Bioinstrumentation - Introduction to Biomedical Engineering - page 574

In Eq. (9.33), by practice, the angle in the exponential is written in radians, and in the
notation is written in degrees. Work in the phasor domain involves the use of complex
algebra in moving between the time and phasor domain, so the rectangular form of the
phasor is also used, given as

V ¼ V m ð cos f þ j sin f Þ ð 9 : 34 Þ

9.12.2 Passive Circuit Elements in the Phasor Domain
To use phasors with passive circuit elements for steady-state solutions, the relation-
ship between voltage and current is needed for the resistor, inductor, and capacitor.

Assume that

For a resistor,

v ¼ IR ¼ RI m cos ð o t þ y Þ

and the phasor of v is

Þ
where . Note that there is no phase shift for the relationship between the phasor cur-
rent and voltage for a resistor.
For an inductor,

ð 9 : 35

v ¼ L di 90 Þ

dt ¼

o LI m sin ð o t þ y Þ¼ o LI m cos ð o t þ y

and the phasor of v is

ð 9 : 36 Þ

Note that inductor current and voltage are out of phase by 90 —that is, current lags behind
voltage by 90 .
For a capacitor, define v ¼ V m cos ð o t þ y Þ and . Now

i ¼ C dv

dt ¼ C d dt V m

ð cos ð o t þ y Þ Þ

90 ¼ CV m o sin o ð

t þ

y Þ¼ CV m o cos o ð

t þ

y Þ