Chapter-7 Active Filters

Example 7.1 - Page 204

In [15]:
from numpy import pi
from __future__ import division
# Given data
f_H= 2*10**3 #cut-off frequency in Hz
C= 0.01*10**-6 # in F
passband_gain= 2.5 
R= 1/(2*pi*f_H*C) # in Ω
R= 8.2 # in kΩ (standard value)
# 1+Rf/R1= passband_gain or Rf should be equal to 1.5*R1 since Rf||R1= R
R1= passband_gain/1.5*R # in kΩ
print "The value of R1 = %0.2f kΩ (Standard value 15 kΩ)" %R1 
Rf= int(1.5*R1) # in kΩ
print "The value of Rf = %0.f kΩ" % Rf

The value of R1 = 13.67 kΩ (Standard value 15 kΩ)
The value of Rf = 20 kΩ

Example 7.2 - Page 207

In [3]:
# Given data
f_H= 2*10**3 #cut-off frequency in Hz
C= 0.033*10**-6 # in F
R= 1/(2*pi*f_H*C) # in Ω
# 2*R= Rf*R1/(Rf+R1)= 0.586*R1**2/(1.586*R1) since Rf= 0.586*R1
R1= 2*R*1.586/0.586 # in Ω
R1= round(R1*10**-3) # in kΩ
print "The value of R1 = %0.f kΩ (Standard value 15 kohm)" %R1
R1= 15 # in kΩ
Rf= R1*0.586 # in kΩ
#Rf= floor(1.5*R1) # in kΩ
print "The value of Rf = %0.2f kΩ" %Rf
print "(The value of Rf may be taken as a pot of 10 kΩ)" 

The value of R1 = 13 kΩ (Standard value 15 kohm)
The value of Rf = 8.79 kΩ
(The value of Rf may be taken as a pot of 10 kΩ)

Example 7.3 - Page 207

In [6]:
# Given data
f_H= 1*10**3 #cut-off frequency in Hz
C= 0.0047*10**-6 # in F
R= 1/(2*pi*f_H*C) # in Ω
R= (R*10**-3) # in kΩ
R1= 30 # in kΩ (assume)
Rf= 0.586*R1 # in kΩ
C= C*10**6 # in µF
print "The value of R'= R = %0.2f kΩ (standard value 33 kΩ)" %R
print "The value of C'= C = %0.4f µF " %C
print "The value of R1= %0.2f kΩ " %R1 
print "The value of Rf= %0.2f kΩ (standard value 20 kΩ pot)" %Rf 

The value of R'= R = 33.86 kΩ (standard value 33 kΩ)
The value of C'= C = 0.0047 µF 
The value of R1= 30.00 kΩ 
The value of Rf= 17.58 kΩ (standard value 20 kΩ pot)

Example 7.4 - Page 212

In [17]:
import math
# Given data
fc= 1*10**3 # in Hz
alpha= 1.414 
C= 0.1*10**-6 # in F (assume)
C_desh= C*alpha**2/4 # in F
C_desh= C_desh*10**6 # in µF
print "The value of C' = %0.2f µF" %C_desh
C_desh= C_desh*10**-6 # in F
R_desh= 1/(2*pi*fc*math.sqrt(C*C_desh)) # in Ω
R_desh= R_desh*10**-3 # in kΩ
print "The value of R' = %0.2f kΩ (standard value 2.2 kΩ)"%R_desh

The value of C' = 0.05 µF
The value of R' = 2.25 kΩ (standard value 2.2 kΩ)

Example 7.5 - Page 213

In [8]:
# Given data
alpha= 1.414 # passband
C= 0.01*10**-6 # in F (assume)
fc= 1*10**3 # in Hz
dc_gain= 6 
R= 1/(2*pi*C*fc) # in Ω
R= R*10**-3 # in kΩ
print "The value of R = %0.1f kΩ (standard value 15 kΩ)" %R
R= 15 # in kΩ
Af= 3-alpha # and Af= 1+Rf/R1 or
# Rf= (Af-1)*R1    (i)
# 2*R= Rf || R1, hence from (i)
R1= 2*R*Af/(Af-1) # in kΩ
print "The value of R1 = %0.1f kΩ (standard value 82 kΩ)" %R1
R1= 82 # in kΩ
Rf= (Af-1)*R1 # in kΩ
print "The value of Rf = %0.f kΩ (standard value 47 kΩ)" %Rf
Aamp= dc_gain/Af 
print "The value of Aamp = %0.2f" %Aamp

The value of R = 15.9 kΩ (standard value 15 kΩ)
The value of R1 = 81.2 kΩ (standard value 82 kΩ)
The value of Rf = 48 kΩ (standard value 47 kΩ)
The value of Aamp = 3.78

