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Transfer Function of Block Diagram

  1. Jan 27, 2015 #1
    1. The problem statement, all variables and given/known data
    The block diagram of a PM DC servo motor with current loop feedback is shown below:

    QWERTY.jpg

    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)


    3. The attempt at a solution

    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.
     
  2. jcsd
  3. Jan 27, 2015 #2
    That seems fine.

    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.

    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.
     
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