---
author: Akbar Rahman
date: \today
title: MMME2051 // Introduction to Alternating Current (AC)
tags: [ alternating_current, ac ]
uuid: 0c90c691-cbf8-43e9-bfa5-7b277c853151
lecture_slides: [ ./lecture_slides/MMME2051EMD_Lecture2B.pdf, ./lecture_slides/MMME2051EMD_Lecture3A.pdf, ./lecture_slides/MMME2051EMD_Lecture3B.pdf ]
---

This section builds on [complex numbers](/permalink?uuid=b9be8780-1ab7-402f-9c67-8cc74a74f7a9).

# Sinusoidal Waves

This module will be using the following format:

$$y(t) = A\cos(\omega t + \Phi)$$

where $A$ is amplitude, $\omega$ is frequency, $\omega t + \Phi$ is phase angle, and $\Phi$ is phase
offset.

[Explore the effects of changing the variables in Desmos](https://www.desmos.com/calculator/dmzytwau2y)

# Phasor

- a phasor is a complex number that represents the initial position of a rotating vector
- use the amplitude ($|V|$) and phase offset ($\Phi$) of a cosine function
- for all AC steady state analysis ($\omega$ is constant), these two variables are the only two needed

#### Example

For voltage $v$ given by

$$v = 150 \cos (50t + 25)$$

it may be represented in the phasor form

$$150 \angle 25$$


#### Example

For current $i$ given by

$$i = 10 \cos \left(50t -\frac{pi}{6}\right)$$

it may be represented in the phasor form

$$10 \angle \frac{pi}{6}$$

## Phasors in Resistive Circuits


![](./images/vimscrot-2023-02-09T12:36:34,648080771+00:00.png)

Convert all variables to phasors or to complex form


![](./images/vimscrot-2023-02-09T12:36:53,528247022+00:00.png)

Apply KCL, KVL, Ohm's Law

\begin{align*}
v &= iR \\
V\angle \Phi = IR \angle\theta \\
I \angle \theta = \frac VR \angle \Phi
\end{align*}

![](./images/vimscrot-2023-02-09T12:38:39,149290641+00:00.png)

## Phasors in Inductive Circuit

![](./images/vimscrot-2023-02-09T12:39:37,770679143+00:00.png)

Ohm's law generalised to incorporate complex resistance, reactance, $X$:

\begin{align*}
v &= iX \\
V\angle\Phi_v &= I\angle\Phi_iX \\
&= I\angle\Phi_ij\omega L \\
\frac{V}{j\omega L}\angle\Phi_v &= I\angle\Phi_i\\
\frac{{V}\omega L}\angle\left(\Phi_v - \frac{\pi}{2}\right) &= I\angle\Phi_i
\end{align*}

# Power

## Resistive Circuits

$$P_\text{avg} = V_\text{rms}I_\text{rms}$$

## Inductive Circuits

$$P = \frac{V^2}{2\omega L}\sin{2\omega t}$$

![A graph which demonstrates that the average power in an inductive circuit is zero.](./images/vimscrot-2023-02-16T11:33:07,279996793+00:00.png)

## Capacitive Circuits

$$P = \frac{\omega CV^2}{2}\sin{2\omega t}$$

![A graph which demonstrates that the average power in a capacitive circuit is zero.](./images/vimscrot-2023-02-16T11:34:50,165565069+00:00.png)

## Real Circuit (Resistive + Reactive)

$$P = V_\text{RMS}I_\text{RMS}(\cos \gamma + \cos{(2\omega t + \gamma)}$$

$$P_\text{avg} = V_\text{RMS}I_\text{RMS}\cos \gamma$$

where $\cos \gamma$ is the power factor (PF) and $\gamma$ is phase deviation between voltage and current.
The PF tells us what fraction of the current does useful work.

![A graph which shows power in a real circuit across multiple cycles.](./images/vimscrot-2023-02-16T11:39:18,047002467+00:00.png)

## Apparent, Active, and Reactive Power

Apparent Power:

$$S = V_\text{RMS}I_\text{RMS}$$

- as power still flows losses still occur
- AC equipment is rated for apparent power as it handles both used and unused power

Active Power:

$$P = S\cos\gamma$$

- this is the real power transferred to the load

Reactive Power:

$$P = S\sin\gamma$$

![](./images/vimscrot-2023-02-16T11:49:59,122138825+00:00.png)A

# Resonance

The inductive load of on a circuit is $Z_C = \frac{1}{j\omega L}$.
If the frequency of the power supply matches $\omega$, you get resonance and the circuit becomes
purely resistive so there is a sharp drop in impedance.