LC circuits in AC
LC Circuits in AC
An LC circuit, also known as a resonant circuit, tank circuit, or tuned circuit, is an idealized electrical circuit consisting of an inductor (L) and a capacitor (C) connected together. When used with an alternating current (AC) source, the behavior of an LC circuit is characterized by its ability to resonate at a specific frequency, known as the resonant frequency. Below, we will explore the principles of LC circuits in AC, their characteristics, and their applications.
Basic Principles
An LC circuit can be connected in series or parallel. In AC circuits, the inductor and capacitor have reactances that oppose the flow of current, but they do so in opposite ways. The inductor's reactance increases with frequency, while the capacitor's reactance decreases with frequency.
Inductive Reactance ($X_L$)
Inductive reactance is the opposition to the change in current provided by an inductor in an AC circuit. It is given by the formula:
$$ X_L = 2\pi f L $$
where:
- $X_L$ is the inductive reactance (measured in ohms, $\Omega$)
- $f$ is the frequency of the AC source (measured in hertz, Hz)
- $L$ is the inductance (measured in henrys, H)
Capacitive Reactance ($X_C$)
Capacitive reactance is the opposition to the change in voltage provided by a capacitor in an AC circuit. It is given by the formula:
$$ X_C = \frac{1}{2\pi f C} $$
where:
- $X_C$ is the capacitive reactance (measured in ohms, $\Omega$)
- $f$ is the frequency of the AC source (measured in hertz, Hz)
- $C$ is the capacitance (measured in farads, F)
Resonance in LC Circuits
At the resonant frequency ($f_0$), the inductive and capacitive reactances are equal in magnitude but opposite in phase, causing them to cancel each other out. The resonant frequency is given by:
$$ f_0 = \frac{1}{2\pi\sqrt{LC}} $$
At resonance, the impedance of the LC circuit is at its minimum (in the case of a series LC circuit) or at its maximum (in the case of a parallel LC circuit). The circuit behaves as if it were purely resistive, with the current and voltage in phase.
Differences Between Series and Parallel LC Circuits
| Property | Series LC Circuit | Parallel LC Circuit |
|---|---|---|
| Basic Configuration | Inductor and capacitor in series | Inductor and capacitor in parallel |
| Impedance at Resonance | Minimum impedance (ideally zero) | Maximum impedance (ideally infinite) |
| Current at Resonance | Maximum current flows through circuit | Minimum current flows through circuit |
| Voltage across Components | Voltage across L and C are equal and opposite | Voltage across L and C are the same |
| Energy Exchange | Energy oscillates between L and C | Energy oscillates between L and C |
| Q Factor | High Q indicates low energy loss | High Q indicates low energy loss |
| Applications | Filters, tuners, oscillators | Filters, impedance matching, sensors |
Examples
Example 1: Calculating Resonant Frequency
Given an LC circuit with an inductor of $1 \text{ mH}$ and a capacitor of $10 \text{ nF}$, calculate the resonant frequency.
$$ f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{1 \times 10^{-3} \cdot 10 \times 10^{-9}}} = \frac{1}{2\pi\sqrt{10^{-11}}} \approx 159.15 \text{ kHz} $$
Example 2: Impedance at Resonance in a Series LC Circuit
Calculate the impedance of a series LC circuit at resonance, where $L = 1 \text{ mH}$ and $C = 10 \text{ nF}$, with a negligible resistance.
$$ Z_{\text{res}} = \sqrt{R^2 + (X_L - X_C)^2} $$
At resonance, $X_L = X_C$, so the impedance is purely resistive:
$$ Z_{\text{res}} = R $$
If the resistance is negligible, then $Z_{\text{res}} \approx 0 \Omega$.
Example 3: Voltage Across Components in a Parallel LC Circuit
In a parallel LC circuit at resonance, the voltage across the inductor and capacitor is the same. If the applied voltage is $V_{\text{source}} = 10 \text{ V}$, then the voltage across each component is also $10 \text{ V}$.
Conclusion
LC circuits in AC play a crucial role in various applications such as filtering signals, tuning radios, and creating oscillators. Understanding the behavior of LC circuits at different frequencies, especially at resonance, is essential for designing and analyzing electronic systems. The ability to calculate resonant frequencies, impedances, and voltages in these circuits is a fundamental skill for students and professionals working with AC circuits.