For the circuit of FIGURE EX32.33, What is the resonance frequency, in both rad/s and Hz?

검증된 단계별 안내

Step 1: Identify the components in the circuit. The circuit consists of a resistor (R = 10 Ω), a capacitor (C = 1.0 μF), and an inductor (L = 1.0 mH) connected in series with an AC voltage source.

Step 2: Recall the formula for the resonance frequency in a series RLC circuit. Resonance occurs when the inductive reactance and capacitive reactance cancel each other out. The resonance angular frequency (ω₀) is given by: ω₀ = 1 / √(L * C).

Step 3: Substitute the given values for L and C into the formula. Convert the units to SI: L = 1.0 mH = 1.0 × 10⁻³ H, and C = 1.0 μF = 1.0 × 10⁻⁶ F.

Step 4: Calculate the resonance angular frequency (ω₀) using the formula ω₀ = 1 / √(L * C). This will give the resonance frequency in radians per second (rad/s).

Step 5: Convert the angular frequency (ω₀) to the resonance frequency in hertz (f₀) using the relationship f₀ = ω₀ / (2π).

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주요 개념

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

Resonance Frequency

Resonance frequency is the frequency at which a system naturally oscillates with maximum amplitude. In an RLC circuit, it occurs when the inductive reactance equals the capacitive reactance, leading to a condition where the impedance is minimized. This frequency can be calculated using the formula f₀ = 1/(2π√(LC)), where L is inductance and C is capacitance.

Inductive Reactance

Inductive reactance (X_L) is the opposition that an inductor presents to alternating current (AC) due to its inductance. It is calculated using the formula X_L = ωL, where ω is the angular frequency in rad/s and L is the inductance in henries. This concept is crucial for determining the behavior of the circuit at different frequencies, especially at resonance.

Capacitive Reactance

Capacitive reactance (X_C) is the opposition that a capacitor presents to AC, which decreases with increasing frequency. It is given by the formula X_C = 1/(ωC), where C is the capacitance in farads. Understanding capacitive reactance is essential for analyzing the resonance condition in RLC circuits, as it directly influences the overall impedance and phase relationships in the circuit.