# User:1sfoerster/sandbox2

simulation showing an energy imbalance at the beginning circuitlab.com for 2R, C, L with both voltage and current sources

## Simulation

Was simulated at circuitlab.com.

Color Voltage Current
Blue ${\displaystyle V_{s}}$ ${\displaystyle i_{1}}$
Orange ${\displaystyle V_{2}}$ ${\displaystyle i_{2}}$
Brown ${\displaystyle V_{L}}$ ${\displaystyle i_{L}}$

## Power Analysis

Power Analysis is rooted in the phasor domain! The phasor domain is limited to sinusoidal driving sources. This is why the discussion of RMS power and it's calculation is not part of this analysis.

${\displaystyle \mathbb {V} _{s}=120{\sqrt {2}}\angle 2.09}$
${\displaystyle \mathbb {I} =15.9\angle 1.73}$
${\displaystyle \mathbb {I} ^{*}=15.9\angle -1.73}$

if :${\displaystyle \mathbb {V} =M_{v}\angle \phi _{v}\quad }$, and ${\displaystyle \quad \mathbb {I} =M_{i}\angle \phi _{i}}$ then

${\displaystyle \mathbb {S} =\mathbb {V} \mathbb {I} ^{*}={\frac {M_{v}M_{i}}{2}}\angle (\phi _{v}-\phi _{i})={\frac {120{\sqrt {2}}*15.9}{2}}\angle (2.09-1.73)=135\angle 0.36=126+47.6j}$

and

${\displaystyle cos(.36)=.936}$

In summary

Value Units Description
${\displaystyle 135}$ volt-ampere va apparent power what utility companies manage: peak power they design for, peak power they have to deliver
${\displaystyle .936}$ unitless power factor, ratio of real power to apparent power, ideally 1
${\displaystyle 126}$ watt W real, average, active power ... what consumers want to pay for (watt-hours)
${\displaystyle 47.6}$ volt-amp-reactive var reactive power ... why not all outlets in a room are on the same circuit breaker

## Intuition

ELI ... voltage leads the current through an inductor

Derivatives cause a lag ... a delay in time ... which is a positive angle in sinusoidal.