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