# How to simplify this block diagram to get the given transfer function?

Given this block diagram, I'm trying to derive the transfer function. I've tried moving the 2 branch to either side of the then getting something like this

But I'm not sure how to simplify it from there. Any guidance would be appreciated. Thank you.

Let $g_1 = {10 \over s+1}, g_2 = {1 \over s}$, and let $w$ be the output of the $g_1$ block.

You get $w = g_1 (u - 2 w -g_2 w)$, which gives $w = {g_1 \over1 + 2 g_1 + g_1 g_2} u$, and so $y = {g_2 g_1 \over1 + 2 g_1 + g_1 g_2} u$, or $y = {1 \over 1+{2 \over g_2}+{1 \over g_1 g_2}} u$ if you would prefer.

Substituting in the values for $g_1, g_2$ gives the transfer function you wanted.

• MaxJax is definitely supported here. It's been kind of flaky the last few days. Sep 16 '14 at 18:04
• @MattYoung: How do I used it? I tried a quick search but found nothing. On some sites I just do $...$ to enclose the formula. Sep 16 '14 at 18:05
• Sep 16 '14 at 18:06
• @JYelton: I am mostly a MSE user, so I am very familiar with that. But it does not work when I write equations above. Sep 16 '14 at 18:08
• @copper It must differ on other sites... Here to use it inline \$ to escape, and to use it on a line of its own, use $\$. Sep 16 '14 at 18:10

Because you want to know if your steps for simplifying the diagram are correct: There is one small - but important - error: If you want to place the block "gain of two" at the signal output you must multiply it with "s" (because now it is integrated) - that`s all. Thus, your second diagram would be correct if you remove the most right line which acts as a short for the block "2s".

As a result, we have H(s)=N(s)/D(s)

with N(s)=10/s(s+1) and D(s)=[1+LG] with loop gain LG=N(s)*2s

Introducing all the corresponding expressions into H(s) results in the given formula.