What is Resonance?

Resonance in a series circuit occurs when the inductive reactance (XL) is equal to the capacitive reactance (XC), i.e., XL = XC. Under this condition, the circuit behaves purely resistive because the reactive components cancel each other out.

Key Characteristics of Resonance:

• The impedance (Z) is at its minimum value, which equals the resistance (R).
• The current in the circuit reaches its maximum value for a given voltage.
• The phase angle between voltage and current becomes zero (φ = 0).

Conditions for Resonance

Resonance occurs when:

ωL = 1 / (ωC)

where:

• ω: Angular frequency (ω = 2πf).
• L: Inductance in henries (H).
• C: Capacitance in farads (F).

Rearranging for resonant frequency (fr):

fr = 1 / (2π√LC)

Impedance and Current at Resonance

At resonance:

• Total impedance (Z): Z = R.
• Current (I): I = V / R, where V is the applied voltage.

Since Z is minimized, the current is at its peak value.

Voltage Across Components at Resonance

Although the overall voltage and current are in phase, the voltages across the inductor (VL) and capacitor (VC) are out of phase with each other. They can be significantly higher than the source voltage due to energy oscillation between the inductor and capacitor.

Voltage Across Inductor: VL = I × XL

Voltage Across Capacitor: VC = I × XC

Since XL = XC at resonance, VL = VC, but they are 180° out of phase.

Power at Resonance

At resonance:

• The power factor (PF) is 1 because the circuit is purely resistive.
• The circuit dissipates maximum power, calculated as:

P = I2R

Applications of Resonance in Series Circuits

Resonance in series circuits is leveraged in many practical applications, including:

• Tuning Circuits: Used in radios, TVs, and communication devices to select desired frequencies.
• Filters: Designed to allow specific frequency ranges while blocking others.
• Oscillators: Generating sinusoidal waveforms in electronic systems.
• Power Systems: Reducing impedance in transmission lines for efficient power delivery.

Problem-Solving Techniques

To analyze resonance in series circuits:

1. Calculate XL and XC using:
   • XL = ωL
   • XC = 1 / (ωC)
2. Determine the resonant frequency (fr):
   • fr = 1 / (2π√LC)
3. Compute impedance (Z) and current (I):
   • At resonance, Z = R and I = V / R.
4. Analyze voltages across the inductor and capacitor:
   • VL = I × XL
   • VC = I × XC

Advantages and Challenges

Advantages:

• Efficient energy transfer at specific frequencies.
• Maximized current for minimal voltage input.
• Essential for signal processing and communications.

Challenges:

• Voltage magnification across the inductor and capacitor can lead to insulation breakdown.
• Sensitive to component tolerances, which can shift the resonant frequency.

Conclusion

Resonance in a series circuit occurs when XL = XC, minimizing impedance and maximizing current. The circuit behaves resistively at resonance, with the power factor equal to 1. Resonance is utilized in applications like tuning circuits, filters, and oscillators.

In the next lesson, we will explore parallel resonance circuits and their characteristics.