0195136047.pdf

114 TIME-DEPENDENT CIRCUIT ANALYSIS

=

(√

2 Irmscosωt

)

L

(

−

√

2 ωIrmssinωt

)

=−Irms^2 XLsin 2ωt (3.1.40)

whereXL=ωLis the inductive reactance.

For a pure capacitorC,

pC(t)=vC(t)i(t)=i

1

C

∫

idt

=

(√

2 Irmscosωt

) 1

C

(√

2

ω

Irmssinωt

)

=−Irms^2 XCsin 2ωt (3.1.41)

whereXC=− 1 /ωCis the capacitive reactance.

The amplitude of power oscillation associated with either inductive or capacitive reactance

is thus seen to be

Q=Irms^2 X=VrmsIrmssinφ (3.1.42)

which is known as thereactive poweror imaginary power. Note thatφis the phase angle between

voltage and current. The unit of real power is watts (W), whereas that of reactive power is reactive

volt amperes (VAR). Thecomplex poweris given by

S ̄=S φ=P+jQ=VRMSIRMS(cosφ+jsinφ)=V ̄rmsI ̄rms∗ (3.1.43)

whereI ̄rms∗ is the complex conjugate ofI ̄. The magnitude ofS ̄, given by

√

P^2 +Q^2 , is known as

theapparent power,with units of volt-amperes (VA). The concept of apower triangleis illustrated

in Figure 3.1.3. The power factor is given byP/S.

The condition formaximum power transferto a load impedanceZ ̄L (=RL+jXL)

connected to a voltage source with an impedance ofZ ̄S (=RS+jXS)(as illustrated in Figure

3.1.4) can be shown to be

Z ̄L=Z ̄∗S or RL=RS and XL=−XS (3.1.44)

When Equation (3.1.44) is satisfied, the load and the source are said to bematched. If source and

load are purely resistive, Equation (3.1.44) reduces toRL=RS.

In the phasor method of analysis, the student should recall and appropriately apply the circuit-

analysis techniques learned in Chapter 2, which include nodal and mesh analyses, Thevenin and ́

Norton equivalents, source transformations, superposition, and wye–delta transformation.

S = VRMSIRMS (VA)

P = VRMSIRMS cos φ (W)

Q = VRMSIRMS sin φ (VAR)

Q

φ = tan P

− (^1) ()
Figure 3.1.3Power triangle.
ZL = ZS*
(RL = RS and XL = −XS)
Source

• −
  Load
  ZS
  VS ZL
  Figure 3.1.4Illustration of maxi-
  mum power transfer.