マイクロストリップ素子のスペクトル領域グリーン関数

マイクロストリップ素子のスペクトル領域グリーン関数の導出

地導体板付き単層誘電体基板の表面に面電流源がある場合

　マイクロストリップ素子として，厚み

d

の単層誘電体基板（比誘電率

ϵ_{r}

）の片面に地導体板を付け（

z = - d

），誘電体と空気の境界面（

z = 0

）に電流源がある場合を考える．このとき，

- d \leq z \leq 0

の誘電体を領域(1)として，スペクトル領域の電界および磁界は，

\begin{array}{rcl} (1) & {\tilde{E}}_{1} (z) & = & {\tilde{E}}_{1}^{+} e^{- j k_{z 1} z} + {\tilde{E}}_{1}^{-} e^{j k_{z 1} z} \\ (2) & {\tilde{H}}_{1} (z) & = & Y_{1} ({\tilde{E}}_{1}^{+} e^{- j k_{z 1} z} - {\tilde{E}}_{1}^{-} e^{j k_{z 1} z}) \end{array}

また，

z \geq 0

の自由空間を領域(2)として，電界および磁界は，

\begin{array}{rcl} (3) & {\tilde{E}}_{2} (z) & = & {\tilde{E}}_{2}^{+} e^{- j k_{z 2} z} \\ (4) & {\tilde{H}}_{2} (z) & = & Y_{2} {\tilde{E}}_{2}^{+} e^{- j k_{z 2} z} \end{array}

地導体板が完全導体のとき，

z = - d

の電界の接線成分はゼロゆえ，

\begin{matrix} (5) & {\tilde{E}}_{1} (- d) = {\tilde{E}}_{1}^{+} e^{j k_{z 1} d} + {\tilde{E}}_{1}^{-} e^{- j k_{z 1} d} = 0 \end{matrix}

よって，

\begin{matrix} (6) & {\tilde{E}}_{1}^{+} = - {\tilde{E}}_{1}^{-} e^{- j 2 k_{z 1} d} \end{matrix}

これより，

\begin{array}{rcl} {\tilde{E}}_{1} (z) & = & - {\tilde{E}}_{1}^{-} e^{- j 2 k_{z 1} d} e^{- j k_{z 1} z} + {\tilde{E}}_{1}^{-} e^{j k_{z 1} z} \\ = & {\tilde{E}}_{1}^{-} e^{- j k_{z 1} z} (- e^{- j k_{z 1} (z + d)} + e^{j k_{z 1} (z + d)}) \\ = & {\tilde{E}}_{1}^{-} e^{- j k_{z 1} z} (j 2) \sin k_{z 1} (z + d) \\ (7) & \equiv & C_{1} \sin k_{z 1} (z + d) \end{array}

また，

\begin{array}{rcl} {\tilde{H}}_{1} (z) & = & Y_{1} (- {\tilde{E}}_{1}^{-} e^{- j 2 k_{z 1} d} e^{- j k_{z} z} - {\tilde{E}}_{1}^{-} e^{j k_{z 1} z}) \\ = & Y_{1} {\tilde{E}}_{1}^{-} e^{- j k_{z 1} z} (- e^{- j k_{z 1} (z + d)} - e^{j k_{z 1} (z + d)}) \\ = & Y_{1} {\tilde{E}}_{1}^{-} e^{- j k_{z 1} z} (- 2) \cos k_{z 1} (z + d) \\ (8) & = & j Y_{1} C_{1} \cos k_{z 1} (z + d) \end{array}

誘電体と空気の境界（

z = 0

）において，

z \geq 0

自由空間を見た入力アドミタンス

Y_{i n}^{(+)}

は，

\begin{matrix} (9) & Y_{i n}^{(+)} = \frac{{\tilde{H}}_{2} (0)}{{\tilde{E}}_{2} (0)} = \frac{Y_{2} {\tilde{E}}_{2}^{-}}{{\tilde{E}}_{2}^{-}} = Y_{2} \end{matrix}

