# Transfer Function of Block Diagram

## Homework Statement

The block diagram of a PM DC servo motor with current loop feedback is shown below: If Ki is adjusted such that J(Ki+R)2 >> 4KTKbL, show that the transfer function may be approximated by

G(s) = (1/Kb) / (τms+1)(τes+1),

where

τm = J(R+Ki) / KTKb

τe = L / (R+Ki)

## The Attempt at a Solution

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I simplified the block diagram and got the transfer function:

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

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

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

Assuming the conditions presented for Ki, I cancelled out the 4KtKbL.

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

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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.