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    Journals >High Power Laser and Particle Beams >Volume 34 >Issue 9 >Page 095010 > Article
    • High Power Laser and Particle Beams
    • Vol. 34, Issue 9, 095010 (2022)
    Analytical method of triplate transmission lines configuration
    Chongyang Mao, Guanqiong Wang, and Xiaoguang Wang
    Author Affiliations
  • Institute of Applied Physics and Computational Mathematics , Beijing 100094, China
    • show less
    DOI: 10.11884/HPLPB202234.210469 Cite this Article
    Chongyang Mao, Guanqiong Wang, Xiaoguang Wang. Analytical method of triplate transmission lines configuration[J]. High Power Laser and Particle Beams, 2022, 34(9): 095010 Copy Citation Text show less

    Abstract

    Triplate transmission line is widely used in huge pulsed power facilities. However, the accurate circuit parameters of the triplate transmission line are difficult to obtain because of three factors. The first one is that the widths of the anode and cathode plates are different, the second one is that the plates’ thickness cannot be ignored, and the third one is that the electromagnetic field near the edges of the plates is seriously distorted. Taking all these three factors into consideration, an analytical method was established to obtain the capacitance per meter, the characteristic impedance, and the electric field magnitudes of the triplate transmission line. In this method, the distorted boundaries are transformed into regular boundaries based on conformal mapping and accurate results are obtained without approximation. Results obtained by this method are the same as those obtained by electromagnetic simulation with a commercial code called Ansoft, which indicates the correctness of this method.

    Keywords

    $ {Z_0} = \dfrac{1}{{vC}} $(1)

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    $ {\textit{z}} = \dfrac{{{d_2} - {d_1}}}{{\displaystyle\int_0^2 {\sqrt {\dfrac{{\left( {p - {t_3}} \right)\left( {p - {t_6}} \right)\left( {p - {t_7}} \right)}}{{\left( {p - {t_2}} \right)p\left( {p - 2} \right)\left( {p - {t_8}} \right)}}} {\rm{d}}p} }}\displaystyle\int_0^t {\sqrt {\dfrac{{\left( {p - {t_3}} \right)\left( {p - {t_6}} \right)\left( {p - {t_7}} \right)}}{{\left( {p - {t_2}} \right)p\left( {p - 2} \right)\left( {p - {t_8}} \right)}}} {\rm{d}}p} + {d_1} $(2)

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    $ \left\{ {∫t2t3(t3−p)(t6−p)(t7−p)(p−t2)(−p)(2−p)(t8−p)dp−d1d2−d1∫02(p−t3)(t6−p)(t7−p)(p−t2)p(2−p)(t8−p)dp=0∫t30(p−t3)(t6−p)(t7−p)(p−t2)(−p)(2−p)(t8−p)dp−h1d2−d1∫02(p−t3)(t6−p)(t7−p)(p−t2)p(2−p)(t8−p)dp=0∫2t6(p−t3)(t6−p)(t7−p)(p−t2)p(p−2)(t8−p)dp−h2d2−d1∫02(p−t3)(t6−p)(t7−p)(p−t2)p(2−p)(t8−p)dp=0∫t6t7(p−t3)(p−t6)(t7−p)(p−t2)p(p−2)(t8−p)dp−d3−d2d2−d1∫02(p−t3)(t6−p)(t7−p)(p−t2)p(2−p)(t8−p)dp=0∫t7t8(p−t3)(p−t6)(p−t7)(p−t2)p(p−2)(t8−p)dp−h2d2−d1∫02(p−t3)(t6−p)(t7−p)(p−t2)p(2−p)(t8−p)dp=0 } \right. $(3)

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    $ t = \dfrac{{a + b\zeta }}{{1 + c\zeta }} $(4)

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    $ \left\{ {a=t2t8+t2t8(t2−2)(t8−2)t2+t8−2b=ac=a−1g=(1−t2)a+t2a−t2 } \right. $(5)

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    $ t = \dfrac{{a + a\zeta }}{{1 + a\zeta - \zeta }} $(6)

