1.3 Alternating Current Fundamentals

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

Alternating Current (AC) 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:

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

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.

3. Three-Phase Systems

In a three-phase electrical system, the equations describe the relationship between voltage, current, and power. Three-phase systems are commonly used in power generation, transmission, and distribution because they provide a more efficient means of delivering electrical energy. Below are the key equations for a three-phase system.

1. Voltage Equations in a Three-Phase System

In a balanced three-phase system, the voltages of the three phases are sinusoidal, with each phase 120 degrees apart from the others. For a line-to-line voltage $V_{LL}$ and line-to-neutral voltage $V_{LN}$, the equations are:

• Line-to-line voltage $V_{LL}$: The relationship between the phase voltage $V_{\text{ph}}$ (line-to-neutral) and the line-to-line voltage is:

  • $V_{LL} = \sqrt{3} \times V_{LN}$

• Line-to-neutral voltage $V_{LN}$: Each phase voltage is represented as a sinusoidal function:

  • $V_{\text{ph}}(t) = V_{LN} \sin(\omega t + \phi)$

  Where:

  • $\omega$ is the angular frequency
  • $\phi$ is the phase angle

2. Current Equations in a Three-Phase System

The current in a balanced three-phase system can be described in a similar manner to voltage. The line current $I_L$ and phase current $I_{\text{ph}}$ are related by:

• Phase current $I_{\text{ph}}$: The current in each phase is sinusoidal and related to the line-to-neutral voltage:

  • $I_{\text{ph}}(t) = \frac{V_{\text{ph}}(t)}{Z}$

  Where $Z$ is the impedance of the load (which could be a resistor, inductor, or a combination).

• Line current $I_L$: In a balanced load, the line current is equal to the phase current:

  • $I_L = I_{\text{ph}}$

3. Power Equations in a Three-Phase System

Power in a three-phase system is calculated using the following key formulas:

• Apparent Power $S$: The total apparent power in a balanced three-phase system is:

  • $S = \sqrt{3} \times V_{LL} \times I_L$

  Where:

  • $V_{LL}$ is the line-to-line voltage
  • $I_L$ is the line current

• Real Power $P$: The real power (active power) in the system is:

  • $P = \sqrt{3} \times V_{LL} \times I_L \times \cos(\phi)$

  Where:

  • $\phi$ is the phase angle between the voltage and current

• Reactive Power $Q$: The reactive power (which does not perform work but is needed to maintain the electric and magnetic fields) is:

  • $Q = \sqrt{3} \times V_{LL} \times I_L \times \sin(\phi)$

• Voltages in a balanced three-phase system are 120 degrees apart.
• Currents in a balanced system are proportional to the voltages and impedances in the load.
• Power is more efficiently transmitted using three-phase systems because the power delivery is continuous and steady, avoiding the pulsations that occur in single-phase systems.

These equations form the basis for understanding the operation and performance of three-phase systems in both power generation and distribution.

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.