Resonance in Series LRC Circuits: Study with Video Lessons, Practice Problems & Examples

Resonance in LRC circuits occurs when the inductive reactance (X_L) equals the capacitive reactance (X_C), minimizing impedance (Z) to the resistance (R). The resonant frequency (f₀) is calculated using the formula $1^{/\sqrt{LC}}$. At resonance, maximum current flows, calculated as I_max = V_max / R. In series circuits, current remains constant across all components.

Resonance in Series LRC Circuits

Hey guys. In this video, we're going to talk about resonance and LRC circuits. Alright. Let's get to it. I've graphed on the upper right corner the impedance, the resistance, the capacitive reactances, and the inductive reactances all as functions of omega. Okay? The impedance depends upon these three things. Now the resistance larger capacitive reactances get larger and larger and larger the smaller omega is. Okay? So the impedance blows up at large or small frequency. But there is a minimum in between there. Recall that the impedance is the square root of r2 + (xl-xc)2, all of that square rooted. Okay? When the 2 impedances, the capacitive and the inductive impedances equal each other, that's when we were at a minimum for the impedance. When the inductive and the capacitance reactances equal one another, Then you lose this term right here and the impedance equals its smallest value which is the resistance. Okay when this occurs we say that the circuit is in resonance. Okay? The resonant frequency of an LRC circuit is given by this equation and this is just found by solving xc = xl for the frequency. Okay? Since resonance occurs when the impedance is smallest the current is going to be largest in the circuit when the circuit is in resonance. Okay? Let's do an example. An AC circuit is composed of a 10 ohm resistor, a 2 Henry inductor, and a 1.2 milli farad capacitor. If it is connected to a power source that operates at 120 volts, what frequency should it operate at to produce the largest possible current in that's in this circuit? What would the value of this current be? Okay. What frequency should operate at is just asking what is the resonant frequency Such that it produces the largest possible current, right? We know that in resonance you have the largest possible current. So the resonant angular frequency is 1LC. So this is 121.210-3 = 20.4 inverse seconds. But if they're asking for a frequency, it's better to give this in terms of the linear frequency in case that's what they're looking for. The linear frequency is just omega over 2 pi and I can call that f0 to imply that it's the resonant frequency, and this is going to be 20.42π = 3.25 Hertz. Okay? That's one answer, done. What is the current in this circuit? The maximum current in resonance. Don't forget that the maximum produced by a source is always going to be the maximum voltage divided by the impedance. In resonance, the impedance just becomes the resistance. So in resonance we have that this is just z equals r. Right? So it's a maximum voltage of 120 volts divided by a 10 ohm resistor is 12 amps. Very easy, very straightforward. You don't have to use that very complicated resonance equation. Okay guys? And the last couple points I wanna make is that in a series LRC circuit, the current is the same throughout the inductor and the capacitor. The current's the same through everything. The resistor, the inductor, and the capacitor. That's what it means to be in series. In resonance since their reactances are the same this must mean that the maximum voltage across the inductor and the capacitor is also the same. Alright guys that wraps up our discussion on the resonance, sorry the resonant frequency and resonance in an LRC circuit. Thanks for watching.

A series LRC circuit is formed with a power source operating at VRMS = 100 V, and is formed with a 15 Ω resistor, a 0.05 H inductor, and a 200 µF capacitor. What is the voltage across the inductor in resonance? The voltage across the capacitor?

• A. V_L = 141 V; V_C = 107 V
• B. V_L = 107 V; V_C = 107 V
• C. V_L = 141 V; V_C = 141 V
• D. V_L = 149 V; V_C = 149 V

Here’s what students ask on this topic:

What is resonance in a series LRC circuit?

Resonance in a series LRC circuit occurs when the inductive reactance (X_L) equals the capacitive reactance (X_C). At this point, the impedance (Z) of the circuit is minimized to the resistance (R), allowing maximum current to flow. The resonant frequency (f₀) can be calculated using the formula:

$\frac{1}{\sqrt{LC}}$

At resonance, the current is at its maximum value, given by I_max = V_max / R. In a series circuit, the current remains constant across all components.

How do you calculate the resonant frequency in an LRC circuit?

The resonant frequency (f₀) in an LRC circuit is calculated using the formula:

$\frac{1}{\sqrt{LC}}$

Here, L is the inductance in henries (H) and C is the capacitance in farads (F). This formula derives from the condition where the inductive reactance (X_L) equals the capacitive reactance (X_C), minimizing the impedance to the resistance (R).

What happens to the impedance of an LRC circuit at resonance?

At resonance, the impedance (Z) of an LRC circuit is minimized to the resistance (R). This occurs because the inductive reactance (X_L) equals the capacitive reactance (X_C), causing their effects to cancel each other out. The impedance formula at resonance simplifies to:

$\sqrt{R^2}$

Since the square root of R² is R, the impedance at resonance is simply the resistance of the circuit.

Why is the current maximum at resonance in an LRC circuit?

The current is maximum at resonance in an LRC circuit because the impedance (Z) is minimized to the resistance (R). At this point, the inductive reactance (X_L) equals the capacitive reactance (X_C), canceling each other out. The formula for maximum current (I_max) is:

$\frac{V}{R}$

Since Z = R at resonance, the current is at its highest possible value, given the maximum voltage (V_max) and the resistance (R).

How do you find the maximum current in a resonant LRC circuit?

To find the maximum current (I_max) in a resonant LRC circuit, you use the formula:

$\frac{V}{R}$

Here, V is the maximum voltage supplied by the source, and R is the resistance of the circuit. At resonance, the impedance (Z) is minimized to the resistance (R), allowing the current to reach its maximum value.