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    $ \zeta = \dfrac{{t - a}}{{a + t - at}} $(7)

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    $ w = \dfrac{m}{{\displaystyle\int_0^1 {\dfrac{1}{{\sqrt {\left( {1 - {q^2}} \right)\left( {1 - {g^2}{q^2}} \right)} }}{\rm{d}}s} }}\displaystyle\int_0^\zeta {\dfrac{1}{{\sqrt {\left( {1 - {q^2}} \right)\left( {1 - {g^2}{q^2}} \right)} }}{\rm{d}}q} $(8)

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    $ m = \displaystyle\int_0^1 {\dfrac{1}{{\sqrt {\left( {1 - {q^2}} \right)\left( {1 - {g^2}{q^2}} \right)} }}{\rm{d}}q} = K\left( g \right) $(9)

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    $ w = \displaystyle\int_0^\zeta {\dfrac{1}{{\sqrt {\left( {1 - {q^2}} \right)\left( {1 - {g^2}{q^2}} \right)} }}{\rm{d}}q} $(10)

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    $ m + {\text{i}}n = \displaystyle\int_0^{{1 \mathord{\left/ {\vphantom {1 g}} \right. } g}} {\dfrac{1}{{\sqrt {\left( {1 - {q^2}} \right)\left( {1 - {g^2}{q^2}} \right)} }}{\rm{d}}q} $(11)

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    $ n = \displaystyle\int_1^{{1 \mathord{\left/ {\vphantom {1 g}} \right. } g}} {\dfrac{1}{{\sqrt {\left( {{q^2} - 1} \right)\left( {1 - {g^2}{q^2}} \right)} }}{\rm{d}}q} = \displaystyle\int_0^1 {\dfrac{1}{{\sqrt {\left( {1 - q{'^2}} \right)\left( {1 - g{'^2}q{'^2}} \right)} }}{\rm{d}}q'} = K\left( {g'} \right) $(12)

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    $ q = \varepsilon En $(13)

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    $ C = \dfrac{{4q}}{V} = \dfrac{{4\varepsilon En}}{V} = \dfrac{{2\varepsilon n}}{m} $(14)

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    $ {Z_0} = \dfrac{1}{{vC}} = \dfrac{1}{{\dfrac{1}{{\sqrt {\mu \varepsilon } }} \cdot \dfrac{{2\varepsilon n}}{m}}} = \dfrac{{\sqrt \mu m}}{{2\sqrt \varepsilon n}} $(15)

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    $ {\boldsymbol{E}} = - \left( {\dfrac{{\partial u}}{{\partial x}}{{\boldsymbol{e}}_x} + \dfrac{{\partial u}}{{\partial y}}{{\boldsymbol{e}}_y}} \right) $(16)

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    $ \left| {\boldsymbol{E}} \right| = \sqrt {{{\left( {\dfrac{{\partial u}}{{\partial x}}} \right)}^2} + {{\left( {\dfrac{{\partial u}}{{\partial y}}} \right)}^2}} $(17)

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    $ \dfrac{{\rm{d}}w}{{\rm{d}{\textit{z}}}} = \dfrac{{\partial u}}{{\partial x}} + {\rm{i}}\dfrac{{\partial v}}{{\partial x}} = \dfrac{{\partial u}}{{\partial x}} - {\rm{i}}\dfrac{{\partial u}}{{\partial y}} $(18)

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    $ \left| {\dfrac{{\rm{d}}w}{{\rm{d}{\textit{z}}}}} \right| = \left| {\dfrac{{\partial u}}{{\partial x}} - {\rm{i}}\dfrac{{\partial u}}{{\partial y}}} \right| = \sqrt {{{\left( {\dfrac{{\partial u}}{{\partial x}}} \right)}^2} + {{\left( {\dfrac{{\partial u}}{{\partial y}}} \right)}^2}} = \left| {\boldsymbol{E}} \right| $(19)

