# Chapter 6: Active Filters

### Example No. 6.1, Page NO: 269

In [17]:
%matplotlib inline

Populating the interactive namespace from numpy and matplotlib

WARNING: pylab import has clobbered these variables: ['f']
%pylab --no-import-all prevents importing * from pylab and numpy

In [18]:
#Variable Declaration:
import math
fH=10.0*10**3                       #Frequency in hertz
f=12.0*10**3                        #Frequency in hertz

#Calculations:
t=(f**2/fH**2)
Hif=1/(math.sqrt(1+t))              #Calculating magnitude of voltage tranfer fuction
Hifdb=20*math.log(Hif)/math.log(10) #Calculating magnitude of voltage tranfer fuction

#Result:
print('Delta Vi=%.2f dB'%Hifdb)

Delta Vi=-3.87 dB


### Example No. 6.2, Page No: 270

In [19]:
#Variable Declaration:
import math
fh=2000.0                    #Cut-off frequency in hertz
A=2.0                        #Gain

#Calculations:
R=1/(2*math.pi*fh*C)         #Calculating value of resistance
R=R/1000.0                   #Calculating resistance value

#Results:
print('R= %.3f kohm'%R)
#Rf/Ri=A-1
print('\n Hence Rf=Ri=10kohm')

R= 7.958 kohm

Hence Rf=Ri=10kohm


### Example No.6.3, Page No: 272

In [20]:
#Variable Declaration:
import math
fh=10.0*10**3             #Cut-off frequency in hertz
f=12.0*10**3              #Signal freuency in hertz
R=200.0*10**3             #Resistance value in ohm

#Calaculations:
RC=1/(2*math.pi*fh)       #Calculating value of resistance
C=RC/R                    #Calculating value of capacitance
C3=1.414*C                #Calculating value of capacitance
C4=0.707*C                #Calculating value of capacitance

t=(f**4/fh**4)
Hif=1/(math.sqrt(1+t))                 #Calculating magnitude of voltage tranfer fuction
Hifdb=20*math.log(Hif)/math.log(10)    #Calculating magnitude of voltage tranfer fuction

#Result:
print('Hif= %.2f dB'%Hifdb)

Hif= -4.88 dB


### Example 6.4, Page No-275

In [21]:
#Variable Declaration:
import math
N=2.0                    #Order of filter
fh=2.0*10**3             #upper cut-off frequency in hertz
alpha=1.414              #Value of damping coefficient

#Calculations:
R=1/(2*math.pi*fh*C)     #Calculating resistance value
Rkohm=R/1000.0           #Calculating resistance value
A=3-alpha                #Calculating passband gain
RfbyRi=A-1               #Calculating ratio Rf/Ri

#Results:
print('R= %.1f kohm'%Rkohm)
print('\nRf/Ri= %.3f'%RfbyRi)
print('\nHence, take Rf=5.86 kohm and Ri=10 kohm')

R= 0.8 kohm

Rf/Ri= 0.586

Hence, take Rf=5.86 kohm and Ri=10 kohm


### Example No.6.5, Page No: 276

In [22]:
#Variable declaration:
import math

fh=10.0*10**3           #cut-off frequency in hertz
f=12.0*10**3            #Signal frequency in hertz

#For third order low pass butterworth filter

#Calculations:
t=(f**6/fh**6)
Hif=1/(math.sqrt(1+t))                 #Calculating magnitude of voltage transfer function
Hifdb=20*math.log(Hif)/math.log(10)    #Calculating magnitude of voltage transfer function

#Result:
print('Hif= %.4f dB'%Hifdb)

#For fourth order low pass butterworth filter

#Calculation:
t=(f**8/fh**8)
Hif=1/(math.sqrt(1+t))
Hifdb=20*math.log(Hif)/math.log(10)

#Result:
print('\nHif= %.2f dB'%Hifdb)

Hif= -6.0054 dB

Hif= -7.24 dB


### Example No. 6.6, Page No: 276

In [23]:
#Variable Declaration:
import math
N=2.0                    #Order of filter
fh=2000.0                #cut-off frequency in hertz
alpha1=0.765             #damping coefficient
alpha2=1.848             #damping coefficient

