# RLC circuit - calculating resonant frequency

If I take a series RLC circuit connected to a battery, the impedance is minimized when $\omega = \frac{1}{\sqrt{LC}}$.

I also know that the series RLC circuit is analogous to a damped driven harmonic oscillator. However, the resonant frequency of a damped driven harmonic oscillator is reduced due to the damping. It is given by $\omega = \sqrt{\omega_0^2 - \gamma^2}$ where $\gamma$ is a damping parameter.

I am unable to see why the analogy fails here: How come the RLC circuit's resonant frequency has no dependence on $R$ but the harmonic oscillator does?

For definition - resonant frequency is the frequency of the driving force (or voltage) that maximized the amplitude (or current).

• For the definition of resonance frequency, see also this Phys.SE post. – Qmechanic Sep 12 '16 at 14:01
• That's useful. Am I then right to say that the driven damped oscillator ought to be driven at $\omega_0$, the undamped resonant frequency but that it oscillates at $\sqrt{\omega_0^2-\gamma^2}$? – user1936752 Sep 12 '16 at 14:39

## 3 Answers

The anomaly is explained when it is realised that there are different types of resonance.
Once steady state has been reached a driver of constant amplitude makes a driven system oscillate at the frequency of the driver.
The frequency at which the response of the driven system is a maximum is called the resonant frequency.
For a mechanical system the maximum response of the driven system can be found by measuring by displacement amplitude, velocity amplitude, energy amplitude.
For an electrical system the responses can be found by measuring change, current, energy.

Often the easiest response to measure for a mechanical system is the displacement amplitude and for this type of resonance - displacement resonance - the peaks occurs at frequencies given by $\omega = \sqrt{\omega_0^2 - \gamma^2}$.
This is also true for the charge resonance of an LCR circuit.

For an electrical circuit it is often easier to measure the current and the current resonance occurs when $\omega = \omega_o$. For the mechanical case this would be called velocity resonance.

For a mechanical system start from the equation for forced motion with damping proportional to speed $$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + k x = F_0 \cos \omega t$$ that can be written in the form $$\frac{d^2x}{dt^2} + \gamma\frac{dx}{dt} + \omega_0^2 x = \frac{F_0}{m} \cos \omega t$$ Looking at a solution of the form $x = A(\omega) \cos (\omega t - \delta)$ gives $A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + (\gamma \omega)^2}}$ and $\delta (\omega)$ can easily be found as $\tan \delta = \frac{\gamma \omega}{\omega_0^2-\omega^2}$. Resonance here is determined by maximising $A(\omega)$ and occurs at $\omega_m \ne \omega_0$. Going to the electrical analogue you have $$L\frac{dI}{dt} + IR + \frac{q}{C} = E_0 \sin \omega t$$ and using $I = \frac{dq}{dt}$ so that the above becomes $$L\frac{d^2q}{dt^2} + R\frac{d q}{dt} + \frac{q}{C} = E_0 \cos \omega t$$ so that $q = Q(\omega) \cos (\omega t - \delta)$ and the maximum $Q(\omega$) occurs for a frequency $\omega_m \ne \omega_0$. However, for circuits you are usually interested in currents rather than charges. If you look at the form for the current, $I(t) = \frac{dq}{dt} = \omega Q(\omega) \sin (\omega t - \delta)$ and the maximum of $\omega Q(\omega)$ occurs at $\omega_0$. OK, this is the mathematic reason, not sure if you'd expect this by careful thought.

One might think the story part (here to be) as irrelevant but please study it fully and you will get a full answer. Recently I was surfing for RLC calculation methods when I found this solution for calculating resonance frequency. I used different values of resistance, capacitance and inductance. Finally, I got the same results you are mentioning here. Now the answer to your questions. Here are the three calculation formulae for Resonance frequency, Series damping and parallel dampings.

1. Resonance frequency = 1/√LC
2. Series damping factor is: Just to answer your question you can see that Resonance depends on the value of Inductance and Capacitance only, whereas Damping factor depends on the value of R as well. You can Google the Wikipedia's article and easily find the technical details behind this. (Photo source Wikipedia screen shot)