# AC Power Analysis

Oh boi this is difficult RMS $V_{rms}=T1 ∫_{0}v_{2}(t)dt a constant$

$V_{rms}=2 V_{m} $

$P_{av}=RV_{rms} =I_{rms}R$

$V_{dc}=V_{rms}$

### Maximum Average Power Transfer

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

$P_{max}=8R_{th}∣V_{th}∣_{2} $

### 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).

$P=21 V_{m}I_{m}cos(θ_{v}−θ_{i})$ $P=V_{rms}I_{rms}cos(θ_{v}−θ_{i})$

$cos(θ_{v}−θ_{i})$ 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.

$Q=21 V_{m}I_{m}sin(θ_{v}−θ_{i})$ $Q=V_{rms}I_{rms}sin(θ_{v}−θ_{i})$

Remember that the $θ$ 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)=Rv_{2}(t) $

### Average Power

It’s just the integral of the instantenous power divided by the time period. $P_{avg}=T1 ∫_{0}p(t)dt=RV_{rms} $

→ 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}=8R_{th}∣V_{th}∣_{2} $

### Complex Power

The complex power for a general element is defined by

$S=21 VI_{∗}=V_{rms}I_{rms}$

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∣=P_{2}+Q_{2} $