Page 79 - Circuit Analysis II with MATLAB Applications

Half-Power Frequencies - Bandwidth

By definition, the half-power frequencies Z 1 and Z 2 in Figure 2.12 are the frequencies at which the
magnitude of the input admittance of a parallel resonant circuit, is greater than the magnitude at res-

onance by a factor of 2 , or equivalently, the frequencies at which the magnitude of the input
impedance of a parallel resonant circuit, is less than the magnitude at resonance by a factor of 2 as
shown above. We observe also, that Z 1 and Z 2 are not exactly equidistant from Z 0 . However, it is
convenient to assume that they are equidistant, and unless otherwise stated, this assumption will be
followed in the subsequent discussion.

We call Z 1 the lower half-power point, and Z 2 the upper half-power point. The difference Z – Z 1 is
2
the half-power bandwidth BW , that is,

Bandwidth = BW = Z – Z (2.29)
2 1
The names half-power frequencies and half-power bandwidth arise from the fact that the power at
2
these frequencies drop to 0.5 since 2 2 = 0.5 .
e
The bandwidth BW can also be expressed in terms of the quality factor as follows:
Q
Consider the admittance

§
1 ·
Y = G + j ZC – -------
© ZL ¹
Z
0 ·
§
Multiplying the term by G ---------- , we get
j
© Z G ¹
0
ZZ C Z
§
0
0
Y = G + jG -------------- – ------------------ ·
© Z G ZZ LG ¹
0
0
Recalling that for parallel resonance
Z C 1
0
Q = ---------- = --------------
0P
G Z LG
0
by substitution we get
Z
------
Y = G 1 + jQ 0P § © ------ – Z 0 · (2.30)
Z
0 Z ¹
and if Z Z = , then
0
Y = G

Circuit Analysis II with MATLAB Applications 2-13
Orchard Publications