Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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Bruce Mayer, PE Engineering 43: Electrical Circuits & Devices

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 43

Prob 5.80

SolutionDual-Source

Complex-PowerBalance



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

imsvms tIItVV cosorcos

irmssvrmss tIItVV cosorcos

rmsmmrms UUUU 2or2

jQP S



BackGround EE (§5.5)

Each Passive Component (R, C, L, Vsrc, Isrc) Generates it’s own Complex Power in an operating Circuit

Furthermore Complex Power is Conserved, just a real power was conserved in purely resistive Circuits; i.e.:

Equating Real & Imag Parts

Components n"" for the01

n

kkS

0&0 kk QP



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)

• IRrms ≡ Current thru R• VRrms ≡ Voltage across R

RVPRIP RrmsRRrmsR22 &

RIP rmsR2



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– XL ≡ Inductive ReActance, ωL

LLrmsLLLrmsL XVQXIQ 22 &



R, C, L vs. P&Q

CAPACITORS create NEGATIVE ReActive (imaginary) Power per Text Eqns (5.70) & (5.72)

• Where– ICrms ≡ Current thru C– VCrms ≡ Voltage across C– XC ≡ Capacitive ReActance,

−1/(ωC)

CVQC

IQ

XVQXIQ

CrmsCCrmsC

CCrmsCCCrmsC

1&1or

&

22

22



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

*

21 VIS

jBAjBA

UU mm

*

*

or

UU

UU



S by ShortHand Calc

If

Then

Also For Sources the Apparent Power:

*

21 VIS

*

*

21ImIm

21ReRe

kkk

kkkk

Q

P

VIS

IVS

SrmsSrmsSmSmSSSS

SSSSSS

SSSrmsSrmsapparent

IVIV

IVP

222222

and 22222

1

as 21

**

***

*

IVIV

IVIVIV

IVS



Apparent Power

Apparent Power Calculations are usually used with SOURCES, but the concept also applies to Impedance Loads

Given an Impedance with Current IZ, and Voltage-Drop VZ then

ZZZapparent

ZZZ

ZZZZrmsZrmsapparent

QPP

jQP

IVP

S

S

IVS

22

*

since also

21



Problem 5.80

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



Confirm: ∑Pavg = 0

Note that the Average Powers Add to ZERO as they must by conservation of Energy



Check ReActive Power

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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