begin notes on ac

uni/mmme/2051_electromechanical_devices/.n2w.yml

@@ -1 +1 @@
-itags: [ mmme2051 ]
+itags: [ mmme2051, electronics, electromechanical_devices ]

uni/mmme/2051_electromechanical_devices/ac_intro.md

@@ -0,0 +1,136 @@
+---
+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.

uni/mmme/2051_electromechanical_devices/ac_power.md

@@ -0,0 +1,51 @@
+---
+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 ]
+---
+
+# 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)
+
+$$|V_\text{line}| = |V_\text{phase}|$$
+
+$$I_\text{line} = \sqrt 3 I_\text{phase}$$

uni/mmme/2051_electromechanical_devices/basic_circuitry.md

@@ -0,0 +1,88 @@
+---
+author: Akbar Rahman
+date: \today
+title: MMME2051 // Basic Circuitry
+tags: []
+uuid: 6767b7f0-705a-43a1-9e02-aeee6b454538
+---
+
+# Symbols & Notations Used in Circuit Diagrams
+
+![](./images/vimscrot-2023-02-09T11:07:33,231545943+00:00.png)
+
+
+![AC Voltage Source](./images/vimscrot-2023-02-09T12:35:04,954248906+00:00.png)
+
+# Series Circuits
+
+Summary of this section in tabular form found in slides p30.
+
+## Resistors
+
+![](./images/vimscrot-2023-02-09T11:25:05,307411437+00:00.png)
+
+\begin{align*}
+I &= I_1 = I_2 = I_3 \\
+V &= V_1 + V_2 + V_3 \\
+R &= R_1 + R_2 + R_3
+\end{align*}
+
+## Inductors
+
+More inductors in series makes is harder for current to change rapidly.
+
+![](./images/vimscrot-2023-02-09T11:26:21,405008860+00:00.png)
+
+\begin{align*}
+I &= I_1 = I_2 = I_3 \\
+V &= V_1 + V_2 + V_3 \\
+L &= L_1 + L_2 + L_3
+\end{align*}
+
+## Capacitors
+
+More capacitors in series makes is easier for voltage to change rapidly.
+
+![](./images/vimscrot-2023-02-09T11:26:43,223213152+00:00.png)
+
+\begin{align*}
+I &= I_1 = I_2 = I_3 \\
+V &= V_1 + V_2 + V_3 \\
+\frac 1C &= \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}
+\end{align*}
+
+# Parallel Circuits
+
+## Resistors
+
+![](./images/vimscrot-2023-02-09T11:27:45,821432874+00:00.png)
+
+\begin{align*}
+I &= I_1 + I_2 + I_3 \\
+V &= V_1 = V_2 = V_3 \\
+\frac 1R &= \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
+\end{align*}
+
+## Inductors
+
+More inductors in parallel makes is easier for current to change rapidly.
+
+![](./images/vimscrot-2023-02-09T11:30:46,501506991+00:00.png)
+
+\begin{align*}
+I &= I_1 + I_2 + I_3 \\
+V &= V_1 = V_2 = V_3 \\
+\frac 1L &= \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3}
+\end{align*}
+
+## Capacitors
+
+More capacitors in parallel makes is harder for voltage to change rapidly.
+
+![](./images/vimscrot-2023-02-09T11:31:31,829222602+00:00.png)
+
+\begin{align*}
+I &= I_1 + I_2 + I_3 \\
+V &= V_1 = V_2 = V_3 \\
+C &= C_1 + C_2 + C_3
+\end{align*}

uni/mmme/2051_electromechanical_devices/fundamentals.md

@@ -32,6 +32,10 @@ where $V$ is voltage across a component, $I$ is current through it, and $R$ is r
 - Impedance is used when there are energy storage elements to a component.
 - Resistance, a special case of impedance, can be used when there is no storage element
 
+## Admittance
+
+$$Y \frac1Z$$
+
 # Kirchhoff's Laws
 
 ## Current
@@ -84,3 +88,11 @@ Capacitors try to minimize changes in voltage.
 
 If a capacitor is shorted, the current through the connecting wires will be extremely high, causing
 the wires to heat up.
+
+# Root Mean Square (RMS)
+
+$$x_{\text{RMS}} = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}}$$
+
+For a sinusoidal wave:
+
+$$x_\text{RMS} = \frac{A}{\sqrt2}$$

uni/mmme/2051_electromechanical_devices/kirchhoff.md

@@ -0,0 +1,59 @@
+---
+author: Akbar Rahman
+date: \today
+title: MMME2051 // Kirchhoff's Current Law, Voltage Law
+tags: [ kirchhoff, kcl, kvl ]
+uuid: 88e2eb6a-7f6a-4ea0-9850-81305028e7b5
+lecture_slides: ./lecture_slides/MMME2051EMD_Lecture2A.pdf
+---
+
+# Application of Kirchhoff's Current/Voltage Laws (KCL, KVL)
+
+(lecture slides 14-21)
+
+![](./images/vimscrot-2023-02-09T11:09:25,300096365+00:00.png)
+
+1. Identify all the loops in the circuit and assign each loop a "loop current" variable:
+
+
+   ![](./images/vimscrot-2023-02-09T11:10:56,126073331+00:00.png)
+
+1. Identify "branch current" values (apply KCL)
+
+
+   ![](./images/vimscrot-2023-02-09T11:11:59,570957109+00:00.png)
+
+1. Apply KVL to each loop:
+
+   Loop 1: $10 - 2 - V_1 - V_2 = 0$  
+   Loop 2: $V_2 - V_4 = 0$  
+   Loop 3: $V_4 - V_3 - V_5 = 0$
+
+   ![](./images/vimscrot-2023-02-09T11:13:18,376344102+00:00.png)
+
+1. Apply Ohm's Law to KVL
+
+   Loop 1 (origin at node A):
+
+   \begin{align*}
+   0 &= 10 - 2 - V_1 - V_2 \\
+     &= 8 - I_1R_1 - (I_1-I_2)R_2 = 0 \\
+   8 &= I_1(R_1+R+2) - I_2R_2 \\
+     &= 6I_1 - I_2
+   \end{align*}
+
+   Loop 2 (origin at node B):
+
+   \begin{align*}
+   0 &= V_2 - v_4 \\
+     &= (I_1-I_2)R_2 - (I_2-I_3)R_4 \\
+     &= I_1 - 3I_2 + 2I_3
+   \end{align*}
+
+   Loop 3 (origin at node C):
+
+   \begin{align*}
+   0 &= V_4 - V_3 - V_5 \\
+     &= (I_2-I_3)R_4 - I_3R_3 - I_3R_5 \\
+     &= 2I_2 - 5I_3
+   \end{align*}