# 偶極子天線

## 直天線的理論分析

${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)\ =\ {\frac {\mu _{0)){4\pi ))\int _{\mathbb {V} '}{\frac {\mathbf {J} (\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|))\,d^{3}\mathbf {r} '}$

### 積分方程法

${\displaystyle {\frac {\partial ^{2}A_{z)){\partial z^{2))}-{\frac {1}{c^{2))}{\frac {\partial ^{2}A_{z)){\partial t^{2))}=-{\frac {1}{c^{2))}{\frac {\partial ^{2}\phi }{\partial z\partial t))-{\frac {1}{c^{2))}{\frac {\partial ^{2}A_{z)){\partial t^{2))}={\frac {1}{c^{2))}{\frac {\partial E_{z)){\partial t))}$

${\displaystyle {\frac {\mu _{0)){4\pi ))\int _{\mathbb {V} '}({\frac {\partial ^{2)){\partial z^{2))}-{\frac {1}{c^{2))}{\frac {\partial ^{2)){\partial t^{2))})({\frac {J_{z}(\mathbf {r} ',\,t_{r})}{|\mathbf {r} -\mathbf {r} '|)))\,d^{3}\mathbf {r} '={\frac {1}{c^{2))}{\frac {\partial E_{z)){\partial t))}$

${\displaystyle {\frac {\mu _{0)){4\pi ))\int _{\mathbb {V} '}J_{z}(\mathbf {r} ')({\frac {\partial ^{2)){\partial z^{2))}+k^{2})({\frac {\exp {(-ik|\mathbf {r} -\mathbf {r} '|))){|\mathbf {r} -\mathbf {r} '|)))\,d^{3}\mathbf {r} '={\frac {ik}{c))E_{z))$

${\displaystyle \int _{\mathbb {V} '}J_{z}(\mathbf {r} ')({\frac {\exp {(-ik|\mathbf {r} -\mathbf {r} '|))){|\mathbf {r} -\mathbf {r} '|)))\,d^{3}\mathbf {r} '=C\cos {(kz)}-i{\frac {\omega \epsilon _{0)){2k))U\sin {(k|z|)))$

### 細空心圓柱形天線

${\displaystyle \int _{-L/2}^{L/2}I(z')({\frac {\exp {(-ik|\mathbf {r} -\mathbf {r} '|))){|\mathbf {r} -\mathbf {r} '|)))\,d^{3}\mathbf {r} '=C\cos {(kz)}-i{\frac {\omega \epsilon _{0)){2k))U\sin {(k|z|)))$

${\displaystyle I(z,t)=I_{0}cos({\frac {2\pi }{\lambda ))(L/2-|z|))\cos(\omega t)}$

${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)\ =\ {\frac {\mu _{0}I_{0}\mathbf {\hat {z)) }{4\pi ))\int _{-L/2}^{L/2}{\frac {\cos(k(L/2-|z|'))\cos(\omega t_{r})}{|\mathbf {r} -\mathbf {r} '|))\,dz'}$

${\displaystyle r>>L}$
${\displaystyle r>>\lambda }$
${\displaystyle r>>L^{2}/\lambda }$

${\displaystyle \mathbf {A} (\mathbf {r} ,\,t)\ =\ {\frac {\mu _{0}I_{0}\mathbf {\hat {z)) }{4\pi r))\int _{-L/2}^{L/2}\cos(k(L/2-|z|'))\cos(\omega t-kr+kz'\cos {\theta })dz'={\frac {\mu _{0}I_{0}\cos(\omega t-kr)\mathbf {\hat {z)) }{2\pi kr))({\frac {\cos {(kL/2\cos {\theta })-\cos {(kL/2)))}{\sin ^{2}{\theta ))})}$

${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} \approx {\frac {\mu _{0}I_{0}\sin {(\omega t-kr)}\mathbf {\hat {\phi )) }{2\pi r))({\frac {\cos {(kL/2\cos {\theta })-cos{(kL/2)))}{\sin {\theta ))})}$

${\displaystyle {\frac {dP}{d\Omega ))={\frac {\mu _{0}cI_{0}^{2)){8\pi ^{2))}({\frac {\cos {(kL/2\cos {\theta })-cos{(kL/2)))}{\sin {\theta ))})^{2))$

${\displaystyle P={\frac {\mu _{0}cI_{0}^{2)){2\pi \sin ^{2}(kL/2)))\{\gamma +\ln(kL)-\operatorname {Ci} (kL)+{\tfrac {1}{2))\sin(kL)\operatorname {Si} (2kL)-2\operatorname {Si} (kL)+{\tfrac {1}{2))\cos(kL)[\gamma +\ln(kL/2)+\operatorname {Ci} (2kL)-2\operatorname {Ci} (kL)]\}=R_{\mathrm {dipole} }I_{0}^{2))$

## 參考文獻

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4. ^ Rouse, Margaret. Dipole Antenna. Online IT Encyclopedia. TechTarget.com. 2003 [April 29, 2013]. （原始內容存檔於2020-10-27）.
5. ^ Balanis, Constantine A. Modern Antenna Handbook. John Wiley & Sons. 2011: 2.3 [2016-07-06]. ISBN 1118209753. （原始內容存檔於2014-07-20）.
6. ^ 約翰·戴維·傑克遜著，朱培豫譯. 经典电动力学. 人民教育出版社. 1979: 444-446.