begin notes on ac

uni/mmme/2051_electromechanical_devices/ac_intro.md

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+---
+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
+
+# 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.