... | ... | @@ -33,7 +33,7 @@ to the lag, latexmath:[\varphi] and set the result to zero: |
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\begin{aligned}
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\mathrm{d}{\Delta E(\varphi,r)}{\varphi} &= -\int_{z_\text{begin}}^{z_\text{end}} (1 + \omega \frac{\partial t(z,\varphi)}{\partial \varphi}) \sin(\omega t(z,\varphi) + \varphi) E_z(z,r)\\
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&= -\cos(\varphi) \int_{z_\text{begin}}^{z_\text{end}} (1 + \omega \frac{\partial t(z,\varphi)}{\partial \varphi}) \sin(\omega t(z,\varphi)) E_z(z,r) dz \\
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&-\sin(\varphi) \int_{z_\text{begin}}^{z_\text{end}} (1 + \omega \frac{\partial t(z,\varphi)}{\partial \varphi}) \cos(\omega t(z,\varphi)) E_z(z,r) dz \stackrel{!}{=} 0.
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&-\sin(\varphi) \int_{z_\text{begin}}^{z_\text{end}} (1 + \omega \frac{\partial t(z,\varphi)}{\partial \varphi}) \cos(\omega t(z,\varphi)) E_z(z,r) dz \equiv 0.
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\end{aligned}
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++++
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... | ... | @@ -157,7 +157,7 @@ For this example we find |
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V_\text{core} &= \frac{V_{0}}{\sin(2.0/3.0 \pi)} = \frac{2 V_{0}}{\sqrt{3.0}}\\
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\varphi_\text{c1} &= \frac{\pi}{6}\\
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\varphi_\text{c2} &= \frac{\pi}{2}\\
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\varphi_\text{ef} &= - 2\pi \cdot(\mathbf{NUMCELLS} - 1) \cdot \mathbf{MODE} = 26\pi
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\varphi_\text{ef} &= - 2\pi \cdot(\mathrm{NUMCELLS} - 1) \cdot \mathrm{MODE} = 26\pi
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\end{aligned}
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++++
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... | ... | @@ -194,7 +194,7 @@ period of latexmath:[3\,s]. We thus find |
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[latexmath]
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++++
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\begin{aligned}
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0 \stackrel{!}{=}& \int_{0}^{1.5\cdot s} \sin\left(\omega \left(\frac{z}{\beta c} + t_{0}\right) + \varphi\right) E_z^{(2)}(z,r)\, dz \\
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0 \equiv & \int_{0}^{1.5\cdot s} \sin\left(\omega \left(\frac{z}{\beta c} + t_{0}\right) + \varphi\right) E_z^{(2)}(z,r)\, dz \\
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&+ \frac{2}{\sqrt{3}} \int_{0}^{39\cdot s}\sin\left(\omega \left(\frac{z + 1.5\,s}{\beta c} + t_{0}\right) + \frac{\pi}{6} + \varphi \right) E_z^{(1)}(z \text{ mod}(3\,s),r)\,dz \\
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&+ \frac{2}{\sqrt{3}} \int_{0}^{39\cdot s}\sin\left(\omega \left(\frac{z + 1.5\,s}{\beta c} + t_{0}\right) + \frac{\pi}{2} + \varphi \right) E_z^{(1)}((z + s) \text{ mod} (3\,s),r)\, dz \\
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&+ \int_{1.5\cdot s}^{3\cdot s} \sin\left(\omega\left(\frac{z + 39\,s}{\beta c} + t_{0}\right) + \varphi\right) E_z^{(2)}(z,r)\, dz
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... | ... | @@ -207,7 +207,7 @@ latexmath:[\omega / \beta c \approx 10\pi]. We then get |
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[latexmath]
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++++
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\begin{aligned}
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0 \stackrel{!}{=}& \int_{0}^{3\cdot s} \sin\left(\omega \left(\frac{z}{\beta c} + t_{0}\right) + \varphi\right) E_z^{(2)}(z)\, dz \\
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0 \equiv & \int_{0}^{3\cdot s} \sin\left(\omega \left(\frac{z}{\beta c} + t_{0}\right) + \varphi\right) E_z^{(2)}(z)\, dz \\
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+ \frac{26}{\sqrt{3}} & \int_{0}^{3\cdot s}\left(\sin\left(\omega \left(\frac{z}{\beta c} + t_{0}\right) + \frac{7\pi}{6} + \varphi \right)
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+ \sin\left(\omega \left(\frac{z}{\beta c} + t_{0}\right) + \frac{5\pi}{6} + \varphi \right)\right) E_z^{(1)}(z)\, dz \\
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= & \int_{0}^{3\cdot s}\sin\left(\omega \left(\frac{z}{\beta c} + t_{0}\right) + \varphi\right) \left(E_z^{(2)} - 26\cdot E_z^{(1)}\right)(z)\,dz\end{aligned}
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... | ... | @@ -231,7 +231,7 @@ Using the convolution theorem we find |
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[latexmath]
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++++
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0 \stackrel{!}{=} \int_{0}^{3\cdot s} g(\xi - z) (G - 26 \cdot H)(z) \, dz =
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0 \equiv \int_{0}^{3\cdot s} g(\xi - z) (G - 26 \cdot H)(z) \, dz =
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\mathcal{F}^{-1}\left(\mathcal{F}(g)\cdot(\mathcal{F}(G) - 26 \cdot \mathcal{F}(H))\right)
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++++
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... | ... | |