Reactance Direction In Phasors

Discussion in 'Homework Help' started by djwinger, May 9, 2012.

  1. djwinger

    Thread Starter New Member

    Jan 3, 2012
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    I have an exam tomorrow and just wanted to clarify something, so any help on this would be great.

    I drew this up and I'm hoping it is correct. Is it?

    The thing I'm confused about is the direction of the reactances. Why do they switch for parallel?

    Is the reason because the reactance is always out of phase with the resistor and therefore must be tied to the same direction as the element* that's not in phase with the resistor?

    *for lack of a better word

    So for instance, if Vc is in phase with Vr, then Xc and Ic are in the same direction.
    If Ic is in phase with Ir, then Xc and Vc are in the same direction.

    Is this the correct way to think about it or have I got the theory all messed up? My notes for this module are rubbish compared to the other modules :(
     
    Last edited: May 9, 2012
  2. WBahn

    Moderator

    Mar 31, 2012
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    What is your math background? Have you had some calculus? Have you worked with complex numbers? Have you worked with complex impedance?

    Once we have this in hand, we can try to explain things consistent with the math you are comfortable with.
     
  3. djwinger

    Thread Starter New Member

    Jan 3, 2012
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    Not really done calculus yet. Math background isn't that strong to be honest. Working with complex numbers and polar form for phasors. I know how to convert between the two etc.

    Don't think we've done complex impedance (unless I have without knowing that's what I was doing).
     
  4. WBahn

    Moderator

    Mar 31, 2012
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    If you've worked with complex numbers for phasors, then you are working with complex impedance.

    The complex impedance for the three basic linear elements are:

    <br />
Resistor: Z_R = R<br />
Capacitor: Z_C = \frac{1}{j\omega C} = -j\frac{1}{\omega C}<br />
Inductor: Z_L = j \omega L<br />

    where

    <br />
j = \sqrt{-1}<br />

    and

    <br />
\omega = 2\pi f<br />

    where w is the 'radian frequency' (radians/sec) and 'f' is the 'cyclical frequency' (cycles per second or hertz).

    For what it's worth, these relations come from taking what is called the Fourier Transform of the differential equations that describe the voltage/current relationship for each component. In doing so, the messy differential equations that would normally describe a circuit being analyzed are transformed into systems of linear equations that can be analyzed using the same methods you used for DC circuits having only resistors. You just simply transform each C and L into its complex impedance and then do the same math, except now you have to deal with complex numbers.

    All the 'phasors' are are graphical plots of the complex numbers on the complex number plane where the x-axis is the real part and the y-axis is the imaginary part.

    So let's consider two cases: series and parallel.

    In the series RLC case, we know that the current in all three devices has to be the same at all times. There's no point comparing the current phasors for each of them (because they are identical since they are in series). So let's instead plot and compare the voltage phasors using a reference current of 1A at watever frequency we are interested in and at zero phase. The voltage across each device is then given by V=IZ. When multiplying complex numbers, the phases add. Since our I is at zero phase, the phase of the voltage is therefore the phase of the impedance, or +π/2 for an inductor, 0 for a resistor, and -π/2 for a capacitor.

    In the parallel RLC case, we know that the voltage across all three devices has to be the same at all times. Similar to before, it makes no sense to compare the voltage phasors, since they are identical. So let's plot and compare the current phasors using a reference voltage of 1V at watever frequency we are interested in and at zero phase. The current through each device is then given by I=V/Z. When dividing complex numbers, the phases subtract. Since our V is at zero phase, the phase of the voltage is therefore the negative of the phase of the impedance, or +π/2 for a capacitor, 0 for a resistor, and -π/2 for an inductor.

    What I strong recommend is that you don't try to memorize a bunch of rules of the "if this, then this" variety, but instead work to gain a level of understanding of the fundamentals so that you can derive the relations you need, confidently and at will.
     
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  5. djwinger

    Thread Starter New Member

    Jan 3, 2012
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    Thanks so much WBahn, this was really helpful. If only my course material had been written by you, rather than a monkey, I would have been an expert long ago!
     
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