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Math Help - Equivalent Impedance

  1. #1
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    Equivalent Impedance

    The equivalent impedance Z of two impedances Z_1 and Z_2 in parallel is given by the the formula

    \frac{1}{Z} = \frac{1}{Z_1} + \frac{1}{Z_2}

    If Z_1 = 3 + j2 and Z_2 = 1 - j3 calculate Z giveing your answer in the form r(\cos\theta + j\sin\theta) where \theta is in radians.

    Any help with this would be highly appreciated.

    Thanks
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  2. #2
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    Hello anothernewbie
    Quote Originally Posted by anothernewbie View Post
    The equivalent impedance Z of two impedances Z_1 and Z_2 in parallel is given by the the formula

    \frac{1}{Z} = \frac{1}{Z_1} + \frac{1}{Z_2}

    If Z_1 = 3 + j2 and Z_2 = 1 - j3 calculate Z giveing your answer in the form r(\cos\theta + j\sin\theta) where \theta is in radians.

    Any help with this would be highly appreciated.

    Thanks
    \frac{1}{Z}= \frac{1}{3+2j}+\frac{1}{1-3j}
    =\frac{1-3j+3+2j}{(3+2j)(1-3j)}

    =\frac{4-j}{9-7j}
    \Rightarrow Z = \frac{9-7j}{4-j}
    =\frac{(9-7j)(4+j)}{4^2+1^2}

    =\frac{1}{17}(43-19j)

    =r(\cos\theta +j\sin\theta)
    where
    r\cos\theta = \frac{43}{17} and r\sin\theta = -\frac{19}{17}
    Square and add: r^2(\cos^2\theta+\sin^2\theta) = \frac{43^2+19^2}{17^2}

    \Rightarrow r = 2.765 (to 4 s.f.)

    Divide: \tan\theta = -\frac{19}{43}

    \Rightarrow \theta = - 0.4161 radians (to 4 s.f.)

    The method is sound, but check my working!

    Grandad
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  3. #3
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    thanks grandad!
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