notes/uni/mmme/2051_electromechanical_devices/ac_power.md

---
author: Akbar Rahman
date: \today
title: MMME2051 // AC Power
tags: [ ac, alternating_current, power ]
uuid: c269b4b7-7835-4b50-8d4f-ff5bc63a8a3d
lecture_slides: [ ./lecture_slides/MMME2051EMD_Lecture3B.pdf ]
exercise_sheets: [ ./exercise_sheets/Exercise Sheet 4 - Power factor and three phase.pdf ]
---

This section builds on [introduction to AC](/permalink?uuid=0c90c691-cbf8-43e9-bfa5-7b277c853151).

# Definitions

- Phase voltage - voltage across any phase
- Line voltage - voltage between two live lines
- Phase current - current through any phase
- Line current - current through any live line

# Three-Phase Load

$$P = \sqrt{3} V_lI_l\cos\gamma$$

![](./images/vimscrot-2023-02-17T13:12:48,739518484+00:00.png)

- 3-phase devices (source and load) are usually balanced, meaning that the impedance in each
  phase is equal ($Z_1 = Z_2 = Z_3$).
- For loads, this means that the voltage across them are the same, apart from the phase angles:

  \begin{align*}
          v_{1N} = V\cos{2\pi ft} \\
          v_{2N} = V\cos{2\pi ft - \frac{2\pi}{3}} \\
          v_{3N} = V\cos{2\pi ft + \frac{2\pi}{3}}
  \end{align*}

- Balanced loads and sources ensure that line/phase currents have equal magnitudes and that the
  neutral current is 0

## Star Load

![](./images/vimscrot-2023-02-17T13:14:49,017883457+00:00.png)

$$|V_\text{line}| = \sqrt 3 |V_\text{phase}|$$

$$I_\text{line} = I_\text{phase}$$

## Delta Load

![](./images/vimscrot-2023-02-17T13:15:12,490943631+00:00.png)

$$|V_\text{line}| = |V_\text{phase}|$$

$$I_\text{line} = \sqrt 3 I_\text{phase}$$

# Power Factor (PF)

$$\text{PF} = \cos{\gamma} = \cos{\left(\Phi_v-\Phi_i\right)}$$