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math_2.md

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A 是dex1, B是dex2

$$ A_x A_y = (A_x + \theta_A E_{in})(A_y - \Delta y )$$ $$ B_x B_y = (B_x - E_{out})(B_y + \theta_B \Delta y ) $$

$$ \Delta y = A_y - \frac{A_x A_y}{A_x + \theta_A E_{in}} $$ $$ E_{out} = B_x - \frac{B_x B_y}{B_y + \theta_B \Delta y } $$

$$ E_{out} = \frac{ B_x \theta_B \Delta y}{ B_y + \theta_B \Delta y } $$

replace $\Delta y$ in $E_{out}$ 2

$$ \Delta y = \frac{A_y \theta_A E_{in} }{A_x + \theta_A E_{in}} $$

$$ E_{out} = \frac{ B_x A_y \theta_B \theta_A E_{in} }{ B_y A_x + B_y \theta_A E_{in} + \theta_B A_y \theta_A E_{in} } $$

replace $\Delta y$ in $E_{out}$ 3

replace $\Delta y$ in $E_{out}$

$$ \Delta y = \frac{A_y \theta_A E_{in} }{A_x + \theta_A E_{in}} $$

$$ E_{out} = B_x - \frac{B_x B_y}{B_y + \theta_B (\frac{A_y \theta_A E_{in} }{A_x + \theta_A E_{in}}) } $$ $$ E_{out} = B_x - \frac{B_x B_y (A_x + \theta_A E_{in})}{B_y (A_x + \theta_A E_{in}) + \theta_B (A_y \theta_A E_{in}) } $$ $$ E_{out} = B_x - \frac{B_x B_y A_x + B_x B_y \theta_A E_{in}}{B_y A_x + B_y \theta_A E_{in} + \theta_B A_y \theta_A E_{in} } $$ $$ E_{out} = \frac{ B_x \theta_B A_y \theta_A E_{in} }{B_y A_x + B_y \theta_A E_{in} + \theta_B A_y \theta_A E_{in} } $$

$$ p = E_{out} - E_{in} = \frac{ (B_x \theta_B A_y \theta_A E_{in}) }{B_y A_x + B_y \theta_A E_{in} + \theta_B A_y \theta_A E_{in} } - E_{in} $$

$$ p = E_{in} (\frac{ B_x \theta_B A_y \theta_A }{B_y A_x + B_y \theta_A E_{in} + \theta_B A_y \theta_A E_{in} } - 1) $$

$$ p = \frac{ B_x \theta_B A_y \theta_A E_{in} - E_{in} B_y A_x - E_{in} B_y \theta_A E_{in} - E_{in}\theta_B A_y \theta_A E_{in} }{B_y A_x + B_y \theta_A E_{in} + \theta_B A_y \theta_A E_{in} } $$

draft

calculate $\Delta y$

$$ A_x + \theta_A E_{in} = \frac{A_x A_y}{A_y - \Delta y} $$ $$ {A_y - \Delta y} = \frac{A_x A_y}{A_x + \theta_A E_{in}} $$

calculate $E_{out}$