Analysis of Circular Polarization of Cylindrically Bent Microstrip Antennas

<table>How to calculate the semi phase centers from the equiphase surface. (a): The astigmatic (not spherical) wave front. (b): The circular wavefront in the H-plane (<svg height="8.4250002" id="M14" style="vertical-align:-0.273pt" version="1.1" viewbox="0 0 18.0875 8.4250002" width="18.0875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g><g transform="matrix(.017,0,0,-.017,9.565,8.013)"><path d="M475 445l-64 -69q-119 -128 -289 -280q27 6 81 -15q66 -26 102 -26q48 0 111 93l23 -19q-41 -84 -75.5 -116.5t-73.5 -32.5q-37 0 -147 53q-43 21 -63 7q-22 -15 -38 -41q-4 -7 -11 -7q-8 10 -8 26q0 38 47 83l292 273q-32 -5 -67 -1q-63 7 -105 7q-32 0 -53.5 -22
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</svg>-plane) emanates from <svg height="15.25" id="M15" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 20.674999 15.25" width="20.674999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" id="x1D443"></path></g>
<g transform="matrix(.012,0,0,-.012,10.875,15.188)"><path d="M729 650v-28q-62 -5 -76 -19.5t-14 -75.5v-404q0 -62 13.5 -76t75.5 -19v-28h-268v28q65 5 79.5 19t14.5 76v199h-341v-199q0 -62 14 -76t76 -19v-28h-263v28q61 5 74.5 19t13.5 76v404q0 62 -14 76t-76 19v28h259v-28q-58 -5 -71 -19t-13 -76v-162h341v162
q0 61 -13.5 75.5t-72.5 19.5v28h261z" id="x48"></path></g>
</svg>. (c): In the E-plane (<svg height="12.5125" id="M16" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 18.4375 12.5125" width="18.4375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,8.013)"><path d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" id="x1D466"></path></g><g transform="matrix(.017,0,0,-.017,9.905,8.013)"><path d="M475 445l-64 -69q-119 -128 -289 -280q27 6 81 -15q66 -26 102 -26q48 0 111 93l23 -19q-41 -84 -75.5 -116.5t-73.5 -32.5q-37 0 -147 53q-43 21 -63 7q-22 -15 -38 -41q-4 -7 -11 -7q-8 10 -8 26q0 38 47 83l292 273q-32 -5 -67 -1q-63 7 -105 7q-32 0 -53.5 -22
t-47.5 -67l-25 13q15 39 38 78q40 66 83 66q57 0 120 -12q52 -10 80 -10q29 0 67 42z" id="x1D467"></path></g>
</svg>-plane), <svg height="15.25" id="M17" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 18.3125 15.25" width="18.3125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
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<g transform="matrix(.012,0,0,-.012,10.875,15.188)"><path d="M517 162l29 -4q-18 -100 -35 -158h-482v28q68 5 83.5 20t15.5 77v398q0 66 -13 80t-73 19v28h446q0 -8 8 -144l-29 -4q-11 51 -32 85q-11 17 -35 23t-78 6h-76q-23 0 -28 -5t-5 -27v-228h94q67 0 83 12t25 63h29v-191h-29q-10 54 -26 65.5t-81 11.5h-95v-189
q0 -65 24 -81q19 -13 104 -13q57 0 83 7t43 27q28 33 50 94z" id="x45"></path></g>
</svg> is the center of the circular wavefront. <svg height="15.25" id="M18" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 20.674999 15.25" width="20.674999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" id="x1D443"></path></g>
<g transform="matrix(.012,0,0,-.012,10.875,15.188)"><path d="M729 650v-28q-62 -5 -76 -19.5t-14 -75.5v-404q0 -62 13.5 -76t75.5 -19v-28h-268v28q65 5 79.5 19t14.5 76v199h-341v-199q0 -62 14 -76t76 -19v-28h-263v28q61 5 74.5 19t13.5 76v404q0 62 -14 76t-76 19v28h259v-28q-58 -5 -71 -19t-13 -76v-162h341v162
q0 61 -13.5 75.5t-72.5 19.5v28h261z" id="x48"></path></g>
</svg> and <svg height="15.25" id="M19" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 18.3125 15.25" width="18.3125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" id="x1D443"></path></g>
<g transform="matrix(.012,0,0,-.012,10.875,15.188)"><path d="M517 162l29 -4q-18 -100 -35 -158h-482v28q68 5 83.5 20t15.5 77v398q0 66 -13 80t-73 19v28h446q0 -8 8 -144l-29 -4q-11 51 -32 85q-11 17 -35 23t-78 6h-76q-23 0 -28 -5t-5 -27v-228h94q67 0 83 12t25 63h29v-191h-29q-10 54 -26 65.5t-81 11.5h-95v-189
q0 -65 24 -81q19 -13 104 -13q57 0 83 7t43 27q28 33 50 94z" id="x45"></path></g>
</svg> do not coincide.</table>

International Journal of Antennas and Propagation

fig3

Figure 3

Figure 3: Analysis of Circular Polarization of Cylindrically Bent Microstrip Antennas 