QUESTION 3: AC THEORY $$ \begin{array}{l} ext { The armature of an alternator rotates at } 3000 ext { rev/min between the poles of a } \ ext { magnet. The maximum voltage of the generated wave is } 35,36 \mathrm{~V} ext { and the } \ ext { waveform is sinusoidal. } \ ext { Calculate the following: } \ \begin{array}{ll}3.1 .1 \quad ext { The frequency } \ 3.1 .2 & ext { The instantaneous } \mathrm{EMF} ext { after } 0,004 ext { seconds } \ 3.1 .3 \quad ext { The time taken from the instant zero to reach } 20 \mathrm{~V} ext { for the first time } & ext { (5) } \ 3.2 & ext { Define the term phasor diagram. }\end{array}\end{array} \begin{array}{l} ext { (2) }\end{array} $$

Question

QUESTION 3: AC THEORY $$ \begin{array}{l}	ext { The armature of an alternator rotates at } 3000 	ext { rev/min between the poles of a } \ 	ext { magnet. The maximum voltage of the generated wave is } 35,36 \mathrm{~V} 	ext { and the } \ 	ext { waveform is sinusoidal. } \ 	ext { Calculate the following: } \ \begin{array}{ll}3.1 .1 \quad 	ext { The frequency } \ 3.1 .2 & 	ext { The instantaneous } \mathrm{EMF} 	ext { after } 0,004 	ext { seconds } \ 3.1 .3 \quad 	ext { The time taken from the instant zero to reach } 20 \mathrm{~V} 	ext { for the first time } & 	ext { (5) } \ 3.2 & 	ext { Define the term phasor diagram. }\end{array}\end{array} \begin{array}{l}	ext { (2) }\end{array} $$

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### **QUESTION 3: AC THEORY**

**Given:**
- **Rotational Speed (N):** 3000 revolutions per minute (rev/min)
- **Maximum Voltage (Eₘₐₓ):** 35.36 V
- **Waveform:** Sinusoidal

Let's address each part of the question step-by-step.

---

### **3.1.1. Calculate the Frequency (f)**

**Formula:**
$$ f = \frac{N \times P}{120} $$

Where:
- $$ N $$ = Rotational speed in rev/min
- $$ P $$ = Number of poles

**Assumption:**
- **Number of Poles (P):** 2 (since the alternator is rotating between the poles of a magnet)

**Calculation:**
$$ f = \frac{3000 \times 2}{120} = \frac{6000}{120} = 50 \, \text{Hz} $$

**Answer:**
$$ \boxed{50 \, \text{Hz}} $$

---

### **3.1.2. Calculate the Instantaneous EMF After 0.004 Seconds**

**Formula for Instantaneous EMF:**
$$ E(t) = Eₘₐₓ \times \sin(2\pi f t) $$

**Given:**
- $$ Eₘₐₓ = 35.36 \, \text{V} $$
- $$ f = 50 \, \text{Hz} $$
- $$ t = 0.004 \, \text{seconds} $$

**Calculation:**
$$ E(t) = 35.36 \times \sin(2\pi \times 50 \times 0.004) $$
$$ E(t) = 35.36 \times \sin(0.4\pi) $$
$$ \sin(0.4\pi) \approx \sin(72^\circ) \approx 0.9511 $$
$$ E(t) \approx 35.36 \times 0.9511 \approx 33.6 \, \text{V} $$

**Answer:**
$$ \boxed{33.6 \, \text{V}} $$

---

### **3.1.3. Calculate the Time Taken to Reach 20 V for the First Time**

**Given:**
- $$ E(t) = 20 \, \text{V} $$
- $$ Eₘₐₓ = 35.36 \, \text{V} $$
- $$ f = 50 \, \text{Hz} $$

**Steps:**
1. Set up the equation:
$$ 35.36 \times \sin(2\pi \times 50 \times t) = 20 $$
2. Solve for the sine term:
$$ \sin(2\pi \times 50 \times t) = \frac{20}{35.36} \approx 0.566 $$
3. Find the angle:
$$ 2\pi \times 50 \times t = \arcsin(0.566) \approx 0.601 \, \text{radians} $$
4. Solve for $$ t $$:
$$ t = \frac{0.601}{2\pi \times 50} \approx \frac{0.601}{314.16} \approx 0.001915 \, \text{seconds} $$
$$ t \approx 1.915 \, \text{milliseconds} $$

**Answer:**
$$ \boxed{1.92 \, \text{milliseconds}} $$

---

### **3.2. Define the Term Phasor Diagram**

**Definition:**

A **phasor diagram** is a graphical representation used in alternating current (AC) circuit analysis to depict the magnitude and phase relationships between sinusoidal voltages or currents. In this diagram:

- **Phasors** are represented as rotating vectors in a two-dimensional plane.
- The **length** of each phasor corresponds to the amplitude (magnitude) of the waveform.
- The **angle** each phasor makes with the reference axis represents the phase angle relative to a chosen reference phasor.

**Purpose:**
Phasor diagrams simplify the analysis of AC circuits by allowing engineers to visualize and calculate the resultant voltages and currents based on their magnitudes and phase angles.

**Answer:**

---
**Phasor Diagram**:

*A graphical representation that uses rotating vectors (phasors) to illustrate the magnitudes and phase angles of sinusoidal voltages or currents in an AC circuit, showing their relative amplitudes and phase relationships.*

---
**Simplified Answers:** 1. **Frequency (f):** $$ \boxed{50 \, \text{Hz}} $$ 2. **Instantaneous EMF After 0.004 Seconds:** $$ \boxed{33.6 \, \text{V}} $$ 3. **Time to Reach 20 V for the First Time:** $$ \boxed{1.92 \, \text{milliseconds}} $$ 4. **Phasor Diagram:** A graphical tool that uses rotating vectors to show the magnitudes and phase angles of sinusoidal voltages or currents in an AC circuit, simplifying the analysis of their relationships.

QUESTION 3: AC THEORY

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QUESTION 3: AC THEORY

3.1.1. Calculate the Frequency (f)

3.1.2. Calculate the Instantaneous EMF After 0.004 Seconds

3.1.3. Calculate the Time Taken to Reach 20 V for the First Time

3.2. Define the Term Phasor Diagram

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