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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

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

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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.

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  • \$\begingroup\$ MaxJax is definitely supported here. It's been kind of flaky the last few days. \$\endgroup\$ – Matt Young Sep 16 '14 at 18:04
  • \$\begingroup\$ @MattYoung: How do I used it? I tried a quick search but found nothing. On some sites I just do $...$ to enclose the formula. \$\endgroup\$ – copper.hat Sep 16 '14 at 18:05
  • \$\begingroup\$ @copper meta.math.stackexchange.com/questions/5020/… \$\endgroup\$ – JYelton Sep 16 '14 at 18:06
  • \$\begingroup\$ @JYelton: I am mostly a MSE user, so I am very familiar with that. But it does not work when I write equations above. \$\endgroup\$ – copper.hat Sep 16 '14 at 18:08
  • \$\begingroup\$ @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 $$. \$\endgroup\$ – JYelton Sep 16 '14 at 18:10
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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.

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