Time Constant

Discussion in 'Homework Help' started by Robert.Adams, Jul 10, 2010.

  1. Robert.Adams

    Thread Starter Active Member

    Feb 16, 2010
    112
    5
    I need help finding the time constant for the circuit attached. I've written the nodal equations at the two significant nodes and have come up with an answer by substituting the non-cap node into the other to come up with a differential equation. I am just seeking clarification.
     
  2. Ghar

    Active Member

    Mar 8, 2010
    655
    72
    Personally I'd find it by doing a delta-wye transformation or by using the extra element theorem... either way the answer is 3.33s

    http://en.wikipedia.org/wiki/Delta_wye

    You'd transform R1, R2 and R3 into a Y connection...

    wye-transform.png

    Now you get:
    Rth = R5 + R7 || (R4 + R6) = 0.667 \Omega

    Then the time constant is of course just RC = 3.33 s

    Plot showing that's correct and they're equivalent:

    wye-transform2.png

    Or you can use the extra element theorem, by removing R1:

    Rth = (R with R1 open)\frac{1 + \frac{R seen by R1 with C1 short}{R1}}{1 + \frac{R seen by R1 with C1 open}{R1}}\\<br />
Rth = R2 || (R3 + R4) \frac{1 + \frac{R3 || R4}{R1}}{1 + \frac{(R2 + R3)||R4}{R1}}<br />
Rth = 2||4.2 \frac{1 + 4||0.2}{1 + 2.2||4} = 0.667 \Omega

    Same answer.

    Edit:


    Orrrrr I could notice that it's much simpler than that, d'oh.

    R2 is strictly in parallel with C therefore you can remove it and think about it later.
    What you're left with is R1 || R4 + R3, giving you:

    Rth = R2 || (R1 || R4 + R3) = 2 || (1||4 + 0.2) = 2 || 1 = 0.667 \Omega
     
    Last edited: Jul 10, 2010
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