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    $ \dfrac{{{\rm{d}}w}}{{{\rm{d}}{\textit{z}}}} = \dfrac{{{\rm{d}}w}}{{{\rm{d}}\zeta }}\dfrac{1}{{\dfrac{{{\rm{d}}t}}{{{\rm{d}}\zeta }}}}\dfrac{1}{{\dfrac{{{\rm{d}}{\textit{z}}}}{{{\rm{d}}t}}}} $(20)

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    $ \dfrac{{{\rm{d}}{\textit{z}}}}{{{\rm{d}}t}} = \dfrac{{{d_2} - {d_1}}}{{\displaystyle\int_0^2 {\sqrt {\dfrac{{\left( {p - {t_3}} \right)\left( {p - {t_6}} \right)\left( {p - {t_7}} \right)}}{{\left( {p - {t_2}} \right)p\left( {p - 2} \right)\left( {p - {t_8}} \right)}}} dp} }}\sqrt {\dfrac{{\left( {t - {t_3}} \right)\left( {t - {t_6}} \right)\left( {t - {t_7}} \right)}}{{\left( {t - {t_2}} \right)t\left( {t - 2} \right)\left( {t - {t_8}} \right)}}} $(21)

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    $ \dfrac{{{\rm{d}}t}}{{{\rm{d}}\zeta }} = \dfrac{{a\left( {2 - a} \right)}}{{{{\left[ {1 + \left( {a - 1} \right)\zeta } \right]}^2}}} $(22)

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    $ \dfrac{{{\rm{d}}w}}{{{\rm{d}}\zeta }} = \dfrac{1}{{\sqrt {\left( {1 - {\zeta ^2}} \right)\left( {1 - {g^2}{\zeta ^2}} \right)} }} $(23)

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    $ \dfrac{{{\rm{d}}w}}{{{\rm{d}}{\textit{z}}}} = \dfrac{{\displaystyle\int_0^2 {\sqrt {\dfrac{{\left( {p - {t_3}} \right)\left( {p - {t_6}} \right)\left( {p - {t_7}} \right)}}{{\left( {p - {t_2}} \right)p\left( {p - 2} \right)\left( {p - {t_8}} \right)}}} {\rm{d}}p} }}{{a\left( {2 - a} \right)\left( {{d_2} - {d_1}} \right)}}\dfrac{{{{\left[ {1 + \left( {a - 1} \right)\zeta } \right]}^2}}}{{\sqrt {\left( {1 - {\zeta ^2}} \right)\left( {1 - {g^2}{\zeta ^2}} \right)} }}\sqrt {\dfrac{{\left( {t - {t_2}} \right)t\left( {t - 2} \right)\left( {t - {t_8}} \right)}}{{\left( {t - {t_3}} \right)\left( {t - {t_6}} \right)\left( {t - {t_7}} \right)}}} $(24)

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    $ \left| {\boldsymbol{E}} \right| = \left| {\dfrac{{\displaystyle\int_0^2 {\sqrt {\dfrac{{\left( {p - {t_3}} \right)\left( {p - {t_6}} \right)\left( {p - {t_7}} \right)}}{{\left( {p - {t_2}} \right)p\left( {p - 2} \right)\left( {p - {t_8}} \right)}}} {\rm{d}}p} }}{{a\left( {2 - a} \right)\left( {{d_2} - {d_1}} \right)}}\dfrac{{{{\left[ {1 + \left( {a - 1} \right)\zeta } \right]}^2}}}{{\sqrt {\left( {1 - {\zeta ^2}} \right)\left( {1 - {g^2}{\zeta ^2}} \right)} }}\sqrt {\dfrac{{\left( {t - {t_2}} \right)t\left( {t - 2} \right)\left( {t - {t_8}} \right)}}{{\left( {t - {t_3}} \right)\left( {t - {t_6}} \right)\left( {t - {t_7}} \right)}}} } \right| $(25)

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    Chongyang Mao, Guanqiong Wang, Xiaoguang Wang. Analytical method of triplate transmission lines configuration[J]. High Power Laser and Particle Beams, 2022, 34(9): 095010
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