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we know that in a series LC circuit the resonance frequency is defined as that frequency in which the total impedance offered by the circuit is zero.

We also know that in a parallel LC circuit the resonance frequency is defined as that frequency in which the total admittance offered by the circuit is zero.

I have seen several times analyzes of much more complex circuits (composed of several series and more parallel elements of L and C) in which the resonance frequency was defined as that frequency in which the total admittance was infinite, for example.

My question is: what is the general definition of "resonance condition"? With what criterion do we define the resonance condition for a given circuit?

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Write differential equations that describe a circuit. Set all sources equal to zero. Find the non-zero functions of time that satisfy the differential equations. For lumped, linear, time invariant circuits, these functions will be complex exponentials. A “resonant frequency” in radians per second is the imaginary part of the argument of the complex exponential. This definition works for all circuits, as well as lumped mechanical systems.

The resonant frequencies are a property of the circuit, and have nothing to do with inputs or outputs.

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  • \$\begingroup\$ Thank you for your answer. Can you explain me the reason of putting all sources equal to zero? I thought that it may indicates that the circuit oscillates also in absence of external sources. Is it true? \$\endgroup\$ – Kinka-Byo Nov 17 '19 at 16:20
  • \$\begingroup\$ A source will add another signal into the response: a signal that has nothing to do with the circuit. The resonance condition is a property of the circuit. Some circuits will not have imaginary parts in the argument of the exponential, ( the argument will be real). You could say that these circuits do not resonate, or you could say that these circuits have “non oscillatory resonances”, I suppose. \$\endgroup\$ – user69795 Nov 17 '19 at 20:31

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