A non-resistive circuit

L D[AR2P#0.0] · Dec 23, 2007

The following circuit has the interesting property that:

If R=SquareRootOf(L/C), then the impedance of the circuit is independent of frequency. Determine the value of this impedance.

(Assume sinusoidal exitation)

Peter Nachtwey · Dec 23, 2007

It has been a long time. It is easy to see that the resistance will be the same when the frequency is 0 and when it is infinity. It is the impedence for the frequencies in between that are the question.

Solving this problem requires knowledge from the sophomore year in college.

L D[AR2P#0.0] · Dec 23, 2007

Peter Nachtwey said:
Solving this problem requires knowledge from the sophomore year in college.

... or lateral thinking, but that would spoil the fun of some complex number algeabra

jlcann0n · Dec 23, 2007

Since there are no numbers to go with the Cap. & Ind. it could be assumed that they would cancel each other out and it would be a purely resistive circuit.

manmeetvirdi · Dec 23, 2007

Peter Nachtwey · Dec 24, 2007

manmeetvirdi said:
2R

No, one should evaluate circuits at DC, 0 Hz, and at infinite Hz. This can be done by inspection. At 0 Hz the capacitor impedance is infinite so all the current must go through the top resistor. However, the inductor impedance goes to 0 at 0 Hz which shorts out the lower resistor. Therefore the impedance at 0 Hz is that of the top resistor R.

At infinite Hz the capacitor impedance is 0 which shorts out the top resistor. The impedance of the top half is 0. The impedance of the inductor is infinite which means the impedance of the lower part is R

If what LD is saying is true, the impedance is always R. I think the challenge is proving it. This problem isn't that hard if you know how the impedance changes as a function of frequency.

Z_total=z₁*z₂/(z₁+z₂)+z₃*z₄/(z₃+z₄)
z₁=R
z₂=????
z₃=R
z₄=????

Pandiani · Dec 24, 2007

I think this can be considered as a solid proof.

http://www.plctalk.net/qanda/uploads/RLC1.jpg

L D[AR2P#0.0] · Dec 24, 2007

Peter Nachtwey said:
If what LD is saying is true, the impedance is always R.

Correct.

The square root of minus one is shown as j in circuit analysis, so

Z_total=z₁*z₂/(z₁+z₂)+z₃*z₄/(z₃+z₄)
z₁=R
z₂=1/jwC
z₃=R
z₄=jwL

leitmotif · Dec 24, 2007

Let us see now

Xl = 2 Pi F L
Xc = 1 over 2 Pi F C

IF impedance is independet of frequency
then Xl = Xc at all frequencies
since they are inversely proportinal to each other and they cancel out.
Therefore L = C but can I really say that (how can two different units ie Farads and Henrys be equal??)

Would not impedance (actually resistance in this case) be 2R??

L D[AR2P#0.0] · Dec 24, 2007

Refer to Pandiani's post - you must formulate an expression for the overall impedance of the network. j notation is the easiest method to adopt for analysis.

Pandiani · Dec 24, 2007

leitmotif said:
Let us see now

Xl = 2 Pi F L
Xc = 1 over 2 Pi F C

IF impedance is independet of frequency
then Xl = Xc at all frequencies
since they are inversely proportinal to each other and they cancel out.
Therefore L = C but can I really say that (how can two different units ie Farads and Henrys be equal??)

Would not impedance (actually resistance in this case) be 2R??

Hehe, your reasoning is wrong because of the following reasons:
1. You don't know that impedance is independent of frequency. You must prove it.
2. If L and C are constant parameters, which they are, it is impossible to be XL = Xc at all frequencies (0 to infinite) isn't it?
3. XL and XC can cancel each other (XL = XC) out only on certian frequency and if they are in series, then you can look at the scheme like they don't exist...

leitmotif · Dec 24, 2007

Pandiani said:
Hehe, your reasoning is wrong because of the following reasons:
1. You don't know that impedance is independent of frequency. You must prove it.
2. If L and C are constant parameters, which they are, it is impossible to be XL = Xc at all frequencies (0 to infinite) isn't it?
3. XL and XC can cancel each other (XL = XC) out only on certian frequency and if they are in series, then you can look at the scheme like they don't exist...

I should not do these after getting off work at midnite.
1. I thought it was a given. After I reread I see where I was wrong.
2. Again tired thinking. At low freq Xl would be a low value and Xc would be high -- at high freq the inverse.
3. Xl and Xc can only be equal values and cancel at only one frequency.

I kept looking at this and thinking it was simple but posted to make sure I was thinking right.
Devils Advocates come in handy.
Thank you.

Dan Bentler

Peter Nachtwey · Dec 24, 2007

L D[AR2 said:
Refer to Pandiani's post - you must formulate an expression for the overall impedance of the network. j notation is the easiest method to adopt for analysis.

What about Laplace transforms? How do you do analysis in tht time domain? I have found jw to only be handy when I need to find the magnitude and phase at a particular frequency.

What would you do if the R in the "another resistive network" thread was replaced by a capcaitor and you needed the voltage as a function of time at each of the nodes?

L D[AR2P#0.0] · Dec 25, 2007

For this particular problem, I would argue that j notation is easiest as demonstrated in Pandiani's post #7.

Of course if you change the problem to analyse transients, then differential equations or Laplace would be required.

A non-resistive circuit

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