Corrections to “Lower Bounds on Q for Finite Size Antennas of Arbitrary Shape”

Oleksiy S. Kim

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    Equations (24) and (25) in [1, Appendix B] should, respectively, read as \begin{align*}&\hspace {-2pc}\int \nolimits _{V_\infty }-(\nabla G_{1}) G_{2}^{*} - {\hat {\boldsymbol {r}}} jk\frac {e^{jk( {\boldsymbol {r}}_{1}- {\boldsymbol {r}}_{2})\cdot {\hat {\boldsymbol {r}}} }}{16\pi ^{2}| {\boldsymbol {r}}|^{2}} {dV} =-\frac { {\boldsymbol {r}}_{12}}{| {\boldsymbol {r}}_{12}|}\frac {\cos (k| {\boldsymbol {r}}_{12}|)}{8\pi } \notag \\&-\, j\frac {2 {\boldsymbol {r}}_{1}}{8\pi k^{2}}\left ({ \frac {\sin (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{3}}-\frac {k\cos (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{2}} }\right )\notag \\&-\, j\frac {| {\boldsymbol {r}}_{1}|^{2}-| {\boldsymbol {r}}_{2}|^{2}}{8\pi k^{2}}\frac { {\boldsymbol {r}}_{12}}{| {\boldsymbol {r}}_{12}|^{2}}\notag \\&\times \left ({ \frac {k^{2}\sin (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|}- 3\left ({\frac {\sin (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{3}}-\frac {k\cos (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{2}} }\right )}\right ) \end{align*} and \begin{align*}&\hspace {-2pc}\int \nolimits _{V_\infty } j(\nabla G_{1}) G_{2}^{*} - {\hat {\boldsymbol {r}}} k\frac {e^{jk( {\boldsymbol {r}}_{1}- {\boldsymbol {r}}_{2})\cdot {\hat {\boldsymbol {r}}} }}{16\pi ^{2}| {\boldsymbol {r}}|^{2}} {dV} =j\frac { {\boldsymbol {r}}_{12}}{| {\boldsymbol {r}}_{12}|}\frac {\cos (k| {\boldsymbol {r}}_{12}|)}{8\pi } \\&{-} \frac { {\boldsymbol {r}}_{12}}{8\pi k^{2}}\left ({ \frac {\sin (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{3}}-\frac {k\cos (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{2}} }\right ) \\&{-} \frac { {\boldsymbol {r}}_{1}+ {\boldsymbol {r}}_{2}}{8\pi k^{2}}\left ({ \frac {\sin (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{3}}-\frac {k\cos (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{2}} }\right ) \\&{-} \frac {| {\boldsymbol {r}}_{1}|^{2}-| {\boldsymbol {r}}_{2}|^{2}}{8\pi k^{2}}\frac { {\boldsymbol {r}}_{12}}{| {\boldsymbol {r}}_{12}|^{2}} \\&\times \left ({ \frac {k^{2}\sin (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|}-3\left ({\frac {\sin (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{3}}-\frac {k\cos (k| {\boldsymbol {r}}_{12}|)}{| {\boldsymbol {r}}_{12}|^{2}} }\right )}\right ) \\=&j\frac { {\boldsymbol {r}}_{12}}{2} \text {Re}\{G_{12}\}-\frac {1}{2 k^{2}} \text {Im}\{\nabla _{1} G_{12}\} \\&{-}\frac { {\boldsymbol {r}}_{1}+ {\boldsymbol {r}}_{2}}{2 k^{2}} \text {Im}\left \{{\nabla _{1} G_{12}\cdot \frac { {\boldsymbol {r}}_{12}}{| {\boldsymbol {r}}_{12}|^{2}}}\right \} \\&{+}\frac {| {\boldsymbol {r}}_{1}|^{2}-| {\boldsymbol {r}}_{2}|^{2}}{2k^{2}| {\boldsymbol {r}}_{12}|^{2}} \text {Im}\{ {\boldsymbol {r}}_{12}k^{2}G_{12}+3\nabla _{1} G_{12}\}. \end{align*}
    Original languageEnglish
    JournalI E E E Transactions on Antennas and Propagation
    Issue number5
    Pages (from-to)2773-2773
    Number of pages1
    Publication statusPublished - 2017

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