How do we define resonant frequency that has arbitrary combination of resistors,capacitors and inductors?

For instance inductors and capacitors may neither be in series or parallel, but may arbitrarily be connected with resistor.

For the given figure, how do we define resonant frequency? enter image description here

  • \$\begingroup\$ You may want to read the Wikipedia page, Natural frequency. \$\endgroup\$ – The Photon Jun 8 '19 at 5:23
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    \$\begingroup\$ Are you sure you mean 'define'. We define something by saying 'MAKE IT SO'. Once we have a definition, we might then proceed to measure that characteristic of a system, or from a circuit schematic, calculate what it's going to be. \$\endgroup\$ – Neil_UK Jun 8 '19 at 5:35
  • \$\begingroup\$ A practical definition of resonant frequency "the frequency at which the circuit resonates when it is excited by an impulse." Calculating it for an arbitrary network seems difficult. I would be tempted to solve it using spice. Please clarify if you are looking for a DEFINITION or a method of computation. \$\endgroup\$ – mkeith Jun 8 '19 at 16:57
  • \$\begingroup\$ I am looking for a numerical definition. Definition as in resonant frequency has to be related to power/current/voltage in some circuit element being maximum when the input is tuned to a certain frequency. So does such type of general definition exist? \$\endgroup\$ – helloworld1e. Jun 8 '19 at 18:02
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    \$\begingroup\$ Neil_UK, I must admit that I cannot follow your comment. Of course, we have a definition for the "resonant frequency". As an example, a crystal has two such frequencies which follow such a definition (phase criterion). \$\endgroup\$ – LvW Jun 10 '19 at 14:39

Here comes the commonly accepted definition for resonance:

The input-output relation (V-I, V-V, I-V or I-I) of a frequency-dependent circuit shows amplitude and phase relations which changes when the frequency is varying.

The frequency, where the phase difference between the two quantities is zero is defined as a "resonant point".

In some cases, this "resonant frequency" gives a maximum or a minimum of the corresponding output-to-input relation - but not always. Therefore, these amplitude considerations cannot be used as a general definition for resonance. It is the phase relation between input and output which is used for this definition.,

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Rewrote my answer because I suggested at first something already said and skipped questioner's demand of sinusoidal test signal.

Resonance = internal buildup of reactive power at certain frequency more than at the frequencies a little off. Some circuits can have several resonant frequencies.

In math there's complex conjugate pole or zero pairs in the one port input impedance functions. If you consider the circuit as 2 port, where your test signal supplies the input and the output signal is the voltage or current of a component, then the resonance appears as a complex conjugate pole pair of the transfer function. That's equivalent to other answer, which presents resonances as (possibly decaying) sinusoidal oscillations after an initiative pulse.

I guess those math theories are not interesting just now, because if you know transfer function math, you also should know those elementaries.

In practice measuring the impedance vs frequency can reveal the resonances. They appear as peaks or notches. But nothing quarantees that you can see all resonances by measuring the impedance, because there can be so low impedance parallel or high-impedance series non-resonating parts that their effect hides the resonance. As well it's possible to construct a RLC circuit where the impedance has minimum or maximum at a frequency, but there's no resonance. So, an existing resonance can be seen, but to be sure, something else is needed.

Measuring the and comparing the AC voltages of the parts at different frequencies do not prove resonances. You can well find a maximum or minimum voltage at certain frequency, but that needs no resonance. Think for ex. a 100Hz RC highpass filter and a 1 kHz RC lowpass filter cascaded. You get max output at some frequency, but that's not a resonant frequency.

The conclusion: The existence of resonances at a certain frequency interval cannot be reliably decided by inputting a sinusoidal test signal to one port RLC circuit and measuring currents and voltages of the parts in the circuit. Better results can be got does there exist sinusoidal oscillations when a voltage or current impulse is inputted, as already told by others. If one could measure frequency domain transfer functions from input to every part voltage and current, then the existence of complex conjugate poles could be calculated and that's a different situation.

Add due the comment about low pass filters and zeros:

The next image has actually 2 different filtering circuits with a common test signal voltage source:

enter image description here

The upper is a 2nd order lowpass filter. Its output is the voltage at node 2. The transfer function has a substantial peaking at 15,9 kHz. That frequency is the resonant frequency of series resonant circuit L1C1. The transfer function has a pair of poles which have imaginary parts at frequencies plusminus 15,9 kHz.

