Finding individual impedance given the magnitude.

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Discussion Overview

The discussion revolves around finding the unknown capacitance in a parallel circuit consisting of a 10-H inductor, a 200-Ω resistor, and a capacitor, given that the magnitude of the total impedance is 125Ω at a frequency of ω=100 rad/s. Participants explore various methods to approach the problem, including impedance calculations and algebraic manipulations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant attempts to calculate the total impedance by combining the inductor and resistor first, then adding the capacitor, but finds the resulting equation complex and unwieldy.
  • Another participant suggests using the formula for total impedance in parallel circuits, but does not provide a specific solution.
  • Concerns are raised about the lack of a phase angle accompanying the magnitude of the impedance, which some participants believe is necessary for solving the problem.
  • There is a discussion about the correct interpretation of the magnitude of the impedance, with one participant clarifying that the magnitude does not equate directly to the impedance itself, which is a complex number.
  • Some participants express confusion over the arithmetic involved in calculating the equivalent impedance and suggest keeping the calculations algebraic until the end for clarity.
  • A later reply acknowledges a mistake in the previous calculations and offers a clearer method for expressing the equivalent impedance in rectangular form.
  • One participant realizes that they were overcomplicating the problem and thanks another for their guidance.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement regarding the methods used to calculate the impedance. While some participants provide corrections and clarifications, there is no consensus on a single approach or solution to the problem.

Contextual Notes

Participants note the complexity of the calculations and the potential for confusion when dealing with complex numbers and magnitudes. There are unresolved steps in the mathematical reasoning, particularly concerning the relationship between the impedance and its magnitude.

Basher1
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A 10-H inductor, 200-Ω resistor and a capacitor C are in parallel. find C if the magnitude of the impedance is 125Ω at ω=100rad/s




Now i attempted to solve it by first doing the parallel combination of the inductor and the resistor. Then i would do the parallel combo of these with the capacitor with the variable C. This left me with a massive equation. I am sure there has to be an easier way. please help
 
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Welcome to pf!

Hi Basher1! Welcome to pf! :smile:

The total impedance is 1/(1/Z1 +1/Z2 + 1/Z3) :wink:

Show us what you get :smile:​
 
hello Tiny tim. I've tried adding the impedance of the inductor and resistior then trying to find the unknown capacitance. but i was given the magnitude of the impedance. I wasn't given anything in rectangular form.
 
Show us what you get. :smile:
 
ZR = 200Ω,
ZL = j1000Ω
ZC = 1/j100CΩ
ω = 100rad/
magnitude of impedance = 125Ω

Zeq = (200 + j1000)/(200.j1000) + j100C = 0.008

= (200 x 10^6) + (40 x 10^6)j/(4 x 10^10) + j100C = 0.008

= (5 x 10^-3) + (1 x 10^-3)j + (j100C) = 0.008

=> (j100C) = 0.003 + j(0.001)


C = (0.003 + j(0.001))/(j100)
clearly a capacitance cannot have an imaginary value. I have no idea what I'm doing. personally i thought you would require a phase angle to accompany the magnitude of the impedance in order to solve so that we could get a reactance and resitive component and thus i could equate coefficients.
 
Last edited:
Basher1 said:
ZR = 200Ω,
ZL = j1000Ω
ZC = 1/j100CΩ
ω = 100rad/
magnitude of impedance = 125Ω

Zeq = (200 + j1000)/(200.j1000) + j100C = 0.008

= (200 x 10^6) + (40 x 10^6)j/(4 x 10^10) + j100C = 0.008

= (5 x 10^-3) + (1 x 10^-3)j + (j100C) = 0.008

=> (j100C) = 0.003 + j(0.001) C = (0.003 + j(0.001))/(j100)
clearly a capacitance cannot have an imaginary value. I have no idea what I'm doing. personally i thought you would require a phase angle to accompany the magnitude of the impedance in order to solve so that we could get a reactance and resitive component and thus i could equate coefficients.

All three of the impedances are in parallel. So why are you adding them as though only two of them are in parallel, and the third is in series with the other two?

EDIT: Hmm, no sorry, my mistake. I see that your operation is correct after all. I'll reply momentarily with further comments.
 
Last edited:
No that's ok.

I would like to hear your feedback
 
It's your second line that's wrong. Zeq-1 is NOT equal to 1/125 (0.008), because 125 Ω is just the *magnitude* of the impedance: |Zeq|. The impedance Zeq is in general a complex number.
 
so (4 x 10^10)/((200 x 10^6) + (40 x 10^6)j + (4 x 10^12)jC) = Zeq?

If i rationalise this I end up with a massive value and a c^2. there has to be an easier way?
 
  • #10
Basher1 said:
so (4 x 10^10)/((200 x 10^6) + (40 x 10^6)j + (4 x 10^12)jC) = Zeq?

If i rationalise this I end up with a massive value and a c^2. there has to be an easier way?

I'm having a lot of trouble following your arithmetic, for instance where the 200e6 and 40e6 come from.

Personally, I'd prefer to keep things algebraic until the very end. Things become a lot less cluttered, and it's just nicer to end up with a general solution for the equivalent impedance of any parallel LRC circuit that you then plug numbers into for your specific situation.

The easiest way I've found so far is as follows (I'll just omit the "eq" subscript and call the equivalent impedance "Z"). So we have:

Z-1 = (1/R) + (1/jωL) + jωC

Note that 1/j = -j, so you can express this as:

Z-1 = (1/R) - j(1/ωL) + jωC

= (1/R) + j[ωC - 1/ωL]

So now you have Z-1 expressed as a complex number in rectangular form (i.e. with a real part and an imaginary part, as opposed to polar form, which has a magnitude and a phase). How would compute the magnitude of this number using the real part and the imaginary part? I.e. how do you find |Z-1|? How is |Z| (which is what you are trying to solve for) related to |Z-1|?
 
  • #11
argh yes i was complicating things. Square the real part, imaginary part, take the root of the entire thing. equate this to 1/125. thankyou cepheid
 

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