PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Alternating Current Analysis 231

where

V

T

RMS vtdt

T

(^12)
0

()

∫

and

I

T

itdt

T

MS pt dt

TT

R 

PAVG

(^12)
00

() (

∫∫

and

1

)

R.3.20 Let the current through and voltage across an arbitrary load be given by

i(t) = Im sin(t)

and

v(t) = Vm sin(t + ) (an RL equivalent circuit since v(t) leads i(t) by )

Then the instantaneous power is given by

p(t) = i(t) ⋅ v(t) = Im Vm sin(t) ⋅ sin(t + )

Using trigonometric identities

pt

VI VI

t

VI

( )
mmcos( ) mmcos( )cos( )mmsin( )sin( t)

22

2

2

  2

where

VI V I

mm m m VIRMS RMS

2 2

2

.

Let VRMS IRMS = A

then, p(t) = A cos(θ) − A cos(θ) cos(2ωt) + A sin(θ) sin(2ωt).

R.3.21 Let us explore the resistive case, where θ = 0°. Then p(t) of R.3.20 becomes

p(t) = A − A cos(2t)

And the average power, often referred as the real power, is given by

PA

VI

AVG

mm
VIRMS RMS

2

(in watts)

PAVG = real(VRMS IRMS*) (the character * denotes the complex conjugate of IRMS)

The energy dissipated by the resistor R, in the form of heat over one full cycle, is

given by

WR = VRMS IRMS T (in joules)