$$i_a=K_s{\displaystyle \frac{Z_f{\cos }^2\beta +S_f}{Z_f\cos \beta }}$$

(6)

$$Z_{fc}={\displaystyle \frac{S_f}{(q/K_s-1){\cos }^2\beta }}$$

(8)

$$\begin{array}{c}I=({\theta }_s-{\theta }_i)Z_s\cos \beta +{\displaystyle \underset{Z_s}{\overset{Z_f}{\int }}(\theta -{\theta }_i)\cos \beta dz}\\ =({\theta }_s-{\theta }_i){\displaystyle \frac{Z_f}{2}}\cos \beta +{\displaystyle \underset{Z_f/2}{\overset{Z_f}{\int }}({\theta }_s-{\theta }_i)\cos \beta \sqrt{1-{(\frac{z-Z_f/2}{Z_f/2})}^2}dz}\\ ={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)}{8}}\cos \beta Z_f\end{array}$$

(9)

$$I_c={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)}{8}}\cos \beta Z_{fc}={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)S_f}{8(q/K_s-1)\cos \beta }}$$

(10)

$$t_{\mathrm{c}}={\displaystyle \frac{I_c}{q\cos \beta }}={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)S_f}{8q(q/K_s-1){\cos }^2\beta }}$$

(11)

$$({\displaystyle \frac{\mathrm{d}{Z}_f}{dt}}{)}_1={\displaystyle \frac{8}{({\theta }_s-{\theta }_i)(4+\pi )\cos \beta }}{({\displaystyle \frac{dI}{dt}})}_1={\displaystyle \frac{8K_s(Z_f{\cos }^2\beta +S_f)}{({\theta }_s-{\theta }_i)(4+\pi )Z_f{\cos }^2\beta }}$$

(12)

$$K_s\sin \beta Z_s\cos \beta dt={\displaystyle \frac{(4+\pi )({\theta }_s-{\theta }_i)}{8}}Ld{Z}_f\cos \beta$$

(13)

$${({\displaystyle \frac{d{Z}_f}{dt}})}_2={\displaystyle \frac{4K_s{Z}_f\sin \beta }{({\theta }_s-{\theta }_i)(4+\pi )L}}$$

(14)

$${\displaystyle \frac{\mathrm{d}{Z}_f}{dt}}={\displaystyle \frac{8K_s}{({\theta }_s-{\theta }_i)(4+\pi )}}({\displaystyle \frac{Z_f{\cos }^2\beta +S_f}{Z_f{\cos }^2\beta }}-{\displaystyle \frac{Z_f\sin \beta }{2L}})$$

(15)

$$F_s={\displaystyle \frac{(G_s+G_t)\cos \beta \tan {\phi }^{\prime }+(u_a-u_w)\tan {\phi }^{b}L+c^{\prime }L}{(G_s+G_t)\sin \beta +P}}$$

(16)

$$\tan {\phi }^b=\Theta \cdot \tan {\phi }^{\prime }$$

(17)

$$F_s={\displaystyle \frac{\tan {\phi }^{\prime }\left[\left(G_s+G_t\right)\cos \beta +(u_a-u_w){\Theta }_{i}L\right]+c^{\prime }L}{(G_s+G_t)\sin \beta +P}}$$

(18)

$$F_s={\displaystyle \frac{G_s\cos \beta \tan {\phi }_s^{\prime }+c_s^{\prime }L}{G_s\sin \beta +P}}$$

(19)

$$F_s={\displaystyle \frac{\left(G_s+G_t+G_n\right)\cos \beta \tan {\phi }^{\prime }+c^{\prime }L}{(G_s+G_t+G_n)\sin \beta +P}}$$

(20)

$$G_i=G_{is}+G_{it}+G_{in}={\gamma }_s{A}_{is}+G_{it}+{\gamma }_i{A}_{in}$$

(21)

$$G_{it}=\{\begin{array}{c}\left[L_1{Z}_f-{\displaystyle \frac{3Z_f^2\sin {\beta }_1}{4}}-{\displaystyle \frac{3Z_f^2\cos {\beta }_1}{4\tan ({\alpha }_1-{\beta }_1)}}\right]\cos {\beta }_1{\gamma }_d\left[{\displaystyle \frac{4(1+{\theta }_i)+\pi ({\theta }_s-{\theta }_i)}{8}}\right],i=1\\ L_i\cos {\beta }_i{\gamma }_d{Z}_f\left[{\displaystyle \frac{4(1+{\theta }_i)+\pi ({\theta }_s-{\theta }_i)}{8}}\right],1<i<n\\ \left[L_n{Z}_f-{\displaystyle \frac{3Z_f^2\sin {\beta }_n}{4}}-{\displaystyle \frac{3Z_f^2\cos {\beta }_n}{4\tan ({\beta }_n-{\alpha }_n)}}\right]\cos {\beta }_n{\gamma }_d\left[{\displaystyle \frac{4(1+{\theta }_i)+\pi ({\theta }_s-{\theta }_i)}{8}}\right],i=n\end{array}$$

(22)

$$P_i={\gamma }_w{A}_{is}\sin {\beta }_i$$

(23)

$$F_s={\displaystyle \frac{\left[G_i\cos {\alpha }_i+P_i\sin ({\beta }_i-{\alpha }_i)+E_{i-1}\sin ({\alpha }_{i-1}-{\alpha }_i)\right]\tan {\phi }^{\prime }+E_i+c^{\prime }{l}_i}{G_i\sin {\alpha }_i+P_i\cos ({\beta }_i-{\alpha }_i)+E_{i-1}\cos ({\alpha }_{i-1}-{\alpha }_i)}}$$

(24)

$$\begin{array}{c}{E}_i=F_s{G}_i\sin {\alpha }_i-(G_i\cos a_i\tan {\phi }^{\prime }+c^{\prime }{l}_i)+P_i\left[\cos ({\beta }_i-{\alpha }_i)-\sin ({\beta }_i-{\alpha }_i)\tan {\phi }^{\prime }\right],i=1\\ E_i=F_s{G}_i\sin {\alpha }_i+E_{i-1}\left[F_s\cos ({\alpha }_{i-1}-{\alpha }_i)-\sin ({\alpha }_{i-1}-{\alpha }_i)\tan {\phi }^{\prime }\right]-(G_i\cos a_i\tan {\phi }^{\prime }+c^{\prime }{l}_i)\\ +P_i\left[\cos ({\beta }_i-{\alpha }_i)-\sin ({\beta }_i-{\alpha }_i)\tan {\phi }^{\prime }\right],i\ge 2\end{array}$$

(25)

$$E_i=\{\begin{array}{c}0,E_i<0\\ E_i,E_i\ge 0\end{array}$$

(26)