Short vs Open Circuit: Is That Correct?

[annamal]

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 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?

 

[Baluncore]

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.

 

[annamal]

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?

 

[gneill]

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?

 

[Baluncore]

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.

 

[FinBurger]

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,

 

[FactChecker]

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.

 

[Babadag]

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

 

[Babadag]

For instance, if Z=j(ω*L-1/ω/Cap) for short-circuit Z=0 and for open circuit 1/Z=0 [Z=∞]

 

[Babadag]

If ω=0 Z=∞ since 1/0/Cap=∞