Short vs Open Circuit: Is That Correct?

In summary, the impedance between points a and b is equivalent to a short circuit when the angular frequency is 0, and equivalent to an open circuit when the angular frequency is infinite. This can be determined by considering the individual impedances of the inductor and capacitor, and observing that for a short circuit, the impedance is 0, while for an open circuit, the impedance is infinite.
  • #1
annamal
381
33
Homework Statement
At what frequency or frequencies is the impedance between a and b equivalent to a short circuit and open circuit in the circuits below?
Relevant Equations
ZL = j*w*L
ZC = -j/(w*C)
Screenshot 2023-04-01 at 9.03.31 PM.png


For the first circuit, Req = ZL + ZC = -j/(w*C) + j*w*L = 0 for short circuit, so w = 0?
For the open circuit case, -j/(w*C) + j*w*L = infinity, so w = infinity?

Is that correct?
 
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  • #2
annamal said:
Homework Statement: At what frequency or frequencies is the impedance between a and b equivalent to a short circuit and open circuit in the circuits below?
Relevant Equations: ZL = j*w*L
ZC = -j*w*C

For the open circuit case, -j*w*C + j*w*L = infinity, so w = infinity?
w is the angular frequency.
Z=R+jX; X may be zero, or infinity, but w is not zero.
Resonance occurs at w necessary for XC+XL = 0, in both the parallel and the series cases.
 
  • #3
Ok, I wrote my equations wrong initially. But I am wondering what frequency is the impedance an open circuit? The impedance would have to equal infinite?
 
  • #4
Take each of the components individually. What happens to the impedance of a capacitor at ω = 0? What happens at ω → ∞? How about the inductor?
 
  • #5
You need to look at the impedance difference, of a series or a parallel circuit.
For DC, in one you sum the resistance, in the other you sum the conductance.
For AC you sum the impedance, or the admittance.
 
  • #6
Annamal,
You need to slow down and do your maths correctly.

For example, let's consider circuit a. The impedance of a series circuit is the sum of the impedances.
So, Z = -j/(wC) + jwL. We set Z=0, do some algebra and get w = 1/sqrt(LC). That'd the frequency where the impedance is zero (a short circuit). When is Z infinite? If w=0, then the impedance of the capacitor is infinite, so that is one answer. If w=infinity, then the impedance of the inductor is infinite, so that is another answer.

You will find that the answers for circuit b are reversed. It looks like a short for w=0 and w=inf., and it looks like an open circuit when w=1/sqrt(LC).

I invite you to graph the impedance as a function of w. It is very instructive.
Regards,
 
  • #7
annamal said:
Ok, I wrote my equations wrong initially. But I am wondering what frequency is the impedance an open circuit? The impedance would have to equal infinite?
Yes, it would have to be infinite. I think they are expecting a qualitative answer like high, low, or medium, rather than an actual frequency. There is no frequency where a real-world inductor is exactly like an open circuit. Likewise for a capacitor being a short circuit. This problem has not specified values for the inductor, or capacitor, nor a tolerance level for the circuit to be considered open or shorted.
 
  • #8
In the first case the two reactance are in series and in the second case the are parallel , as you already said. So, as you know, the series is the sum and the parallel it is the division of the product by sum.

For short-circuit the result is 0 and for open has to be infinite. In order to find infinite, you have to consider 1/Z=0 and calculate the ω=2*π*f=x
 
  • #9
For instance, if Z=j(ω*L-1/ω/Cap) for short-circuit Z=0 and for open circuit 1/Z=0 [Z=∞]
 
  • #10
If ω=0 Z=∞ since 1/0/Cap=∞
 

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