Chapter 1: Operational Amplifier Fundamentals

Example 1.1, Page 4

In [1]:
import math

#Variable declaration

R0 = 1.0       #ohm
Ri = 100.0     #kilo ohm
Aoc = 100.0    #volts per volts
Rs=0.0             #kilo ohm
Rl=0.0             #ohm
gain=0.0
input_load=0.0
output_load=0.0

#Calculation

def calculate():                       #returns gain
    global input_load, output_load
    input_load = (Ri/(Rs+Ri))
    output_load = (Rl/(R0+Rl))
    ans=input_load*Aoc*output_load     # in V/V
    return ans

#answer part (a)

Rs=25.0
Rl=3.0
gain=calculate()
print "a)"
print "The overall gain is ",round(gain,1),"V/V"
print "The input load is ",input_load*100,"% of it's unloaded value"
print "The output load is ",output_load*100,"% of it's unloaded value"

#answer part (b)

Rs=50.0
Rl=4.0
gain=calculate()
print "b)"
print "The overall gain is ",round(gain,1),"V/V"
print "The input load is ",round(input_load*100,1),"% of it's unloaded value"
print "The output load is ",round(output_load*100,1),"% of it's unloaded value"

a)
The overall gain is  60.0 V/V
The input load is  80.0 % of it's unloaded value
The output load is  75.0 % of it's unloaded value
b)
The overall gain is  53.3 V/V
The input load is  66.7 % of it's unloaded value
The output load is  80.0 % of it's unloaded value

Example 1.2, Page 9

In [2]:
import math

#Variable declaration

vt = 1.0     # in volt
R1 = 2.0     # in kilo ohm
R2 = 18.0    #in kilo ohm

#Calculation

def calculate(a):               #returns Vo
    global vt,R1,R2
    ans=vt*(1+(R2/R1))/(1+((R2/R1)/a))         #equation 1.11
    return ans

#answer

print "a)Vo = ",round(calculate(10**2),5),"V"
print "b)Vo = ",round(calculate(10**4),5),"V"
print "c)Vo = ",round(calculate(10**6),5),"V"

#textbook contains precision error

a)Vo =  9.17431 V
b)Vo =  9.99101 V
c)Vo =  9.99991 V

Example 1.3, Page 14

In [3]:
import math

#Variable Declaration

R1=10*10**3            #ohm
R2=100*10**3           #ohm

#Calculation

Ri=R1            # Input Resistance 
Ro=0             #Output Resistance 
A=-(R2/R1)       #Ideal Overall Gain

#answer

print "Ri =",round(Ri/1000,2),"kilo ohm"
print "Ro =",round(Ro),"ohm"
print "A =",round(A,2),"V/V"

Ri = 10.0 kilo ohm
Ro = 0.0 ohm
A = -10.0 V/V

Example 1.4, Page 18

In [4]:
#Variable declaration

rf1 = 3     # coefficient of V1
rf2 = 4     # coefficient of V2
rf3 = 2     # coefficient of V3

#Calculations

rf1*=2      # Common factor 2
rf2*=2      # Common factor 2
rf3*=2      # Common factor 2
r1=20       # assumption
rf=r1*rf1
r2=rf/rf2
r3=rf/rf3

#answer

print "R1 = ",r1,"kilo ohm"
print "R2 = ",r2,"kilo ohm"
print "R3 = ",r3,"kilo ohm"
print "Rf = ",rf,"kilo ohm"

R1 =  20 kilo ohm
R2 =  15 kilo ohm
R3 =  30 kilo ohm
Rf =  120 kilo ohm

Example 1.5, Page 18

In [5]:
#Variable declaration

r1,r2,rf          #vo=10*v1+5=-(rf/r1*v1)-rf/r2*(-15)

#Calculation

r1=10
rf=10*r1;       #-rf/r1*v1=10*v1
r2=rf*15/5      #-rf/r2*(-15)=5

#answer

print "R1 = ",r1,"kilo ohm"
print "R2 = ",r2,"kilo ohm"
print "Rf = ",rf,"kilo ohm"

