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Unformatted text preview: ction of any transfer by properly
[ adding these b asic responses. Note t hat if a particular term is in t he n umerator, its
phase is a dded, b ut if t he t erm is in t he d enominator, its phase is s ubtracted. This
makes it easy t o p lot t he p hase function 1. H (jw) as a function of w. C omputation
of IH (jw )1, u nlike t hat of t he p hase function, however, involves t he m ultiplication
a nd division of various terms. This is a formidable task, especially when we have
t o p lot this funct ion for t he e ntire range of w (0 t o 00).
We know t hat a log operation converts multiplication a nd division t o a ddition
a nd s ubtraction. So, i nstead of plotting IH(jw)1, why not plot log IH(jw)1 t o
simplify our t ask? We can take advantage of the fact t hat logarithmic units are
desirable in several applications, where t he variables considered have a very large
range of variation. This is p articularly t rue in frequency response plots, where we
m ay have t o p lot frequency response over a frequency range s tarting from a very low
frequency near 0 t o a very high frequency in t he r ange of lO lD or higher. A linear plot
for such a large r ange will bury much of t he useful information. Also, t he a mplitude
response may h ave a very large dynamic range from a low of 1 0- 6 t o a high o f 10 .
A linear plot would be unsuitable for such a situation. Therefore, logarithmic plots
not only simplify o ur t ask of plotting, b ut, fortunately, they are also desirable in
this situation. T he logarithmic u nit is t he d ecibel a nd is equal t o t wenty times t he
l ogarithm of t he q uantity (log t o t he base 10). Therefore, 20log lO IH(jw)1 is simply
the log a mplitude in decibels (dB). Thus, instead of plotting IH (jw) I, we shall plot
20l0g lO IH(jw)1 a s a function of w. T hese plots (log amplitude a nd phase) are called
B ode p lots. F or t he t ransfer function in Eq. (7.12a), the l og a mplitude is This function can be p lotted as a function of w. However, we c an effect further
simplification b y using t he logarithmic scale for t he variable w itself. Let us define
a new variable U such t hat (jw)2\
+T (7.15a) The log-amplitude function - 20u is p lotted as a function of u in Fig. 7.3a. This
is a s traight line with a slope of - 20. I t crosses t he u-axis a t u=O. T he w-scale
(u = logw) also appears in Fig. 7.3a. Semilog g raphs can be conveniently used for
plotting, a nd we can directly plot w on semilog paper. A ratio of 10 is a d ecade
a nd a ratio of 2 is known as a n o ctave. F urthermore, a decade along t he w-scale
is equivalent t o 1 u nit along t he u-scale. We can also show t hat a r atio of 2 (an
octave) along t he w-scale equals t o 0.3010 (which is loglO 2) along t he u-scale.t J 30 t.,
~-. I ~~ 1--I i II~r---.... I I !
I! - _. 20 10 1 --.. ~ 0 :t: .2' 2010gIH(jw)1 = 20 log K aIa2 + 2010g 11 + j w 1+ 2010 g 11 + jw 1 - 2010gliwl
20 log 1 + b ;- (7.14) - 2010gw = - 20u 0 - = logw U Hence N jW\
- 20 log 1 + ~
\ 479 7.2 Bode P lots 7 Frequency Response a nd Analog Filters 478 I II -10 .I j!...
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