How does a parallel resonant circuit exhibit high impedance at its resonant frequency?
1 Answer
A parallel resonant circuit, also known as a tank circuit or a parallel LC circuit, consists of an inductor (L) and a capacitor (C) connected in parallel. At the resonant frequency of the circuit, the impedance reaches its highest value. To understand why this happens, let's explore the behavior of the components at and around the resonant frequency.
Inductor behavior:
An inductor's impedance (ZL) is proportional to the frequency (f) of the AC signal passing through it. The impedance of an inductor is given by the formula: ZL = jωL, where j is the imaginary unit (√(-1)), ω is the angular frequency (2πf), and L is the inductance in henries (H).
At low frequencies, the impedance of an inductor is relatively small, and it mainly opposes rapid changes in current. As the frequency increases, the impedance of the inductor also increases linearly. However, at the resonant frequency (f_res), the inductive reactance (XL) reaches its maximum value, and the inductor behaves as a high impedance element. The resonant frequency can be calculated using the formula: f_res = 1 / (2π√(LC)).
Capacitor behavior:
A capacitor's impedance (ZC) is inversely proportional to the frequency of the AC signal. The impedance of a capacitor is given by the formula: ZC = 1 / (jωC).
At high frequencies, the impedance of a capacitor is relatively small, and it mainly opposes rapid changes in voltage. As the frequency decreases, the impedance of the capacitor increases. At the resonant frequency, the capacitive reactance (XC) equals the inductive reactance (XL), and they cancel each other out. As a result, the impedance of the capacitor and inductor combination becomes very high, effectively blocking the flow of current through the circuit.
Impedance at resonant frequency:
When the inductive reactance (XL) and capacitive reactance (XC) become equal and opposite in phase, their contributions to the overall impedance cancel out. The net impedance (Z) of the parallel resonant circuit at the resonant frequency is determined primarily by the resistance (R) in the circuit.
The formula for the impedance of a parallel resonant circuit is given by:
Z = R || (XL - XC),
where "||" represents the parallel combination of two impedances.
At the resonant frequency, XL - XC = 0, and thus the impedance reduces to:
Z_resonant = R || 0 = R.
Since the capacitive and inductive reactances cancel each other out, the impedance at the resonant frequency is purely resistive and equal to the resistance in the circuit. This means the circuit appears highly resistive or has high impedance at resonance, limiting the flow of current through the circuit. As a result, the current in the circuit is minimized, and the voltage across the components reaches its maximum value.
Inductor behavior:
An inductor's impedance (ZL) is proportional to the frequency (f) of the AC signal passing through it. The impedance of an inductor is given by the formula: ZL = jωL, where j is the imaginary unit (√(-1)), ω is the angular frequency (2πf), and L is the inductance in henries (H).
At low frequencies, the impedance of an inductor is relatively small, and it mainly opposes rapid changes in current. As the frequency increases, the impedance of the inductor also increases linearly. However, at the resonant frequency (f_res), the inductive reactance (XL) reaches its maximum value, and the inductor behaves as a high impedance element. The resonant frequency can be calculated using the formula: f_res = 1 / (2π√(LC)).
Capacitor behavior:
A capacitor's impedance (ZC) is inversely proportional to the frequency of the AC signal. The impedance of a capacitor is given by the formula: ZC = 1 / (jωC).
At high frequencies, the impedance of a capacitor is relatively small, and it mainly opposes rapid changes in voltage. As the frequency decreases, the impedance of the capacitor increases. At the resonant frequency, the capacitive reactance (XC) equals the inductive reactance (XL), and they cancel each other out. As a result, the impedance of the capacitor and inductor combination becomes very high, effectively blocking the flow of current through the circuit.
Impedance at resonant frequency:
When the inductive reactance (XL) and capacitive reactance (XC) become equal and opposite in phase, their contributions to the overall impedance cancel out. The net impedance (Z) of the parallel resonant circuit at the resonant frequency is determined primarily by the resistance (R) in the circuit.
The formula for the impedance of a parallel resonant circuit is given by:
Z = R || (XL - XC),
where "||" represents the parallel combination of two impedances.
At the resonant frequency, XL - XC = 0, and thus the impedance reduces to:
Z_resonant = R || 0 = R.
Since the capacitive and inductive reactances cancel each other out, the impedance at the resonant frequency is purely resistive and equal to the resistance in the circuit. This means the circuit appears highly resistive or has high impedance at resonance, limiting the flow of current through the circuit. As a result, the current in the circuit is minimized, and the voltage across the components reaches its maximum value.