Series and parallel resonance circuits are key players in AC systems. They're like the dynamic duo of frequency-dependent circuits, each with its own superpower. Series RLC resonates when inductive and capacitive reactances cancel out, while parallel RLC shines when branch currents balance.

These circuits are the heart of many electronic applications. They can filter signals, tune radios, and even help in power factor correction. Understanding their behavior at different frequencies is crucial for designing efficient and effective electrical systems.

Series RLC Resonance

Circuit Components and Behavior

• Series RLC circuit combines resistor, inductor, and capacitor in a single path
• Current flows through all components equally in a series circuit
• Voltage across each component varies based on its impedance
• Total impedance of the circuit results from the sum of individual component impedances
• Reactance of inductor increases with frequency while capacitor reactance decreases

Resonance Frequency and Conditions

• Resonance frequency occurs when inductive and capacitive reactances are equal
• Calculated using the formula $f_r = \frac{1}{2\pi\sqrt{LC}}$
• At resonance, circuit impedance becomes purely resistive
• Current reaches its maximum value at resonance frequency
• Voltage across inductor and capacitor can exceed source voltage (voltage magnification)

Impedance and Phase Relationships

• Impedance magnitude at resonance equals the resistance value
• Phase angle between voltage and current becomes zero at resonance
• Below resonance frequency, circuit exhibits capacitive behavior
• Above resonance frequency, circuit demonstrates inductive characteristics
• Quality factor (Q) influences the sharpness of the resonance peak

Parallel RLC Resonance

Circuit Configuration and Principles

• Parallel RLC circuit connects resistor, inductor, and capacitor in parallel branches
• Voltage across all components remains the same in a parallel circuit
• Current divides among the branches based on their individual admittances
• Total admittance of the circuit results from the sum of branch admittances
• Susceptance of inductor decreases with frequency while capacitor susceptance increases

Resonance Frequency and Conditions

• Resonance frequency in parallel RLC matches that of series RLC: $f_r = \frac{1}{2\pi\sqrt{LC}}$
• At resonance, inductive and capacitive susceptances cancel each other out
• Circuit admittance becomes purely conductive at resonance
• Total current reaches its minimum value at resonance frequency
• Branch currents can exceed the total current (current magnification)

Admittance and Current Relationships

• Admittance magnitude at resonance equals the conductance value
• Phase angle between voltage and total current becomes zero at resonance
• Below resonance frequency, circuit exhibits inductive behavior
• Above resonance frequency, circuit demonstrates capacitive characteristics
• Quality factor (Q) affects the width of the resonance dip in the admittance curve

Frequency Response Characteristics

Amplitude Response Analysis

• Frequency response describes circuit behavior across a range of frequencies
• Gain (or attenuation) varies with frequency in resonant circuits
• Bandwidth defined as the frequency range where response is within -3dB of peak
• Quality factor (Q) inversely related to bandwidth
• Selectivity of the circuit improves with higher Q values

Impedance Magnitude Variations

• Impedance magnitude in series RLC reaches minimum at resonance
• Parallel RLC exhibits maximum impedance magnitude at resonance
• Shape of impedance curve depends on circuit Q factor
• High Q circuits have sharper impedance peaks or dips
• Impedance magnitude used to determine power transfer characteristics

Phase Angle Behavior

• Phase angle between voltage and current changes with frequency
• Series RLC phase angle shifts from +90° to -90° as frequency increases
• Parallel RLC phase angle shifts from -90° to +90° as frequency increases
• Phase angle becomes zero at resonance for both series and parallel circuits
• Rate of phase change around resonance depends on circuit Q factor