1.3 Alternating Current Fundamentals

1. Principle of Alternating Voltage and Current Generation, Equations, and Waveforms

• Alternating Current (AC): It is an electric current that reverses its direction periodically, as opposed to direct current (DC), where the flow of electric charge is in one direction only.

• Generation of AC: AC is typically generated using alternators or synchronous generators, where mechanical energy (e.g., from a turbine) is converted into electrical energy. The most common method of generation is through electromagnetic induction, where a conductor moves through a magnetic field.

• AC Waveforms: The most basic waveform for AC is a sine wave, which represents a smooth, periodic oscillation. A typical AC waveform is defined by the following parameters:

  • Peak Value (Maximum Value): The highest value of the waveform (voltage or current).

  • RMS (Root Mean Square) Value: The effective value of the waveform. For a sinusoidal AC, the RMS value is the peak value divided by √2.

  • Average Value: The average of all instantaneous values in one complete cycle, often zero for symmetric sinusoidal waveforms.

Equation for a sinusoidal AC waveform:

$$v(t) = V_{\text{max}} \sin(\omega t + \phi)$$

Where:

$$v(t) = \text{instantaneous voltage}, \\ V_{\text{max}} = \text{peak voltage}, \\ \omega = \text{angular frequency} \quad (\omega = 2\pi f, \text{where } f \text{ is the frequency}), \\ t = \text{time}, \\ \phi = \text{phase angle}$$

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2. Average, Peak, and RMS Values

1. Peak Value:

   • The peak value (also known as the maximum value) is the highest point reached by the voltage or current in one cycle.

   • For a sinusoidal AC, the peak value is denoted as $( V_{\text{peak}} )$or $( I_{\text{peak}} )$.

2. RMS (Root Mean Square) Value:

   • The RMS value is a measure of the effective value of an AC waveform. It is the equivalent DC value that would produce the same power dissipation in a resistive load.

   • For a sinusoidal waveform:

     $$V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}}$$
   • This means that the RMS value is approximately 0.707 times the peak value for a sinusoidal waveform.

3. Average Value:

   • The average value is the arithmetic mean of the values of the waveform over one complete cycle. For a pure sinusoidal waveform, the average value is zero (due to the symmetrical nature of the waveform). However, the average absolute value (or the rectified average value) is often used:

     $$V_{\text{avg}} = \frac{2}{\pi} V_{\text{peak}} \approx 0.637 \times V_{\text{peak}}$$
   • For half-wave rectified signals, the average value is non-zero.

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3. Three-Phase Systems

• Three-Phase Power: This is a method used to generate and distribute alternating current. In a three-phase system, three conductors carry three sinusoidal voltages or currents that are out of phase with each other by $( 120^\circ )$. It is the most common method for industrial and commercial power distribution.

• Advantages of Three-Phase Systems:

  • Constant Power: In a three-phase system, the power delivered is more constant compared to a single-phase system, which results in smoother operation of motors.

  • More Efficient: Three-phase systems are more efficient for transmitting electrical power over long distances.

  • Smaller Conductors: For the same amount of power, a three-phase system requires less conductor material than a single-phase system.

Equations for Three-Phase Systems:

For line-to-line voltage:

$$V_L = \sqrt{3} V_{\text{phase}} \quad \text{(line-to-line voltage)}$$

Where:

$$V_L = \text{line-to-line voltage} \\ V_{\text{phase}} = \text{phase voltage (voltage across each phase)}$$

For power in three-phase systems:

• Balanced Load Power:

$$P = \sqrt{3} V_L I_L \cos(\theta) \quad \text{(Balanced Load Power)}$$

Where:

$$P = \text{total power} \\ V_L = \text{line-to-line voltage} \\ I_L = \text{line current} \\ \theta = \text{phase angle between voltage and current}$$

• Total Apparent Power:

$$S = \sqrt{3} V_L I_L \quad \text{(Total Apparent Power)}$$

• Reactive Power:

$$Q = \sqrt{3} V_L I_L \sin(\theta) \quad \text{(Reactive Power)}$$

The key point is that in a balanced three-phase system, the sum of the instantaneous powers in all three phases is constant.

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Conclusion

• AC is an electrical current that reverses direction periodically, generated through electromagnetic induction.

• Key AC parameters: peak value, RMS value (effective value), and average value.

• Three-phase systems provide more constant and efficient power, requiring less conductor material compared to single-phase systems.