Explicit Calculation of Q Factor of Loaded Tuned Circuits

[The Tortoise-Man]

I would like to discuss in detail an interesting aspect dealing with general considerations and methods to calculate the Q factor of oscillating systems on the example of this regenerative receiver.

Althought it is closely related to the discussion here discussion here (especially Baluncore's contributions) I think that it requires to be discussed in separate manner on it's own right since here I want to focus of general ways and possibilities to determine Q factor of oscillating system whichs slightly exceeds the complexity of usual serial and parallel
RLC- circuits (see explicit formulas here: https://en.wikipedia.org/wiki/Q_factor#RLC_circuits )Baluncore wrote in #3:

What you call a matching box is actually a resonant tuned circuit
that selects the frequency you want. If the tuned circuit is too
heavily loaded, it will have a lower Q, so less sensitivity and
less selectivity.

And my question is just how to check mathematically that the Q of heavily loaded tuned circuit
has indeed a relatively low Q. That might intuitively make sense, but sometimes require an exact
calculation. And my question is how to perform it. I nowhere found any techniques, only for
elementary cases of RLC- circuits as I said before. Are there methods for more complicated
circuits known?

For example which formulas & techniques for calculation of Q were used here in the case of regenerative receiver in the linked thread but also for general loaded tuned circuits to analyse the behavior of their Q in dependence of the parameters (resistence, reactance) of added load. Although this problem seemingly arises quite naturally, I nowhere found a formally clear approach from mathematical point of view to calculate Q in such
situations.

Could anybody give maybe a kind of overview how to calculate to Q
of a loaded tuned circuit in dependence of the
load?

 

[DaveE]

What you will do here is to model that heavily loaded tuned circuit as a network of ideal circuit elements by simplifying and only including the salient components. Then the precise calculation will look like the simple cases that you already know. This is assuming that the concept of Q makes sense for your network.

The precise mathematical version of Q relates only to the simple harmonic oscillator differential equation. It is an expression of the root locations of the quadratic characteristic equation of the SHO; $1+\frac{1}{Q}(\frac{s}{\omega_o})+(\frac{s}{\omega_o})^2$ . That is all. The math isn't hard, modelling real world circuits well is often hard.

 

[DaveE]

Also, a comment on "heavily damped", which, of course, means different things to different people.

The roots of the characteristic polynomial $Q(s) = 1+\frac{1}{Q}(\frac{s}{\omega_o})+(\frac{s}{\omega_o})^2$ are $s_{1,2}=-\frac{\omega_o}{2Q} (1 \pm \sqrt{1-4Q^2})$ When $Q<\frac{1}{2}$ the roots are real and we can factor the polynomial into the product of simple poles. This is approximately $Q(s) \approx [1+Q(\frac{s}{\omega_o})][1+\frac{1}{Q}(\frac{s}{\omega_o})]$ . Which is much easier to deal with than $s_{1,2}$ . This approximation allows us to ignore Q for heavily damped systems.

 

[tech99]

Most losses in the LC circuit will be in the coil. You cannot easily calculate the Q of a coil but will have to rely on other people's measurements. However, if the coil is believed to be efficient, then the applied load might be dominant and allow a calculation of loaded Q. For instance, let us suppose we have a parallel RC circuit and we expect a coil Q of about 100. Now we apply a load having a resistance of 10 Ohms in series with the coil. If the reactance of the coil at that frequency is 100 Ohms, then the loaded Q is X/R = 100/10 = 10 approximately. Alternatively, if we connect an antenna with a resistance of 1,000 Ohms in parallel with our coil, then the loaded Q will be R/X =1,000/100 = 10 approximately.
If the circuit is provided with regeneration, or positive feedback, this will cancel out some of the losses, so Q will be raised. In effect, regeneration adds negative resistance to the circuit. It is very difficult in practice to calculate the effect as it depends on the adjustment. If we increase regeneration so that oscillation commences, the Q multiplying effect is immediately lost and we have a Q equal to that without regeneration. Nevertheless, oscillating detectors are usually very sensitive, because the oscillation swings the device right over its square law region, increasing the efficiency of detection. If using an oscillating detector, it can be followed with an audio filter to obtain the desired selectivity.

 

[The Tortoise-Man]

What you will do here is to model that heavily loaded tuned circuit as a network of ideal circuit elements by simplifying and only including the salient components. Then the precise calculation will look like the simple cases that you already know. This is assuming that the concept of Q makes sense for your network.

The precise mathematical version of Q relates only to the simple harmonic oscillator differential equation. It is an expression of the root locations of the quadratic characteristic equation of the SHO; $1+\frac{1}{Q}(\frac{s}{\omega_o})+(\frac{s}{\omega_o})^2$ . That is all. The math isn't hard, modelling real world circuits well is often hard.

Conjecture: can it be stated that if we start with any arbitrary network then Q makes sense if and only
if it's possible to model the network as SHO, (and therefore having such model fixed,
we can associate to it this quadratic characteristic equation
(which if welldefined for every SHO) and then we are in business by doing a "coefficient
comparison" between the quadratic equation we associated to your network
and the stadard one $1+\frac{1}{Q}(\frac{s}{\omega_o})+(\frac{s}{\omega_o})^2$.
In that way we can determine Q.
In summary: Can we associate to our network a quadratic characteristic equation, then formally
also the Q in the way I tried to explain,right?