I quess there's nothing surprising. See the two upper plots. The upmost is the absolute value of the transfer function and in the middle there's the phase shift. It really has 90 degrees lag at the resonant frequency.

enter image description here

The bottom plot belongs to the lower filtering circuit. It's a bandstop filter which outputs the voltage of node 3. It kills fully the LC resonant frequency 15.9 kHz. There's no complex conjugate pole with imaginary part frequency=plusminus 15.9 kHz, there's a pair of zeros.

A contradiction?

No, because if the output signal is considered to be the current of C2, the circuit is a bandpass filter and it has a complex conjugate pole pair with imaginary part frequencies plusminus 15,9 kHz which is the resonant frequency.

What gives to me a right to make a turn and change the output signal only to argue against a commentator? I reserved that right in the beginning. The questioner asked about one port LC circuits. I said that the resonances appear as poles of the transfer function when the output is considered to be a voltage or a current of some component in the circuit. In this case I happened to took the current of C2. But as well I could have selected the voltage over C2 or the voltage over R2 or L2


This is to show that "impedance reactance part=0" is insufficient criteria for resonance:

enter image description here

enter image description here

Inside the circuit there's 15.9 kHz parallel resonator L2C2. In the uppermost plot we see at 15,9 kHz the resonator has voltage maximum. The blue plot is the total impedance seen by V1 and the green plot is the reactance part of that impedance.

The resonance happens in the resonant circuit at 15,9kHz, but it cannot be seen by observing the input impedance.

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  • \$\begingroup\$ Quote: "...the resonance appears as a complex conjugate pole pair of the transfer function...." I think, this is not correct. Counter example: Each second-order lowpass with a Q-value Qp>0.5 has a conjugate-complex pole pair (defined by the pole frequency). But, of course, this property has nothing to do with the effect we call "resonance". In contrary, the phase shift between input and output at the pole frequency will be 90 deg. \$\endgroup\$ – LvW Jun 10 '19 at 19:43
  • \$\begingroup\$ @LvW That's definitely a resonance, a series resonant circuit+resistors, if it happens to be a LC filter. The same buildup can happen, of course, in an amp with RC feedback(=active filter) BTW In my ancient music synth the Q factor adjustment of the LOWpass filter had parameter name "Resonance" \$\endgroup\$ – user287001 Jun 10 '19 at 19:50
  • \$\begingroup\$ Simple question: Don`t you agree that a conjugate-complex pole pair in the s-plane defines the "pole frequency" of a second-order filter with a phase shift of 90 deg? This is independend on realization alternatives (active or passiv, LC or RC) \$\endgroup\$ – LvW Jun 10 '19 at 19:57
  • \$\begingroup\$ What's the nominator? Zeros affect, too. \$\endgroup\$ – user287001 Jun 10 '19 at 20:03
  • \$\begingroup\$ Yes - and thats the reason, you cannot use the complex pole pair ALONE for definig the case of resonance!! See my counter example. If I am wrong, the counter example must be wrong.Is it wrong? \$\endgroup\$ – LvW Jun 10 '19 at 20:08

IN CONCEPT: for any linear circuit there are signals that, when applied to the circuit, will get really really big as time goes on, For the normal type of circuits, any signal that has this property will be a sinusoidal type of wave, that has an exponentially falling envelope. If you pretend that there exist ideal capacitors and inductors, then constant amplitude sinusoidal waves are also included in this group. Since a lot of times, circuits don’t have much loss, the idea of resonant “frequency” makes sense..for example, if a sinusoidal current at the resonant frequency is applied, the voltage will just keep getting bigger and bigger. For circuits with a lot of loss, the falling exponential often dominates, and the sinusoidal part is small or nonexistent. So, in this case, the “frequency” word is confusing. Traditionally, engineers still use “resonant frequency” even in these cases where the falling exponential dominates .

So, for example, you may have a circuit with a single resistor and a single capacitor. A falling exponential signal with a time constant r times c, when applied to this circuit, will result in a really big response. This is the “resonance” for that circuit, even though the falling exponential has no repeating, frequency-like part.

IN MATH TERMS: All lumped, linear circuits are described by linear, constant coefficient, ordinary differential equations. All differential equations like this have a homogenous solution that are sums of time signals of the form A*exp(st), where A and s are complex numbers, and t is time. These signals are the resonances of the circuit. A is called the amplitude of the resonance and s is called the frequency of the resonance. The imaginary part of s gives rise to sinusoidal shaped resonances, and the real part of s gives rise to exponentially falling shaped resonances. Engineers call s the “resonant frequency” of the circuit.

