Question about the energy stored in an RLC parallel resonant network

[cianfa72]

Yep. The existence of the Voltage source is upsetting the normal RLC situation because Energy can be supplied and removed by the source, independently of the Voltages across the L and C.

In the case now under analysis we're considering a current source and not a Voltage source. Even in this case the stored energy is not constant in time.

 

[cianfa72]

This all seems lost in the semantic sea.
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May I ask the OP: what is the purpose of defining this particular parameter in this particular case? In my experience, the parallel RLC circuit (which is not exactly what you have sketched) is used mostly as a notch filter. As such the working definition of $Q_{filter}$ is typically related to the ratio of the bandwidth to the center freguency of the filter notch. Hence the "Q" of the filter. This definition is related to but not identical to the Q of a corresponding damped simple harmonic oscillator and instead indicates a ratio of dissipation.

Sure, you are using basically the Q definition from the "standard" form of the transfer function denominator.

 

[DaveE]

I did the job to derive the expression for the total energy stored in post #15 network on the right (i.e. sinusoidal current source $I_0$)

$E_L(t) = \frac {1} {2} L_sI^2_L(t)$, $~E_C(t) = \frac {1} {2}CV^2_C(t)$, $~I_L(t) = I_L sin (\omega t)$

$I_L = \frac {I_0} {(1 - ~\omega^2L_sC) ~+~ j\omega R_sC}$

$V_C(t) = I_L |R_s + j\omega L_s| sin (\omega t + \phi)$, $~tan(\phi) = \frac {\omega L_s} {R_s}$

$E_L(t) = \frac {1} {2} \frac {I^2_0} {(1-~\omega^2 L_sC) ~+~ \omega^2 R^2_s C^2}L_ssin^2(\omega t)$

$E_C(t) = \frac {1} {2} \frac {I^2_0} {(1-~\omega^2 L_sC) ~+~ \omega^2 R^2_s C^2}C(R^2_s + \omega^2L^2_s)sin^2(\omega t + \phi)$

thus the total energy stored at time t is:

$E(t) = \frac {1} {2} \frac {I^2_0} {(1-~\omega^2 L_sC) ~+~ \omega^2 R^2_s C^2}[L_ssin^2(\omega t) + C(R^2_s + \omega^2L^2_s)sin^2(\omega t + \phi)]$

starting from this expression I do not think there exists actually an angular frequency $\omega$ the total stored energy stays constant in time.

Maybe, perhaps I'll look at it later. Of course you want to create $cos^2(\omega t) +sin^2(\omega t)$ by solving for a phase shift of $\frac{\pi}{2}$ and the amplitudes equal. It does look over determined since in those two constraints you only have ω to adjust.

 

[cianfa72]

Of course you want to create $cos^2(\omega t) +sin^2(\omega t)$ by solving for a phase shift of $\frac{\pi}{2}$ and the amplitudes equal. It does look over determined since in those two constraints you only have ω to adjust.

Solving for a phase shift of $\frac{\pi}{2}$ gives $\omega \rightarrow \infty$ because $\phi = tan^{-1} (\frac {\omega L_s} {R_s})$ so I believe a solution for $\omega$ actually does not exist.

Reasoning again about it I believe the point to highlight is the following: consider an RLC network for which the total energy stored in reactive elements (inductor, capacitor) at AC steady-state does not stay constant in time (e.g. the network from the previous posts).

Then I believe we cannot apply the first definition of Q factor here, we need to use the second one involving real (dissipated) power and the reactive one:

$Q(\omega) = \omega \times \frac {\text{maximum energy stored}} {\text{power loss}}$

Using this definition and assuming for "maximum energy stored" the maximum of the energy stored on each reactive component (either inductor or capacitor) separately , we can show that Q factor is just the ratio of the reactive power (evaluated either for the inductor or the capacitor) to the real (dissipated) power.

In particular when evaluated at the resonance $\omega_r$ of the RLC network (defined as the frequency for zero total reactive power) the two kind of reactive power actually take the same value thus providing the same Q value.

 

[DaveE]

On further thought, it's kind of obvious to me now that the circuit in the left side of post #15 can't have constant stored energy. No calculations required. Let's just consider the capacitor voltage, somehow created by the source current. That voltage defines the energy stored on the capacitor, in particular it's phase. In order to compensate for the capacitor energy oscillation, the energy stored in the inductor must be 180^o out of phase, which means it's current must be 180^o out of phase with the capacitor voltage. Yet it is the capacitor voltage that determines the inductor current (flowing through the series resistor and inductor). The phase shift created by the series resistor makes it impossible to make the inductor current out of phase w.r.t capacitor voltage at any frequency.

 

[cianfa72]

In order to compensate for the capacitor energy oscillation, the energy stored in the inductor must be 180^o out of phase, which means it's current must be 180^o out of phase with the capacitor voltage.

It should be 90^o out of phase (think of $sin^2(x) + cos^2(x) = 1$)

Yet it is the capacitor voltage that determines the inductor current (flowing through the series resistor and inductor). The phase shift created by the series resistor makes it impossible to make the inductor current out of phase w.r.t capacitor voltage at any frequency.

That's right. For the same reason this conclusion also holds for post#15 circuit on the right (current source) -- just consider as V_c the voltage established across the capacitor at AC steady-state frequency $\omega$ and apply the same reasoning as above.

 

[cianfa72]

I believe the take-home message for circuits in post#15 is the following: in both cases the RLC network acts at frequency $\omega_r$ in AC steady-state as a 'tank' circuit: the instantaneous power from the source (either voltage or current source) is always $\geq 0$ meaning the flow of energy in time is always from the source to the RLC network (zero the other way around). However, the instantaneous power at a given time t dissipated on the resistor is not the same as the instantaneous power from the source (at the same t) !

Consequently that RLC network basically acts storing the possibly "excess" energy from the source not dissipated on the resistor at the same time t, that "excess" energy will be dissipated on the resistor later on in the cycle.

Note that in both "classic" RLC series and parallel networks, instead, at a given time t there is not any "excess" of energy stored because now the instantaneous power from the source at a givent time t is exactly the same as the instantaneous power dissipated on the resistor at the same time t.

 

[hutchphd]

I believe that is a very fine way to look at it...good that you can reach a satisfactory internal conclusion!

 

[PaulWDent]

I think you have come across the issue which we discussed on this forum a few weeks back, that for the parallel resonant circuit (using R and L in series), max impedance and zero phase angle do not coincide. The resonant frequency is that where phase angle between applied voltage and current is zero. In this case, at switch-on the stored energy will build up asymptotically over several cycles and then remain constant, and the generator will then only supply the resistor. At switch-off, the stored energy will be transferred to the resistor over several cycles.

Of course it's not constant with time if there is an R in the circuit, dissipating the power. The energy will decay exponentially.
It's only constant when R=0, as with electrons flying around an atom's nucleus, which forms an infinite Q lossless resonator.

In a crystal, many such identical resonators are coupled together. With no loss, any coupling is overcoupling leading to multiple resonant peaks. The probably of two resonant peaks being the same frequency is zero. This the basis of Pauli's exclusion principle.

 

[hutchphd]

The question refers to a driven RLC network. It is always good to read the question!