# Resonant frequency and maximum signal width

I've seen resonant frequency being defined as the point where the input signal is in phase with the output signal (So reactance/susceptance is 0).

However conceptually, teaching material seems to link it mostly with maximized output signal width (It's never explicitly stated, just heavily implied: See band-pass filters).

In short, is there any actual link between the resonant frequency and the signal width? Or is the link only there for the special case of RLC circuits?

EDIT: I'm trying to get a general intuition, hence the lack of specifics - but here is the example that got me on this train of thought for context:

  I
x=>-x--x
+   |  |
R1 R2
V   |  |
C  L
-   |  |
x---x--x


Excuse the ascii art for now, I'll look for something better. (Input: V, Ouput: I)

I was trying to computer the resonance frequency on the above (I have no concrete values), and was wandering if the maximization of the output width (|I|) happens when I and V are in phase; same as a parallel RLC circuit.

• what is signal width? pulse width? bandwidth? pls add a reference to "signal width" to give us some context for the question. – user_1818839 Jun 30 '20 at 12:28
• Are you exclusively talking about band-pass filters? I ask because resonance is also associated with low-pass and high-pass filters. – Andy aka Jun 30 '20 at 12:36
• @BrianDrummond By signal width, I mean the length of the phasor. As for the rest, this is a theoretical question, so no answer there. See the edit for context. – gtsiam Jun 30 '20 at 12:51
• @Andyaka I'm looking into band-pass filters, but mostly trying to make sense of my textbook. See the circuit in the question for context. I guess the question can be phrased as whether or not this can function as a band-pass filter, and how can I tell. – gtsiam Jun 30 '20 at 12:54
• What phasor? I think you need to add a phasor diagram so we can tell what you're really asking. – user_1818839 Jun 30 '20 at 12:55

Using the term "resonance", one should - at first - know what it means and how it is defined.

I am pretty sure that "resonance" is commonly understood as the state of a frequency-dependent part or network when there is no phase shift between voltage and current. In many cases, this is identical with the voltage maximum at the ouput node of the network. But this is not always the case. When there is no frequency that can fulfill this "resonant condition", the corresponding network has no resonant point.

Classical networks: Bandpass and band stop filters.

However, lowpass and highpass netwoks can produce resonant points when they are combined with inverting amplifiers. This is the working principle for phase-shift oscillators: A third-order network produces (at one single frequency) a phase shift of 180deg, which - together with the inverting property of the amplifier - results in zero deg phase shift (resonance) .

• The first line of the question defines resonance the same way. – gtsiam Jun 30 '20 at 13:27
• "But this is not always the case", this is what I was trying to verify, but I had an error in my initial calculations, so I thought it was the case for my example. I was wrong. (see my answer below). – gtsiam Jun 30 '20 at 13:28
• As for the latter, I'm still working towards more complex circuits, but I'll keep that in mind when I come across them. Thank you very much. – gtsiam Jun 30 '20 at 13:31

Well this is a little embarrassing. After some more tinkering on Geogebra, turns out the example I provided was a good counter-example. The vertical line is the resonance frequency and the graphs are $$\|Y(jω)|\$$ (blue) and $$\Arg(Y(jω))\$$ (red):

The values used are (I know they are unrealistic, they're just for illustration purposes): $$\R_1 = 0.7, R_2 = 0.7, L = 1, C = 0.1\$$

• What are the parts values you have used? As Y(jw) is a frequency-dependent function - how does the phase looks like? – LvW Jun 30 '20 at 13:32
• @Lvw See the edited answer – gtsiam Jun 30 '20 at 13:38