A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled and, at the same time, the inductor value is halved?

검증된 단계별 안내

Identify the formula for the resonance frequency of an RLC circuit, which is given by \( f = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is the inductance, \( C \) is the capacitance, and \( f \) is the resonance frequency.

Note the changes in the circuit components: the capacitor value is doubled (\( C' = 2C \)) and the inductor value is halved (\( L' = \frac{L}{2} \)).

Substitute the new values of \( L' \) and \( C' \) into the resonance frequency formula to find the new resonance frequency \( f' \). Use \( f' = \frac{1}{2\pi\sqrt{L' \cdot C'}} \).

Simplify the expression by substituting \( L' = \frac{L}{2} \) and \( C' = 2C \) into the formula, resulting in \( f' = \frac{1}{2\pi\sqrt{\frac{L}{2} \cdot 2C}} \).

Further simplify the expression to find that the new resonance frequency \( f' \) is equal to the original resonance frequency \( f \), because the changes in \( L \) and \( C \) counteract each other.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Resonance Frequency in RLC Circuits

The resonance frequency of a series RLC circuit is the frequency at which the inductive reactance equals the capacitive reactance, resulting in maximum current flow. It is given by the formula f₀ = 1 / (2π√(LC)), where L is the inductance and C is the capacitance. At resonance, the impedance of the circuit is minimized, and the circuit can oscillate freely.

Effect of Capacitance on Resonance Frequency

Doubling the capacitance in an RLC circuit affects the resonance frequency inversely. According to the resonance frequency formula, increasing capacitance decreases the resonance frequency, as it is in the denominator of the square root. This means that if the capacitance is doubled, the resonance frequency will decrease, leading to a lower frequency of oscillation.

Effect of Inductance on Resonance Frequency

Halving the inductance in an RLC circuit also influences the resonance frequency, but in the opposite direction. Since inductance is also in the denominator of the square root in the resonance frequency formula, reducing the inductance increases the resonance frequency. Therefore, if the inductance is halved, the resonance frequency will increase, resulting in a higher frequency of oscillation.