Relating resonant frequency to electrical impedance

[coquelicot]

If a self oscillating electrical (passive) system is excited at two nodes A and B by a sinusoidal current, and if this system has one degree of freedom, then the response of the system is maximal at the resonance frequency. Quantitatively, this means that the ratio of the exciting complex voltage by the exciting complex current is maximal at the resonance frequency. In other words, 1/|Z(w)| is minimal at the resonance frequency w, with Z the complex impedance of the system seen as a circuit A-B (I believe this is common knowledge but I have no source). For example, the impedance of an LC tank circuit is Z = j w L/(1 - w²LC), hence 1/|Z| is minimal whenever w²LC = 1.
I wonder if there is not a more general statement for electrical systems with several degree of freedom. Something like: the frequency of the vibrational modes are the local minimum of the function 1/|Z(w)|, or what is the same, of 1/|Z(w)|². Any insight, references?

 

[sophiecentaur]

The filter theory that's taught to Electrical Engineers uses this basic idea. There are many sources of information about this (standard textbooks have it but I don't have one to recommend). This link is the sort of thing I'm suggesting.

 

[coquelicot]

The filter theory that's taught to Electrical Engineers uses this basic idea. There are many sources of information about this (standard textbooks have it but I don't have one to recommend). This link is the sort of thing I'm suggesting.

Humm, a priori, a filter is not a self oscillating system, despite I think you are probably right to say that filter techniques and material may be applied to the determination of the modes of such a system.

 

[davenn]

I wonder if there is not a more general statement for electrical systems with several degree of freedom. Something like: the frequency of the vibrational modes are the local minimum of the function 1/|Z(w)|, or what is the same, of 1/|Z(w)|2. Any insight, references?

see if this is any closer to what you are after ...

https://www.allaboutcircuits.com/textbook/alternating-current/chpt-6/parallel-tank-circuit-resonance/Dave

 

[coquelicot]

see if this is any closer to what you are after ...

https://www.allaboutcircuits.com/textbook/alternating-current/chpt-6/parallel-tank-circuit-resonance/Dave

Unfortunately, this is too elementary and does not much help me.

 

[sophiecentaur]

Humm, a priori, a filter is not a self oscillating system,

I'm not sure what you mean. Filters consist of element groups that may resonate. Your requirement is for analysis of a passive system so where is the difference and how is what you want different from standard filter theory?

 

[marcusl]

You might look into the field that is generally called "electrical network theory" or "electrical network analysis." Kirchoff's laws and Laplace transforms generate a system of equations that describes the network, which can then be analyzed in a number of ways. One can look at delta function excitation (impulse response), Heaviside excitation (step response), use eigenvalue decomposition to find the normal modes (this will directly give the resonances, if there are any), etc. There is the state variable approach, as well. I'm not an EE so I don't know the standard texts to recommend. I picked up an old text from a clearance rack some decades ago--Gupta, Transform and State Variable Methods in Linear Systems--that covers this material, and, in any case, there should be dozens of references for you to choose from.

BTW, your analysis of a "self oscillating circuit" with maximal Z on resonance assumes a parallel tank. There is a dual resonator, consisting of a series LC (more generally LCR), for which Z is minimum at resonance.

 

[coquelicot]

I'm not sure what you mean. Filters consist of element groups that may resonate. Your requirement is for analysis of a passive system so where is the difference and how is what you want different from standard filter theory?

Indeed, I have some problem to formulate this. In fact, the term resonance is applied in various situations that are rather different one to the other. Let me try to formulate what I want to mean: imagine an electrical network with passive R, L, C elements connected one to the other in a possibly complicated manner. Let me call such a network partially closed if it contains at least one loop. For example, a series LC circuit is not partially closed. Such a network is susceptible of natural dumping oscillations, if it is exited by an impulse current between two nodes A and B of some of his loops. If this is the case, let w be the angular frequency of these natural oscillations and call w the resonance frequency between A and B. Also, Let Z be the impedance of the circuit between A and B. If a sinusoidal current is applied between A and B, I believe that the response of the circuit will be maximal at the resonance frequency w, and this means that w fulfils 1/|Z(w)| = min. But I'm not sure. There is probably a more clever, general and formal way to formulate these kind of problems, and general theorems.

 

[Bart McCoy]

So it sounds like you want to be able to make a statement about impedance for the general case of multiple resonances in structures, if I understand correctly. If you are talking about multiple resonances from electrically short lumped circuits, that is one thing-- a much simpler case which you can calculate in a spreadsheet or MATLAB. If you are talking about microwave (E or H field) cavity resonances in physical structures (e.g. 3D microwave cavities or 2D microwave cavities or filters), then that's something completely different. More complex, but not unthinkable.

I wrote a paper that used these principles-- I collected impedance measurements of 2D resonant structures. However, I simply borrowed from the theory/work of K. Gupta (K. Gupta, "Analysis and Design of Planar Microwave Components"-- IEEE press, 1994, on Amazon). He shows that impedance of such resonant structures (with multiple degrees of resonant freedom) depends on WHERE you measure it and he gives the general recipe for how to compute the impedance (pp. 49-73, for simpler rectangular cavities). Calculations of impedance vs frequency were a great match to measurements. Alternatively, you can just simulate it with 3D EM tools like HFSS to measure the impedance.

 

[Bart McCoy]

I didn't see your last comment, about the problem you are trying to formulate.

You are talking about a very complex situation, if you have multiple energy storage elements (L & C) arranged in different loops & complicated ways.
In that case, you may see any combination of "series" and "parallel" resonances. With series resonance, (energy sloshing back and forth, THROUGH nodes A and B), you would observe a low impedance. By definition, a series resonance happens when energy storage elements are out of phase and slosh energy back and forth with little series loss. With parallel resonances, point A and/or point B might be tapped in between 2 energy storage elements sloshing energy back and forth, but not be in the series path. In that case, the impedance is very high, like any LC parallel tank.

When combined, you could see many complex things. In traditional EE analysis, you would see BOTH poles (maximal response) and zeros (minimal response) over frequency, if you plot things on a Bode plot. The tendency of such networks to dampen or accentuate those resonances has been described by many books on control theory (e.g. Ogata, "Modern Control Engineering"), where you plot poles/zeros in a complex plane and analyze. It's hard to say something specific about the impedance-vs-frequency unless you know the entire complicated network attached to points A and B.