1. The bode plot of the system is:
The zeros are located at:
f=0, 500 Hz
The poles are located at:
f=3.4, 50, 2120 Hz
2. Converting Hz to rad/s,
Zeros:
ωz1=0, ωz2=2π500=1000π
Poles:
ωp1=2π3.4=21.36, ωp2=2π50=100π, ωp3=2π2120=4240π
3. The form of the transfer function is:
Hs=Kss+1000π(s+21.36)(s+100π)(s+4240π)
Hs=j2.136=1=Kj2.136j2.136+1000π(j2.136+21.36)(j2.136+100π)(j2.136+4240π)
1K=7.47×10-5→K=13387
The transfer function becomes:
Hs=13387ss+1000πs+21.36s+100πs+4240π
=13387ss1000π+11000πs21.36+1s100π+1s4240π+121.36100π4240π
H(s)=0.4705ss1000π+1s21.36+1s100π+1s4240π+1
4. The frequency response of the transfer function is generated in MATLAB using the following code:
clc
s = tf('s');
wz = [1000*pi];
wp = [21.36 100*pi 4240*pi];
K = 0.4705;
Tf = (K*s*(s/wz+1))/((s/wp(1)+1)*(s/wp(2)+1)*(s/wp(3)+1));
bode(Tf)
The output graph is shown in the following figure:
The magnitude plot is similar to the given Bode plot. The expected locations of the cut-off frequencies are achieved (at 3.4 Hz and 50 Hz). The latter zero and pole expected at frequencies 500 Hz and 2120 Hz are also observed in the Bode plot.
5. The band-pass circuit with the following circuit schematic is used:
This is essentially a combination of a low-pass and a high-pass filter. The cut-off frequencies are located at f = 3.4 Hz and f = 50 Hz. The center frequency is estimated at:
f0=3.4×50=13Hz
6. The DC gain is set to k = 105. For a Q = 0.0354, the resistors can be computed as:
0.0354=12R2R1→R2R1=0.005
k=105=R20.005R1
Let R2=1MΩ. Hence, R1=1M0.005×105=2000Ω.
Let the capacitor values equal each other, C1=C2=C. The capacitor values can be computed as:
f0=12πCR1R2→13=12πC2k×1M→C=273.75 nF
7. The capacitance is adjusted in order to achieve a more equivalent response. The schematic built in MULTISIM is:
The Bode plot resulting from AC sweep of the system is:
The transitions of this frequency response are a little steeper than the previous ones. The bandwidth of the filter safely covers the 3.4 Hz and 50 Hz cut-off frequencies. Since this filter contains only 2 zeros and 2 poles, the zero and pole at the 500 Hz and 2120 Hz frequency points, respectively, cannot be included in the system.