R1 =  10 kilo ohm
R2 =  300 kilo ohm
Rf =  100 kilo ohm

Example 1.6, Page 20

In [6]:
#Variable declaration

ri1=100       # in kilo ohm
ri2=100      # in kilo ohm

#Calculation

r1=ri1;
r2=3*r1;    #r2/r1=3
# r3 + r4 = ri2 and (1+r1/r2)/(1+r3/r4)=1
#Solving the above two
r3=ri2/4;
r4=ri2-r3

#answer

print "R1 = ",r1,"kilo ohm"
print "R2 = ",r2,"kilo ohm"
print "R3 = ",r3,"kilo ohm"
print "R4 = ",r4,"kilo ohm"

#in textbook r3 and r4 values are reversed which doesn't satisfy the equations

R1 =  100 kilo ohm
R2 =  300 kilo ohm
R3 =  25 kilo ohm
R4 =  75 kilo ohm

Example 1.7, Page 25

In [7]:
#Variable Declaration

A=100 
accuracy=0.1

#Calcualtion

T=100/accuracy
beta=1.0/100.0   # A_ideal=i/beta=100
a=(10**3)/beta
beta=(a/100-1)/a    # A=a/(1+(a*beta))

#answer

print "a)T >= ",int(T)
print "b)a >= ",int(a)
print "a)Beta = ",beta

a)T >=  1000
b)a >=  100000
a)Beta =  0.00999

Example 1.8, Page 26

In [8]:
import math

#Variable Declaration

a = 10**5       
beta
T

#Calculation

def calculate():
    global a,beta,T
    T=a*beta
    ans=10.0/(1+T)    # for a +- 10% change in a
    return ans

#answer

beta=10**(-3)                                                    #given
desensitivity_factor=calculate();                                # stores the answer
print "a) A changes by (+-)",round(desensitivity_factor,6),"%"   #part a

beta=1                                                           #given
desensitivity_factor=calculate();
print "b) A changes by (+-)",round(desensitivity_factor,6),"%"   #part b

a) A changes by (+-) 0.09901 %
b) A changes by (+-) 0.0001 %

Example 1.9, Page 33

In [9]:
import math

#Variable Declaration

rd = 2.0    # Mega ohm
ro = 75.0   # ohm
a = 200000.0   # V/V

#Calculation

def calculate(R1,R2):
    global a,ro,rd
    beta=R1/(R1+R2)
    if(R1==float("inf")):      # for infinty
        beta=1
    T=a*beta
    A=(1+(R2/R1))/(1+(1/T))    # equation 1.55
    if(R1==float("inf")):      # for infinity
        A=1/(1+(1/T))
    Ro=ro/(1+T)                # equation 1.61
    Ri=rd*(1+T)                # equation 1.59
    print "  A = ",round(A,6),"V/V"
    print "  Ro = ",round(Ro*(10**3),3),"mili ohm"
    print "  Ri = ", round(Ri,3),"Mega ohm"

#answer

print "a)"
calculate(1.0,999)
print "b)"
calculate(float("inf"),1)

a)
  A =  995.024876 V/V
  Ro =  373.134 mili ohm
  Ri =  402.0 Mega ohm
b)
  A =  0.999995 V/V
  Ro =  0.375 mili ohm
  Ri =  400002.0 Mega ohm

Example 1.10, Page 35

In [10]:
import math

#Variable Declaration

a = 200000.0    # V/V
ro = 75         # ohm

#Calculating function

def calculate(R1,R2):
    global a,ro
    T=a*(R1/(R1+R2)) 
    A=(-1)*(R2/R1)/(1+(1/T))   # equation 1.63
    Rn=R2/(1+a)               # equation 1.67b
    Ri=R1                     # equation 1.68
    Ro=ro/(1+T)
    print " A = ",round(A,5),"V/V"
    print " Rn = ",round(Rn,5),"ohm"
    print " Ri = ",round(Ri,5),"ohm"
    print " Ro = ",round(Ro,5),"ohm"
                    
