9.10:
Parallel Resonance
Consider an AM radio tuner comprising a parallel connection of a resistor, capacitor, and inductor, which collectively contribute to the circuit's input admittance.
At resonance, the imaginary part of the admittance is zero, causing the resonant frequency to vary inversely with the square root of the product of inductance and capacitance.
At the resonant frequency, the parallel combination of the inductor and capacitor acts as an open circuit. As a result, the resistor draws minimal current, and energy oscillates between the inductor's magnetic field and the capacitor's electric field.
The frequency response indicates that the current magnitude initially decreases, reaches a minimum at the resonance frequency, and increases as the frequency increases.
At the half-power frequencies, the current is 1.4142 times the minimum current.
The bandwidth is calculated by determining the difference between the higher and lower half-power frequencies.
The quality factor, which indicates the sharpness of the resonance curve, can be expressed in terms of the reactances.
A higher quality factor yields half-power frequencies symmetrically distributed around the resonant frequency and a narrower bandwidth.
9.10:
Parallel Resonance
The parallel RLC circuit is an arrangement where the resistor (R), inductor (L), and capacitor (C) are all connected to the same nodes and, as a result, share the same voltage across them. The parallel RLC circuit is analyzed in terms of admittance (Y), which reflects the ease with which current can flow. The admittance is given by:
Resonance in a parallel RLC circuit occurs when the net reactance is zero, meaning the capacitive and inductive effects cancel each other out. This condition is achieved when:
Solving for the resonant frequency gives:
This resonant frequency is where the circuit will exhibit purely resistive behavior, and the current through the resistor will be at its maximum. The power dissipated in the circuit is maximum at resonance due to the maximum current flow. At the half-power point frequencies, the current is approximately 0.707 of the maximum current, leading to half the maximum power dissipation. The bandwidth of the parallel RLC circuit is the difference between these half-power frequencies and is found using:
The quality factor (Q) is a dimensionless parameter that compares the resonant frequency to the bandwidth, indicating the selectivity or sharpness of the resonance peak. In high-quality circuits where Q≥10, the half-power frequencies can be approximated using:
A higher Q factor signifies that the circuit is highly selective, resonating strongly at a narrow range of frequencies around the resonance frequency. This property is particularly beneficial in radio communications, allowing for filtering unwanted frequencies and minimizing interference. Parallel resonance circuits are particularly useful in filtering applications acting as band-stop or notch filters, blocking a specific frequency range while allowing others to pass. This makes them valuable in signal processing for eliminating unwanted frequencies or noise.