Resonance in AC circuits
Resonance in AC Circuits
Resonance in AC circuits is a phenomenon that occurs when the inductive reactance and capacitive reactance in a circuit are equal in magnitude but opposite in phase, causing the circuit to oscillate at a particular frequency known as the resonant frequency. At this frequency, the impedance of the circuit is minimized, and the circuit can theoretically oscillate indefinitely with no external energy input. This is because the energy stored in the inductor and capacitor is transferred back and forth between them without being dissipated as heat.
Understanding Reactance and Impedance
Before diving into resonance, it's important to understand the concepts of reactance and impedance:
Reactance (X): It is the opposition offered by a circuit element to the flow of alternating current (AC) due to inductance or capacitance. It is measured in ohms (Ω).
- Inductive Reactance (X_L): (X_L = 2\pi f L), where (f) is the frequency and (L) is the inductance.
- Capacitive Reactance (X_C): (X_C = \frac{1}{2\pi f C}), where (C) is the capacitance.
Impedance (Z): It is the total opposition to the flow of AC and is a combination of resistance (R) and reactance (X). It is also measured in ohms (Ω).
- Impedance in a series RLC circuit: (Z = \sqrt{R^2 + (X_L - X_C)^2})
Resonant Frequency
The resonant frequency ((f_0)) of a circuit is the frequency at which the inductive reactance equals the capacitive reactance ((X_L = X_C)). The formula for finding the resonant frequency is:
[ f_0 = \frac{1}{2\pi\sqrt{LC}} ]
where (L) is the inductance and (C) is the capacitance.
Characteristics of Resonance
At resonance:
- The impedance of the circuit is purely resistive, meaning (Z = R).
- The circuit consumes only real power, and the power factor is unity.
- The voltage across the inductor is equal and opposite to the voltage across the capacitor, resulting in a high circulating current within the LC loop.
- The current in the circuit is at its maximum value for a given voltage.
Differences and Important Points
Here is a table summarizing the differences between inductive reactance, capacitive reactance, and impedance at resonance:
| Property | Inductive Reactance (X_L) | Capacitive Reactance (X_C) | Impedance (Z) at Resonance |
|---|---|---|---|
| Definition | Opposition due to inductance | Opposition due to capacitance | Total opposition to AC at resonance |
| Formula | (X_L = 2\pi f L) | (X_C = \frac{1}{2\pi f C}) | (Z = R) |
| Unit | Ohms (Ω) | Ohms (Ω) | Ohms (Ω) |
| At Resonance | (X_L = X_C) | (X_C = X_L) | (Z = R) (Minimum) |
| Frequency Dependence | Increases with frequency | Decreases with frequency | Independent of frequency |
Examples to Explain Important Points
Example 1: Finding the Resonant Frequency
Given an AC circuit with an inductor of 10 mH and a capacitor of 100 μF, find the resonant frequency.
Using the formula for resonant frequency:
[ f_0 = \frac{1}{2\pi\sqrt{LC}} ]
[ f_0 = \frac{1}{2\pi\sqrt{10 \times 10^{-3} \cdot 100 \times 10^{-6}}} ]
[ f_0 = \frac{1}{2\pi\sqrt{10^{-3}}} ]
[ f_0 = \frac{1}{2\pi\cdot 0.03162} ]
[ f_0 \approx 159.15 \text{ Hz} ]
Example 2: Impedance at Resonance
Consider a series RLC circuit with (R = 50 \Omega), (L = 25 \text{ mH}), and (C = 200 \text{ μF}). Calculate the impedance at the resonant frequency.
First, find the resonant frequency:
[ f_0 = \frac{1}{2\pi\sqrt{LC}} ]
[ f_0 = \frac{1}{2\pi\sqrt{25 \times 10^{-3} \cdot 200 \times 10^{-6}}} ]
[ f_0 = \frac{1}{2\pi\sqrt{5 \times 10^{-3}}} ]
[ f_0 \approx 71.34 \text{ Hz} ]
At resonance, (X_L = X_C), so the impedance is purely resistive:
[ Z = R = 50 \Omega ]
This means that at 71.34 Hz, the impedance of the circuit is 50 ohms.
Conclusion
Resonance in AC circuits is a fundamental concept that has applications in various fields, including radio, telecommunications, and power systems. Understanding resonance allows engineers to design circuits that can filter specific frequencies, maximize power transfer, and minimize energy losses.