To analyze this series circuit, let's determine the following:

1. Impedances

   • The impedance of the inductor (coil) consists of resistance $R$ and inductive reactance $X_L$:
     $Z_L = R + jX_L = 30 + j40 \, \Omega$
   • The capacitive reactance $X_C$ is calculated using: $X_C = \frac{1}{2\pi f C}$ Given $f = 60$ Hz and $C = 15$ μF ($15 \times 10^{-6}$ F): $X_C = \frac{1}{2\pi (60) (15 \times 10^{-6})}$

2. Total Impedance
   The total impedance $Z_{total}$ is given by: $Z_{total} = Z_L - jX_C = (30 + j40) - jX_C$

3. Current in the Circuit
   The current $I$ is found using Ohm’s Law: $I = \frac{V}{Z_{total}}$ where $V = 120V$.

4. Power Factor and Phase Angle
   The power factor (PF) and phase angle $\theta$ can be determined from: $\cos\theta = \frac{R}{|Z_{total}|}$ where $|Z_{total}|$ is the magnitude of the total impedance.

5. Voltage Drops Across Components

   • Voltage across the coil: $V_L = I Z_L$
   • Voltage across the capacitor: $V_C = I (-jX_C)$

Let's Compute These Values

I'll first calculate the capacitive reactance and proceed with the calculations. math

Given values

V = 120 # Voltage in Volts f = 60 # Frequency in Hz R = 30 # Resistance in Ohms X_L = 40 # Inductive reactance in Ohms C = 15e-6 # Capacitance in Farads

Compute capacitive reactance

X_C = 1 / (2 * math.pi * f * C)

Compute total impedance Z_total = (R + jX_L) - jX_C = R + j(X_L - X_C)

X_total = X_L - X_C # Net reactance Z_total_magnitude = math.sqrt(R2 + X_total2) # Magnitude of total impedance

Compute current magnitude using Ohm's Law

I_magnitude = V / Z_total_magnitude

Compute power factor and phase angle

power_factor = R / Z_total_magnitude phase_angle_deg = math.degrees(math.acos(power_factor))

Compute voltage drops

V_L_magnitude = I_magnitude * X_L # Voltage across inductor V_C_magnitude = I_magnitude * X_C # Voltage across capacitor

Output results

X_C, Z_total_magnitude, I_magnitude, power_factor, phase_angle_deg, V_L_magnitude, V_C_magnitude### Computed Results:

1. Capacitive Reactance:
   $X_C = 176.84 \, \Omega$

2. Total Impedance Magnitude:
   $|Z_{total}| = 140.09 \, \Omega$

3. Current Magnitude:
   $I = 0.857 \text{ A}$

4. Power Factor:
   $\text{PF} = 0.214$

5. Phase Angle:
   $\theta = 77.63^\circ$

6. Voltage Drops:

   • Across the inductor:
     $V_L = 34.26V$
   • Across the capacitor:
     $V_C = 151.48V$

Would you like a step-by-step explanation for any part?

Related Questions:

1. How does resonance affect this circuit?
2. What would happen if the frequency were increased?
3. How can the power factor be improved in this circuit?
4. What is the real and reactive power consumed by the circuit?
5. How does changing the capacitance impact the circuit behavior?

Tip:

In AC circuits, power factor correction is often done using capacitors or inductors to reduce energy losses.