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	$V_{s}$	$i_{1}$
Orange	$V_{2}$	$i_{2}$
Brown	$V_{L}$	$i_{L}$

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.

$\mathbb {V} _{s}=120{\sqrt {2}}\angle 2.09$

$\mathbb {I} =15.9\angle 1.73$

$\mathbb {I} ^{*}=15.9\angle -1.73$

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

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

and

$cos(.36)=.936$

In summary

Value	Units	Description
$135$	volt-ampere va	apparent power what utility companies manage: peak power they design for, peak power they have to deliver
$.936$	unitless	power factor, ratio of real power to apparent power, ideally 1
$126$	watt W	real, average, active power ... what consumers want to pay for (watt-hours)
$47.6$	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.