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
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\$\begingroup\$ Please show us what you have come up with so far. Where exactly are you getting stuck? \$\endgroup\$ – Elliot Alderson Dec 14 '18 at 1:30
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\$\begingroup\$ 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. \$\endgroup\$ – Newbie010 Dec 14 '18 at 1:46
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\$\begingroup\$ 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 \$\endgroup\$ – Newbie010 Dec 14 '18 at 1:47
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\$\begingroup\$ C=Y/H5 since Y=C*H5 \$\endgroup\$ – Tony Stewart EE75 Dec 14 '18 at 3:05
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1\$\begingroup\$ The best and classical way is: Block diagram reduction technique. Start to replace local feedback loops by one block only! \$\endgroup\$ – LvW Dec 14 '18 at 8:54
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My answer comes a little late, but I think it should still be interesting.
I took the block diagram from you (@Newbie010) as a basis and added some labels which are helpful for the calculation. My method gets along here without any simplification.
At this point I ask you to check if what I have done is correct. I did not find any mistake in my calculation, but a critical consideration would be appropriate.
Block diagram with new labels:
Based on this picture, the following equations can now be determined:
$$X(s)+C(s)=B(s)=A(s) \tag{1}$$ $$D(s) = B(s) - H_1A(s) \tag{2}$$ $$E(s) = H_4Y(s)+H_1A(s) \tag{3} $$ $$F(s) = E(s) + H_3D(s) \tag{4}$$ $$Y(s) = H_5F(s) \tag{5} $$
Other important and useful equations:
$$ F(s) = \frac{Y(s)}{H_5} \tag{6}$$ $$ C(s) = H_2F(s) \tag{7}$$ $$ C(s) = H_2\frac{Y(s)}{H_5} \tag{8}$$
The aim now is to have only inputs and outputs in its transfer function $H(s)$. Important here is this connection $H(s) = \frac{Y(s)}{X(s)}$
I now begin to generate $H(s)$ by starting with equation 5 and using the relations to other equations in turn. This is straight forward:
$$ \begin{align} Y(s) &= H_5F(s) \\ &= H_5\left(E(s)+H_3D(s)\right) \\ &= H_5\left(H_4Y(s)+H_1A(s)+H_3D(s)\right) \\ &= H_5\left(H_4Y(s)+H_1A(s)+H_3B(s) - H_3H_1A(s)\right) \\ &= H_5\left(H_4Y(s)+H_1\left(X(s)+C(s)\right)+H_3\left(X(s)+C(s)\right) - H_1H_3\left(X(s)+C(s)\right)\right) \\ &= H_5\left(H_4Y(s)+H_1\left(X(s)+H_2\frac{Y(s)}{H_5}\right)+H_3\left(X(s)+H_2\frac{Y(s)}{H_5}\right) - H_1H_3\left(X(s)+H_2\frac{Y(s)}{H_5}\right)\right) \\ &= H_5H_4Y(s)+H_5H_1X(s)+H_5H_1H_2\frac{Y(s)}{H_5}+H_5H_3X(s)+H_5H_3H_2\frac{Y(s)}{H_5}-H_5H_1H_3X(s)-H_5H_1H_3H_2\frac{Y(s)}{H_5} \\ &=Y(s)\left(H_5H_4+H_1H_2+H_3H_2-H_1H_3H_2\right)+X(s)\left(H_5H_1+H_5H_3-H_5H_1H_3\right) \end{align} $$
Therefor:
$$ Y(s)(1-H_5H_4-H_1H_2-H_3H_2+H_1H_3H_2) = X(s)(H_5H_1+H_5H_3-H_5H_1H_3) $$
It follows: $$ H(s) = \frac{Y(s)}{X(s)} = \frac{H_5H_1+H_5H_3-H_5H_1H_3}{1-H_5H_4-H_1H_2-H_3H_2+H_1H_3H_2} $$
Now I wanted to test if what I have done can be confirmed by a simulation. Therefore I converted the block diagram in Matlab. Also I realized my generated transfer function as a block in Matlab, the results were the same!
For the simulation I used the following blocks: $$ H_1 = 5, H_2 = \frac{1}{s}, H_3 = \frac{1}{s+2} $$ $$H_4 = 7, H_5 = \frac{1}{s}$$
If I now insert these values into my TF and let the overall expression be simplified once with the use of Wolframalpha I get the following for $H(s)$:
$$\frac{5s+6}{s^2-10s-20}, s\neq 0, s \neq -2$$
In Matlab I implemented your block diagram with the corresponding values and my simplified transfer function. Judging by the simulation, the results are the same.
I hope that my answer meets the requirements and hope that I could help a little bit with this. Thanks a lot!
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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?
