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Simplifying a resistor network using pi-T (Y-delta) conversion

  1. Jul 8, 2012 #1

    JJBladester

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    1. The problem statement, all variables and given/known data

    Reduce the resistor network to a single resistor. Go step-by-step and indicate the series or parallel combinations being reduced.

    pi-t%2520conversion.jpg

    2. Relevant equations

    For series resistors: [itex]R_T=R_1+R_2+R_3+\cdot \cdot \cdot +R_N[/itex]
    For parallel resistors: [itex]R_T=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\cdot \cdot \cdot +\frac{1}{R_N}}[/itex]

    ∏-T Conversion:
    [tex]R_1=\frac{R_BR_C}{R_A+R_B+R_C}[/tex]
    [tex]R_2=\frac{R_AR_C}{R_A+R_B+R_C}[/tex]
    [tex]R_3=\frac{R_AR_B}{R_A+R_B+R_C}[/tex]

    3. The attempt at a solution

    The first thing I noticed is that RA, RB, and RC are not in series and they're not in parallel. This led me to the ∏-T (Y-Δ) conversion. After the conversion, I was able to make further simplifications in steps (2) and (3).

    In step (4), I get stuck because I don't know how to simplify the circuit given the way R5 is hooked up. Any pointers?
     
  2. jcsd
  3. Jul 8, 2012 #2

    JJBladester

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    My thinking, assuming I drew everything correctly, is that R5 gets bypassed because a practically resistance-free path exists (the wire on the left-leg of the triangle above R5).

    Therefore the equivalent resistance would be:

    Req=R3+R2||(R1+R4) = 10.57Ω + 58.92Ω ≈ 69.5Ω

    Does this make sense?
     
  4. Jul 9, 2012 #3

    lewando

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    Yes. You can also think of the wire as a 0-ohm resistor and use the parallel resistor formula.
     
  5. Jul 9, 2012 #4
    I guess cable across Rc and R4
    So mark all the points as zero voltage.
    Now we have a parallel circuit of
    1. Ra-150Ω
    2. Series of Rb and parallel resistors of Rc and R4-129Ω

    Equivalent resistance=69.35Ω
     
    Last edited: Jul 9, 2012
  6. Jul 9, 2012 #5

    JJBladester

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    This makes sense.

    So, [itex]R_T=\frac{1}{\frac{1}{\sim 0 \Omega }+\frac{1}{220\Omega }}\approx 0 \Omega [/itex]

    Thanks lewando.
     
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