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-30 I i 0.05 0.01 ( u=-2) ~ I'-t- II . .-- _~rr . ,, ! " ............. I 0.1 I! 0.5 ~I I 1i I I! ! I ; I lea ! jI I ; , il 1 W- l~ r-J 00=1
(u 0 0_ II
!i 100 50 10 (u = 0) ( u=-I) .. -, " 1I I I'
I I. , ' .-., j = I) (u = 2) (7.13) T he t erm 2 0Iog(KaIa2/bIb3) is a c onstant. We observe t hat t he log amplitude is
a s um of four basic terms corresponding t o (i) a constant, (ii) a pole or zero a t
t he origin (201og Ijwl)' (iii) a first-order pole or zero (20 log 11 + j w/all, a nd (iv)
complex c onjugate poles o r zeros (20 log 11 + jwb2/b3 + (jw)2 /b31l· We can sketch
these four basic t erms as functions of w and use t hem t o c onstruct t he log-amplitude
plot of a ny d esired transfer function. Let us discuss each of t he terms. , 1 50"1==+=+=+++=I+I1C==+=+==l=rnr:m==+==+=+=mm==+=+=::r=m:m
II . .....
90"b_ . . _~.~~-;-1-;j;;m=±±hmfh-:dddd~!db=b!-::-:k~:Hj
-i 1I --r
T 11"--~ 5 0"·---+-+-I--HH+i+--- - (b ~ -,- --- -r ------ --- -- Il
~ - 50"f--·...........
- --.. ~-t-L+ ~ 0• • "._ I + j I
- -. -I
.. _ISO" ! ---..... , 0.05 0.01
( u=-2) 0.1 0.5 ( u=-I) 00=1
( u=O) + :ft·=
10 i --!- 50 ( u= I ) 100
(u = 2) Fig. 7 .3 Amplitude and phase response of a pole or a zero a t the origin.
T he log a mplitude of this t erm is also a constant, 20 l og(K aIa2/bIb3). T he
p hase c ontribution from this t erm is zero. 2. P ole (or Z ero) a t t he origin
Log M agnitude t This p oint c an b e s hown as follows: Let Wj a nd W2 a long t he w-scale correspond t o Uj a nd
along t he u ·scale. T hen l ogwj = Uj, a nd logw2 = U 2. T hen
U 2 - Uj = loglO W 2 - logJQwj = logJQ(w2/Wj)
T hus, if
(w2/wIJ = 10 (which is a decade)
U 2 - Uj = loglO 10 = 1
a nd if
(W2/WIJ 2 (which is a n o ctave)
U 2 - Uj = loglO 2 = 0.3010 = Such a pole gives rise t o t he t erm - 20 log Ijwl, which can be expressed as
- 20logljwl = - 2010gw U2 7 Frequency Response and Analog Filters 4BO Note t hat equal increments in u are equivalent t o equal ratios on the w-scale. Thus,
one unit along t he u-scale is t he same as one decade along the w-scale. This means
t hat t he a mplitude plot has a slope of - 20 d B/decade or - 20(0.3010) = - 6.02
d B/octave ( commonly s tated as 6 dB/octave). Moreover, the amplitude plot crosses
t he w-axis a t w = 1, since u = 10glQw = 0 when w = 1.
For t he c ase of a zero a t t he origin, t he log-amplitude term is 20 log w. T his
is a straight line passing through w = 1 a nd having a slope of 20 d B/decade (or 6
dB/octave). T his plot is a mirror image about t he w-axis of the plot for a pole a t
t he origin a nd is shown dotted in Fig. 7.3a.
Phase 7.2 Bode P lots 481 18
6 o - 12 T he phase function corresponding to the pole a t t he origin is - Ljw [see Eq.
(7.12b)]. Thus -18 (7.15b) L H(jw) = - Ljw = - 90° O .Ola T he phase is c onstant ( - 90°) for all values of w , as depicted in Fig....
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