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Complex numbers rectangular form

  1. Jul 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Given the equivalent impedance of a circuit can be calculated by the expression:

    Z = Z1 X Z2 / Z1 + Z2

    If Z1 = 4 + j10 and Z2 = 12 - j3, calculate the impedance Z in both rectangular and polar form.


    2. Relevant equations

    Multiplication and division of complex numbers.


    3. The attempt at a solution

    I want to solve the rectangular first. As I want to totally understand how this is done. My attempt so far:

    Z1 + Z2 = 4 + j10 + 12 + j3
    = 4 + 12 + J10 - J3
    Z1 + Z2 = 16 + j7

    Z1 x Z2 = 4 + j10 x 12 - j3
    = (4 x 12) + (12 x j10) + (4 x -j3) + (j10 x j3)
    = 48 + j120 - j12 - j^2 30

    Since j^2 = -1
    Then
    Z1 x Z2 = 48 + j108 - (-1) 30
    = 48 + j108 +30
    Z1 x Z2 = 78 + j108

    Z = 78 + j108 multiply by conjugate 16 - j7
    ----------- -------
    16 + j7 16 - j7

    Z = -1248 - j1728 - j546 - j^2 756
    ----------------------------------
    16^2 + 7^2

    -1248 - 756 = 2004
    j1728 - j546 = j1182

    Z = -2004 - j1182
    ---------------
    305

    -2004 / 305 = -6.570491803
    j1182 / 305 = 3.875409836

    Z = 6.5705 ± j3.8754
     
  2. jcsd
  3. Jul 9, 2013 #2
    Looking at this again I think the end result should be:

    Z = -6.5705 + j3.8754

    ??
     
  4. Jul 9, 2013 #3

    Mark44

    Staff: Mentor

    I'm sure you really mean this:
    Z = (Z1Z2)/(Z1 + Z2)

    Knowledgeable folks would read the right side you wrote as
    $$Z_1 X (\frac{Z_2}{Z_1}) + Z_2$$

    When you write fractions with a sum in the numerator or denominator, USE PARENTHESES!!!

    So far, so good.
    Correct here as well.
    This is the right approach.
    Your arithmetic is off here. 78 * 16 should be a positive number and 16 * 108j should be a positive number times j.
    There shouldn't be ±. It's one or the other.

     
  5. Jul 9, 2013 #4
    Thanks, I am very close then!
    Need to go through the signs again and see what I come out with.
    Then do polar and see if they correspond I guess?
     
  6. Jul 9, 2013 #5
    Thanks again, nights sleep and went through it again quickly.
    Now have Z = 6.5705 + j3.8754

    I think this is right?

    Will confirm in polar form today!
     
  7. Jul 10, 2013 #6
    Just did the much simpler polar calculation and got the exact result as above :-)

    Thanks Mentor for your pointing out my sign errors. I must of been tired!!
     
  8. Jul 10, 2013 #7
    Just did the much simpler polar calculation and got the exact result as above ( after converting to rectangular)
     
  9. Apr 1, 2017 #8
    great thread, thought the signs were off too. were filling it with negatives and it was confusing me as I had a positive result. some other examples I have seen are finding 78*(-j7) as a positive and was throwing all my basic ideas off. glad you got to it in the end and put in the work. nice!
     
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