+251 \(T(s) = \frac{G(s)}{1+N(s)G(s)}\)
if \(G(s) = \frac{10s}{(s+3)(s-2)}\)and \(N(s)=s+2\)
simplify the expression for T(s)
I'm in 2 minds about this...
I just want to cancel the G functions out on the top and bottom and then be left with, \(\frac{1}{1+(s+2)}\)
however my other brain is saying I need to do the arithmetic and see what I get.
+118667 Mmm
\(T(s) = \frac{G(s)}{1+N(s)G(s)}\)
Immediately I a restraint \(N(s)G(s)\neq -1\)
so
\((s+2)*\frac{10s}{(s+3)(s-2)}-1\ne0\\ 10s(s+2)-(s+3)(s-2)\ne0\\ 10s^2+20s-(s^2+s-6)\ne0\\ 10s^2+20s-s^2-s+6\ne0\\ 9s^2+19s+6\ne0\\ s\ne \frac{-19\pm\sqrt{145}}{18}\)
\(G(s) = \frac{10s}{(s+3)(s-2)}\\so\\ s\ne-3,\quad s\ne2 \\ N(s)=s+2 \)
\(T(s)=\frac{\frac{10s}{(s+3)(s-2)}}{ 1+(s+2)\frac{10s}{(s+3)(s-2)}}\\ T(s)=\frac{\frac{10s}{(s+3)(s-2)}}{ 1+(s+2)\frac{10s}{(s+3)(s-2)}}\times \frac{(s+3)(s-2)}{(s+3)(s-2)}\\ T(s)=\frac{10s}{(s+3)(s-2)+(s+2)10s}\\ T(s)=\frac{10s}{s^2+s-6+10s^2+20s}\\ T(s)=\frac{10s}{11s^2+21s-6}\quad where \;\;s\neq 2,-3,\frac{-21\pm\sqrt{705}}{22},\;or\;\frac{-19\pm \sqrt{145}}{18}\)