　一方，誘電体と空気の境界（

z = 0

）において，

z \leq 0

の誘電体を見た入力アドミタンス

Y_{i n}^{(-)}

は，

z = - d

での電磁界の連続条件を考慮すると，

\begin{array}{rcl} Y_{i n}^{(-)} & = & \frac{- {\tilde{H}}_{1} (0)}{{\tilde{E}}_{1} (0)} \\ = & - \frac{- {\tilde{E}}_{1} (- d) j Y_{1} \sin k_{z 1} d + {\tilde{H}}_{1} (- d) \cos k_{z 1} d}{{\tilde{E}}_{1} (- d) \cos k_{z 1} d - {\tilde{H}}_{1} (- d) j Z_{1} \sin k_{z 1} d} \\ = & \frac{\cos k_{z 1} d}{j Z_{1} \sin k_{z 1} d} \\ (10) & = & - j Y_{1} \cot k_{z 1} d \end{array}

スペクトル領域の電界型ダイアディック・グリーン関数

z = 0

におけるスペクトル領域の電流を

\tilde{J}

とすると，

z = 0

の電磁界の連続条件より，

\begin{array}{rcl} (11) & {\tilde{E}}_{2} (0) - {\tilde{E}}_{1} (0) & = & 0 \\ (12) & {\tilde{H}}_{2} (0) - {\tilde{H}}_{1} (0) & = & - \tilde{J} \end{array}

これより，

\begin{matrix} (13) & Y_{i n}^{(+)} {\tilde{E}}_{2} (0) + Y_{i n}^{(-)} {\tilde{E}}_{2} (0) = - \tilde{J} \end{matrix}

よって，

\begin{matrix} (14) & {\tilde{E}}_{2} (0) = {\tilde{E}}_{1} (0) = - \frac{\tilde{J}}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} \equiv {\tilde{Z}}^{(0)} \tilde{J} \end{matrix}

ただし，

\begin{array}{rcl} {\tilde{Z}}^{(0)} & = & - \frac{1}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} \\ (15) & = & - \frac{1}{Y_{2} - j Y_{1} \cot k_{z 1} d} \end{array}

これより，

\begin{array}{rcl} Z_{_{T E}}^{(0)} & = & - \frac{1}{Y_{2_{T E}} - j Y_{1_{T E}} \cot k_{z 1} d} \\ (16) & = & - \frac{1}{\frac{k_{z 2}}{ω μ_{2}} - j \frac{k_{z 1}}{ω μ_{1}} \cot k_{z 1} d} \end{array}

ここで，

μ_{2} = μ_{1} = μ_{0}

より，

\begin{array}{rcl} {\tilde{Z}}_{_{T E}}^{(0)} & = & - \frac{ω μ_{0} \sin k_{z 1} d}{k_{z 2} \sin k_{z 1} d - j k_{z 1} \cos k_{z 1} d} \\ = & - \frac{j ω μ_{0} \sin k_{z 1} d}{k_{z 1} \cos k_{z 1} d + j k_{z 2} \sin k_{z 1} d} \\ (17) & = & - \frac{j ω μ_{0} \sin k_{z 1} d}{T_{e}} \end{array}

ただし，

\begin{matrix} (18) & T_{e} \equiv k_{z 1} \cos k_{z 1} d + j k_{z 2} \sin k_{z 1} d \end{matrix}

また，

\begin{array}{rcl} {\tilde{Z}}_{_{T M}}^{(0)} & = & - \frac{1}{Y_{2_{T M}} - j Y_{1_{T M}} \cot k_{z 1} d} \\ (19) & = & - \frac{1}{\frac{ω ϵ_{2}}{k_{z 2}} - j \frac{ω ϵ_{1}}{k_{z 1}} \cot k_{z 1} d} \end{array}

