AC Power Analysis

Oh boi this is difficult RMS $V_{rms}=\sqrt{\frac{1}{T}{\int }_0^T{v}^2(t)dt}\text{a constant}$

$V_{rms}=\frac{V_m}{\sqrt{2}}$

$P_{av}=\frac{V_{rms}^2}{R}=I_{rms}^{2}R$

$V_{dc}=V_{rms}$

Maximum Average Power Transfer

The maximum average power absorbed by the load $Z_L$ occurs when $R_L=Z_{th}$

$P_{max}=\frac{|{\tilde{V}}_{th}{|}^2}{8R_{th}}$

Active (Real) Power $P$

The active/real power is constant and represents the portion of the power that is transformed from electric energy to non-electric energy (ex: heat).

$P=\frac{1}{2}{V}_m{I}_m\cos ({\theta }_v-{\theta }_i)$ $P=V_{rms}{I}_{rms}\cos ({\theta }_v-{\theta }_i)$

$\cos ({\theta }_v-{\theta }_i)$ is known as the Power Factor

Reactive Power $Q$

The reactive power of $Q$ is also constant and represents the portion of the power that is NOT transformed into non-electric energy, but rather it is exchanged between circuit elements such as capacitors, inductors, and sources.

$Q=\frac{1}{2}{V}_m{I}_m\sin ({\theta }_v-{\theta }_i)$ $Q=V_{rms}{I}_{rms}\sin ({\theta }_v-{\theta }_i)$

Remember that the $\theta$ is written in terms of voltage. We know how Capacitors and Inductors behave (90 degrees between current and voltage), so you can make these final assumptions $Q_C=-V_{rms}{I}_{rms}$ $Q_L=V_{rms}{I}_{rms}$

Instantaneous Power

$p(t)=i(t)v(t)=\frac{v^2(t)}{R}$

Average Power

It’s just the integral of the instantenous power divided by the time period. $P_{avg}=\frac{1}{T}{\int }_0^{T}p(t)dt=\frac{V_{rms}^2}{R}$

→ Apparently, when solving circuits, the average power is the the real power $P$.

Maximum Average Power Transfer

THe results found for DC Circuits for Maximum Power Transfer are extended. $P_{max}=\frac{|{\tilde{V}}_{th}{|}^2}{8R_{th}}$

Complex Power

The complex power for a general element is defined by

$S=\frac{1}{2}\tilde{V}{\tilde{I}}^{\ast }={\tilde{V}}_{rms}{\tilde{I}}_{rms}^{\ast }$

Complex power is the combination of Active Power $P$ and Reactive Power $P$: $S=P+jQ$

Apparent Power

Apparent power is just the norm of the Complex power.

$P_{app}=V_{rms}{I}_{rms}=|S|=\sqrt{P^2+Q^2}$

Conventions

title: Some Conventions for the Units of Power
In the exam, if you see different units, this is how to tell apart different powers. 
 
| Type of Power | Unit | 
| :---: | :---: |
| Active Power $P$ | Watt (W) | 
| Reactive Power $Q$ | Volt-Ampere-Reactive (VAR) | 
| Complex Power $S$| Volt-Ampere (VA) | 
|Apparent Power $P_{app} = \vert S\vert$ | Volt-Ampere (VA) |

Related

• Power Factor