Example 7.7 - Page 221

In [9]:
# Given data
R= 2.1*10**3 # in kΩ
C= 0.05*10**-6 # in F
R1= 20*10**3 # in Ω
Rf= 60*10**3 # in Ω
# Low cut-off frequency,
f_L= 1/(2*pi*R*C) # in Hz
f_L= f_L*10**-3 # in kHz
print "The cut-off frequency = %0.3f kHz" %f_L

The cut-off frequency = 1.516 kHz

Example 7.8 - Page 225

In [14]:
import math
from __future__ import division
# Given data
R1= 2*10**3 # in Ω
R2= 2/3*10**3 # in Ω
R3= 200*10**3 # in Ω
C= 0.1*10**-6 # in F
Af= R3/(2*R1) # gain
print "The value of Af = %0.f" %Af
# R1= Q/(2*pi*f_C*C*Af)           (i)
# R2= Q/(2*pi*f_C*C*(2*Q**2-Af))  (ii)
# R3= Q/(pi*f_C*C)                (iii)
Q= math.sqrt((R3/(2*R2)+Af)/2) # from (ii) and (iii)
print "The value of Q = %0.f" %Q
f_C= Q/(R3*pi*C) # in Hz  (from (iii))
print "The value of f_C = %0.f Hz" %f_C
omega_0= 2*pi*f_C # in radians/second
print "The value of omega_0 = %0.f radians/seconds" %omega_0

The value of Af = 50
The value of Q = 10
The value of f_C = 159 Hz
The value of omega_0 = 1000 radians/seconds

Example 7.10 - Page 227

In [22]:
# Given data
f_L= 2*10**3 # in Hz
f_H= 2.5*10**3 # in Hz
Af= -5 
f_C= math.sqrt(f_L*f_H) # centre frequency in Hz
del_f= f_H-f_L #bandwidth in Hz
Q= f_C/del_f # selectivity
# Assume C1= C2= C= 0.01µF
C= 0.01*10**-6 # in F
R3= 1/(pi*del_f*C) # in Ω
R3= R3*10**-3 # in kΩ
print "The value of R3 = %0.2f kΩ (standard value 64 kΩ)" %R3
R3= 64 # in kΩ
R3= R3*10**3 # in Ω
R1= -R3/(2*Af) #in Ω
R2= R1/(4*pi**2*f_C**2*R1*R3*C**2-.1)
R1= R1*10**-3 # in kΩ
C=C*10**6 # in µF
print "The value of R1 = %0.1f kΩ " %R1
print "The value of R2 = %0.f kΩ (standard value 800Ω)" %R2
print "The value of C = %0.2f µF " %C

The value of R3 = 63.66 kΩ (standard value 64 kΩ)
The value of R1 = 6.4 kΩ 
The value of R2 = 801 kΩ (standard value 800Ω)
The value of C = 0.01 µF

Example 7.11 - Page 228

In [23]:
# Given data
f_C= 1*10**3 #centre frequency in Hz
f_C_desh= 2*10**3 #new centre frequency in Hz
Q= 5 # selectivity
Af= -8 
C= 0.01*10**-6 # in F (assume)
R3= Q/(pi*f_C*C) #in Ω
R3= R3*10**-3 # in kΩ
print "The value of R3 = %0.1f kΩ (160 kΩ (approx))" %R3
R1= round(-R3/(2*Af)) # in kΩ
print "The value of R1 = %0.f kΩ" %R1
R2= R1*10**3/(4*pi**2*f_C**2*R1*10**3*R3*10**3*C**2-1) # in Ω
R2= R2*10**-3 # in kΩ
print "The value of R2 = %0.2f kΩ (2 kΩ (approx))" %R2
R2= 2 # in kΩ (approx)
R2_desh= R2*(f_C/f_C_desh)**2 # in kΩ
R2_desh= R2_desh*10**3 # in Ω
print "The value of R2' = %0.f Ω" %R2_desh

The value of R3 = 159.2 kΩ (160 kΩ (approx))
The value of R1 = 10 kΩ
The value of R2 = 1.89 kΩ (2 kΩ (approx))
The value of R2' = 500 Ω

Example 7.12 - Page 229

In [24]:
# Given data
R= 10*10**3 # in Ω
C1= 0.1*10**-6 # in F 
C2= 0.0025*10**-6 # in F 
f_H= 1/(2*pi*R*C2) #higher cut-off frequency in Hz
f_H= f_H*10**-3 # in kHz
f_L= 1/(2*pi*R*C1) #lower cut-off frequency in Hz
BW= f_H-f_L*10**-3 # bandwidth in kHz
print "The higher cut-off frequency = %0.3f kHz" %f_H
print "The lower cut-off frequency = %0.2f Hz" %f_L
print "The bandwidth = %0.1f kHz" %BW

The higher cut-off frequency = 6.366 kHz
The lower cut-off frequency = 159.15 Hz
The bandwidth = 6.2 kHz

