response. At f_{1}, the TM_{10} mode is dominant as half wave length variation in surface currents is present along S. However due to the finite L_{v}, the surface current length is perturbed and the frequency is reduced. At f_{2}, the horizontal slots of the U-slot modify the TM_{11} mode resonance frequency and this frequency is governed by modified TM_{11} mode. The formulation of resonant length at f_{1} is given by modifying S_{e} with respect to the U-slot dimensions as given in Equations (17) to (20).

For f_{1},

, m = 1, n = 0 for TM_{10} mode (20)

For narrow slots, (L_{v} < S/4), the perturbation in surface current length is lesser. In those cases, using B = 0 and A = 1, a closer prediction in resonant length is obtained. For L_{v} > S/4, the perturbation in surface current length increases and hence by using B = 0.6 and A = 2, a closer prediction in the length and the resonance frequency is obtained. The f_{1} calculated by using Equation (20) and E calculated by using Equation (12) are plotted in Figures 9(a-f). Using the proposed formulations, for all the slot dimensions, an E of less than 5% is obtained. At f_{2}, the formulation in resonant length is obtained by using Equations (21) to (23).

For f_{2},

, m = 1, n = 1 for TM_{11} mode (23)

Similar to pair of slots cut ETMSA, the factor ‘2’ in Equation (21) is to account for the circulation of surface currents around the slot length. To account for the current variation in TM_{11} mode a sinusoidal function in L_{v} is used in Equation (21). The A is the weighting function and the relation for A is derived based on variation in frequency with respect to U-slot dimensions for TM_{11} mode. The frequency and E are calculated by using Equations (23) and (16), respectively and they are plotted in Figures 10(a-f). For the entire range, frequencies calculated using the proposed formulations agrees closely with the simulated values. For designing the U-slot cut ETMSA, first the value of L_{v} is selected. For the dual frequencies of 900 and 1300 MHz, using L_{v} = 1.0 cm and w = 0.2 cm, the horizontal U-slot length obtained by using frequency plots in Figures 9 and 10 is, L_{h} = 3.0. This ETMSA with U-slot is simulated using IE3D and the

frequencies and BW’s are 883, 1245 MHz and 15, 18 MHz, respectively as shown in ^{0} and 90^{0}, respectively. At f_{2}, a higher cross polarization is present. The cross polar level decreases with the increase in U-slot dimensions. These proposed formulations are also validated on RTduroid substrate. Using duroid substrate, for to have

TM_{10} resonance frequency nearly equal to 950 MHz, the S equals 14 cm. For pair of rectangular slots cut ETMSA, the f_{1} and f_{2} obtained using IE3D and the proposed formulations and E between them are shown in Figures 12 and 13. For duroid substrate also a closer agreement between the two results is realized. Similar results are obtained for U-slot cut ETMSA.

The resonance frequency calculation for slot mode in pairs of rectangular slots and U-slot cut ETMSAs using quarter and half wavelength approximations are discussed. It was observed that they give closer match only for some values of slot lengths and their positions inside the patch. An in-depth modal analysis for pair of rectangular slots cut and U-slot cut ETMSA was carried out and it is observed that the slots does not introduce any mode but reduces the higher order TM_{11} resonance frequency of ETMSA and along with the fundamental TM_{10} mode realizes dual frequency response. The slots affects the current distribution on the patch at TM_{11} mode and thus with increasing slot length the radiation pattern at the frequency due to the slot becomes in the broadside direction. The cross polarization level reduces with the slot length. By studying the surface current distributions at the dual frequencies, the formulation in resonant length at dual frequencies is proposed for these slotted ETMSAs and the frequencies obtained using these formulations agree closely with the simulated as well as measured results.