PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Alternating Current Analysis 245

and

Q = |S|sin( ) = (^150) * 0.809 = 121.2 var
R.3.49 Let us revisit the RLC series circuit shown in Figure 3.18. The RLC circuit is said
to be at resonance if the phase angle between the current i(t) and its voltage v(t) is
zero. The network is then purely resistive (R) at this frequency and this particular
frequency is referred to as the resonant frequency.
ANALYTICAL Solution
The total impedance of the series RLC circuit is given by
ZRjL
T C
()


 
^1



and at resonant ωL = 1/ωC or ωR = 1/ √

---

LC , and fr = ωR/2π or fR = 1/(2π √

---

LC ), where
fR is the resonant frequency, a phenomenon fi rst observed by Thomson around 1853.
R.3.50 At the resonant frequency fR, a series RLC circuit presents the following
characteristics:
a. ZT is purely resistive (phase angle between I and V is zero).
b. ZT is at a minimum.
c. Current is at a maximum.
d. The effective power is at its maximum (P = I^2 R = V^2 /R).
R.3.51 The quality factor Q (do not be confused with reactive power) is a dimensionless
ratio of the energy stored in the inductor to the average energy dissipated by the
resistor, given by

Q

inductive reactive power

average power

The quality factor Q for an RLC series circuit is then given by

Q

L

R

L

LC

X

S R

RL

(^2)
and the dissipation factor D is defi ned as D = 1/Q or D = ____ωR
RL
or D = ωRRC.
R.3.52 The BW and cutoff frequencies for the series resonant RLC circuit can be evaluated
by the following equations:





1

2

1

1

22

 R

S

R

QQS









rad/s

i(t)

v(t)

+ L –

+ R–

+ C –

+ –

FIGURE 3.18

RLC series circuit diagram of R.3.49.