#Calculations:
R=1/(2*math.pi*fh*C)     #Calculating resistance value
Rkohm=R/1000.0           #Calculating resistance value

A1=3-alpha1              #Calculating passband gain
A2=3-alpha2              #Calculating passband gain

Rf1byRi1=A1-1            #Calculating Rf1/Ri1 ratio
Rf2byRi2=A2-1            #Calculating Rf2/Ri2 ratio

#Results:
print('R= %.1f kohm'%Rkohm)
print('\nRf1/Ri1= %.3f'%Rf1byRi1)
print('\nHence, take Rf1=12.35 kohm and Ri1=10 kohm')
print('\nRf2/Ri2= %.3f'%Rf2byRi2)
print('\nHence, take Rf2=15.2 kohm and Ri2=100 kohm')

R= 0.8 kohm

Rf1/Ri1= 1.235

Hence, take Rf1=12.35 kohm and Ri1=10 kohm

Rf2/Ri2= 0.152

Hence, take Rf2=15.2 kohm and Ri2=100 kohm


### Example No. 6.7, Page No: 279

In [24]:
#Variable Declaration:
import math
A=2.0                     #Gain
fL=2.0*10**3              #Frequency in hertz

#Calculations:
R=1/(2*math.pi*fL*C)      #Calculating value of resistance
Rkohm=R/1000.0            #Calculating value of resistance
RfbyRi=A-1                #Calculating ratio Rf/Ri

#Results:
print('R= %.1f kohm'%Rkohm)
print('\nRf/Ri= %.3f'%RfbyRi)
print('\nHence, take Rf=10 kohm and Ri=10 kohm')

R= 8.0 kohm

Rf/Ri= 1.000

Hence, take Rf=10 kohm and Ri=10 kohm


### Example No: 6.8, Page No: 282

In [25]:
#Variable Declaration:
import math

R2=16.0*10**3           #Resistance in ohm
R3=16.0*10**3           #Resistance in ohm
Rf=15.8*10**3           #Resistance in ohm
Ri=27.0*10**3           #Resistance in ohm

#Calculations:
fL=1/(2*math.pi*math.sqrt(R2*R3*C2*C3))  #Calculating lower cut-off frequency
fL=fL/1000.0                             #Calculating lower cut-off frequency
A=1+Rf/Ri                                #Calculating passband gain

#Results:
print('\nfL= %.1f kHz'%fL)
print('\nA= %.3f'%A)

a=[-35.99,-23.96,-3.12,1.00,3.95,4.00,4.01,4.01,4.01]
b=[log10(100),log10(200),log10(700),log10(1000),log10(3000),log10(7000),log10(10000),log10(30000),log10(100000)]
plot(b,a,'b')
plot(log10(100),-35.99,'o')
plot(log10(200),-23.96,'o')
plot(log10(700),-3.12,'o')
plot(log10(1000),1.00,'o')
plot(log10(3000),3.95,'o')
plot(log10(7000),4.00,'o')
plot(log10(10000),4.01,'o')
plot(log10(30000),4.01,'o')
plot(log10(100000),4.01,'o')
m=arange(-40,1,0.0005)
x5=(3*m)/m
plot(x5,m,'--')
k = arange(2.0001,3.5, 0.0005)
plot(k,4.006*k/k,'--')
text(1.6,4.006,'+4.006')
text(3.3,-38,'Passband')
text(2.3,-38,'Stopband')
text(3,2,'3dB')
title('Frequency response of ssecond order high-pass filter')
ylabel('Voltage gain in dB')
xlabel('frequency index n where n=log(f)')

fL= 1.0 kHz

A= 1.585

Out[25]:
<matplotlib.text.Text at 0x8410ba8>

### Example No. 6.9, Page No: 284

In [26]:
#Variable Declaration:
import math
fh=50000.0             #frequency in hertz

#Calculations:
R=1/(2.0*math.pi*fh*C)      #Calculating value of resistance
Rkohm=R/1000.0              #Calculating value of resistance

