Definition and Overview
Resonance Concept
Resonance: condition in AC circuits where inductive reactance equals capacitive reactance (XL = XC). Result: net reactance zero, impedance purely resistive.
Physical Interpretation
Energy oscillates between magnetic field in inductor and electric field in capacitor with minimal loss. Circuit oscillates at natural frequency.
Significance
Allows maximum current/voltage at resonant frequency, frequency selectivity. Basis for filters, oscillators, impedance matching.
"Resonance is the amplification of response in a system at a particular frequency where energy exchange is most efficient." -- David J. Griffiths
Types of Resonance
Series Resonance
Occurs in series RLC circuit: resistor (R), inductor (L), capacitor (C) connected sequentially. Minimum impedance at resonance.
Parallel Resonance
Occurs in parallel RLC circuit: components connected in parallel. Maximum impedance at resonance.
Other Resonance Forms
Mechanical, acoustic, nuclear resonance; focus here is electrical resonance in AC circuits.
Resonant Frequency
Formula
Determined by inductance and capacitance values:
f0 = 1 / (2π√(LC))Frequency Dependence
Resonance frequency independent of resistance; changes with L or C variations.
Angular Frequency
Angular resonant frequency ω0 = 2πf0 = 1 / √(LC)
Series Resonance Circuit
Circuit Description
Components R, L, C connected in series with AC source.
Impedance at Resonance
Impedance Z minimized, equals R only: XL = XC, Z = R.
Current Behavior
Current amplitude maximized at resonance, limited by R only.
Voltage Across Components
Voltages across L and C can exceed source voltage due to energy exchange.
Parallel Resonance Circuit
Circuit Description
R, L, C connected in parallel to AC source.
Impedance at Resonance
Impedance Z maximized; circuit behaves as open circuit at resonance.
Current Characteristics
Input current minimized; circulating current between L and C components.
Voltage Behavior
Voltage across parallel branches equal to source voltage.
Impedance Characteristics
General Expression
Impedance Z = R + j(XL - XC) where XL = ωL, XC = 1/ωC.
At Resonance
XL = XC, imaginary parts cancel, Z purely resistive.
Frequency Variation
Below resonance: circuit capacitive (XC > XL), above resonance: inductive (XL > XC).
Graphical Representation
Impedance vs frequency curve shows minimum (series) or maximum (parallel) at resonance.
| Frequency Range | Impedance Behavior | Circuit Type |
|---|---|---|
| Below Resonance | Capacitive Reactance Dominates | Series and Parallel |
| At Resonance | Impedance Minimum (Series), Maximum (Parallel) | Series and Parallel |
| Above Resonance | Inductive Reactance Dominates | Series and Parallel |
Quality Factor (Q)
Definition
Q = ratio of reactive power to resistive power; measure of sharpness of resonance peak.
Series Circuit Q
Q = (1/R)√(L/C) = ω0L/R = 1/(ω0CR)
Parallel Circuit Q
Q = R√(C/L) = R/(ω0L) = ω0CR
Interpretation
Higher Q: narrower bandwidth, higher selectivity, lower energy loss.
Typical Values
Q ranges from few units (low selectivity) to thousands (high selectivity in radio frequency circuits).
Bandwidth and Selectivity
Bandwidth (Δf)
Frequency range where power ≥ half maximum; Δf = f2 - f1, f1 and f2 are half-power frequencies.
Relation to Q
Q = f0 / Δf
Selectivity
Narrow bandwidth means high selectivity; important in filtering and tuning applications.
Effect of Resistance
Increased R increases bandwidth, decreases Q, reduces selectivity.
Phase Angle Behavior
Definition
Phase angle θ between voltage and current defined by tan θ = (XL - XC)/R.
At Resonance
θ = 0°, current and voltage in phase.
Below Resonance
θ negative, circuit capacitive, current leads voltage.
Above Resonance
θ positive, circuit inductive, current lags voltage.
Energy Storage and Exchange
Energy in Inductor
Stored magnetic energy WL = ½ L I2, varies with current.
Energy in Capacitor
Stored electric energy WC = ½ C V2, varies with voltage.
Energy Exchange
At resonance, energy oscillates between L and C with minimal loss; sustains oscillations.
Role of Resistance
Resistance dissipates energy, reduces amplitude, broadens resonance peak.
Applications of Resonance
Radio Tuning
Resonance selects desired frequency signals by tuning L and C.
Filters
Bandpass and notch filters based on resonance frequency for signal processing.
Oscillators
Resonant circuits provide frequency stabilization in oscillators.
Impedance Matching
Resonance used to minimize reflections and maximize power transfer.
Medical Applications
Magnetic resonance imaging (MRI) exploits nuclear resonance principles.
Mathematical Analysis
Series RLC Circuit Impedance
Z = R + j(ωL - 1/ωC)Resonant Frequency Condition
ω0L = 1 / (ω0C)Quality Factor Formula
Q = ω0L / R = 1 / (ω0CR)Bandwidth Relation
Δω = ω2 - ω1 = ω0 / QPower Factor
PF = cos θ = R / |Z|, unity at resonance.
| Parameter | Expression | Description |
|---|---|---|
| Resonant Frequency | f0 = 1/(2π√LC) | Frequency of reactance cancellation |
| Quality Factor (Series) | Q = (1/R)√(L/C) | Sharpness of resonance |
| Bandwidth | Δf = f0/Q | Frequency range of resonance |
| Impedance at Resonance | Z = R | Purely resistive |
References
- Hayt, W. H., & Kemmerly, J. E. "Engineering Circuit Analysis," McGraw-Hill, 8th Edition, 2012, pp. 500-540.
- Nilsson, J. W., & Riedel, S. A. "Electric Circuits," Pearson, 10th Edition, 2014, pp. 320-360.
- Alexander, C. K., & Sadiku, M. N. O. "Fundamentals of Electric Circuits," McGraw-Hill, 6th Edition, 2016, pp. 410-450.
- Griffiths, D. J. "Introduction to Electrodynamics," Pearson, 4th Edition, 2013, pp. 250-270.
- Franco, S. "Design with Operational Amplifiers and Analog Integrated Circuits," McGraw-Hill, 3rd Edition, 2002, pp. 120-145.