#answer

print "a)"
calculate(100000.0,100000.0)
print "b)"
calculate(1000.0,1000000.0)

a)
 A =  -0.99999 V/V
 Rn =  0.5 ohm
 Ri =  100000.0 ohm
 Ro =  0.00075 ohm
b)
 A =  -995.01993 V/V
 Rn =  4.99998 ohm
 Ri =  1000.0 ohm
 Ro =  0.37351 ohm

Example 1.11, Page 38

In [11]:
import math

#Variable Declaration

R1 = 1000000.0     # ohm
R2 = 1000000.0     # ohm
R3 = 100000.0      # ohm
R4 = 1000.0        # ohm
RL = 2000.0        # ohm
rd = 1000000.0     #ohm
a = 10**5        # V/V
ro = 100.0         # ohm

#Calculation

A_ideal = (-1)*(R2/R1)*(1+(R3/R2)+(R3/R4))       # ideal op-amp and summing  currents at node  v1
T = a/(1+(R2/R1)+(R2/rd))/(1+(ro/(R2+(R1*rd/(R1+rd))))+(ro/RL))/100  #equation 1.73
A = A_ideal/(1+(1/T))                   
dev=(A_ideal-A)/A_ideal*100

#answer

print "a) A_ideal =",A_ideal,"V/V"
print "b) A =",round(A,2),"V/V"
print "Deviation from ideal =",round(dev,2),"%"

#book example has precision error so answer is 0.32%

a) A_ideal = -101.1 V/V
b) A = -100.78 V/V
Deviation from ideal = 0.31 %

Example 1.12, Page 40

In [12]:
import math

#Variable Declaration

rd = 1000.0          # kilo ohm
a = 10**4            # V/V
ro = 100.0           #ohm
R1 = 10.0            # kilo ohm
R2 = 20.0            # kilo ohm
R3 = 30.0            # kilo ohm
R4 = 300.0           # kilo ohm
RL = 2.0             # kilo ohm

#Calculation

def parallel(a,b):
    ans=a*b/(a+b)
    return ans

Ra = parallel(R1,parallel(R2,parallel(R3,rd)))
Rb=Ra+R4
Rc=parallel(Rb,RL)                      #After suppressing all input sources
Rd=Rc+ro/1000                           #replacing the op-amp with it's terminal resistances
Vn=Rb/Ra                                #and applying a test voltage and analysing the circuit
Vt=Rd/Rc
beta=1/Vn/Vt
T=a*beta
v1=R4/R1
v2=R4/R2
v3=R4/R3
A=1/(1+1/T)

#answer

print "a)"
print "  Beta =",round(beta,6),"V/V"
print "  T =",round(T,1)
print "b)"
print "  Vo= -(",round(A*v1,2),"V1 +",round(A*v2,2),"V2 +",round(A*v3,2),"V3 )"

a)
  Beta = 0.016911 V/V
  T = 169.1
b)
  Vo= -( 29.82 V1 + 14.91 V2 + 9.94 V3 )

Example 1.13, Page 41

In [13]:
import math

#Variable Declaration

rd = 100.0        # kilo ohm
ro = 100.0        # ohm
R1 = 30.0         # kilo ohm
R2 = 20.0         # kilo ohm
R3 = 10.0         # kilo ohm

#Calculation

def parallel(a,b):
    ans=a*b/(a+b)
    return ans

beta_n = (parallel(R1,rd)+R1)/((ro/1000)+R2+parallel(R1,rd)+R3)    # from circuit 1.35 after appyling
beta_p = R3/((ro/1000)+R2+parallel(R1,rd)+R3)                      # voltage divide formula twice
beta=beta_n-beta_p                  #equation 1.76

#answer

print "Beta =",round(beta,4),"V/V"

# beta_n calculation in book is wrong

Beta = 0.8101 V/V

Example 1.14, Page 43

In [14]:
import math

#Variable Declaration 

R1 = 10     #kilo ohm
R2 = 20     #kilo ohm
V1 = 3      # V
Iq = 0.5    # mA
RL = 2      #kilo ohm

