# total transfer function of a LTI System with feedback

I have calculated the transfer function of such systems many times, but now i feel like I`m going around in circles. I have drawn a few help-signals to help me solve it, but they are dependent of one another and no matter how many times I replace the values, I just finish where I started. I must calculate the total transfer function of the system from the attached image. The coloured lines are the help-signals that I added and their description is below. Any help is appreciated]1

• Please show us what you have come up with so far. Where exactly are you getting stuck? – Elliot Alderson Dec 14 '18 at 1:30
• I tried to replace the variables with what is after the equal sign. for example A with X+H2*C in the second and fourth relationship. The thing is, there is no end to this replacing. So let's say we replace A B would become X+H2*C-XH1-H1*H2*C. D would become H4*Y+XH1+H1*H2*C And C would become H4*Y+XH1+H1*H2*C+XH3+H2*H3*C-X*H1*H3-H1*H2*H3*C. Now i have a relationship between Y,X and C and i somehow need to find Y/X. – Newbie010 Dec 14 '18 at 1:46
• And if get Y/X out of that relationship with C, i would have to replace C once again, getting into another loop, because C is dependent on the other signals – Newbie010 Dec 14 '18 at 1:47
• C=Y/H5 since Y=C*H5 – Sunnyskyguy EE75 Dec 14 '18 at 3:05
• The best and classical way is: Block diagram reduction technique. Start to replace local feedback loops by one block only! – LvW Dec 14 '18 at 8:54

Not a full answer; just some assistance....

Just one simple block diagram re-arrangement gets you an improvement (as per LvW's comment to your answer): -

Notice the added red block contans $$\1-H_1(s)\$$ and this removes the summing node just in front of $$\H_3(s)\$$. I'm assuming that was an intentional negative sign on the summing block I removed hence the negative sign in the new block I added.

Sometimes you have to add stuff to simplify but other folk may choose a different starting point.

The next thing to do (in my opinion) would be to re-arrange things like this: -

Hopefully I've not screwed up anywhere but I'm sure if i have there'll be someone eager to point it out.

Can you take it from here?