LCR Circuits in AC

An LCR circuit is an electrical circuit consisting of an inductor (L), a capacitor (C), and a resistor (R), connected in series or parallel with each other. When an alternating current (AC) is applied to such a circuit, the voltage and current vary sinusoidally with time. The behavior of an LCR circuit in AC is determined by the impedance of each component and their phase relationship with the applied voltage.

Components of an LCR Circuit

Before diving into the behavior of LCR circuits, let's briefly review the properties of the individual components:

• Inductor (L): An inductor stores energy in its magnetic field when current flows through it. Its impedance increases with frequency, and it causes the current to lag behind the voltage by 90 degrees (π/2 radians).

• Capacitor (C): A capacitor stores energy in its electric field when a voltage is applied across it. Its impedance decreases with frequency, and it causes the current to lead the voltage by 90 degrees (π/2 radians).

• Resistor (R): A resistor dissipates energy in the form of heat. It has a constant impedance regardless of frequency, and the current and voltage are in phase.

Impedance and Phase Angle

The impedance (Z) of an LCR circuit is the total opposition to the flow of AC and is a combination of resistance (R), inductive reactance (XL), and capacitive reactance (XC). The impedance can be calculated using the following formula:

$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

where:

• $X_L = 2\pi fL$ is the inductive reactance
• $X_C = \frac{1}{2\pi fC}$ is the capacitive reactance
• $f$ is the frequency of the AC source

The phase angle (φ) between the voltage and current is given by:

$$\tan(\phi) = \frac{X_L - X_C}{R}$$

Depending on the values of $X_L$ and $X_C$, the circuit can be inductive ($X_L > X_C$), capacitive ($X_C > X_L$), or purely resistive ($X_L = X_C$).

Resonance in LCR Circuits

At a certain frequency, known as the resonant frequency ($f_0$), the inductive reactance equals the capacitive reactance ($X_L = X_C$), and the impedance of the circuit is purely resistive. The resonant frequency is given by:

$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$

At resonance, the circuit exhibits the following characteristics:

• The impedance is at its minimum, equal to the resistance (Z = R).
• The current is at its maximum and in phase with the voltage.
• The voltages across the inductor and capacitor are equal and opposite, resulting in a cancelation of their reactive effects.

Differences and Important Points

Here is a table summarizing the differences between the components in an LCR circuit:

Component	Impedance	Phase Relationship	Frequency Dependency
Inductor (L)	$X_L = 2\pi fL$	Current lags voltage by 90°	Increases with frequency
Capacitor (C)	$X_C = \frac{1}{2\pi fC}$	Current leads voltage by 90°	Decreases with frequency
Resistor (R)	R (constant)	Current in phase with voltage	Independent of frequency

Examples

Example 1: Calculating Impedance and Phase Angle

Consider an LCR circuit with $L = 0.1 \text{ H}$, $C = 10 \text{ µF}$, $R = 50 \text{ Ω}$, and an AC source with a frequency of $50 \text{ Hz}$. Calculate the impedance and phase angle.

First, calculate the reactances:

$$X_L = 2\pi fL = 2\pi \times 50 \times 0.1 = 31.4 \text{ Ω}$$ $$X_C = \frac{1}{2\pi fC} = \frac{1}{2\pi \times 50 \times 10 \times 10^{-6}} = 318.3 \text{ Ω}$$

Now, calculate the impedance:

$$Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{50^2 + (31.4 - 318.3)^2} \approx 290.1 \text{ Ω}$$

Finally, calculate the phase angle:

$$\tan(\phi) = \frac{X_L - X_C}{R} = \frac{31.4 - 318.3}{50} \approx -5.78$$ $$\phi = \arctan(-5.78) \approx -80.2^\circ$$

The negative phase angle indicates that the circuit is capacitive, and the current leads the voltage.

Example 2: Resonance Condition

Using the same values for $L$ and $C$ as in Example 1, find the resonant frequency.

$$f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{0.1 \times 10 \times 10^{-6}}} \approx 159.2 \text{ Hz}$$

At this frequency, the circuit will be purely resistive with an impedance equal to $R = 50 \text{ Ω}$.

Understanding LCR circuits in AC is crucial for designing and analyzing AC power systems, radio transmitters and receivers, and various electronic devices. The interplay between inductance, capacitance, and resistance defines the circuit's response to an AC signal, and resonance plays a key role in maximizing or minimizing current flow at specific frequencies.