Steady-state analysis and impedance-admittance calculations

Discussion in 'Homework Help' started by lemon, Dec 24, 2011.

  1. lemon

    Thread Starter Member

    Jan 28, 2010
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    Hi:
    Please assume I know nothing of the topic title above and have to complete the following question. To start with, could somebody please look at the question and guide me to the relevant reading material on this site so that I don't waste time sifting through pages and pages of unrelated material.
    Thank you.

    The question is given as:
    Using the complex plane, use all the voltages and currents for the circuit shown in Figure Q3-1 (please see attached image).
     
  2. Zazoo

    Member

    Jul 27, 2011
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    Volume II (AC), chapters 1-5 will help you solve this.
    Depending on your familiarity witch circuit analysis in general, Volume I (DC), chapters 2, 5, 6, 7 and 10 are also important.
    It's hard to narrow it down any further if we assume that you have no knowlede of the topic (since so many concepts build upon other more basic concepts.)

    "phasor circuit analysis" would be the most relevant search term to use.
     
    Last edited: Dec 24, 2011
  3. lemon

    Thread Starter Member

    Jan 28, 2010
    125
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    wow! That was a good read. These pages are written extremely well, I must say.
    I can see that I need to solve a Series-parallel R, L, and C circuit. But there are a couple of things I am confused about. So, my first order of business must be to determine values of impedance (Z) for all components based on the frequency of the AC power source. But where is the frequency here. How do I get it from the trig. equation given for voltage?

    Also, the question says to use the complex plane. Does that mean I should use rectangular notion and not complex numbers in my calculations?
     
  4. Zazoo

    Member

    Jul 27, 2011
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    For a sinusoid, the form is: A cos (ωt ± θ)

    A is amplitude
    ω is frequency in radians/sec
    θ is phase

    Points on the complex plane can be represented in both polar and rectangular form. The y axis is the imaginary (j) axis, and the x axis is the real axis.

    Rectangular form: x + jy

    "Using the complex plane" refers to solving the problem using a frequency domain transformation (phasors, Laplace, etc.) vs. leaving the circuit elements in the time domain and solving the circuit with a differential equation.
     
  5. lemon

    Thread Starter Member

    Jan 28, 2010
    125
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    yeah I'm struggling with this one - could somebody please get me started.

    the image attached is what i think the 8cos(t+pi/4) but not sure what i am doing
     
    Last edited: Dec 31, 2011
  6. Zazoo

    Member

    Jul 27, 2011
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    Start by converting all circuit elements and quantities into phasors and complex impedances.

    For capacitors, impedance is: 1/(jωC)
    For inductors, impedance is: jωL
    For resistors, impedance is simply: R (i.e. no change)

    You can represent the unknown currents and volatges as phasors by writing them in uppercase (e.g. v2 becomes V2)

    Known AC voltage and currents (e.g. sources) can be converted to phasors:

    Acos(ωt+θ) in the time domain becomes A<θ. (here the less than sign is actually meant to be the "angle" symbol)

    Once everything is in phasor form, you can solve the circuit using any the standard network analysis theorems and tools (e.g. nodal/mesh, Thevenin, etc.)

    After solving for unknown phasor quantites (currents and voltages) you can convert them back to sinusoids by just reversing the process used to conver them to phasors. Note that the frequency (ω) for every quantity in the circuit will always be the same, it is only the phase and magnitude that can change.

    Searching "phasor circuit analysis" on google gives some good example problems. This one walks you through each step:
    http://www.usna.edu/MathDept/CDP/ComplexNum/Module_6/EXERCISE6.htm
     
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  7. lemon

    Thread Starter Member

    Jan 28, 2010
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  8. Zazoo

    Member

    Jul 27, 2011
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    The frequency, in radians, is whatever factor is multiplying "t" when in the form Acos(ωt+θ). In this case ω=1.
     
  9. lemon

    Thread Starter Member

    Jan 28, 2010
    125
    2
    Before I solve the circuit using any the standard network analysis theorems and tools (e.g. nodal/mesh, Thevenin, etc.), could you please check to see if I have done the conversions to phasor quantities correctly, please?

    I have these:
    Resistors are just as they are - no change
    Reactive capacitance(Xc) = 0.3183Ω
    Reactive inductance(Xl) = 6.2832Ω
    Conversion of capacitor to phasor domain, Zc = 0-j2.0Ω
    Conversion of inductor to phasor domain, Zl = 0+1.0Ω
    Voltage source V1 = 8<π/4° or 8<0.7854°

    thanks
     
  10. Zazoo

    Member

    Jul 27, 2011
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    These all look good. I'm guessing Zl was a typo (meant to be 0+j1.0Ω)

    It looks like you used 2∏ in calculating these. The frequency is already in radians (1 radian/sec), so you wouldn't use the 2∏ factor.
    If frequency is in radians (ω), then Xl = ωL
    if frequency is in hertz (f), then Xl = 2∏fL
    In otherwords, ω=2∏f

    Xc and Xl are just the imaginary parts of the complex impedances Zc and Zl, i.e. Xc = -2.0 and Zl = 1.0
     
  11. lemon

    Thread Starter Member

    Jan 28, 2010
    125
    2
    these calculations are criminal and i don't know how to do them with a calculator. could someone please check and let me know I'm on track

    thank you
     
  12. Zazoo

    Member

    Jul 27, 2011
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    Complex algebra can be very tedious without a calculator. The TI graphing calculators can easily handle these if your instructor doesn't require you to show all the detail.

    Since the question asks for all unknown currents and voltages, it's easier to leave the parallel branches as is (rather than combining them.) I would combine the R3 and L1 components as you did however.
    With R3 and L1 joined you can set up a node equation with only one unknown node (and thus only one equation to solve for V1.) Once you have V1 you can just use Ohms law to get all of the branch currents (V/Z=I)
     
  13. thatoneguy

    AAC Fanatic!

    Feb 19, 2009
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    HP RPN Scientific calculators can do math with polar and rectangular coordinates without a problem as well.

    Though the TI-89 seems to be the calculator of the decade.
     
  14. lemon

    Thread Starter Member

    Jan 28, 2010
    125
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    you mean use the Node Voltage Method? I hope not. I'm not understanding that very well but I suppose it will be good practice
     
  15. lemon

    Thread Starter Member

    Jan 28, 2010
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  16. Zazoo

    Member

    Jul 27, 2011
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    Yes. We can help you anywhere you get stuck or make a mistake.
     
  17. lemon

    Thread Starter Member

    Jan 28, 2010
    125
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    @Zazoo
    Is V1 the voltage across (r3 -- L)?
    Or the 8cos(t + pi/4)?
     
  18. Zazoo

    Member

    Jul 27, 2011
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    If you choose the bottom of the circuit as your reference point (your zero, or ground), then V1 is the node voltage at the node where I, I1, I2, and I3 meet. V1 was just an arbitrary variable I used since the node isn't named in the schematic.
     
  19. lemon

    Thread Starter Member

    Jan 28, 2010
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    ok. I'm confused by the first step. In the page explaining Node Voltage Method the first step says:
    A voltage source in series with a resistance must be replaced by an equivalent current source in parallel with the resistance. We will write KCL equations for each node. The right hand side of the equation is the value of the current source feeding the node.
    No problem. If you have a voltage source that is an integer not a trig equation. How do I divide 8cos(t+pi/4) by 2.0 ohms?
     
  20. lemon

    Thread Starter Member

    Jan 28, 2010
    125
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    oh wait wait!! I think we covered that right. V1 is 8V.
    So:
    I1 = E1/R1 = 8/2 = 4A
    Am I good?
     
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