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electromechanical_devices ] diff --git a/uni/mmme/2051_electromechanical_devices/ac_intro.md b/uni/mmme/2051_electromechanical_devices/ac_intro.md new file mode 100755 index 0000000..d680ee1 --- /dev/null +++ b/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. diff --git a/uni/mmme/2051_electromechanical_devices/ac_power.md b/uni/mmme/2051_electromechanical_devices/ac_power.md new file mode 100755 index 0000000..4993b74 --- /dev/null +++ b/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) + +$$|V_\text{line}| = |V_\text{phase}|$$ + +$$I_\text{line} = \sqrt 3 I_\text{phase}$$ diff --git a/uni/mmme/2051_electromechanical_devices/basic_circuitry.md b/uni/mmme/2051_electromechanical_devices/basic_circuitry.md new file mode 100755 index 0000000..e6e79a4 --- /dev/null +++ b/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*} diff --git a/uni/mmme/2051_electromechanical_devices/exercise_sheets/Exercise Sheet 1.pdf b/uni/mmme/2051_electromechanical_devices/exercise_sheets/Exercise Sheet 1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..571ada7ffb73d56a2595ed5f2f785816944eaeeb GIT binary patch literal 153114 zcmdqJ1zc6z);GM75(Md#2Bjr8EeJ?UcZ+m)mx?q<2?)~N-4fCzUDDm%z4;b+yytlD zIrq8G`#tgAckBACxz?I<%`xU2bBzBOb1WXqybz{mddiNB__!SSAtN%8Fp^llHbF+@ z;bBm4w$f)1)UnesvovIo(J|DwC1D00%Q3vvu`s04x1bjhpeA9Ex3jU=wNtRs*O#-j zv?F2WxT_+mV`D02VPFZo=C-J>M*>vS=jCOP)VI^o1Kvr(#LRhDQ^{E0QQt>#(K8uBrLbBNSN*j&?Na| z^V=u?&>Ub1=n!ar+l)a@-`3LJ2IwB3SjfpvMBYxvP9G?X$b(71cQr)hIY{o_4R9fB zX<-LE6lNk}yL%?gO!9*pVHOf5*4tY^B^Bw{Ci=R!^~B79(PLsG0sjAZBx7T#E3a>- z&H#)*gMz-3-H$2)KTd)_PGYxA00X9HtRrCQqz*J?Bmr}RpRzEMaIi8xW 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a/uni/mmme/2051_electromechanical_devices/kirchhoff.md b/uni/mmme/2051_electromechanical_devices/kirchhoff.md new file mode 100755 index 0000000..a31b125 --- /dev/null +++ b/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*} diff --git a/uni/mmme/2051_electromechanical_devices/lecture_slides/MMME2051EMD_Lecture1A.pdf b/uni/mmme/2051_electromechanical_devices/lecture_slides/MMME2051EMD_Lecture1A.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cbf686714c131e6d8a044e56f4323931475d1787 GIT binary patch literal 769516 zcmce-2Ut_j(l~rVfB@2qB+>#w37r5!=)L#QiwcAyNJ}VE1Of;O#?TcJBS=S4KtRx+ z3Q|G`0TpSYfQU4u3IhIv{_cD4d;jl!p6~sh@A(3oJ?HG~?#!IonVs3mNo##WX_SmI 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