#Calculation

V0 = (-1)*R2/R1*V1
It = abs(V0)/RL                     # Currents through R1,R2,Rt are i1,i2,It respectively
i1 = It/R1
i2 = i1                             # applying voltage divider rule
i0 = i2+It
icc = Iq
iee = icc+ i0
Poa = 30*Iq+((V0+15)*i0)            #Whenever current passes through voltage drop, power = vi

#answer

print "a)"
print "  Icc =",icc,"mA"
print "  Iee =",iee,"mA"
print "  I0 =",i0,"mA"
print "b)"
print "  Power Poa =",Poa,"mW"

a)
  Icc = 0.5 mA
  Iee = 3.5 mA
  I0 = 3 mA
b)
  Power Poa = 42.0 mW

Example 1.15, Page 43

In [15]:
#Variable Declaration

ro = 75.0      #kilo ohm
T = 200000.0
Vs = 10.0      # V
Rl = 1.0       #kilo ohm

#Calculation

iL = Vs/Rl
Ro = ro/(1+T)
del_v = Ro*10*(10**(-3))

#answer

print "b)"
print "  Change in v =",round(del_v*(10**6),2),"micro Volt -> quite a small change"

b)
  Change in v = 3.75 micro Volt -> quite a small change

Example 1.16, Page 46

In [16]:
%matplotlib inline

import matplotlib.pyplot as plt
import scipy as np
import math

#Variable Declaration

A = -2               #V/V
peak = 10           # V

#Calculation 

output = np.absolute(A) * peak

#answer

print "The op-amp saturates at Vo=+-13 V"
print "With Vn= 20/3-13/3 =",round((20.0/3)-(13.0/3),4),"V"

#Graphs

t1 = np.arange(0,1,.0005)               #  Triangular waveform
t2 = np.arange(1,3,.0005)
t3 = np.arange(3,5,.0005)

m1 = np.arange(0,0.65,.0005)
m2 = np.arange(.65,1.35,.0005)
m3 = np.arange(1.35,2.65,.0005)            # Output Vo wave
m4 = np.arange(2.65,3.35,.0005)
m5 = np.arange(3.35,4.65,.0005)
m6 = np.arange(4.65,5,.0005)                # Output Vn wave
m7 = np.arange(0.65,1,.0005)
m8 = np.arange(1,1.35,.0005)
m9 = np.arange(2.65,3,.0005)
m10 = np.arange(3, 3.35, .0005)

plt.subplot(2,1,1)

plt.suptitle("Vt (Blue), Vo (Red) and Vn (Green) Graphs")
plt.xlim(0,4.5)
plt.xlabel("time->")
plt.ylabel("V->")
plt.plot(t1,peak*t1,"b",)
plt.plot(t2,(-1)*peak*t2+2*(peak),"b",)
plt.plot(t3,peak*t3-4*(peak),"b",)
plt.grid(True)

plt.subplot(2,1,2)

plt.xlim(0,4.5)
plt.xlabel("time->")
plt.ylabel("V->")
plt.plot(m1,-20*m1,"r")
plt.plot(m2,np.full(len(m2),-13),"r")
plt.plot(m3,20*m3-40,"r")
plt.plot(m4,np.full(len(m4),13),"r")
plt.plot(m5,-20*m5+80,"r")
plt.plot(m6,np.full(len(m6),-13),"r")

plt.plot(m1,np.full(len(m1),0),"g",)
plt.plot(m7,6.665*m7-4.4,"g")
plt.plot(m8,-6.665*m8+8.8,"g")
plt.plot(m3,np.full(len(m3),0),"g")
plt.plot(m9,6.665*m9-17.6,"g")
plt.plot(m10,-6.665*m10+22.4,"g")
plt.plot(m5,np.full(len(m5),0),"g")
plt.plot(m6,np.full(len(m6),0),"g")
plt.grid(True)

The op-amp saturates at Vo=+-13 V
With Vn= 20/3-13/3 = 2.3333 V