--- 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  Convert all variables to phasors or to complex form  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*}  ## Phasors in Inductive Circuit  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 $$P = \frac{V^2}{2\omega L}\sin{2\omega t}$$  ## Capacitive Circuits $$P = \frac{\omega CV^2}{2}\sin{2\omega t}$$  ## 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.  ## 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: $$P = S\sin\gamma$$ 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.