Let's solve the step by step--- ## **Step 1: Understanding the Circuit** The circuit is a **series RLC filter** with a resistive load \( R_L \) connected in parallel with the LC part. **Given:** - \( R_L = 200\,\Omega \) - The circuit is loaded \( R_L \). --- ## **Step 2: Redraw the Circuit** The circuit can be redrawn as: - Input voltage \( v_i \) - Series resistor \( R \) - Parallel combination of \( L \), \( C \), and \( R_L \) at the output --- ## **Step 3: Calculate the Equivalent Parallel Resistance** The total resistance in parallel with the LC branch is just \( R_L \) (since only \( R_L \) is present in parallel). --- ## **Step 4: Find the Center Frequency (\( f_ \) or \( \omega_ \))** For a parallel RLC circuit, the **center frequency** (or resonant frequency) is: \[ \omega_ = \frac{1}{\sqrt{LC}} \] The presence of \( R_L \) **does not affect** the resonant frequency, since it's in parallel with \( LC \). - **Center frequency remains the same:** \[ f_ = \frac{1}{2\pi\sqrt{LC}} \] --- ## **Step 5: Find the New Quality Factor (\( Q \))** For a **series RLC circuit loaded with a parallel resistor** (\( R_L \)), the total resistance seen at resonance is: \[ R_{total} = R + \frac{1}{1/R_L} \] But since \( R_L \) is in parallel with the output, it loads the circuit. The **equivalent resistance** seen by the LC tank (ignoring \( R \)) is \( R_L \). The **Q-factor** for a parallel RLC circuit is: \[ Q = \frac{R_{eq}}{\omega_ L} \] Where \( R_{eq} \) is the parallel combination of all resistors across LC. If \( R \) is small compared to \( R_L \), we can use \( R_L \) for \( R_{eq} \). **But if you have a series resistance \( R \) and a parallel \( R_L \),** the **total damping resistance** is: \[ R_{total} = R + \left( \frac{\omega_^2 L^2}{R_L} \right) \] But for the Q-factor, the **parallel resistance damps the resonance**: \[ Q = \frac{1}{R_{total}} \sqrt{\frac{L}{C}} \] But the correct Q for a parallel RLC with a load is: \[ Q = R_{parallel} \sqrt{\frac{C}{L}} \] Where \( R_{parallel} \) is the equivalent parallel resistance across LC (here, just \( R_L \)). --- ## **Step 6: Final Answers** ### **New Center Frequency** \[ f_ = \frac{1}{2\pi\sqrt{LC}} \] ### **New Q-factor** \[ Q = R_L \sqrt{\frac{C}{L}} \] --- ## **Step 7: Plug in Given Values (if any)** If you have values for \( L \) and \( C \), plug them in along with \( R_L = 200\,\Omega \). --- ## **Summary Table** | Parameter | Formula | Value (if L, C given) | |-------------------|--------------------------------------|-----------------------| | Center Frequency | \( f_ = \frac{1}{2\pi\sqrt{LC}} \) | | | Q-Factor | \( Q = R_L \sqrt{\frac{C}{L}} \) | | **If you provide values for \( L \) and \( C \), I can compute the exact numbers!**