XL and XC refer to the reactances of an inductor and a capacitor, respectively, in an AC circuit. Reactance is the opposition that these components offer to the flow of alternating current, similar to resistance in a DC circuit.
In an AC circuit, the reactance of an inductor (XL) is given by the equation XL = 2πfL, where f represents the frequency of the AC signal and L is the inductance of the inductor. On the other hand, the reactance of a capacitor (XC) is given by the equation XC = 1/(2πfC), where C represents the capacitance of the capacitor.
At resonance, the frequency of the AC signal is such that the reactances of the inductor and capacitor are equal. This means that XL = XC. This occurs when the angular frequency ω (equal to 2πf) is equal to 1/√(LC), where L is the inductance and C is the capacitance of the circuit.
To better understand this concept, let’s consider a practical example. Imagine a circuit consisting of an inductor and a capacitor connected in series. At resonance, the reactances of the inductor and capacitor cancel each other out, resulting in a net reactance of zero. This means that the impedance of the circuit is solely determined by the resistance (R) in the circuit.
At resonance, the impedance (Z) is reduced to R because the reactances of the inductor and capacitor cancel each other out. This implies that the impedance is at its lowest possible value at resonance. As a result, the current flowing through the circuit will be the largest value possible since the impedance is minimized.
To illustrate this further, let’s consider the analogy of a swing. Imagine pushing a swing at its natural frequency. If you time your pushes correctly, the swing will reach its highest amplitude, indicating that the energy transfer is most efficient. Similarly, at resonance in an AC circuit, the current reaches its maximum value because the system is perfectly matched, allowing for maximum energy transfer.
At resonance, XL and XC are equal, resulting in a net reactance of zero. The impedance of the circuit is solely determined by the resistance (R), and it is at its lowest possible value at resonance. Consequently, the current flowing through the circuit is at its maximum value. This phenomenon can be understood through the analogy of a swing, where maximum energy transfer occurs when the system is at its natural frequency.