Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

12.10.4 
Bandwidth Versus quality Factor of Series 
RLC
 circuit 
The frequency-response at the band-pass output of series RLC circuit (i.e., the voltage across the 
resistor) was obtained as 
V j
V j
j
j
R
S
n
n
n
n
n
n
(
)
(
)
(
)
(
)
w
w
xww
w
w
xw w
xww
w
w
x w w
=
−
+
=
−
+
2
2
2
4
2
2
2
2 2
2
2
2
∠
∠
= −
−
−
f
f
p
xw w
w
w
R
R
n
n
where
rad
2
2
1
2
2
tan
(12.10-1)
It was shown that this output has a maximum gain of unity and that occurs at 
w
=
w
n
. We wish to 
develop a measure for the frequency selectivity exhibited by this output in this subsection.
The resistor gets all of the input voltage if input frequency is resonant frequency. Therefore, 
the resistor receives maximum power for fixed amplitude input if the input frequency is resonant 
frequency. The power dissipated at all other frequencies will be less than this value. There will be 
one or more frequencies at which the power dissipated in the resistor is exactly 50% of the power that 
will be dissipated if input is at resonant frequency (assuming the input amplitude is kept fixed). These 
frequencies have been used traditionally to define measures of frequency selectivity in band-pass 
circuits. They are called half-power frequencies for obvious reason.
Power dissipated in a resistor becomes 50% when voltage developed across it becomes 1/
√
2 
≈
0.707. Therefore, half-power frequencies are the angular frequencies at which the magnitude response 
of the circuit output become 70.7% of some reference gain value. The reference gain value in the case 
of low-pass circuit is the DC gain, in the case of band-pass circuit it is the maximum gain and in the 
case of high-pass circuit it is the gain as 
w
 
→
∞
.
The frequency at which the gain of a band-pass circuit reaches maximum is termed centre frequency 
in filter studies. A typical band-pass response is shown in Fig. 12.10-6. The half power frequencies 
w
1
and 
w
2
are marked in the magnitude plot.

---

ResonanceinSeries
RLC
Circuit

12.35
1
Gain
0.707
ω
0.8
0.6
0.4
0.2
ω
1
ω
2
ω
n
Phase
(rad)
0.5
1
1.5
–0.5
–1
–1.5
ω
1
ω
2
ω
n
ω
π
4
π
4
–
Fig. 12.10-6 
Atypicalband-passcircuitfrequency-response
The difference between the two half-power frequencies is called the bandwidth of the band-pass 
circuit. We develop an expression for bandwidth of a narrow band-pass circuit in the following 
and develop interesting insight into the relation between the bandwidth and quality factor of the
circuit.
2
4
1
2
2
4
1
2
2
2 2
2
2
2
2
2
2 2
2
2
2
xww
w
w
x w w
xww
w
w
x w w
n
n
n
n
n
n
(
)
(
)
−
+
=
∴
(
)
−
+
=
Lett
Then,
4
4
4
2
2
2
x
x
x
x
x
x
n
=
−
+
=
∴ −
−
= ⇒ −
w
w
x
x
x
.
(
)
(
)
(
2
2 2
2
2 2
2
1
1
2
1
0
1
xx
x
x
x
x
2
2
2
2
2
1 0
1
)
= ±
∴ ±
− = ⇒ =
±
+
x
x
x
x
∓
Taking only positive valuees for
2,1
w w
x
x w
xw
,
(
)
=
+
±
⇒
=
1
2
2
n
n
bw
∴
=
=
Centre frequency
Bandwidth
1
2
x
Q  
(12.10-2)
This Q factor (equivalently, damping factor) has indeed turned out to be an important parameter for 
series RLC circuit. Eqn. 12.10-2 is the third interpretation for Q. We had seen earlier that, in a weakly 
damped series RLC circuit, the fractional loss of total stored energy in the circuit over one cycle of 
oscillation is given by 4
px
. Since Q 
=
1/2
x
,
Q
=
2
p
Total stored energy in the source-free circuit
Eneergy lost in one cycle of free response
.
Second interpretation is based on the same energy ratio under sinusoidal steady-state conditions at 
resonant frequency. Let the circuit be at resonance with 1 V amplitude input. Then,

---

12.36


SeriesandParallel
RLC
Circuits
v t
t
i t
R
t
v t
RC
t
S
n
n
C
n
n
( )
sin
,
( )
sin
( )
cos
=
∴
=
=
−
1
1
1
w
w
w
w
and
Total sttored energy
=
+
=
+
Li t
Cv t
L
R
t
C
R C
C
n
n
n
( )
( )
sin
cos
2
2
2
2
2
2
2
2
2
2
2
2
w
w
w
tt
L
R
LC
R
n
n
=
=
=
×
2
1
1
2
2
(
)
sin
∵
w
w
Energy dissipated in one cycle
1
R
w
w
w
p
w
p
n
n
n
t
d
t
R




