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6
Figure 1-1 Equivalent circuits for inductors and capacitors.
The complex impedance of a capacitor is
C j R jX R Z

/
   
(series equivalent circuit) (9a)
 
 
2
1
1
1 / 1
/ 1
p p
p p p
p p
p
p p
p p
R C
R C j R
R C j
R
C j R
C j R


 









(parallel equivalent) (9b)
Quality Factors
Originally, the quality factor Q was defined for an inductor as a measure of the efficiency
of energy storage in the inductor when an AC current is passed through it. Mathematical-
ly, the definition is
1
Q

2

(max. energy stored)

(energy dissipated per Hz) (10a)

2

f
(max. energy stored)

(average power dissipated) (10b)
Since the average power dissipated in the inductor with series resistance R is | I |
2
R and
the maximum energy stored in the inductor is
L
|
I
|
2
, the quality factor for an inductor is giv-
en by
Q

L / R . (11a)
By equating (8a) and (8b), the series equivalent circuit parameters
R
and
L
can be ex-
pressed in terms of the parallel parameters
R
p
and
L
p
. When that is done and substituted
in equation (10a), we find that the quality factor also is written
Q

R
p
/

L
p
. (11b)
While the concept of the quality factor was originally applied to inductors, it may be
extended so that the efficiency of energy storage in a capacitor may be expressed in
terms of the circuit components and frequency. Thus, if the series resistance and capaci-
tance of a capacitor are, respectively,
R and C as in Figure 1-1, then (10b) is evaluated to
be
Q

1 /

CR . (12a)
By equating (9a) and (9b), the series equivalent circuit parameters
R
and
C
can be ex-
pressed in terms of the parallel parameters R
p
and C
p
. When that is done and substituted
in equation (12a), we find that the quality factor for a capacitor also is written
1
W. L. Everett and G. E. Anner, Communication Engineering , McGraw-Hill, New York, 1956.