ここで，

ϵ_{2} = ϵ_{0}

，

ϵ_{1} = ϵ_{r} ϵ_{0}

より，

\begin{array}{rcl} {\tilde{Z}}_{_{T M}}^{(0)} & = & - \frac{k_{z 1} k_{z 2}}{ω ϵ_{0}} \cdot \frac{\sin k_{z 1} d}{k_{z 1} \sin k_{z 1} d - j k_{z 2} ϵ_{r} \cos k_{z 1} d} \\ = & - \frac{k_{z 1} k_{z 2}}{ω ϵ_{0}} \cdot \frac{j \sin k_{z 1} d}{k_{z 2} ϵ_{r} \cos k_{z 1} d + j k_{z 1} \sin k_{z 1} d} \\ (20) & = & - \frac{j k_{z 1} k_{z 2} \sin k_{z 1} d}{ω ϵ_{0} T_{m}} \end{array}

ただし，

\begin{matrix} (21) & T_{m} \equiv k_{z 2} ϵ_{r} \cos k_{z 1} d + j k_{z 1} \sin k_{z 1} d \end{matrix}

よって，ダイアディック・グリーン関数の成分

{\tilde{G}}_{x x}^{^{E J}}

は，

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{E J}} & = & \frac{1}{k_{t}^{2}} (k_{y}^{2} {\tilde{Z}}_{_{T E}}^{(0)} + k_{x}^{2} {\tilde{Z}}_{_{T M}}^{(0)}) \\ = & \frac{1}{k_{t}^{2}} (- k_{y}^{2} \frac{j ω μ_{0} \sin k_{z 1} d}{T_{e}} - k_{x}^{2} \frac{j k_{z 1} k_{z 2} \sin k_{z 1} d}{ω ϵ_{0} T_{m}}) \\ = & - \frac{j}{ω ϵ_{0}} \frac{\sin k_{z 1} d}{k_{t}^{2}} (\frac{k_{y}^{2} k_{0}^{2}}{T_{e}} + \frac{k_{x}^{2} k_{z 1} k_{z 2}}{T_{m}}) \\ = & \frac{- j}{ω ϵ_{0}} \cdot \frac{\sin k_{z 1} d}{k_{t}^{2}} \cdot \frac{k_{y}^{2} k_{0}^{2} T_{m} + k_{x}^{2} k_{z 1} k_{z 2} T_{e}}{T_{e} T_{m}} \\ (22) & = & \frac{- j}{ω ϵ_{0}} \frac{\sin k_{z 1} d}{k_{t}^{2}} \frac{k_{z 2} (k_{x}^{2} k_{z 1}^{2} + k_{y} k_{0}^{2} ϵ_{r}) \cos k_{z 1} d + k_{z 1} (k_{x}^{2} k_{z 2}^{2} + k_{y}^{2} k_{0}^{2}) j \sin k_{z 1} d}{T_{e} T_{m}} \end{array}

ここで，

\begin{array}{rcl} k_{x}^{2} k_{z 1}^{2} + k_{y}^{2} k_{0}^{2} ϵ_{r} & = & k_{x}^{2} (k_{0}^{2} ϵ_{r} - k_{t}^{2}) + (k_{t}^{2} - k_{x}^{2}) k_{0}^{2} ϵ_{r} \\ (23) & = & k_{t}^{2} (- k_{x}^{2} + k_{0}^{2} ϵ_{r}) \end{array}

\begin{array}{rcl} k_{x}^{2} k_{z 2}^{2} + k_{y}^{2} k_{0}^{2} & = & k_{x}^{2} (k_{0}^{2} - k_{t}^{2}) + (k_{t}^{2} - k_{x}^{2}) k_{0}^{2} \\ (24) & = & k_{t}^{2} (- k_{x}^{2} + k_{0}^{2}) \end{array}

\begin{array}{rcl} \frac{1}{ω ϵ_{0}} & = & \frac{1}{ω \sqrt{ϵ_{0} μ_{0}}} \sqrt{\frac{μ_{0}}{ϵ_{0}}} \\ (25) & = & \frac{Z_{0}}{k_{0}} \end{array}