R1=R/1.082                  #Calculating value of resistance
R2=R/0.9241                 #Calculating value of resistance
R3=R/2.613                  #Calculating value of resistance
R4=R/0.3825                 #Calculating value of resistance
Hif=0.02
s=(Hif**2)/(1-Hif**2)
s1=s**0.125
f=fh/1.6815 * s1            #Calculating value of frequency
fkhz=f/1000.0               #Calculating value of frequency

#Results:
print('R= %.3f kohm'%Rkohm)
print('\nR1= %.3f kohm'% (R1/1000))
print('\nR2= %.3f kohm'% (R2/1000))
print('\nR3= %.3f kohm'% (R3/1000))
print('\nR4= %.3f kohm'% (R4/1000))
print('\nf= %.2f kHz'% fkhz)

R= 3.183 kohm

R1= 2.942 kohm

R2= 3.445 kohm

R3= 1.218 kohm

R4= 8.322 kohm

f= 11.18 kHz


### Example No. 6.10, Page No: 286

In [27]:
#Variable Declaration:
import math
fh=2500.0                   #Frequency in hertz
fL=250.0                    #Frequency in hertz

#Calculations:
B=fh-fL                     #Calculating bandwidth
fr=math.sqrt(fh*fL)         #Calculating frequency
fc=(fL+fh)/2                #Calculating frequency

#Results:
print('Bandwdth B= %d Hz'%B)
print('\nResonant Frequency fr= %.2f Hz'%fr)
print('\nCenter Frequency fr= %d Hz'%fc)
print('\nHence, resonant frequency is always less than center frequency')

Bandwdth B= 2250 Hz

Resonant Frequency fr= 790.57 Hz

Center Frequency fr= 1375 Hz

Hence, resonant frequency is always less than center frequency


### Example No. 6.11, Page No: 286

In [28]:
#Variable Declaration:
import math
#Part A
fr=1000.0           #Frequency in hertz
B=3000.0            #Bandwidth in hertz

#Calculations:
Q=fr/B              #Calculation Quality factor

#Part B
fL= math.sqrt((B*B/4)+fr**2) - B/2 #Calculating lower cut-off frequency

#Part C
fh=fL+B                            #Calculating high cut-off frequency

#Results:
print('Quality factor Q= %.2f '%Q)
print('Since Q<0.5, this is a wideband filter')
print('\nfL= %.2f Hz'%fL)
print('\nfh= %.2f Hz'%fh)

Quality factor Q= 0.33
Since Q<0.5, this is a wideband filter

fL= 302.78 Hz

fh= 3302.78 Hz


### Example No: 6.12, Page No: 288

In [29]:
#Variable Declaration:
import math
# Part A
#For a bandpass filter
R=20000.0                          #Resistance in ohm
Rr=2700.0                          #Resistance in ohm

#Calculations:
fr=0.1125*(math.sqrt(1+R/Rr))/(R*C)   #Calculating resonant frequency

#Part B
B=0.1591/(R*C)                        #Calculating Bandwidth

#Results:
print('Resonant frequency= %.1f Hz'%fr)
print('\nBandwidth= %.1f Hz'%B)

Resonant frequency= 1631.0 Hz

Bandwidth= 795.5 Hz


### Example NO. 6.13, Page No: 289

In [30]:
#Variable Declaration:
fr=200.0                     #Resonant frequency in hertz
B=20.0                       #Bandwidth in hertz

#Calculations:
Q=fr/B                       #Calculating quality factor

R=0.1591/(B*C)               #Calculating resistance value
Rr=R/(2.0*Q*Q-1)             #Calculating resistance value
R=R/1000.0                   #Calculating resistance value

#Results:
print('\nR= %.1f kohm'%R)
print('\nRr= %.1f ohm'% Rr)

R= 24.1 kohm

Rr= 121.1 ohm


### Example No. 6.14, Page NO: 307

In [31]:
#Variable Declaration:
R=1.0*10**6                   #Resistance in ohm
C=40.0*10**-12                #Capacitance value

#Calculations:
fck=1/(R*C)                   #Calculating clock frequency
fck=fck/1000                  #Calculating clock frequency

#Result:
print('Fck= %.1f kHz'%fck)

Fck= 25.0 kHz