=
∴
∫
2
0
2
(
)
Total stored energy
Energy disssipated in one cycle
To
=
=
=
=
∴ =
1
2
1
2
1
2
1
2
2
2
p
w
p
p x
p
p
n
L
R
L C
R
Q
Q
/
ttal stored energy under resonance condition
Energy dissipatted in one cycle under resonance condition
12.10.5 
quality Factor of Inductor and capacitor
We are already familiar with the concept of Q factor for a series RLC circuit. We discuss a similar 
factor for the elements themselves in this section.
We had assumed till now that the inductor and capacitor used in the series RLC circuit are ideal 
elements and that they have no parasitic elements associated with them. This is not true in practice. 
The non-zero parasitic elements associated with inductor and capacitor will affect the performance of 
RLC circuits considerably in narrow band-pass circuit applications.
An inductor has a non-zero wire resistance that goes along with its inductance in series. Further, 
if the inductor uses iron core, there will be hysteresis and eddy current losses in the iron core due 
to time-varying magnetic fields in the core. These losses are strongly dependent on frequency of 
operation and flux level in the core. Core loss is usually modelled by a resistance in parallel to 
the inductance. However, due to its complex dependence on frequency of operation, it can not be 
satisfactorily modelled by a single value of resistance at all frequencies.
In addition, a physical inductor will have distributed capacitance of winding shunting its inductance 
value. Thus, a practical inductor is more like the equivalent circuit shown in Fig. 12.10-7 (a).
However, in frequency-response studies, the distributed capacitance C
p
is usually ignored since 
its value is generally too small to affect the circuit performance. Circuit equivalent in (b) is usually 
employed in studying the effect of losses in the inductor on the Q factor of a circuit employing 
this inductor. R
p
can, at the best, represent the core losses in the inductor only for a small band of 
frequencies around a frequency value at which it was measured. 
R
p
R
s
C
p
L
(a)
R
p
R
s
L
(b)
L
(c)
R
s
'
'
Fig. 12.10-7 
Equivalentcircuitsforapracticalinductor

---

ResonanceinSeries
RLC
Circuit

12.37
The circuit in (b) can not be reduced to the circuit in (c) such that (c) remains equivalent to (b) at 
all frequencies. However, (b) can be reduced to (c) – i.e., a value of L
′
and R
s
′
can be found such that 
(c) will have same phasor impedance as that of (b) - at some particular frequency. Usually, the value 
of L
’
will be close to L and it is approximated that way in practice. R
s
′
will include the effects of R
p
and 
R
s
together. Since, in any case, a specific value of R
p
is valid only over a small band around a specific 
frequency, the circuit in (b) can be equivalenced to circuit in (c) subject to the condition that it can be 
expected to give reasonably accurate results only over a small band of frequencies around the specific 
frequency at which R
p
and R
s
′
 
are measured or calculated. This is satisfactory in the case of resonance 
studies in under-damped circuits since the frequency range of interest is a small band of frequencies 
around 
w
n
. The value of R
s
′
is usually indicated in an indirect manner by specifying it through a ratio. 
That ratio is the Q factor of Inductor and it is defined as the ratio of reactance of the inductor at 
w
to 
the resistance value R
s
′
relevant to that frequency. Q factor of an inductor will change with frequency. 
Therefore,
Q factor of an inductor at 
measured at 
w
w
w
=
′
L
R
s
A commercial Q-Meter that is available in any well-equipped laboratory will have features that 
permit measurement of Q of inductors at various frequencies.
A practical capacitor also has three parasitic elements associated with it. The foil resistance and 
lead inductance come in series with the capacitance. The leakage current that flows through the 
imperfect dielectric employed in the capacitor is modelled by a resistance in parallel to the capacitor. 
The loss mechanisms in the capacitor are also frequency dependent and hence an equivalent circuit 
for a practical capacitor will be valid only for a small band of frequencies around a specific frequency 
at which the measurement is carried out.
R
p
R
s
C
(b)
R
p
L
s
R
s
C
(a)
C 
′
(c)
R
p
′
Fig. 12.10-8 
Equivalentcircuitsforapracticalcapacitor
The series inductance L
s
in the detailed equivalent circuit in Fig. 12.10-8 (a) is usually ignored 
(or absorbed along with the inductor in series) in studies on under-damped resonant circuits. And, 
the equivalent circuit in (b) is approximated by the circuit in (c) with C
’
≈
C and R
p
’
measured by a 
Q-meter. The equivalent circuit in (c) is understood to be valid only for a small band (
≈
±
20% max) 
around the frequency at which R
p
′
was measured. The Q-meter measures it indirectly and displays the 
Quality factor of capacitor. Q factor for a capacitor is defined as the ratio of resistance value R
p
′
at 
w
to the reactance of the capacitor at 
w
. Q factor of a capacitor will change with frequency. Therefore, 
Q
C R
P
−
=
× ′
factor of an capacitor at
measured at 
w w
w
.