Find the equivalent transfer function, T(s) = C(s) / R(s), for the system shown in figure using Maso's Rule and block diagram.
$Here,\phantom{\rule{0ex}{0ex}}Forwardpathgain\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Path1-\frac{1}{{s}^{2}}x\frac{50}{s+1}xs\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Path2-\frac{1}{{s}^{2}}x\frac{50}{s+1}x2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Individualloopgain\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Loop1-\frac{1}{{s}^{2}}x\frac{50}{s+1}xsx(-1)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Loop2-\frac{50}{s+1}x\frac{2}{s}x(-1)$
$Accordingtomason\text{'}sgainformula\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{C\left(s\right)}{R\left(s\right)}=\frac{Path1+Path2}{1-Loop1-Loop2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{\left(\frac{1}{{s}^{2}}x\frac{50}{s+1}xs\right)+\left(\frac{1}{{s}^{2}}x\frac{50}{s+1}x2\right)}{1-\left(\frac{1}{{s}^{2}}x\frac{50}{s+1}xsx(-1)\right)-\left(\frac{50}{s+1}x\frac{2}{s}x(-1)\right)}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{50}\mathbf{s}\mathbf{+}\mathbf{100}}{{\mathbf{s}}^{\mathbf{3}}\mathbf{+}{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{50}\mathbf{s}\mathbf{+}\mathbf{100}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$