AC voltage in a series LCR circuit

A continuous voltage source, such as a battery, is used to generate current, which is then allowed to flow. The created current flows in just one direction and has a consistent amplitude and a single direction. In general, current flows from the negative terminal to the positive terminal. The phrase “alternating current” refers to a scenario in which the current direction alternates across the resistor. This handout addresses some of the issues that may arise when using RLC series circuits and potential solutions to these issues.

In this article, you will briefly understand AC voltage to an LCR circuit, and in an LCR series ac circuit, the voltage across each component.

What is an LCR circuit, and how does it function?

The capacity of the LCR circuit to tune or resonate is widely recognized. It refers to an electrical circuit that consists of three components: an inductor (L), a resistor (R) and a capacitor (C). In this circuit, the inductor, resistor, and capacitor are connected in series so that the same amount of current flows through it. 

In general, an RLC circuit’s differential equation is analogous to the differential equation of a driven, damped oscillator. Impedance, for example, is employed in the formulation of the LCR circuit equations. Impedance is the resistance associated with the LCR series circuit. It is composed of the resistance given by the inductor, resistor, and capacitor, among other components.

Z represents impedance, which can be determined using the following equation:

Z = (R2 + (XC – XL)2) is a mathematical formula.

Its metric unit is represented by the symbol (ohm).

LCR circuit development

Consider the following electrical circuit, which connects an inductor, a capacitor, and a resistor in series. Assume the circuit receives an alternating current (AC) voltage.

The circuit has an inductor (L), a capacitor (C), and a resistor (R), all connected in series.

Assuming the following source voltage values:

Vm = Vm sin (t)

Here,

The symbol Vm represents the amplitude of the applied voltage.

The letter w represents the frequency at which the voltage is administered.

Assume that q is the charge on the linked capacitor, and when Kirchhoff’s loop rule is applied to this circuit, the following conclusion is obtained:

L (dI/dt) + IR + q/C = v

Circuit Resonance in an LCR

Electric circuits, such as the LCR, have the ability to resonate at a specific frequency, known as the resonance frequency, f0. Hertz is the standard unit of measurement for all frequencies. Because it is more technically convenient, the angular frequency 0 is employed in calculations. It is important to understand that angular frequency is measured in radians per second, not hertz. The equation for angular and resonant frequency is as follows:

ω=2πf0

Energy can be stored in both an electric and magnetic field when the capacitor is charged, and the current flows through the inductor device, resulting in resonance. Both of these stored energies may be transferred and oscillatory in nature from one device to the other inside the circuit. When released, a weight on a spring causes an oscillation, which is a mechanical analogy. It should be noted that the analogy between a weight oscillating on a spring and an LCR circuit is not coincidental, as the LCR circuit can be described by the same second-order differential equation that describes the weight oscillation. 

For a spring–weight system, friction serves as an approximation to the resistor’s role in a circuit. In the absence of an external force, frictional force will eventually bring an oscillation to a halt. Similar to the LCR circuit, if there is no AC power source in the circuit, the oscillation will be “dampened” (slowed down) by the resistance in the circuit.

The frequency at which the impedance (resistance to the flow of current) of the circuit is at its lowest can be explained as his resonance frequency. In other words, the frequency at which the impedance is purely resistive can be defined as the frequency. Resonance is caused when the inductor and capacitor have equal but opposite phase impedances, which cancel each other out.

Conclusion

We have discussed that current can be generated by connecting a continuous source of voltage or battery to a resistor. The generated current only travels in one direction and has a fixed amplitude. Generally, the battery’s negative and positive ends are connected. Alternating current is referred to when the direction of the current across a resistor changes on a regular basis. 

As we’ve seen, an LCR circuit is often referred to as a tuned or resonant one. An inductor (L), a resistor (R), and a capacitor (C) are all parts of the circuit. The same amount of current passes through the circuit since the resistor, inductor, and capacitor are linked in series. A driven and damped oscillator is analogous to the RLC circuit’s differential equation. The LCR circuit equations are derived from notions like impedance, which are essential. The resistance in the LCR series circuit is referred to as impedance.