Impedance (Inductor-Capacitor)

  1. Statement:
    If we have a circuit consisting of an inductor and capacitor in series, the impedance is defined by the following:
    [tex]Z = L + C = \sqrt{(X_{L} - X_{C})^{2}} = \sqrt{(X_{C} - X_{L})^{2}}[/tex]

    My question:
    Howcome the impedance (or reactance) isn't defined by the following equation:
    [tex]Z = L + C = \sqrt{X_{L}^{2} + X_{C}^{2}} = \sqrt{X_{C}^{2} + X_{L}^{2}}?[/tex]

    Reasoning:
    I would think that we would add the impedance, even for inductors and capacitors since from my knowledge, impedance for circuits in series is added together.

    Thanks,

    JL
     
    Last edited: Aug 16, 2009
  2. jcsd
  3. rock.freak667

    rock.freak667 6,231
    Homework Helper

    Inductor: XL=2πfLj
    Capacitor:XC= -j/2πfC

    Add them together and you get Z=(XL-XC)j.

    so |Z|=√(XL-XC)2
     
  4. That makes sense, except when I looked up the definition of impedance for inductors and capacitors I found the following:

    [tex]Z = X_{L} = [(\omega)L]j = [(2 \pi f)L]j[/tex]
    [tex]Z = X_{C} = [\frac{1}{(\omega)C}]j = [\frac{1}{(2 \pi f L)C}]j[/tex]

    Therefore,
    [tex]X_{L} + X_{C} = [(2 \pi f)L]j + [\frac{1}{C(2 \pi f L)}]j = ...?[/tex]
     
    Last edited: Aug 16, 2009
  5. rock.freak667

    rock.freak667 6,231
    Homework Helper

    Wikipedia says it is [itex]X_C=\frac{-j}{2\pi f C}[/itex]. The voltage through the capacitor is 90 degrees out of phase with the voltage through an inductor.
     
  6. diazona

    diazona 2,156
    Homework Helper

    Actually I believe it's 180 degrees out of phase. The capacitor voltage is 90 degrees out of phase with the voltage through a resistor (or would be, if there were a resistor), and the inductor voltage is 90 degrees out of phase with the resistor in the other direction. For a net difference of 180 degrees.

    That's why the impedances are defined as
    [tex]X_L = i \omega L[/tex]
    and
    [tex]X_C = -\frac{i}{\omega C}[/tex]
    Those two quantities are 180 degrees out of phase with each other.
     
  7. 1.)
    What if there were no resistors in the circuit, and it was only an inductor and capacitor (along with the sinusoid power source)?



    2.)
    Could someone elaborate more why the two quantities are 180 degrees out of phase (I'm guessing that's why the inductor has a phase [tex]\frac{\pi}{2}[/tex], while the capacitor has a phase of -[tex]\frac{\pi}{2}[/tex])?



    3.)
    I looked on the Wikipedia page and found my definition on the "Device Examples" section:
    [tex]
    Z = X_{L} = [(\omega)L]j = [(2 \pi f)L]j
    [/tex]
    [tex]
    Z = X_{C} = [\frac{1}{(\omega)C}]j = [\frac{1}{(2 \pi f L)C}]j
    [/tex]

    But further down I also find your definition,
    [tex]X_C = -\frac{j}{\omega C}.[/tex]
    How does this definition help us to conclude our result,
    [tex]
    Z = L + C = \sqrt{(X_{L} - X_{C})^{2}} = \sqrt{(X_{C} - X_{L})^{2}} ?
    [/tex]



    4.) On the same Wikipedia page,
    I understand the following line of equation,
    [tex]-j = cos(-\frac{\pi}{2}) + jsin(\frac{\pi}{2}) = e^{j(- \frac{\pi}{2})}[/tex]

    But I don't understand how [tex]\frac{1}{j} = -j ?[/tex]

    Thanks,


    JL
     
    Last edited: Aug 16, 2009
  8. vk6kro

    vk6kro 4,059
    Science Advisor

    But I don't understand how 1 / j = -j ?

    A common trick when solving these complex number problems is to multiply by J / J to bring the J to the top line of a fraction.

    j = √-1

    1 / j = 1 / √-1

    Multiply by J / J or √-1 / √-1

    Gives you 1 / J = (√-1) / -1 or
    1 / J = - (√-1)
    1 / J = -J

    So, 1 / J = -J
     
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