The voltage 8.00 sin (400t) is applied to a series RLC circuit, with R = 200 Ω, L = 0.100 H, and C = 1.00 µF. What are the impedance Z and phase angle φ?

1. Break Down the Problem

To find the impedance $Z$ and the phase angle $\phi$ in a series RLC circuit, we will follow these steps:

1. Calculate the inductive reactance $X_L$.
2. Calculate the capacitive reactance $X_C$.
3. Use the values of $R$, $X_L$, and $X_C$ to find the impedance $Z$.
4. Calculate the phase angle $\phi$ using the impedance values.

2. Relevant Concepts

1. The inductive reactance $X_L$ is given by: $X_L = \omega L$ where $\omega$ is the angular frequency, $L$ is the inductance.

2. The capacitive reactance $X_C$ is given by: $X_C = \frac{1}{\omega C}$ where $C$ is the capacitance.

3. The impedance $Z$ for a series RLC circuit is given by: $Z = \sqrt{R^2 + (X_L - X_C)^2}$

4. The phase angle $\phi$ can be calculated using: $\tan(\phi) = \frac{X_L - X_C}{R}$

3. Analysis and Detail

Step 1: Calculate the angular frequency $\omega$

• Given the voltage expression $8.00 \sin(400t)$, we have: $\omega = 400 \, \text{rad/s}$

Step 2: Calculate the inductive reactance $X_L$

• Given $L = 0.100 \, \text{H}$: $X_L = \omega L = 400 \times 0.100 = 40 \, \Omega$

Step 3: Calculate the capacitive reactance $X_C$

• Given $C = 1.00 \, \mu\text{F} = 1.00 \times 10^{-6} \, \text{F}$: $X_C = \frac{1}{\omega C} = \frac{1}{400 \times 1.00 \times 10^{-6}} = 2500 \, \Omega$

Step 4: Calculate the impedance $Z$

• Given $R = 200 \, \Omega$: $Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{200^2 + (40 - 2500)^2}$ $Z = \sqrt{40000 + (-2460)^2} = \sqrt{40000 + 6057600} = \sqrt{6097600} \approx 2470.37 \, \Omega$

Step 5: Calculate the phase angle $\phi$

• Using: $\tan(\phi) = \frac{X_L - X_C}{R} = \frac{40 - 2500}{200} = \frac{-2460}{200} = -12.3$ $\phi = \tan^{-1}(-12.3) \approx -85.3^\circ$

4. Verify and Summarize

• The calculations have been performed consistently and accurately for $Z$ and $\phi$.

Final Answer

• The impedance $Z \approx 2470.37 \, \Omega$
• The phase angle $\phi \approx -85.3^\circ$