Impedance - RC Circuits (Beginner)

Discussion in 'General Electronics Chat' started by bogart_sci, May 2, 2013.

  1. bogart_sci

    Thread Starter New Member

    May 2, 2013
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    Hello, I am fairly new to electronics and am currently studying a BTEC in the subject. Was wondering if anyone can help with my confusion over Impedance in RC / RL circuits?
    I am wondering when I should be using complex numbers when calculating impedance. For example, if asked to find the impedance in an RC circuit should I express the answer in terms of Z = R + jX? eg: 2 - j0.001. Or should I treat the Reactance as a regular number? eg: Z = 2 - 0.001 = 1.999. In some of the practice questions the number for X seems an insignificant amount and I end up rounding back up to the value of R.
    For example, given the following values...

    C = 200uF
    R = 2M Ω
    f = 50Hz

    From this I work out Xc = 1 / (2*pi*f*C) = 15.915

    Then I have used Z = √(R2 + Xc2) where the 2s mean squared.
    This gives me a massive 4 billion for R2 and 253.287 for Xc2. Add them together and take the square root yeilds an answer not too far from the value of R in the first place.

    I am not sure if I should be expressing Xc as -j15.915 and thus...

    Z = 2000000 - j15.915

    Is there a right way to express this or is either way acceptable? I have some questions like this in a test paper I have to do so obviously want to present the correct way.

    Hope someone can help.

    Cheers,
     
  2. Six_Shooter

    Member

    Nov 10, 2012
    33
    1
    Impedance is expressed in ohms, so in the end, the value you end up with after going through the formula, and then rounding to the number of significant digits you use is fine, usually then referred to as impedance, even if the complex value (X) is small.

    Sometimes this value will come up close to what the R value is, especially when using an RLC circuit at resonance, where the XL and XC are at polar opposites and essential cancel each other out, leaving only R. You will find other situations where X becomes a significant amount.

    Also to signify a squared or something to a larger power you can use the "^" So 3^2, is "3 to the power of 2".
     
  3. Papabravo

    Expert

    Feb 24, 2006
    10,135
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    The rules are that:
    1. Reactance is a real number
    2. Reactance is the imaginary part of impedance
    3. Impedance should be expressed as a complex number with BOTH a real part and an imaginary part.
    4. The MAGNITUDE of imedance is the square root of the sum of the squares of the real and imaginary parts.
    5. The ARG of the impedance is the arctan(Im(z)/Re(z)), that is the inverse tangent of the Reactance over the Resistance.
     
  4. bogart_sci

    Thread Starter New Member

    May 2, 2013
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    Thanks guys. Much appreciated. Pretty sure I have answered the question correct now, thanks to your advice. Cheers :)
     
  5. bogart_sci

    Thread Starter New Member

    May 2, 2013
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    Thanks guys for answering my question. I feel more confident I have answered my test question correctly, which was in ohms. Cheers
     
  6. MrChips

    Moderator

    Oct 2, 2009
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    One point worth stressing is that imaginary numbers are real, that is, this is something that exists in the real world.

    The impedance of a circuit Z = R + jX

    is a mathematical construct to indicate that R and X are 90-degrees out of phase.

    Another way of expressing Z is with the use of polar coordinates, i.e. the magnitude of Z and the phase angle.
     
  7. #12

    Expert

    Nov 30, 2010
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    There is a sticky on this site about "LaTex" that shows how to do fancy stuff with math symbols, but I just use the way it's done in Basic or Fortran programming (like Six_Shooter said) and seem to be understood.
     
  8. WBahn

    Moderator

    Mar 31, 2012
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    I think you may have a typo. In (1) you say that reactance is a real number, but in (2) you say it is the imaginary part of impedance.

    There are about three consistent ways of saying this, but I'm not sure which you were thinking of and it all comes down to semantics.

    Given 1+2j, if you use the term "imaginary part" to mean "2j", then you have a problem. If you use it to mean "2", then I think you are consistent all the way through. I'm pretty sure that is what you mean.

    I just did some quick searching and it appears that most references use the term as you did, namely that the "imaginary part" is Im{z}, which is a real number. But I did find several that said that it was the portion of a complex number that has sqrt(-1) as a factor, meaning that their "imaginary part" has to be an imaginary number.

    I've gotten in the habit of calling "2" the "imaginary coefficient" since I think that is pretty unambiguous, but it does leave me with a tendancy to think of "imaginary part" as being an imaginary number. Come to think of it, I don't recall what the common term would be for "2j". The "imaginary component" of (1+2j)?

    Just one more example of the need to not assume (this is a general observational comment and is most definitely NOT directed at Papabravo) that everyone is always in agreement on what even common terms used in the parlance mean but, instead, try to gain a feel for as many of the interpretations that are out there as possible and then always be willing to filter our interpretation of a particular discussion through them instead of forcing it through the one interpretation we happen to know.
     
  9. Papabravo

    Expert

    Feb 24, 2006
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    My quick research on the matter in Mathematics, Physics, and Engineering texts is consistent with my usage, that the Im(z) is a real number, and that a complex number is constructed of two real numbers with one of them multiplied by the imaginary unit. It is also my experience that you can define the algebra of complex numbers without using the imaginary unit by defining the rules for an algebra on ordered pairs.

    That is if z = (Re(z), Im(z)) you can do all of the algebra of complex numbers witout using the j=SQRT(-1)
     
  10. WBahn

    Moderator

    Mar 31, 2012
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    Oh, I definitely agree that Im{z} is a real number. Certainly didn't mean to imply otherwise. And I also agree that you can define a complex number as being an ordered pair and define the operations on them without resorting to the imaginary unit.

    I was talking about specifically the meaning of the term the "imaginary part" and whether that included the imaginary unit or not. I have found references both ways, but the preponderance favor not including it.

    But that means that it is not true that a complex number is made up of a real part and an imaginary part because both are real and, thus, something is missing (if the definition of a complex number in use is the sum of a real number and an imaginary number).

    So what do YOU use for xxx in the following sentence: Given a complex number z that is equal to (4+5j), the xxx of z is 5j.

    I tend to favor having that be the "imaginary part" and having Im{z} be called the imaginary coefficient. That makes the claim that a complex number is the sum of a real part and an imaginary part a true statement. But, I realize that the dominant use is not consistent with that (or, rather, that the term "imaginary part" is used inconsistently to mean either Im{z} or j*Im{z} depending on context). But I dislike using context-sensitive terminology where reasonably avoidable, so I try to use the term "imaginary component" for j*Im{z}.
     
  11. MrChips

    Moderator

    Oct 2, 2009
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    How about

    "Reactance is the amplitude of the imaginary component of impedance".
     
  12. WBahn

    Moderator

    Mar 31, 2012
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    That is definitely unambiguous as far as what reactance is because it doesn't matter how the term "imaginary component" is interpretted, but it sheds no light on how "imaginary component" should be interpreted, either.

    I know I'm being nit picky, but part of the reason is that I want to use the terminology as correctly as I can when speaking to students. This particular case just became particularly relevant because I was just notified a few minutes ago that I will be teaching Transform Methods in the Fall. Of course, unless there is good reason not to, I will use the terminology used by the author. But, even so, I want to at least make the students aware in passing of other common usage.
     
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