# In a RLC Circuit, what is the effect of Resonance?

The book I am reading does not go into greater detail of what Resonance within a circuit really is. The only definition I am given is that of the Resonant Angular Frequency. Which is the angular frequency for maximum oscillation.

$$\omega L= \frac{1}{\omega C}$$ $$\omega_0 = \frac{1}{\sqrt{LC}}$$

Where L is Inductance and C is capacitance. I can see here that the resonance depends on the values for capacitance and inductor's.

From the definition alone, I'd guess that it'd mean the frequency at which you're getting a maximum current when ever the maximum electro-motive force is held constant.

$$I = \frac{E}{Z}$$Where Z is the impedance of the RLC circuit. And E is EMF max.

So... the current is at it's max when the Inductance and Capacitance are both zero. I'd guess that this would mean that a circuit would be seen as if only the Resistors are in effect.

But what does this conceptually imply? And when would it matter in use?

• Please use mathjax. – DanielSank Aug 8 '16 at 6:45
• @DanielSank, Thankyou, I was wondering where the reference for that was. – moonshineTheleocat Aug 8 '16 at 7:40
• It's not clear to me what you're asking - what do you mean "when would it matter in use"? Why would physical facts have to "matter in use", or have further conceptual implications? Please try to make clear what exactly is confusing you here. – ACuriousMind Aug 8 '16 at 21:35
• In other words, when would this knowledge have a practical application. – moonshineTheleocat Aug 9 '16 at 7:03

You are incorrect when you say "... the current is at it's max when the Inductance and Capacitance are both zero." If the capacitance were zero, the effect would be that you have a break in the circuit and no current would flow. If the voltage across the circuit is $V(t) = V_0 \sin(\omega t)$ then for an RLC series circuit the current through the circuit is given as $$I(t) = \frac{V_0}{|Z|} \sin (\omega t - \phi)$$ where the impedance, $Z$, is given as $Z = \sqrt{R^2 + (X_L - X_C)^2}$, with $X_L = \omega L, X_C = \frac{1}{\omega C}$ and $\phi$ is the phase between the current and the applied voltage. For given vales of $R, L, C$ and varying $\omega$, this has a minimum value of $R$ when $X_L = X_C$ meaning a maximum current will flow. You can determine the resonant frequency from $\omega_0 L = \frac{1}{\omega_0 C}$.

Consider your LCR series circuit as the driveN system and the power supply as the driveR.
The driveR (power supply) will make the current in the driveN system (LCR) oscillate at the frequency of the driveR and these are called forced oscillations.
For a fixed amplitude voltage power supply the response (current) in the driveN system depends on the frequency of the driveR.

At one particular frequency the response of the driveN system (current) is a maximum this is called resonance and occurs at the resonant frequency.

At the resonant frequency the circuit behaves as though it is only a resistor, the impedance of the circuit is equal to the resistance of the resistor, and so the current in the circuit is the supply voltage divided by the resistance of the resistor.

Because it is a series circuit the currents through each of the components are in phase with one another but the voltages across them are not, with the voltage across the inductor leading the current by $90^\circ$, the voltage across the resistor in phase with the current and the voltage across the capacitor lagging the current by $90^\circ$.
At resonance the voltages across the inductor and the capacitor are equal in magnitude but exactly $180^\circ$ out of phase with one another.
The impedance of the inductor $\omega L$ is equal to the impedance of the capacitor $\frac{1}{\omega C}$. Thus the supply voltage equals the voltage across the resistor.

There are many instances in electronics where resonance matters in use and one example is a simple radio receiver called the crystal set.

The radio signal is picked up by an aerial and fed into a tuned circuit comprising an inductor and a capacitor in parallel.
If the frequency of the radio signal (driveR) is the same as the the resonant frequency of the tuned circuit (driveN) the voltage at this frequency across capacitor is very large whereas all the other frequency signals picked up by the aerial produce very much smaller voltages across the capacitor and so are undetected by the rest of the circuit which convert the radio signal into sound.
So the tuned circuit selects the signal from the radio station you wish to listen to and rejects all the other signals. The resonant frequency of the tuned circuit in this circuit is controlled by altering the capacitance of the capacitor.

It's primarily relevant in signal transmission, especially across distances large relative to the wavelength. Frequencies close to the resonant frequency will be best transmitted; all others will be relatively dampened (which will also tend to distort practical signals.)

Probably the simplest use is in transmission of AC current, I'd imagine, though you'd have to ask someone who works in that industry for the details.

For further reading, you may be interested in the concept of electrical impedance.

• 1. This does not address the question. 2. Signal transmission is only one of many many situations where the topic is relevant. – garyp Aug 8 '16 at 11:51