Q-Factor in Series-Parallel Shunt Circuits

[phantomvommand]

The above content discusses the conversion of RL (in series) to RL (in parallel). Importantly, the context is that the RL (in series) was in parallel with a capacitor.

I am interested to know the following:

Q1. Why is $\frac {\omega L} {R} = Q$ true for this circuit, given that this formula was derived for the pure series RLC circuit (and is not applicable to this RL||C circuit unless we are assuming R-L-C is approximately equal to R-L||C)?

Q2. Notably, the conversion between RL to R||L is not technically true, since it takes $R_L >> R$ and therefore ignores R. Is there a physical interpretation for the technical impossibility of converting from RL to R||L? Related to this, what is the physical meaning of a circuit in a situation where $R_L$ is not much larger than R, and therefore cannot be converted from series to parallel?

 

[DaveE]

Notably, the conversion between RL to R||L is not technically true

Yes, It all sounds like BS to me. Although I didn't review it carefully. To say that a series R, L branch can be substituted with parallel R, L equivalent makes no sense to me. Those impedances are only equivalent at a single frequency.

In fact, the dual circuit for a series R, L branch is a parallel G, C loop. But that's a slightly different subject.

My advice: find a better text book; there are many.

 

[phantomvommand]

Yes, It all sounds like BS to me. Although I didn't review it carefully. To say that a series R, L branch can be substituted with parallel R, L equivalent makes no sense to me. Those impedances are only equivalent at a single frequency.

In fact, the dual circuit for a series R, L branch is a parallel G, C loop. But that's a slightly different subject.

My advice: find a better text book; there are many.

I should note the text is assuming everything is at resonance. The math seems to work out, but I am interested in understanding the physical significance of being able to perform this switch at resonance for $R_L >> R$.

 

[tech99]

It is very common when analysing circuits to convert back and forth between series and parallel resistances. An approx method for values of Q above about 5 is to notice that the Q of the component is X/Rs, and also Rp/X. So for a given Q, we can say X/Rs=Rp/X. Actually I think the exact method is to use Q+1 instead of Q.
Converting between series and parallel allows us to add circuit resistances either in parallel or series, as the configuration dictates. It also allows impedance transformation to be obtained, as in the well known L-match circuit.
For a parallel circuit where there is resistance in series with the inductor, it is found for a circuit with low Q that the maximum voltage developed across the circuit and zero phase angle do not quite coincide. I think you would see this by drawing the phasor diagram.

 

[DaveE]

The source impedance, $R_l$, isn't used anywhere, except in the schematic. So I suppose we can assume it's zero? It, and the source transformation mentioned, seem irrelevant.

Series resonance and parallel resonance are fundamentally different topologies. But, if you restrict your analysis to any single frequency, resonance in this case, you can pretend they are the same. They both have a quadratic term (high Q) somewhere in their factored pole-zero response (impedance, transfer function, etc.).

The canonical form of this term will be $[1 + (\frac{1}{Q})(\frac{s}{\omega_o}) + (\frac{s}{\omega_o})^2 ]$ for $Q>\frac{1}{2}$
With the associated definitions:
$s = j \omega$ , for steady state or single frequency analysis. s is the Laplace variable used in transient analysis.
$\omega_o = \frac{1}{ \sqrt{LC}}$ , the resonant frequency.
$Zc=\sqrt{\frac{L}{C}}$ , the characteristic impedance.
$Q=\frac{Zc}{R}$ , for series resonance, or $Q=\frac{R}{Zc}$ , for parallel resonance

Personally, I always found this format easy to remember. It may be worth memorizing.

For example,
The impedance of a series RLC branch is $Z(s)=sL + R +\frac{1}{sC} = \frac{s^2LC+sRC+1}{sC}$
The admittance of a parallel RLC section is $Y(s)=sC + \frac{1}{R} +\frac{1}{sL} = \frac{s^2LC+s\frac{L}{R}+1}{sL}$

I doubt that there is any physical significance, it's just that there are useful similarities in the math. There are about a million ways that EEs talk about resonance, if you don't like this approach there are many others on the web. Many are pretty sloppy with underlying assumptions. I prefer to go back to the math, personally.

You may also be interested in exploring the idea of dual circuits. These are different topologically but have the same solutions under a set of duality transformations, which, of course, is applicable here too.
https://www.ams.org/publicoutreach/feature-column/fc-2019-05

PS: Also note that at resonance $\frac{s}{\omega_o} = j$, which makes the math easy.

 

[DaveE]

If you can't ignore one of the resistors, then it is neither a series or parallel resonance and the math is messy:

 

[tech99]

I should mention that it is easy to convert an inductor with a physical series resistance to one with a hypothetical parallel resistance, and vice versa. Similarly with a capacitor. The values vary with frequency but there is no requirement for it to be in a resonant circuit. If the component is placed in a black box, at one frequency we cannot tell which form it is.
If you then wish to create a resonant circuit by adding another reactance outside your black box, then use the series values for a series reactance and parallel values for a parallel reactance. Notice that the terms parallel and series refer to the location of the generator, otherwise the two circuits are the same.
Series and parallel conversions:
Let Q = Xs/Rs = Rp/Xp
Xs = Q x Rs
Xp=Rp/Q
Rs=Rp/(Q^2+1)
Rp=Rs(Q^2+1)
And as I mentioned previously, where Q is greater than about 5 it is possible for most engineering purposes to use Q^2 instead of Q^2+1.
Where you have a messy circuit, such as parallel LC with parallel R but also having resistor in series with the combination, then convert the LCR resonant circuit to series values and then place the external resistor in series with it by simple addition of the two resistances.