Example 7.13 - Page 230

In [25]:
# Given data
f_L= 200 # in Hz
f_H= 1*10**3 # in Hz
alpha=4 # passband gain
C_desh= 0.01*10**-6 # in F (assume)
R_desh= 1/(2*pi*f_H*C_desh) # in Ω
R_desh= R_desh*10**-3 # in kΩ
print "The value of R' = %0.1f kΩ (Standard value 20 kΩ)" %R_desh
R_desh= 20 # in kΩ (standard value)
# First Order High-Pass Filter
C= 0.05*10**-6 # in F (assume)
R= 1/(2*pi*f_L*C) # in Ω
R= R*10**-3 # in kΩ
R1= 10 # in kΩ
Rf= R1 # in kΩ
C_desh= C_desh*10**6 # in µF
C= C*10**6 # in µF
print "The value of R = %0.1f kΩ (Standard value 20 kΩ)" %R
R= 20 # in kΩ (standard value)
print "The value of R1 and Rf = %0.f kΩ" %R1
print "The value of C' = %0.2f µF" %C_desh
print "The value of C = %0.2f µF" %C

The value of R' = 15.9 kΩ (Standard value 20 kΩ)
The value of R = 15.9 kΩ (Standard value 20 kΩ)
The value of R1 and Rf = 10 kΩ
The value of C' = 0.01 µF
The value of C = 0.05 µF

Example 7.14 - Page 231

In [28]:
# Given data
f_H= 200 # in Hz
f_L= 2*10**3 # in Hz
C= 0.05*10**-6 # in F
# For low-pass filter,
R_desh= 1/(2*pi*f_H*C) # in Ω
R_desh= R_desh*10**-3 # in kΩ
print "The value of R' = %0.1f kΩ (standard value 20 kΩ)" %R_desh
# For high-pass filter,
R= 1/(2*pi*f_L*C) # in Ω
R= R*10**-3 # in kΩ
print "The value of R = %0.2f kΩ (standard value 2 kΩ)" %R

The value of R' = 15.9 kΩ (standard value 20 kΩ)
The value of R = 1.59 kΩ (standard value 2 kΩ)

Example 7.15 - Page 232

In [2]:
from numpy import pi
# Given data
C= 0.068*10**-6 # in F
f_N= 50 # in Hz
R= 1/(2*pi*f_N*C) # in Ω
R= R*10**-3 # in kΩ
print "The value of R = %0.1f kΩ (standard value 50 kΩ)" %R

The value of R = 46.8 kΩ (standard value 50 kΩ)

Example 7.16 - Page 233

In [45]:
import math
# Given data
R= 15*10**3 # in Ω
C= 0.01*10**-6 # in F
f= 2*10**3 # in Hz
PhaseShift= -2*(math.atan(2*pi*f*R*C))*180/pi # in °
print "The phase shift = %0.f°" %PhaseShift
print "i.e. %0.f° (lagging)" %abs(PhaseShift)

The phase shift = -124°
i.e. 124° (lagging)

Example 7.18 - Page 240

In [47]:
# Given data
R_A= 2.2*10**3 # in Ω
R_B= 1.2*10**3 # in Ω
Rf= 4.7*10**3 # in Ω
C= 0.01*10**-6 # in F
k_lp= 1.238 
k_hp= 1/k_lp 
# Part (i)
alpha= 3*R_B/(R_A+R_B) 
print "Part (i) : The value of alpha = %0.3f" %alpha 
print "Given filter is 1db peak Chebyshev" 

# Part (ii)
f_0= 1/(2*pi*Rf*C) #critical frequency in Hz
f_0= f_0*10**-3 # in kHz
f_low_pass= f_0*k_lp # in kHz
print "\nPart (ii) : The low-pass frequency = %0.3f kHz" %f_low_pass
f_high_pass= f_0*k_hp # in kHz
print "The high-pass frequency = %0.3f kHz" %f_high_pass

# Part (iii)
fc= f_0 # bandpass centre frequency in kHz
print "Part (iii) : The bandpass centre frequency = %0.3f kHz" %fc

# Part (iv)
# Formula used : delta_f= fc/Q= fc/(1/alpha)
delta_f= fc/(1/alpha) # in kHz
print "Part (iv) : The bandpass width = %0.3f kHz" %delta_f

# Part (v)
A0= 1/alpha # bandpass gain at centre frequency
print "Part (v) : The bandpass gain at centre frequency = %0.4f" %A0

Part (i) : The value of alpha = 1.059
Given filter is 1db peak Chebyshev

Part (ii) : The low-pass frequency = 4.192 kHz
The high-pass frequency = 2.735 kHz
Part (iii) : The bandpass centre frequency = 3.386 kHz
Part (iv) : The bandpass width = 3.585 kHz
Part (v) : The bandpass gain at centre frequency = 0.9444