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Problem 6.07PP
Mixed real and complex poles. Sketch the asymptotes of the Bode plot magnitude and phase for 
each of the listed open-loop transfer functions. Embellish the asymptote plots with a rough 
estimate of the transitions for each break point. After completing the hand sketches, verify your 
result with Matlab. Turn in your hand sketches and the Matlab results on the same scales..
i (j + 1 oW + 2 j + 2)
^ ( j + 10 )(J+ 6 i+ 2 5 )
(c) r ( x \ = ______
^ (j+ 1 0 )(j^+ 6 s+ 2 5 )
(c) r ( x \ = ____ f’ + p ^ ______
^ (j+ 1 0 )(j^+ 6 s+ 2 5 )
' ' j^(j + I0 ) ( ŝ + 4 j +85)
(e) U s ) = +
j i ( l+ 2 M l+ 3 )
S tep -by-s tep s o lu tio n
Step 1 of 20
(a). L (s) =
(s+2)
L (J cd) = 
L(Jm) =
s(r+10)(s’+2s-t2) 
(Jm+2)
J(D( Jo&t-l 0) (-(D̂ -t-2J(Brf2)
U J
■ fe)’+ Jo+ l
Break or comer frequenci es: = ->J2 rad/sec
gq>2=2 rad/sec 
is^~10 rad/sec
Step 2 of 20
Magnitude plot:
(a) The constant term '0.1' causes an increase in magnitude of 20 log 0.1— -20 dB.
(b) The initial low frequency slope due to pole at tiie origin is -20dB/decade, 
andttiis slope intersects the OdB line at 0) — Irad/sec
(c) A t o = J2 rad/sec, the slope changes from -20dB /decade to -60dB /decade
due to presence of
■teJ
+Ja5rl-1 in die denominator.
Since 2^gd̂ = 2 and cd̂ = -^ ,
I.-. ; = 0.707]
(d) A t ixi= 2 rad/sec, the slope changesfrom -60dB/decadeto -40dB /decade
(Jot \
due to presence of I “̂ ■*'11 erator.
(e) At iX)= 10 rad/sec, the slope changes from -40dB /decade to -60dB /decade
(J ^
due to the presence of I — +11 in the denominator.
Step 3 of 20
Phase plot:
0(rad/sec)
0.1 -93.42°
1 -132.58°
2 7.13°
5 -24.86°
10 44.77°
100 -84.29°
Step 4 of 20 ^
BooeoKQion
W - L ( 0 =
(s+2)
Step 5 of 20
L(Jco) = 
L(Jcj) =
s^(s+10)(s^+6s+25) 
(Jarf2)
-3=10 rad/sec
Step 6 of 20
Magnitude plot:
(a) The constant term *0.008' causes an increase in m s ^ tu d e of 20 log 0.008= -42 dB.
(b) The initial low frequency slope due to the presence o f two poles at the origin is -40 
dB/ decade. And this asyrnptote intersects the OdB line at ® = Irad/sec
(c) At CD= 2 rad/sec, the slope changes from -40dB /decade to -20dB /decade
{Ja, \
due to presence of I — +11 in the numerator.
At Q0= 5 rad/sec, the slope changes from -20dB /decade to -60dB /decade
{due to presence of "I y j +0.24JCO+1 in the denominator.
Since 2 1 ^ = 6 and cc^=5 ,
M = 0 . 6 |
(e) At CD= 10 rad/sec, the slope changes from -60dB /decade to -80dB /decade 
due to the presence of I — +11 in the denominator.
Phase plot:
Step 7 of 20
0(rad/sec)
0.1 -179°
1 -173.15°
5 -228.36°
10 -107.65°
100 -172°
Step 8 of 20
Booe otagnm
(c).
step 9 of 20
L (0 = 
L(Jm) =
(s+2)^
s ̂(s+10) (s^+6s+25) 
(-fl
l i o J [ J
Step 14 of 20 ^
Magnitude plot:
(a) The constant term *0.16'causes an increase in magnitude of 20 log 0.16=-16 dB.
(b) The initial low frequency slope due to tiie presence of two poles at the origin is 
-40dB/decade,and this ass3nnptote intersects the OdB line at o = Irad/sec
(c) At CD= 2 rad/sec, the slope changes from -40dB /decade to -20dB /decade
/■j® ^
due to presence of I — +11 in the numerator.
At 0 = 8.24 rad/sec, the slope changes from -20dB /decade to +20dB /decade
due to presence of -f—TU . 24J
+0.06J1S+1 in the numerator.
Since 24o^ = 4 and 13^=8.24 , 
I.-. 4 = 0.2431
(e) At 0D= 9.22 rad/sec, the slope changes from +20dB /decade to -20dB /decade
due to presence of 05JO+1 in dte denominator.
Since 2413^=4 and o^=9.22 ,
I.-. 4 = 0.2171
(e) At 0D= 10 rad/sec, the slope changes from -20dB /decade to -40dB /decade 
{Ja
due to tile presence of I — +11 in the denominator.
Phase plot:
Step 15 of 20
0(rad/sec)
0.1 -177.64°
1 -158.46°
8.24 -111.06°
9.22 -301.55°
100 -175.44°
Step 16 of 20
BodedKQKm
Step 17 of 20
w. L (s ) =
'■ ' s^s+2)(s+3)
L (J cd)
_ (-a)*+2JGi+2)
~ -m’ (J(ii+3)(Jai+2)
L(Jm) = ■
r ̂V 1
0.33
comer frequencies:
02=2 rad/sec 
03=3 rad/sec
Step 18 of 20
Magnitude plot:
(a) The constant term *0.33'causes an increase in magnitude of 20 log 0.33=-9.63 dB.
(b) The initial low frequency slope due to tiie prasence of two poles at the origin is 
•40dB/decade, and tiii s slope intersects tiie OdB line at 0 = Irad/sec
(c) At 0D= -72 rad/sec, the slope changes from -40dB /decade to OdB /decade
due to presence of ■ te)' +J©+1 in the numerator.
Since 24co ̂ = 2 and 12 -150.5°
5 -24.86°
10 44.77°
100 -84.29°
Step 20 of 20