https://file.aiquickdraw.com/mathbot/file/173068622217193400i6q.jpg

Let's solve each part of this question step-by-step. The circuit has a resistor, an inductor, and a capacitor in series, with values as follows:

- Resistor, \( R = 3 \, \Omega \)
- Inductor, \( j6 \, \Omega \)
- Capacitor, \( -j2 \, \Omega \)
- Voltage source, \( V = 60 \angle 0^\circ \, \text{V} \)

### (a) Calculate total impedance \( Z_T \) and current \( I_T \) in phasor domain.

1. **Impedance Calculation:**
   - The total impedance, \( Z_T \), in a series RLC circuit is the sum of the impedances:
     \[
     Z_T = R + jX_L + jX_C
     \]
     where \( X_L = 6 \, \Omega \) (inductive reactance) and \( X_C = -2 \, \Omega \) (capacitive reactance).
   - Plugging in values:
     \[
     Z_T = 3 + j6 - j2 = 3 + j4 \, \Omega
     \]

2. **Magnitude and Phase of Impedance:**
   - The magnitude of \( Z_T \) is:
     \[
     |Z_T| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \Omega
     \]
   - The phase angle \( \theta \) of \( Z_T \) is:
     \[
     \theta = \tan^{-1} \left( \frac{4}{3} \right) \approx 53.13^\circ
     \]

3. **Current \( I_T \):**
   - Using Ohm's Law in phasor form, \( I_T = \frac{V}{Z_T} \).
   - Given \( V = 60 \angle 0^\circ \, \text{V} \):
     \[
     I_T = \frac{60 \angle 0^\circ}{5 \angle 53.13^\circ} = 12 \angle -53.13^\circ \, \text{A}
     \]

### (b) Draw the phasor diagram of the voltage \( V \) and the current \( I_T \).

The phasor diagram would show:

- The voltage \( V \) as a reference vector at \( 0^\circ \).
- The current \( I_T \) at an angle of \( -53.13^\circ \) relative to the voltage, meaning it lags the voltage by \( 53.13^\circ \).

### (c) Determine:
   - (i) **The real power \( P \)**:
     \[
     P = V_{\text{rms}} \times I_{\text{rms}} \times \cos \theta
     \]
     where \( \cos \theta = \cos 53.13^\circ \approx 0.6 \).
     \[
     P = 60 \times 12 \times 0.6 = 432 \, \text{W}
     \]

   - (ii) **The reactive power \( Q \)**:
     \[
     Q = V_{\text{rms}} \times I_{\text{rms}} \times \sin \theta
     \]
     where \( \sin \theta = \sin 53.13^\circ \approx 0.8 \).
     \[
     Q = 60 \times 12 \times 0.8 = 576 \, \text{VAR}
     \]

   - (iii) **The apparent power \( S \)**:
     \[
     S = V_{\text{rms}} \times I_{\text{rms}} = 60 \times 12 = 720 \, \text{VA}
     \]

   - (iv) **The phase angle \( \theta \)**, which is \( 53.13^\circ \) (lagging).

### (d) Draw the power triangle.

The power triangle would include:

- **Real Power (P)** = 432 W (horizontal side)
- **Reactive Power (Q)** = 576 VAR (vertical side)
- **Apparent Power (S)** = 720 VA (hypotenuse)
- **Phase Angle \( \theta \)** = 53.13° (indicating lagging power factor)

### (e) Determine the power factor and state if it is leading or lagging.

- **Power Factor (PF)** is given by:
  \[
  \text{PF} = \cos \theta = 0.6
  \]
- Since the current lags the voltage by \( 53.13^\circ \), this is a **lagging power factor**.

Would you like a more detailed explanation on any part, or have further questions?

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### Related Questions:
1. How does impedance affect the phase difference between voltage and current?
2. Why does an inductive component cause the current to lag in an AC circuit?
3. How can you determine if a circuit has a leading or lagging power factor?
4. What is the significance of apparent power in AC circuits?
5. How does reactive power influence the overall efficiency of a power system?

### Tip:
When working with phasors, always remember to calculate both magnitude and phase angle, as they provide complete information about the circuit's behavior.

Referring to the series RLC circuit in Fig. Q1, answer the following questions:
(a) Calculate total impedance Z_T and current I_T in phasor domain.
(b) Draw the phasor diagram of the voltage V and the current I_T.
(c) Determine:
   (i) the real power;
   (ii) the reactive power;
   (iii) the apparent power; and
   (iv) the phase angle.
(d) Draw the power triangle (clearly label the values of the phase angle, real power, reactive power, and apparent power in your diagram).
(e) Determine the power factor in part (d), and state whether it is leading or lagging.

Solve a series RLC circuit problem by calculating total impedance, current, and phasor diagram for voltage and current. Learn to find real, reactive, and apparent power along with phase angle, and draw a power triangle. Determine the power factor and identify if it's leading or lagging. Ideal for electrical engineering students.

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