Basic Electronics for Scientists and Engineers

2.6 Using complex numbers in electronics 49
Hencewehave
Î =
ˆV
R + 1
jωC
. (2.75)
This equation is very similar to Eq. (2.67), and the “massaging” is identical to what
we did there. The result is
Î =
ωC ˆV
√
(ωRC) 2 + 1 e−jθ , (2.76)
where, as before, θ is given by
( ) −1
θ = tan −1 . (2.77)
ωRC
In most cases, we have a drive voltage of the form V p cos(ωt) or V p sin(ωt). In
either case, we can plug into Eq. (2.76) a complex voltage V p e jωt with the proviso
that, at the end, we will take the real part of the answer for the cosine drive or the
imaginary part of the answer for the sine drive. Thus
Î =
ωCV p
√
(ωRC) 2 + 1 e j(ωt−θ) (2.78)
and, for a cosine drive, we obtain our former result:
I = Re(Î) =
ωCV p
√ cos(ωt − θ). (2.79)
(ωRC) 2 + 1
2.6.5.2 Series LR circuit
Now let’s apply the technique to a circuit we have not studied before, the series
LR circuit shown in Fig. 2.18. The total impedance of this series combination of a
resistor and inductor is Ẑ tot = R + jωL. Hence,
Î =
ˆV
R + jωL =
ˆV
√
R 2 + (ωL) 2 e jθ (2.80)
V in
I
L
R
Figure 2.18 Series LR circuit.