9.9:
Characteristics of Series Resonant Circuit
Consider a radio transmitter unit that incorporates an RLC series resonance band-pass filter.
This circuit's frequency response indicates that the current magnitude initially increases, peaks at the resonance frequency, and decreases as the frequency increases.
The power dissipation, corresponding to the maximum current value, is highest at resonance.
At the half-power frequencies, the current is 0.707 times the maximum current, and the power dissipated is half the maximum.
The resonant frequency is the geometric mean of these half-power frequencies.
The bandwidth, defined as the frequency range between the half-power frequencies, equals the ratio of resistance to inductance.
The quality factor, representing the sharpness of the resonance curve, relates the maximum energy stored in the circuit to the energy dissipated per oscillation cycle.
At resonance, reactive energy oscillates between the reactive elements, yielding an expression of the quality factor in terms of reactances.
This factor can be expressed as the ratio of the resonant frequency to the bandwidth.
A higher quality factor implies a narrower bandwidth, thereby increasing selectivity.
9.9:
Characteristics of Series Resonant Circuit
Series resonance occurs in a circuit containing inductive (L), capacitive (C), and resistive (R) elements connected sequentially. At the resonance frequency, the inductive and capacitive reactances are equal in magnitude but opposite in sign, effectively canceling each other. This causes the circuit's impedance is minimal, primarily determined by the resistance R. The resonant frequency of an RLC circuit is defined as:
The power dissipation in the resistor is proportional to the square of the current. This shows that the power dissipation is also maximum when the current is maximum in a resonance condition. This maximum power is given by:
where Imax is the maximum current at resonance.
The bandwidth of a circuit is defined as the frequency range over which the power dissipated decreases to half its maximum value. This occurs at the half-power frequencies, where the current reduces to about 70.7% of its maximum level. Bandwidth is calculated as the difference between the higher and lower half-power frequencies.
At resonance, reactive power does not dissipate but instead oscillates between the inductor and the capacitor, while resistive power is dissipated in the resistor. The quality factor relates to the maximum energy stored in the circuit versus the energy dissipated per cycle. In application, a radio transmitter with a higher Q factor for its RLC filter can better isolate a desired signal from nearby frequency noise. The trade-off is bandwidth: a higher Q reduces bandwidth, which could limit the filter's applicability in systems requiring a more comprehensive frequency range.
An RLC series resonance circuit exemplifies precision engineering in radio transmission by functioning as an effective band-pass filter. This circuit is precisely engineered to allow a peak in current magnitude at the resonance frequency, enabling selective frequency transmission and reception.