Complex Power and Reactive Components in Electrical Circuits, Slides of Electrical Circuit Analysis

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Prob 5.

Solution

Dual-Source

Complex-Power

Balance

BackGround EE (§5.5)

• Consider Sinusoidal Sources of the form
• Alternatively
• Recall for a Sinusoid
• Complex Power
  • Where
    • S ≡ Complex Power in V∙A
    • P ≡ Real Power in Watts
    • Q ≡ ReActive Power in VAR

Vs = Vmcos (ω t +θv) or Is = Imcos(ω t+ θi)

Vs = Vrms cos (ω t +θv) or Is = Irmscos(ω t+ θi)

U (^) rms = Um 2 or Um = 2 ⋅U rms S = P + jQ

R, C, L vs. P&Q

• Note that the passive sign Convention Applies to Complex Power a well - Positive Pwr → ABSORBED or DISSIPATED - Negative Pwr → SUPPLIED or GENERATED
• RESISTORS can only DISSIPATE Real Power per Text Eqns (5.69) & (5.71) - I (^) Rrms ≡ Current thru R - VRrms ≡ Voltage across R

PR = IRrms^2 R & PR =VRrms^2 R

PR =Irms^2 R

R, C, L vs. P&Q

• INDUCTORS create POSITIVE ReActive (imaginary) Power per Text Eqns (5.70) & (5.72) - Where - ILrms ≡ Current thru L - VLrms ≡ Voltage across L - X (^) L ≡ Inductive ReActance, ωL

QL ILrmsXL QL VLrms X L =^2 & =^2

S by ShortHand Calc

• Alternatively Calculate S for any Component using the COMPLEX CONJUGATE of the CURRENT Thru the device
• Recall Conjugation for Imaginary of Phasor Values

2

1 S = VI

A jB A jB

U (^) m Um

= + ⇒ = −

= ∠ ⇒ = ∠ −

or U U

U φ U φ

S by ShortHand Calc

• If
• Then
• Also For Sources the Apparent Power :

2

1 S = VI



  

= = 

 

  

= = 

2

Im Im^1

2

Re Re^1

k k k

k k k k

Q

P

S VI

S V I

Srms Srms S S S S Sm Sm

S S S S S S

apparent Srms Srms S S

V I V I

P V I

⋅ = ⋅ = =

= = ⋅

= = =

2 2 2 2 2 2

and (^22222)

1

as 2

1

V I V^ I

V I V I V I

S V I

Problem 5.

• Given Circuit
• Find The Power Condition for each element, the Quantity, Nature, and if the element is Absorbing/Supplying Power
• Note that this is SINGLE LOOP Circuit which thus has ONE Current

Check ReActive Power