AC voltage source driving an inductor and a capacitor

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't understand why would the current be zero in any of the branches . May be I am missing something very fundamental .

Please help me understand this question .

 

[cnh1995]

May be I am missing something very fundamental .

The question should be 'which ammeter may/could read zero ampere?'.
There is a specific value of ω for which one of the meters would read zero.

 

[Jahnavi]

The question should be 'which ammeter may/could read zero ampere?'.
There is a specific value of ω for which one of the meters would read zero.

OK .

Could you please elaborate .

 

[cnh1995]

Could you please elaborate .

Have you studied electrical resonance?

 

[Jahnavi]

Have you studied electrical resonance?

In series LCR circuit . But this is different .

 

[cnh1995]

In series LCR circuit . But this is different .

What will happen at resonant frequency in this circuit? What is the effective impedance of this circuit at resonance?

 

[Jahnavi]

Sorry ! As I said I know only series LCR circuit .

 

[cnh1995]

Sorry ! As I said I know only series LCR circuit .

Resonance occurs at a particular frequency where X_c=X_L. In this circuit, what would be the equivalent reactance at resonance?

 

[Jahnavi]

In this circuit, what would be the equivalent reactance at resonance?

Equivalent reactance should be 0 . The net impedance should be equal to the resistance but since there is no resistance R in this circuit , net impedance Z will also be 0 .

 

[cnh1995]

Equivalent reactance should be 0 . The net impedance should be equal to the resistance but since there is no resistance R in this circuit , net impedance Z will also be 0 .

No, the reactances are in parallel here.

 

[Jahnavi]

No, the reactances are in parallel here.

Should the equivalent reactance be infinite ?

 

[cnh1995]

Should the equivalent reactance be infinite ?

Yes.

 

[Jahnavi]

Yes.

OK . In that case current in A₃ will be zero at resonance ?

 

[cnh1995]

OK . In that case current in A₃ will be zero at resonance ?

Yes.

 

[Jahnavi]

Yes.

OK .

Since the problem is done , could you explain what happens to the current in the other two branches at resonance .

 

[cnh1995]

Here you can see that the source doesn't supply any current. This means the energy keeps oscillating between L and C and no power is drawn from the source.

 

[Jahnavi]

Here you can see that the source doesn't supply any current. This means the energy keeps oscillating between L and C and no power is drawn from the source.

Is the frequency ω altered continuously till the circuit resonates at a particular frequency ?

I am asking this because some current has to flow through A₃ initially so that energy could be stored in L/C .

 

[cnh1995]

Is the frequency altered continuously till the circuit resonates at a particular frequency ?

I am asking this because some current has to flow through A₃ initially so that energy could be stored in L/C .

Yes.
This is the sinusoidal steady-state scenario.
You don't have to know how the energy "gets into" the system. To analyse that, you need to do the transient analysis of the circuit using DEs and initial conditions, which is out of the scope of your curriculum (and this problem).

 

[Jahnavi]

OK .

Perhaps one last question

What if the circuit is initially operated at the resonant frequency ? Would you say there will be no current in the circuit anytime anywhere ?

 

[cnh1995]

Would you say there will be no current in the circuit anytime anywhere

That you need to see using transient analysis.
For the inductor,
Ldi/dt=V_msin(ωt)
If you close the switch at t=0, you will get an undamped transient for the inductor current.
The transient depends on the instant of switching and magnitudes of R and L.
See
https://www.physicsforums.com/posts/5815132/
https://www.physicsforums.com/posts/5797346/

For the capacitor, there won't be any transient current for switching at t=0.

Overall you'll get the sum of these two currents as the resultant source current. I can't say much about its waveform or magnitude unless I do the math.

 

[Jahnavi]

That means , in the transient period , there will be non zero current in A₃ even at resonant frequency ?

 

[cnh1995]

That means , in the transient period , there will be non zero current in A₃ even at resonant frequency ?

Yes.
I'll try to find the exact equation for current once I reach home.

 

[cnh1995]

Yes.
I'll try to find the exact equation for current once I reach home.

Here, after doing some math, you can see
i_L(t)=Vm/ωL-Vmcos(ωt)/ωL

and
I_c(t)=V_mcos(ωt)/ωL
...(since ωL=1/ωC).
Therefore, the source current
I_s(t)=I_L+I_c
=Vm/ωL (or V_mωC).

Thus, you can see if you connect the source at t=0 (i.e. at zero crossing of the input voltage), the current drawn from the source is dc.

 

[cnh1995]

Here is a simulation result of the same situation. It agrees with the math in #23.
For the values of L and C used, 159Hz is the resonant frequency.
You can see the source current(red) is dc.

 

[Jahnavi]

Thanks !

 

[cnh1995]

One more thing I forgot to add regarding one of your earlier posts:
To get A3 to read zero, you have to first connect the source to "put some energy into the LC network", then disconnect the source at a particular instant so that the energy will start oscillating between L and C, and then reconnect the source at a particular instant such that no current is drawn from the source.
Altering the frequency continuously till you hit resonance won't make A3 read zero, because whatever transient starts after you connect the source, lasts forever as there is no resistive damping. This is why I used the term "undamped transient" earlier somewhere. (Though if it lasts forever, it's no longer "transient", but I don't know what I should call it).

All the ac circuits you have in your curriculum are assumed to be in sinusoidal steady state and concepts like phasors (and phasor algebra), impedance, real/reactive power etc are applicable only to the circuits in sinusoidal steady state.

 

[Jahnavi]

I was waiting for this post . This is why I hadn't marked it as solved

Thank you very much !