# Block diagram reduction

I have the following block diagram

I know that it's solution is:

$$\dfrac{10 K_p}{Js^2+(B+2K_pK_v)s+10K_p}$$

But when I try to derive the solution with reduction rules I get different result.

Assumptions:

$5$ and $K_vs$ are in parallel: $(5+K_vs)$

$K_p \cdot \dfrac{2}{(Js+B)s}$

result: $$\dfrac{K_p \cdot \dfrac{2}{(Js+B)s}}{1 + K_p \cdot \dfrac{2}{(Js+B)s} + (5+K_vs)}$$

What am I doing wrong?

I believe that it will be worth for many people to do it from scratch. Let's call the signal after input and feedback summation $E(s)$:

Then question arises what $E(s)$ is equal to? Well that's simple:

$$E(s)=5R(s)-(5+K_vs)C(s)$$

So we have our backward path, now the forward one. What $C(s)$ is equal to?

$$C(s)=E(s)K_p \dfrac{2}{(Js+B)s}$$

Let's substitute previously calculated $E(s)$ to the above equation:

$$C(s)=\dfrac{2}{(Js+B)s}K_p[5R(s)-(5+K_vs)C(s)]$$

Because we have our output $C(s)$ on both sides of the equation, we must eliminate that. Expand the right side of the equation:

$$C(s)=\dfrac{2}{(Js+B)s}5K_p\cdot R(s)-\dfrac{2}{(Js+B)s}K_p(5+K_vs)C(s)$$

Now move the part with $C(s)$ to the left side and move it before parenthesis, this yields:

$$C(s) \left[1+ \dfrac{2}{(Js+B)s}K_p(5+K_vs)\right]=\dfrac{2}{(Js+B)s}5K_p\cdot R(s)$$

Now you can divide right side of the equation by the factor from the left side:

$$C(s) = \dfrac{\dfrac{2}{(Js+B)s}5K_p}{1+ \dfrac{2}{(Js+B)s}K_p(5+K_vs)}\cdot R(s)$$

Great! Now we have direct relation between the input and the output. Let's make this fraction more pleasant:

$$C(s) = \dfrac{\dfrac{10K_p}{(Js+B)s}}{1+ \dfrac{10K_p+2K_pK_vs}{(Js+B)s}}\cdot R(s)$$

Now expand the one in denominator to: $\dfrac{(Js+B)s}{(Js+B)s}$, and add it:

$$C(s) = \dfrac{\dfrac{10K_p}{(Js+B)s}}{\dfrac{(Js+B)s + 10K_p+2K_pK_vs}{(Js+B)s}}\cdot R(s)$$

We can now remove common $(Js+B)s$:

$$C(s) = \dfrac{10K_p}{(Js+B)s + 10K_p+2K_pK_vs}\cdot R(s)$$

Finally let's do one last multiplication:

$$C(s) = \dfrac{10K_p}{Js^2+Bs + 10K_p+2K_pK_vs}\cdot R(s)$$

And voila!

$$C(s) = \dfrac{10K_p}{Js^2+ s(B +2K_pK_v)+ 10K_p}\cdot R(s)$$