# Calculate the frequency corresponding to a given current or power attenuation in an LRC circuit

For a series LRC circuit, I'm trying to calculate the frequency corresponding to a given current (or power) attenuation to either side of the resonance frequency peak according to the formula for current:

$$I = \frac {V_{rms}}{\sqrt{R^2 + 2\pi FL - \frac {1}{(2\pi FC)^2 }}}$$

Is there a way to factor out / isolate F? I can't find anything beyond the usual 3dB spec.

I'm not totally sure what you are trying to do, but you can just plot the amplitude of I vs. frequency.

As an example (R = 20Ω, L = 26.5mH, C = 10µF, and E = 1V peak sinusoid) below is a plot of steady-state current amplitude vs. frequency using a simulation tool (ATPDraw in this case).

EDIT: You can curve fit to get equation for frequency. I just did using Curve Expert:

name: Polynomial Regression (degree=7)

equation: a + bx + cx^2 + dx^3 + ex^4 + fx^5 + gx^6 + h*x^7

Parameters: a = -2.99513035735665E-05 b = 6.72435685497032E-05 c = -1.48237074557974E-07 d = 1.92511125408853E-09 e = -1.21174827540529E-11 f = 3.11454508578765E-14 g = -3.63866095853704E-17 h = 1.61817957432187E-20

• I'm looking for frequency as a function of I, which I expect is complex, presumably involving a pole on either side of the resonance peak. I'm looking to calculate it rather than infer it from a drawn graph. Oct 11, 2020 at 19:52
• How to find the x dB frequency? Oct 11, 2020 at 20:06

For a series LRC circuit.....

Try this online sim tool - it gives you the output voltage across the capacitor for a series connected R, L and C. That of course translates to current because the simulator shows the voltage across the capacitor and, the voltage across the capacitor is related to the input voltage by the transfer function. The graph shows the magnitude of the transfer function amplitude: -

There are buttons you can use to advance the cursor across the response. But, the page also gives you the formulas for magnitude of the voltage across the capacitor relative to the input voltage: -

It should be fairly easy to convert this to current (versus frequency) and to determine frequency (for a given current) requires solving a quadratic equation.

This seems to work:

at x dB, e = SQRT(10^(x/10) - 1)

F1 = (2 pi C e R – sqrt(4 pi^2 C^2 e^2 R^2 + 16 pi^2 L C)) / (-8 pi^2 L C)

F2 = -(2 pi C e R + sqrt(4 pi^2 C^2 e^2 R^2 + 16 pi^2 L C)) / (-8 pi^2 L C)

Thanks for responding!