# Resonance in an LCR circuit

What's the significance of resonance in an LCR circuit? My book says something like this

"Every system has a tendency to oscillate at a particular frequency. This frequency is known as the system's natural frequency".

What is this natural frequency?

• You just quoted the definition yourself. The natural frequency is the frequency at which the system tends to oscillate. – The Photon Feb 19 '17 at 17:14
• Why does it tend to oscillate at this natural frequency. – Kunal Pawar Feb 19 '17 at 17:16
• It depends on the system. If you're talking about a network of springs and weights, the answer is different than if you're talking about an electrical RLC circuit, and different than if you're talking about air pressure in an organ pipe, or a bell being struck by a hammer. But whatever the system is, if it has a tendency to oscillate at some frequency, we call that frequency the natural frequency of the system. – The Photon Feb 19 '17 at 17:21
• +The Photon What's so special about this natural frequency. The system will oscillate at other frequencies too right? – Kunal Pawar Feb 19 '17 at 17:24
• No. It will respond if forced at another frequency, but generally only sustain oscillation at one frequency. When you ring a particular bell, it always plays the same note. – The Photon Feb 19 '17 at 17:27

From Wikipedia,

In this circuit, the three components are all in series with the voltage source. The governing differential equation can be found by substituting into Kirchhoff's voltage law the constitutive equation for each of the three elements. From the KVL,

$$V_R + V_L + V_C = V(t)\,,$$

where $V_R$, $V_L$ and $V_C$ are the voltages across R, L and C respectively and $V(t)$ is the time varying voltage from the source. Substituting in the constitutive equations,

$$RI(t) + L \frac{dI}{dt} + \frac{1}{C} \int_{-\infty}^t I(\tau)\, d\tau = V(t)\, .$$

For the case where the source is an unchanging voltage, differentiating and dividing by $L$ leads to the second order differential equation:

$$\frac{d^2}{dt^2}I(t) + \frac{R}{L} \frac{d}{dt}I(t) + \frac{1}{LC} I(t) = 0\,.$$

The solution to this differential equation is a damped oscillation of frequency $\omega_0 = \frac{1}{\sqrt{LC}}$, the "natural frequency".

Wiki has resonant circuit page. Impedance is X=ωL (inductor) and X= 1/ωC (capacitor). For any values of L and C the impedance of these can be equal at a certain ω. When they are equal impedance is at a max when they are in parallel or min when they are in series.

Depending how the circuit is configured with a resistor maximum signal can be selected at the natural or resonant frequency.