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ƒ r
straight line intersecting the origin with a slope equal to the inductance
of the coil. The mathematical expression for any straight line in a twodimensional
plane is given by
y � mx � b
Thus, for the coil,
X L � 2pfL � 0 � 2pL f � 0
y � m x � b
(where 2pL is the slope), producing the results shown in Fig. 20.9.
For the capacitor,
1
1
XC �� or XC f � �
2pfC
2pC
which becomes yx � k, the equation for a hyperbola, where
y (variable) � XC x (variable) � f
1
k (constant) � �
2pC
The hyperbolic curve for XC( f ) is plotted in Fig. 20.10. In particular,
note its very large magnitude at low frequencies and its rapid dropoff
as the frequency increases.
If we place Figs. 20.9 and 20.10 on the same set of axes, we obtain
the curves of Fig. 20.11. The condition of resonance is now clearly
defined by the point of intersection, where XL � XC. For frequencies
less than fs, it is also quite clear that the network is primarily capacitive
(XC > XL). For frequencies above the resonant condition, XL > XC, and
the network is inductive.
Applying
ZT ( f ) � �[R�(�f �] )
2 � �� [�X� L(�f )
� �[R�(�f �] )
2 � �� [�X�(�f �] )
2 �
� �� X� C�(�f )
�] 2 �
to the curves of Fig. 20.11, where X( f ) � X L( f ) � X C( f ), we obtain
the curve for Z T ( f ) as shown in Fig. 20.12. The minimum impedance
occurs at the resonant frequency and is equal to the resistance R. Note
that the curve is not symmetrical about the resonant frequency (especially
at higher values of Z T).
The phase angle associated with the total impedance is
v � tan �1 � (XL � XC) �
R
(20.15)
For the tan �1 x function (resulting when X L > X C), the larger x is, the
larger the angle v (closer to 90°). However, for regions where X C > X L,
one must also be aware that
tan �1 (�x) � �tan �1 x
(20.16)
At low frequencies, X C > X L, and v will approach �90° (capacitive),
as shown in Fig. 20.13, whereas at high frequencies, X L > X C, and v will
approach 90°. In general, therefore, for a series resonant circuit:
X
0
0
0
X L ( f )
Z T VERSUS FREQUENCY ⏐⏐⏐ 893
X L = 2pfL
∆x
∆y
2pL =
∆y
∆x
= m
FIG. 20.9
Inductive reactance versus frequency.
X C ( f )
X C =
1
2pfC
FIG. 20.10
Capacitive reactance versus frequency.
X C
X C > X L
f s
X L > X C
X L
FIG. 20.11
Placing the frequency response of the
inductive and capacitive reactance of a
series R-L-C circuit on the same set of axes.
Z T
R
0
a
f s
b ≠ a
f
Z T ( f )
FIG. 20.12
Z T versus frequency for the series resonant
circuit.
f
f
f