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Problem 6.04PP
Real poles and zeros. Sketch the asymptotes of the Bode plot magnitude and phase for each of 
the listed open-loop transfer functions. After completing the hand sketches, verify your result 
using Matlab. Turn in your hand sketches and the Matiab results on the same scales.
(a)
= ii i+ T T § T W + T D 5
" - KJ-l-l)(J+S(a+10)
(d) L (s ) = _tJ+2 )(a+ 4 )
Step-by-step solution
step 1 of 25
L ( a ) - a(j + lXa-l-SXa + 10)
t ( / 0.02
y ) in dB is,
k>g|£.(yV»)| = 2 0 Io g |0 .0 2 |-2 0 Io g |/ o g [ f ( ^ ]
step 3 of 25
Follow the steps to draw the magnitude plot.
(i) The constant term 0.02 causes an increase in magnitude of -33.97 dB.
(N) The Initial low frequency slope due to pole at the ohgin is -20dB/decade -
(ill) At at=1 lad/sec ■ ^Idpe changes from .20 dPfdfradf to -40 dB/decade due to the
presence of (ya>+l) in the denominator.
(iv) At a r= S rad/sec >tde slope changes from -40 dB/decade t o -60 dB/decade due to the 
presence of (y'ffl+5) in the denominator.
(v) At dt=10rad/sec.dteslopechangesfrom -60 dB/decade t o -80 dB/decade due to the
presence of (yat+lO) in the denominator.
Step 4 of 25
Consider the phase values.
oXrad/sec) *
0.1 -97.428“
1 -152“
5 -240.26°
10 -282.T
100 -350.8°
IK -359.13
Step 5 of 25
Draw the magnitude and phase plots as shown in figure 1.
Bodeouopom
Step 6 of 25
Execute the following MATtAB code; 
num=1;
den=[1 16 65 50 0];
sys=tf(num,den);
bode(sys)
Step 7 of 25
Obtain the magnitude and phase plots as shown In figure 2.
Step 8 of 25
(b )
i ( s ) = i(j+ 1X4 + 5X4 + 10)
L(jd,) - 
i(ya»)=
O'ar+2)
yaX/aH-l)0'a>+S)(/ar+10)
0 . 0 4 ^ ; ^ + l j
‘̂̂ '^'>(f+')(f+')
Break o r com er fiequencies: cs, = I rad/sec
a>2-2 rad/sec 
ai,=S rad/sec 
a i,= 10 rad/sec
step 9 of 25 ^
Follow the steps to draw the magnitude plot.
(i) The constant term 0.04 causes an increase in magnitude of -28 dB.
(ii) The Initial low frequency slope due to pole at the ohgin is -20dB/dccade -
(Hi) At u>=1 rad/sec > Ibe slope changes from -20 dB/decade to -40 dB/decade due to the 
presence of ( y ® + i) in the denominator.
(iv) At ^ = 2 rad/sec >lbe slope changes from -40 dB/decade t o -20 dB/decade due to the 
presence of (y Ibe slope changes from -20 dB/decade t o -40 dB/decade due to the
presence of (ya>+5) in the denominator.
(vi) At ^ = 1 0 rad/sec >lbe slope changes from -40 dB/decade t o -60 dB/decade due to the 
presence of (yVu+lO) in the denominator.
Step 10 of 25
Consider the phase values.
oXrad/sec) *
0.1 -94.56°
1 -125.44°
5 -172°
10 -204°
100 -261.94°
IK -269.24
Step 11 of 25
Draw the magnitude and phase plots as shown in figure 3.
Bodeefoonn
Step 12 of 25
Execute the following MATtAB code;
num=[1 2];
den=[1 16 65 50 0];
sys=tf(num,den);
bode(sys)
Step 13 of 25 A
Obtain the magnitude and phase plots as shown In figure 4.
Step 14 of 25
(c )
L U ) _____
' ’ 4(4 + 1X4 + 5X4 + 10)
L(ja>) - (/ar+2X/flH6)
yaX/oH-1 )(/ru+5)(/ru+10)
Break o r com er fiequencies: to, * 1 rad/sec
a>,=2 rad/sec 
 Ibe slope changes from -20 dB/decade to -40 dB/decade due
presence of (yV»+l) in the denominator.
(iv) At at=2 rad/sec, the slope changes from .40 dB/dfcadf to -20 dB/decade due
presence of (y ru+ 2 ) in the numerator.
(v) At