Transfer Function of Block Diagram

[ThLiOp]

Homework Statement

The block diagram of a PM DC servo motor with current loop feedback is shown below:

If K_i is adjusted such that J(K_i+R)² >> 4K_TK_bL, show that the transfer function may be approximated by

G(s) = (1/K_b) / (τ_ms+1)(τ_es+1),

where

τ_m = J(R+K_i) / K_TK_b

τ_e = L / (R+K_i)

The Attempt at a Solution

[/B]
I simplified the block diagram and got the transfer function:

G(s) = K_t / (JLs² + J(R + K_i)s + K_tK_b)

Then I tried to factor the denominator. When using the quadratic formula, I found:

(-JR - JK_i) +/- sqrt( J[ J(R+K_i)² - 4K_tK_bL]) / 2JL

Assuming the conditions presented for K_i, I canceled out the 4K_tK_bL.

Am I on the right path? In the end, I still couldn't get the transfer function presented in the homework.

 

[milesyoung]

I simplified the block diagram and got the transfer function:

G(s) = Kt / (JLs2 + J(R + Ki)s + KtKb)

That seems fine.

Then I tried to factor the denominator. When using the quadratic formula, I found:

(-JR - JKi) +/- sqrt( J[ J(R+Ki)2 - 4KtKbL]) / 2JL

Those are just the roots, though. If $ax^2 + bx + c$ is your polynomial, then its factored form is $a(x - x_1)(x - x_2)$, where $x_1$ and $x_2$ are its roots.

Am I on the right path? In the end, I still couldn't get the transfer function presented in the homework.

I can't either. Going your route, and I think the approximation shown is highly suggestive of that, I get the form:
$$
\frac{\dot{\theta}(s)}{V(s)} = \frac{\frac{1}{K_b}}{\tau_m s (\tau_e s + 1)}
$$
Sort of looks like a typo, but maybe there's another route I'm just not seeing.