32.10
Resonance in an AC Circuit
The property of an inductor makes it resist any change in the current passing through it, while the property of a capacitor is to build up the charge…
Let a resistor, an inductor, and a capacitor in series be supplied with a sinusoidal voltage. The circuit's impedance depends on the source's angular frequency. The inductor's reactance increases with the frequency, whereas the capacitor's reactance decreases.
The current is maximum if these reactances are equal. On equating them, the source's frequency which permits this is obtained.
This phenomenon is called the resonance in a series R-L-C circuit. At this frequency, the phase difference between the current and voltage, proportional to the difference between the inductor and capacitor's reactances, is zero.
At the resonance frequency, the impedance is simply the resistance. Hence, the current peaks sharply if the resistance is lower.
The resonance bandwidth is defined as the range of angular frequencies over which the average power is more than half of the maximum value, which is the ratio R over L.
The quality factor is defined as the ratio of the resonance frequency and the bandwidth. The higher the quality factor, the sharper the resonance.
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Q1: What happens to impedance at the resonance frequency in an RLC circuit?
At resonance frequency, the inductive and capacitive reactances cancel each other completely, so impedance equals only the resistance value. This occurs when the inductor's reactance and capacitor's reactance are equal. As a result, the current and voltage are exactly in phase, and current reaches its maximum amplitude. Lower resistance produces sharper current peaks at resonance.
Q2: How do inductive and capacitive reactance differ at different frequencies?
At high frequencies, inductive reactance dominates while capacitive reactance becomes negligible. At low frequencies, the capacitor offers high reactance while the inductor offers minimal reactance. At the resonance frequency, both reactances are equal. This frequency-dependent behavior determines the circuit's phase difference and overall impedance characteristics.
Q3: What is the quality factor and how does it relate to resonance sharpness?
The quality factor is the ratio of resonance frequency to bandwidth. A higher quality factor indicates a sharper resonance peak and narrower bandwidth, meaning power is concentrated more tightly around the resonance frequency. Lower resistance values produce higher quality factors and sharper peaks, while higher resistance broadens the resonance response.
Q4: Why does resistance act as a damping term in an RLC circuit?
Resistance dissipates power as thermal energy during every cycle of oscillation. This energy loss dampens the circuit's response, reducing the current amplitude at resonance and broadening the resonance peak. The greater the resistance, the lower the current amplitude at resonance frequency and the less sharp the power versus frequency curve becomes.
Q5: What is the phase relationship between current and voltage above and below resonance?
At frequencies above resonance, inductive reactance exceeds capacitive reactance, so voltage leads current with positive phase difference. Below resonance, capacitive reactance exceeds inductive reactance, so current leads voltage with negative phase difference. At exactly the resonance frequency, current and voltage are in phase with zero phase difference.
Q6: How is resonance bandwidth defined in an RLC circuit?
Resonance bandwidth is the range of angular frequencies over which average power exceeds half the maximum power value. It is calculated as the ratio of resistance to inductance (R/L). A narrower bandwidth indicates a sharper resonance peak, while a wider bandwidth shows a broader power distribution around the resonance frequency.
Q7: What practical applications use resonance tuning in RLC circuits?
By adjusting capacitance, inductance, and resistance values, circuits can be tuned to dissipate power selectively at different frequencies. This tuning capability is essential for radio frequency transmission and reception, allowing systems to isolate and amplify signals at specific frequencies while rejecting unwanted frequencies.