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  • \$\begingroup\$ Did I understand it right? A "single resistor and a single capacitor" will show a resonance effect in form of an exponenetial signal? Ist this your personal definition? Or ddi you perhaps mix the two terms "resonance" and "response"? \$\endgroup\$ – LvW Jun 10 '19 at 19:29
  • \$\begingroup\$ The term "resonance" arose from people observing vibrations. When electricity was discovered, the scientists used the term to describe oscillating electrical responses. In both cases, the phenomenon is the result of excitation of the "natural" or unforced response of the dynamic system. So, what about systems that have a natural response that is not a vibration? I suppose we could say "these system have no resonance", or we could say "the resonance of these systems is non-oscillatory". Either way of saying it is fine with me. I just tried to answer the question using the latter view. \$\endgroup\$ – user69795 Jun 11 '19 at 13:35
  • \$\begingroup\$ OK - I agree with you: WE COULD SAY ! Why not? But I am not sure, if this is the way commonly used in textbooks and other knowledge sources. (Insofar, it is your personal definition. Good enough). \$\endgroup\$ – LvW Jun 11 '19 at 13:53
  • \$\begingroup\$ In the steady state, what is the phase difference between the current and voltage of an RC circuit under DC excitation? If the phase difference is zero, then, according to LvW, it has resonance at DC. I am just joking around here. I do not think it is meaningful to talk about the resonant frequency of an RC circuit. \$\endgroup\$ – mkeith Jun 11 '19 at 23:09
  • \$\begingroup\$ Lvw: actually, this way of looking at things is not my invention... it is the standard way of understanding linear, lumped systems (including circuits). Classic circuit synthesis as invented by Brune, Cauer, Foster, and Darlington is all about "poles at DC" , or "poles at infinity". In this literature, natural frequencies of a system can be anywhere on the complex plane, and real ones (no oscillations) are not treated any differently than complex ones (that have oscillations). Poles at zero (for example, the impedance of a capacitor) indicate that the "natural frequency" is DC. \$\endgroup\$ – user69795 Jun 12 '19 at 14:36

Well the definition doesn't change with different circuits

The simplest form of resonance i can explain---

If a circuit is operated at resonance frequency

You: Hey source, what's connected across you ?

Source: Resistor

Let's say the circuit has arbitrary combination of resistors, capacitors and inductors and the circuit is operated at the resonance frequency fr,

Usually in lectures we are told calculate the impedance and make the imaginary part zero, the thing is at resonance the reactive components of current cancel each other and the current drawn from the source is in phase with the source hence source thinks that a resistor is connected across it, which in turn gives you the phasor-

enter image description here

You see in resonance maximum power is transferred because

{Every point is same}

  1. inductive and capacitive nature cancels each other to make reactance zero and achieve minimum resistance

  2. the maximum current you can get is when inductive and capacitive current cancel each other making impedance minimum...

  3. the effective source voltage across L and C is zero and is distributed across the resistors only


And for the solving part of the arbitrary circuit you say, just calculate the impedance/admittance of the series/branches and make the reactance/susceptance zero (imaginary part zero).

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  • \$\begingroup\$ In the figure, you have shown a simple example for the resonance effect - and in this basic example the resonant frequency is identical with the amplitude maximum - which is NOT always the case. Therefore, finally you have given the general definition: Imaginary part zero (which is identical to the criterion as mentioned in my answer: zero phase). \$\endgroup\$ – LvW Jun 10 '19 at 19:35

If you know how to calculate the resistance of a network of resistors - series, parallel, and converting a Y pattern to a Δ pattern and vice versa - and if you know how to calculate with complex numbers, the answer is easy.

The resistance of a resistor is R

The resistance of an inductor is iLf.

The resistance of a capacitor is -iC/f.

So for instance a capacitor in series with an inductor has a resistance iLf-iC/f, which is zero at the resonant frequency.

A real electrical engineer will tell you where I should have started saying reactance instead of resistance and also if I should have interchanged the signs of i in the capacity and the inductor.

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  • \$\begingroup\$ If you don’t know the answer, why guess? \$\endgroup\$ – Chu Jun 8 '19 at 10:40

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