Question

Draw vector diagram of loaded alternator at lagging power factor.

Answer 1

To draw the vector diagram of a loaded alternator at a lagging power factor, we need to represent the phasor relationships between the **stator current (I)**, **induced EMF (E)**, and **terminal voltage (V)**, taking into account the lagging power factor. Here's a breakdown of the steps:

### Step-by-step Guide to Drawing the Vector Diagram:

1. **Start with the terminal voltage (V)**:
   - Draw a horizontal vector to represent the terminal voltage (V). This is the voltage measured at the output of the alternator.

2. **Stator current (I)**:
   - Since the power factor is lagging, the current will **lag the terminal voltage**. Draw the current (I) vector at an angle **φ** (the power factor angle) behind the voltage vector (V). This angle is determined by the power factor (cos φ).

3. **Voltage Drop Across the Synchronous Impedance**:
   - The synchronous impedance (Zs) of the alternator consists of resistance (Ra) and reactance (Xs), so the voltage drop across the impedance will have both resistive and reactive components.
   
   - **Resistive Drop (I * Ra)**: This component is in phase with the current (I). Draw a small vector in the same direction as the current vector (I).
   
   - **Reactive Drop (I * Xs)**: Since the synchronous reactance (Xs) is inductive, the voltage drop across it will lead the current by 90°. Draw a vector perpendicular to the current vector (I) to represent the reactive drop.

4. **Induced EMF (E)**:
   - The induced EMF (E) is the vector sum of the terminal voltage (V) and the voltage drop across the synchronous impedance (I * Zs). You can calculate this by adding the resistive drop and reactive drop vectors to the terminal voltage (V). The resultant vector gives the magnitude and direction of the induced EMF (E).

### Phasor Diagram Description:

- **V (Terminal Voltage)**: Horizontal reference vector.
- **I (Stator Current)**: Lags the terminal voltage by an angle **φ**.
- **I * Ra (Resistive Drop)**: In phase with the current (I).
- **I * Xs (Reactive Drop)**: Leads the current (I) by 90°, drawn perpendicular to the current vector.
- **E (Induced EMF)**: Resultant of the voltage drop and the terminal voltage.

The alternator's vector diagram for a lagging power factor will show the induced EMF ahead of the terminal voltage due to the inductive nature of the load.

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### I can create a detailed diagram of this phasor relationship for better visualization if you'd like.

Answer 2

To draw the vector diagram of a loaded alternator operating at a lagging power factor, follow these steps:

### 1. **Identify the Components**

For an alternator, you need to consider the following components in the vector diagram:

- **E**: The generated voltage or electromotive force (EMF).
- **V**: The terminal voltage of the alternator.
- **I**: The load current.
- **X_s**: The synchronous reactance of the alternator.
- **R_s**: The stator resistance (optional, but if considered, it’s in series with the synchronous reactance).

### 2. **Drawing the Vector Diagram**

1. **Draw the EMF Vector (E):**
   - Draw a horizontal line to represent the EMF (\( E \)).
   - This line should be horizontal, pointing to the right.

2. **Draw the Terminal Voltage Vector (V):**
   - The terminal voltage (\( V \)) is usually slightly lagging behind the EMF due to the reactance of the alternator.
   - Draw the vector \( V \) at an angle to the right of \( E \), reflecting the lagging nature. The angle between \( E \) and \( V \) represents the phase drop across the reactance of the alternator.

3. **Draw the Load Current Vector (I):**
   - The current (\( I \)) lags the terminal voltage (\( V \)) by the power factor angle.
   - Draw the current vector (\( I \)) at an angle lagging \( V \). The angle between \( V \) and \( I \) represents the power factor angle. The lagging power factor means the current vector will be to the right of \( V \).

4. **Draw the Reactance Drop Vector (I X_s):**
   - The reactance drop (\( I X_s \)) is the voltage drop across the synchronous reactance.
   - Draw a vector \( I X_s \) in the same direction as the EMF (\( E \)) but starting from the terminal voltage (\( V \)).

5. **Draw the Resistor Voltage Drop Vector (if applicable):**
   - If you include stator resistance, the voltage drop across \( R_s \) is \( I R_s \).
   - Draw this voltage drop in series with \( I X_s \) and starting from the terminal voltage.

### 3. **Label the Diagram**

- **E**: Generated EMF.
- **V**: Terminal voltage.
- **I**: Load current.
- **I X_s**: Voltage drop across synchronous reactance.
- **I R_s**: Voltage drop across stator resistance (if considered).

### Example Vector Diagram:

1. Draw \( E \) as a horizontal line.
2. Draw \( V \) slightly lagging \( E \) (downward at an angle).
3. Draw \( I \) further lagging \( V \) (downward at a greater angle).
4. Draw \( I X_s \) starting from \( V \) and extending horizontally (aligned with \( E \)).
5. If considering \( R_s \), draw \( I R_s \) in series with \( I X_s \).

This results in a vector diagram where the EMF \( E \) is the reference, the terminal voltage \( V \) lags \( E \), and the load current \( I \) lags \( V \), creating a visual representation of the phase relationships in the alternator.

Feel free to use graphing tools or software to create a more precise diagram if needed!