Unit 4 Notes in English, FEEE 2004, Polytechnic 2nd Semester

 

4. A.C. CIRCUITS

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Syllabus of this Unit:

1. Introduction to AC Waveform and Terminology 1.1. Cycle
   1.2. Frequency
   1.3. Time Period
   1.4. Amplitude
   1.5. Angular Velocity
   1.6. RMS Value
   1.7. Average Value
   1.8. Form Factor

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1. Introduction to AC Waveform and Terminology

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1.1 Cycle:

• Definition: A complete waveform of an AC signal is called a cycle, consisting of both positive and negative half-waves.
• Example:
  • An AC signal with a frequency of 50 Hz completes 50 cycles in one second.

Diagram:

+  |                      /\          /\
    |                    /    \      /    \
    |                  /        \  /        \
---|---------------------------------------------------
                        Cycle 1      Cycle 2

• Importance: A cycle represents the full oscillation or waveform that AC completes, going from positive to negative and back.

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1.2 Frequency:

• Definition: Frequency is the number of cycles completed by an AC wave in one second. It is measured in Hertz (Hz).

• Formula:

  $$f = \frac{1}{T}$$

  where f is the frequency, and T is the time period.

• Example: If the frequency is 50 Hz, it means 50 cycles occur every second.

Diagram:

50 Hz = 50 cycles per second

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1.3 Time Period:

• Definition: Time period refers to the time taken to complete one cycle of the AC wave. It is measured in seconds and is the reciprocal of frequency.

• Formula:

  $$T = \frac{1}{f}$$

  where T is the time period and f is the frequency.

• Example: If the frequency is 50 Hz, the time period is:

  $$T = \frac{1}{50} = 0.02 \, \text{seconds}$$
  

Diagram:

    ---- Time Period ----> 0.02 seconds

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1.4 Amplitude:

• Definition: The maximum value of the AC waveform, either in positive or negative direction. It is denoted as V_m and is measured in volts (V).
• Example: If the amplitude of a sinusoidal wave is 10 V, it means the wave reaches up to 10 volts above and below the zero axis.

Diagram:

  +10V
    |        /\  
    |       /  \  
    |      /    \  
    |     /      \ 
----|----------------
           Amplitude 

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1.5 Angular Velocity:

• Definition: Angular velocity refers to the rate at which an object completes a full cycle in radians. It is denoted by ω (omega).

• Formula:

  $$\omega = 2 \pi f$$
  

  where f is the frequency and ω is the angular velocity.

• Example: If the frequency is 50 Hz, then the angular velocity is:

  $$\omega = 2 \pi \times 50 = 314.16 \, \text{radians/second}$$
  

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1.6 RMS Value:

• Definition: The RMS (Root Mean Square) value is the effective value of an AC signal, which represents the value that would produce the same power in a resistive load as a DC signal.

• Formula:

  $$I_{\text{RMS}} = \frac{I_m}{\sqrt{2}}$$

  where I_m is the maximum current.

• Example: If the maximum current is 10 A, then the RMS current is:

  $$I_{\text{RMS}} = \frac{10}{\sqrt{2}} = 7.07 \, \text{A}$$
  

Diagram:

    Amplitude: I_m = 10A
    I_RMS = 7.07A

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1.7 Average Value:

• Definition: The average value of an AC signal is the mean value of the signal over one complete cycle. For a sinusoidal wave, it is 0.637 times the maximum value.

• Formula:

  $$I_{\text{avg}} = \frac{2 I_m}{\pi}$$

• Example: If the maximum current is 10 A, then the average current is:

  $$I_{\text{avg}} = \frac{2 \times 10}{\pi} = 6.37 \, \text{A}$$
  

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1.8 Form Factor:

• Definition: The form factor is the ratio of the RMS value to the average value of an AC signal.

• Formula:

  $$\text{Form Factor} = \frac{I_{\text{RMS}}}{I_{\text{avg}}}$$

• Example: If the RMS value is 7.07 A and the average value is 6.37 A, then the form factor is:

  $$\text{Form Factor} = \frac{7.07}{6.37} = 1.11$$
  

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MCQ Questions:

1. When does an AC signal complete one cycle?

   • a) Only positive half-wave
   • b) Only negative half-wave
   • c) Both positive and negative half-waves
   • d) None of the above
   • Answer: c) Both positive and negative half-waves

2. What is the unit of frequency?

   • a) Second
   • b) Hertz (Hz)
   • c) Volt
   • d) Ampere
   • Answer: b) Hertz (Hz)

3. What is the relationship between time period (T) and frequency (f)?

   • a) T = f
   • b) T = 1/f
   • c) f = 1/T
   • d) Both b and c
   • Answer: d) Both b and c

4. What does the RMS value of an AC signal represent?

   • a) Maximum value
   • b) Average value
   • c) Effective value
   • d) Form factor
   • Answer: c) Effective value

5. What is the average value of an AC wave (in terms of its maximum value)?

   • a) Maximum value
   • b) 0.637 times the maximum value
   • c) 0.707 times the maximum value
   • d) 1.414 times the maximum value
   • Answer: b) 0.637 times the maximum value

6. What is the formula for angular velocity (ω)?

   • a) ω = 2πT
   • b) ω = 2πf
   • c) ω = f/2π
   • d) ω = πf
   • Answer: b) ω = 2πf

7. What is the form factor of a sinusoidal AC wave?

   • a) 0.707
   • b) 1.11
   • c) 1.00
   • d) 0.632
   • Answer: b) 1.11

8. What does the amplitude of an AC wave represent?

   • a) Maximum voltage or current
   • b) Average current
   • c) RMS current
   • d) Angular velocity
   • Answer: a) Maximum voltage or current

9. What is the ratio of RMS value to average value called?

   • a) Amplitude
   • b) Form Factor
   • c) Time Period
   • d) Frequency
   • Answer: b) Form Factor

10. If an AC signal has a frequency of 60 Hz, what will be its time period?

• a) 0.0167 seconds
• b) 0.05 seconds
• c) 0.1 seconds
• d) 0.25 seconds
• Answer: a) 0.0167 seconds

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Important Practice Questions:

1. Explain the relationship between the cycle, frequency, and time period of an AC signal.

   • Answer: A cycle is the complete waveform, consisting of both positive and negative half-waves. Frequency refers to the number of cycles per second, and time period is the time taken for one complete cycle. They are inversely related: $f = \frac{1}{T}$ and $T = \frac{1}{f}$.

2. Differentiate between RMS value and average value of an AC signal.

   • Answer: RMS value is the effective value that represents the equivalent DC value for the same power dissipation, while average value represents the mean value over one complete cycle. For sinusoidal waves, RMS value is 0.707 times the peak value, while the average value is 0.637 times the peak value.

3. What is the form factor and its significance?

   • Answer: Form factor is the ratio of the RMS value to the average value of an AC signal. It is a measure of how “peaked” the waveform is. For a sine wave, the form factor is approximately 1.11.

4. How is angular velocity (ω) calculated, and what is its significance in an AC circuit?

   • Answer: Angular velocity is the rate at which the AC signal completes a cycle in radians, given by the formula $\omega =2\pi \mathrm{f,\; where }$$f$ is the frequency. It helps in understanding the rate of oscillation of the AC signal in radians per second.

5. How is the time period of an AC signal calculated from its frequency, and why is it important?

   • Answer: The time period is the reciprocal of the frequency, calculated as $T = \frac{1}{f}$. It determines how much time is required for one complete cycle of the AC wave, which is important for understanding the behavior of the wave in different circuits.