これより，

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{E J}} & = & \frac{- j Z_{0}}{k_{0}} \frac{\sin k_{z 1} d}{k_{t}^{2}} \frac{k_{z 2} k_{t}^{2} (ϵ_{r} k_{0}^{2} - k_{x}^{2}) \cos k_{z 1} d + k_{z 1} k_{t}^{2} (k_{0}^{2} - k_{x}^{2}) j \sin k_{z 1} d}{T_{e} T_{m}} \\ (26) & = & \frac{- j Z_{0}}{k_{0}} \frac{k_{z 2} (ϵ_{r} k_{0}^{2} - k_{x}^{2}) \cos k_{z 1} d + j k_{z 1} (k_{0}^{2} - k_{x}^{2}) \sin k_{z 1} d}{T_{e} T_{m}} \sin k_{z 1} d \end{array}

また，

\begin{array}{rcl} {\tilde{G}}_{x y}^{^{E J}} & = & {\tilde{G}}_{y x}^{^{E J}} = \frac{k_{x} k_{y}}{k_{t}^{2}} ({\tilde{Z}}_{_{T M}}^{(0)} - {\tilde{Z}}_{_{T E}}^{(0)}) \\ = & \frac{k_{x} k_{y}}{k_{t}^{2}} (- \frac{j k_{z 1} k_{z 2} \sin k_{z 1} d}{ω ϵ_{0} T_{m}} + \frac{j ω μ_{0} \sin k_{z 1} d}{T_{e}}) \\ = & - \frac{j}{ω ϵ_{0}} \frac{k_{x} k_{y} \sin k_{z 1} d}{k_{t}^{2}} (\frac{k_{z 1} k_{z 2}}{T_{m}} - \frac{k_{0}^{2}}{T_{e}}) \\ = & \frac{- j}{ω ϵ_{0}} \cdot \frac{k_{x} k_{y} \sin k_{z 1} d}{k_{t}^{2}} \cdot \frac{k_{z 1} k_{z 2} T_{e} - k_{0}^{2} T_{m}}{T_{e} T_{m}} \\ = & \frac{- j}{ω ϵ_{0}} \cdot \frac{k_{x} k_{y} \sin k_{z 1} d}{k_{t}^{2}} \cdot \frac{k_{z 2} (k_{z 1}^{2} - k_{0}^{2} ϵ_{r}) \cos k_{z 1} d + k_{z 1} (k_{z 2}^{2} - k_{0}^{2}) j \sin k_{z 1} d}{T_{e} T_{m}} \\ = & \frac{- j}{ω ϵ_{0}} \cdot \frac{k_{x} k_{y} \sin k_{z 1} d}{k_{t}^{2}} \cdot \frac{k_{z 2} (- k_{t}^{2}) \cos k_{z 1} d + j k_{z 1} (- k_{t}^{2}) \sin k_{z 1} d}{T_{e} T_{m}} \\ (27) & = & \frac{j Z_{0}}{k_{0}} \cdot \frac{k_{x} k_{y} (k_{z 2} \cos k_{z 1} d + j k_{z 1} \sin k_{z 1} d)}{T_{e} T_{m}} \sin k_{z 1} d \end{array}

{\tilde{G}}_{x x}^{^{E J}}

の計算と同様にして，

\begin{array}{rcl} {\tilde{G}}_{y y}^{^{E J}} & = & \frac{1}{k_{t}^{2}} (k_{x}^{2} {\tilde{Z}}_{_{T E}}^{(0)} + k_{y}^{2} {\tilde{Z}}_{_{T M}}^{(0)}) \\ = & \frac{1}{k_{t}^{2}} (- k_{x}^{2} \frac{j ω μ_{0} \sin k_{z 1} d}{T_{e}} - k_{y}^{2} \frac{j k_{z 1} k_{z 2} \sin k_{z 1} d}{ω ϵ_{0} T_{m}}) \\ = & - \frac{j}{ω ϵ_{0}} \frac{\sin k_{z 1} d}{k_{t}^{2}} (\frac{k_{x}^{2} k_{0}^{2}}{T_{e}} + \frac{k_{y}^{2} k_{z 1} k_{z 2}}{T_{m}}) \\ (28) & = & \frac{- j Z_{0}}{k_{0}} \frac{k_{z 2} (ϵ_{r} k_{0}^{2} - k_{y}^{2}) \cos k_{z 1} d + j k_{z 1} (k_{0}^{2} - k_{y}^{2}) \sin k_{z 1} d}{T_{e} T_{m}} \sin k_{z 1} d \end{array}