If what I have written in last paragraph is correct, the next interesting question which arises naturally is how we should start to attack the question if we can associate to a given network a
quadratic characteristic equation.

Are there any criteria or ways to check it?

Let consider following relatively common situation: Assume that we know the input and output signals (as functions of time) of our network.

Can we associate naturally to it some function (my natural guess would be the complex valued "transfer function"= quotient of Laplace transform of input and output signal)
and then study the behavior of this function in order to deduce further results (my hope: especially the SHO story).

If my intuition not jokes on me (that's what I hope/conjecture, but not know; so please correct me
if I missing totally the point), then the "dream" would be that studing the behavour of this
transfer function might give us information if the network would behave
like SHO.

A brave hope: Maybe in the case the network can be modeled as SHO the transfer function closely related
to the quadratic characteristic equation describing the SHO. At least both functions are complex valued :)

Does the idea make sense or is it just a dreamer's hope to relate some functions which can naturaly be associated to networks to each other to obtain a deep result?

 

[DaveE]

Conjecture: can it be stated that if we start with any arbitrary network then Q makes sense if and only
if it's possible to model the network as SHO, (and therefore having such model fixed,
we can associate to it this quadratic characteristic equation
(which if welldefined for every SHO) and then we are in business by doing a "coefficient
comparison" between the quadratic equation we associated to your network
and the stadard one $1+\frac{1}{Q}(\frac{s}{\omega_o})+(\frac{s}{\omega_o})^2$.
In that way we can determine Q.
In summary: Can we associate to our network a quadratic characteristic equation, then formally
also the Q in the way I tried to explain,right?

If what I have written in last paragraph is correct, the next interesting question which arises naturally is how we should start to attack the question if we can associate to a given network a
quadratic characteristic equation.

Are there any criteria or ways to check it?

Let consider following relatively common situation: Assume that we know the input and output signals (as functions of time) of our network.

Can we associate naturally to it some function (my natural guess would be the complex valued "transfer function"= quotient of Laplace transform of input and output signal)
and then study the behavior of this function in order to deduce further results (my hope: especially the SHO story).

If my intuition not jokes on me (that's what I hope/conjecture, but not know; so please correct me
if I missing totally the point), then the "dream" would be that studing the behavour of this
transfer function might give us information if the network would behave
like SHO.

A brave hope: Maybe in the case the network can be modeled as SHO the transfer function closely related
to the quadratic characteristic equation describing the SHO. At least both functions are complex valued :)

Does the idea make sense or is it just a dreamer's hope to relate some functions which can naturaly be associated to networks to each other to obtain a deep result?

Yes, that is basically correct from the "explicit calculation" point of view. IMO, that is the only rigorous way that Q makes sense.

Still you need to make reasonable approximations to model complex systems with simpler canonical forms. It is that modelling that is usually the tricky step. There is no simple answer, in general, to the question "does the model work?" Every model should be evaluated in light of its inherent limitations. Every model is wrong in some way, but they can be very useful for understanding by reducing complexity and focusing attention on the most salient features of interest.

"Engineering is the art of approximation." - R. D. Middlebrook

 

[The Tortoise-Man]

Still you need to make reasonable approximations to model complex systems with simpler canonical forms. It is that modelling that is usually the tricky step. There is no simple answer, in general, to the question "does the model work?" Every model should be evaluated in light of its inherent limitations. Every model is wrong in some way, but they can be very useful for understanding by reducing complexity and focusing attention on the most salient features of interest.

"Engineering is the art of approximation." - R. D. Middlebrook

Yes, of course. That's was just my sloppe atempt to understand the rough idea. Principally I was just wondering if the "transfer function" is exactly the "right" thing which we should try to study if we want to check that out network can be modeled as SHO (and so Q is welldefined).

The transfer function was just the only function which can in my mind that we can always ad hoc associate to a networks with known input and output without thinking to much.

So it was just a guess, but as far as understand your previous answer correctly indeed the transfer function is exactly THE right function should be analyzed and is exactly THAT one which carry (of course after reasonable simplifications of the real network) the information about if the network can be regarded as SHO (again, of course after appropriate simplifications).

Assume now our network indeed allows such modelation as SHO (and therefore obtains a "quadratic characteristic equation") and we can also calculate the transfer function as quotient of Laplace transform of input and output signal.

Is in this special case precisely known how the transfer function is exactly related to the quadratic characteristic equation from the SHO-model of this network?

Are they related to each other via a special equation?

 

[DaveE]

Is in this special case precisely known how the transfer function is exactly related to the quadratic characteristic equation from the SHO-model of this network?

yes

 

[The Tortoise-Man]

yes

Could you write down this relation between transfer function and quadratic characteristic equation for the special case above or give a reference where it can be looked up?

 

[DaveE]

Could you write down this relation between transfer function and quadratic characteristic equation for the special case above or give a reference where it can be looked up?

OK, here's one I chose mostly at random from a google search: https://lpsa.swarthmore.edu/Representations/SysRepTF.html

This is pretty basic stuff in the study of Laplace transforms and linear systems. You can find it in lots of places.