Resonance Frequency and Bandwidth

The resonance frequency of an RLC resonant circuit is the frequency at which the circuit exhibits maximum response. It is determined by the values of the inductance (L) and capacitance (C) in the circuit. In this case, the resonance frequency is given as 1.5 Hz.

The bandwidth of a resonant circuit is the range of frequencies over which the circuit's response is considered significant. It is determined by the quality factor (Q) of the circuit. A higher Q value indicates a narrower bandwidth, while a lower Q value indicates a wider bandwidth. In this case, the bandwidth is given as 10 kHz.

Calculating Effective Resistance

To calculate the effective resistance (R) of the circuit, we need to use the formula:

R = Q / (2πfC)

Where:
Q is the quality factor of the circuit
f is the resonance frequency
C is the capacitance

Given that the resonance frequency is 1.5 Hz and the capacitance is 150 pF, we still need to determine the quality factor (Q).

Calculating Quality Factor (Q)

The quality factor (Q) can be calculated using the formula:

Q = f / Δf

Where:
f is the resonance frequency
Δf is the bandwidth

Substituting the given values, we have:

Q = 1.5 Hz / 10 kHz

Converting kHz to Hz, we get:

Q = 1.5 Hz / 10,000 Hz

Q ≈ 0.00015

Calculating Effective Resistance (R)

Now that we have the value of Q, we can substitute it into the formula for R:

R = 0.00015 / (2π * 1.5 Hz * 150 pF)

Converting pF to F, we get:

R = 0.00015 / (2π * 1.5 Hz * 150 * 10^(-12) F)

R ≈ 0.00015 / (9.42 * 10^(-12) F)

R ≈ 15,873 ohms

Therefore, the effective resistance of the circuit is approximately 15,873 ohms.

106103.29 ohm for parallel RLC