User:1sfoerster/sandbox2

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

Simulation[edit]

Was simulated at circuitlab.com.

Color Voltage Current
Blue V_s i_1
Orange V_2 i_2
Brown V_L i_L

wikibook circuit theory example 11 simulation output from circuitlab.com for 2R, C, L with both voltage and current sources


Power Analysis[edit]

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_vM_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[edit]

ELI ... voltage leads the current through an inductor

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