> For the complete documentation index, see [llms.txt](https://nec-license.gitbook.io/books/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nec-license.gitbook.io/books/1.-concept-of-basic-electrical-and-electronics-engineering/1.3-alternating-current-fundamentals.md).

# 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.