Block diagrams to transfer functions

[Dustinsfl]

Homework Statement

I am trying to write a block diagram as a transfer function.

Homework Equations

The Attempt at a Solution

Let $G = \frac{1}{s}$, $H = \frac{K}{Js + a}$, and $L= K_f$. Then wouldn't the closed loop transfer function be written as
$$
\frac{\frac{HLG}{1+HL}}{1 + \frac{HLG}{1+HL}} = \frac{KK_f}{s^2J + (KK_f + a)s + KK_f}
$$
I only ask because the book has it as
$$
\frac{K}{s^2J + (KK_f + a)s + K}
$$
and I don't see how that was obtained.

 

[milesyoung]

The transfer function for your inner feedback loop is $\frac{H(s)}{1 + H(s) L(s)}$, not $\frac{H(s) L(s)}{1 + H(s) L(s)}$.

I think that should sort it out.

 

[Dustinsfl]

The transfer function for your inner feedback loop is $\frac{H(s)}{1 + H(s) L(s)}$, not $\frac{H(s) L(s)}{1 + H(s) L(s)}$.

I think that should sort it out.

Can you explain why that is?

 

[milesyoung]

Can you explain why that is?

It's an easy mistake to make. The transfer function in the forward path appears in both the numerator and denominator of the closed-loop transfer function but the transfer function in the feedback path does not:
http://en.wikipedia.org/wiki/Closed-loop_transfer_function

 

[rezeew]

I appreciate your attempt at solving the problem and your use of block diagrams and transfer functions to represent a system. However, it is important to note that the closed loop transfer function can be written in different forms, as long as they are mathematically equivalent. In this case, your solution and the one provided by the book are both correct representations of the closed loop transfer function. It is possible that the author of the book chose to simplify the transfer function by factoring out the K term, which does not change the overall behavior of the system. It is also possible that they used a different method to derive the transfer function. Therefore, it is important to understand the concept and principles behind block diagrams and transfer functions, rather than just focusing on the specific form of the equation. Keep up the good work!