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# simplify

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$$T(s) = \frac{G(s)}{1+N(s)G(s)}$$

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.

Mar 29, 2017

### 2+0 Answers

#1
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Mmm

$$T(s) = \frac{G(s)}{1+N(s)G(s)}$$

Immediately I a restraint  $$N(s)G(s)\neq -1$$

so

$$(s+2)*\frac{10s}{(s+3)(s-2)}-1\ne0\\ 10s(s+2)-(s+3)(s-2)\ne0\\ 10s^2+20s-(s^2+s-6)\ne0\\ 10s^2+20s-s^2-s+6\ne0\\ 9s^2+19s+6\ne0\\ s\ne \frac{-19\pm\sqrt{145}}{18}$$

$$G(s) = \frac{10s}{(s+3)(s-2)}\\so\\ s\ne-3,\quad s\ne2 \\ N(s)=s+2$$

$$T(s)=\frac{\frac{10s}{(s+3)(s-2)}}{ 1+(s+2)\frac{10s}{(s+3)(s-2)}}\\ T(s)=\frac{\frac{10s}{(s+3)(s-2)}}{ 1+(s+2)\frac{10s}{(s+3)(s-2)}}\times \frac{(s+3)(s-2)}{(s+3)(s-2)}\\ T(s)=\frac{10s}{(s+3)(s-2)+(s+2)10s}\\ T(s)=\frac{10s}{s^2+s-6+10s^2+20s}\\ T(s)=\frac{10s}{11s^2+21s-6}\quad where \;\;s\neq 2,-3,\frac{-21\pm\sqrt{705}}{22},\;or\;\frac{-19\pm \sqrt{145}}{18}$$

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Mar 29, 2017
#2
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thank you melody, I appreciate it

Mar 29, 2017