スペクトル領域の磁界型ダイアディック・グリーン関数

　一方，

z = - d

の地導体面上では，

\begin{matrix} (29) & {\tilde{E}}_{1} (- d) = 0 \end{matrix}

また，

\begin{array}{rcl} {\tilde{H}}_{1} (- d) & = & {\tilde{E}}_{1} (0) j Y_{1} \sin k_{z 1} d + {\tilde{H}}_{1} (0) \cos k_{z 1} d \\ = & {\tilde{E}}_{1} (0) j Y_{1} \sin k_{z 1} d - Y_{i n}^{(-)} {\tilde{E}}_{1} (0) \cos k_{z 1} d \\ = & - \frac{\tilde{J}}{Y_{i n}^{(+)} + Y_{i n}^{(-)}} (j Y_{1} \sin k_{z 1} d - Y_{i n}^{(+)} \cos k_{z 1} d) \\ = & - \frac{\tilde{J}}{Y_{2} - j Y_{1} \cot k_{z 1} d} (j Y_{1} \sin k_{z 1} d + j Y_{1} \cot k_{z 1} d \cos k_{z 1} d) \\ = & - \frac{j Y_{1} \tilde{J}}{Y_{2} \sin k_{z 1} d - j Y_{1} \cos k_{z 1} d} \\ (30) & = & \frac{Y_{1} \tilde{J}}{Y_{1} \cos k_{z 1} d + j Y_{2} \sin k_{z 1} d} \equiv {\tilde{P}}^{(- d)} \tilde{J} \end{array}

ここで，

\begin{matrix} (31) & {\tilde{P}}^{(- d)} = \frac{Y_{1}}{Y_{1} \cos k_{z 1} d + j Y_{2} \sin k_{z 1} d} \end{matrix}

これより（

μ_{2} = μ_{1} = μ_{0}

），

\begin{array}{rcl} {\tilde{P}}_{_{T E}}^{(- d)} & = & \frac{Y_{1_{T E}}}{Y_{1_{T E}} \cos k_{z, 1} d + j Y_{2_{T E}} \sin k_{z, 1} d} \\ = & \frac{\frac{k_{z 1}}{ω μ_{1}}}{\frac{k_{z 1}}{ω μ_{1}} \cos k_{z 1} d + j \frac{k_{z 2}}{ω μ_{2}} \sin k_{z 1} d} \\ = & \frac{k_{z 1}}{k_{z 1} \cos k_{z 1} d + j k_{z 2} \sin k_{z 1} d} \\ (32) & = & \frac{k_{z 1}}{T_{e}} \end{array}

また（

ϵ_{2} = ϵ_{0}

，

ϵ_{1} = ϵ_{r} ϵ_{0}

），

\begin{array}{rcl} {\tilde{P}}_{_{T M}}^{(- d)} & = & \frac{Y_{1_{T M}}}{Y_{1_{T M}} \cos k_{z 1} d + j Y_{2_{T M}} \sin k_{z 1} d} \\ = & \frac{\frac{ω ϵ_{1}}{k_{z 1}}}{\frac{ω ϵ_{1}}{k_{z 1}} \cos k_{z 1} d + j \frac{ω ϵ_{2}}{k_{z 2}} \sin k_{z 1} d} \\ = & \frac{k_{z 2} ϵ_{r}}{k_{z 2} ϵ_{r} \cos k_{z 1} d + j k_{z 1} \sin k_{z 1} d} \\ (33) & = & \frac{k_{z 2} ϵ_{r}}{T_{m}} \end{array}

よって，磁界型ダイアディック・グリーン関数の各成分は，

\begin{array}{rcl} {\tilde{G}}_{x x}^{^{H J}} & = & - {\tilde{G}}_{y y}^{^{H J}} = \frac{k_{x} k_{y}}{k_{t}^{2}} ({\tilde{P}}_{_{T E}}^{(- d)} - {\tilde{P}}_{_{T M}}^{(- d)}) \\ = & \frac{k_{x} k_{y}}{k_{t}^{2}} (\frac{k_{z 1}}{T_{e}} - \frac{k_{z 2} ϵ_{r}}{T_{m}}) \\ = & \frac{k_{x} k_{y}}{k_{t}^{2}} \cdot \frac{k_{z 1} T_{m} - ϵ_{r} k_{z 2} T_{e}}{T_{e} T_{m}} \\ = & \frac{k_{x} k_{y}}{k_{t}^{2}} \frac{k_{z 1} (k_{z 2} ϵ_{r} \cos k_{z 1} d + j k_{z 1} \sin k_{z 1} d) - ϵ_{r} k_{2} (k_{1} \cos k_{1} d + j k_{2} \sin k_{1} d)}{T_{e} T_{m}} \\ = & \frac{k_{x} k_{y}}{k_{t}^{2}} \frac{(j k_{z 1}^{2} - j ϵ_{r} k_{z 2}^{2}) \sin k_{z 1} d}{T_{e} T_{m}} \\ = & \frac{j k_{x} k_{y}}{k_{t}^{2}} \frac{{(k_{0}^{2} ϵ_{r} - k_{t}^{2}) - ϵ_{r} (k_{0}^{2} - k_{t}^{2})} \sin k_{z 1} d}{T_{e} T_{m}} \\ (34) & = & \frac{j k_{x} k_{y} (ϵ_{r} - 1) \sin k_{z 1} d}{T_{e} T_{m}} \end{array}

また，

\begin{array}{rcl} {\tilde{G}}_{x y}^{^{H J}} & = & - \frac{1}{k_{t}^{2}} (k_{x}^{2} {\tilde{P}}_{_{T E}}^{(- d)} + k_{y}^{2} {\tilde{P}}_{_{T M}}^{(- d)}) = - \frac{1}{k_{t}^{2}} (\frac{k_{x}^{2} k_{z 1}}{T_{e}} + \frac{ϵ_{r} k_{y}^{2} k_{z 2}}{T_{m}}) \\ = & - \frac{1}{k_{t}^{2}} {\frac{(k_{t}^{2} - k_{y}^{2}) k_{z 1}}{T_{e}} + \frac{ϵ_{r} k_{y}^{2} k_{z 2}}{T_{m}}} \\ = & \frac{k_{z 1}}{T_{e}} - \frac{k_{y}^{2}}{k_{t}^{2}} (\frac{k_{z 1}}{T_{e}} - \frac{ϵ_{r} k_{z 2}}{T_{m}}) \\ (35) & = & \frac{k_{z 1}}{T_{e}} - \frac{j k_{y}^{2} (ϵ_{r} - 1) \sin k_{z 1} d}{T_{e} T_{m}} \end{array}

同様にして，

\begin{array}{rcl} {\tilde{G}}_{y x}^{^{H J}} & = & \frac{1}{k_{t}^{2}} (k_{y}^{2} {\tilde{P}}_{_{T E}}^{(- d)} + k_{x}^{2} {\tilde{P}}_{_{T M}}^{(- d)}) = \frac{1}{k_{t}^{2}} (\frac{k_{y}^{2} k_{z 1}}{T_{e}} + \frac{ϵ_{r} k_{x}^{2} k_{z 2}}{T_{m}}) \\ = & \frac{1}{k_{t}^{2}} {\frac{(k_{t}^{2} - k_{x}^{2}) k_{z 1}}{T_{e}} + \frac{ϵ_{r} k_{x}^{2} k_{z 2}}{T_{m}}} \\ = & - \frac{k_{z 1}}{T_{e}} + \frac{k_{x}^{2}}{k_{t}^{2}} (\frac{k_{z 1}}{T_{e}} - \frac{ϵ_{r} k_{z 2}}{T_{m}}) \\ (36) & = & - \frac{k_{z 1}}{T_{e}} + \frac{j k_{x}^{2} (ϵ_{r} - 1) \sin k_{z 1} d}{T_{e} T_{m}} \end{array}