Analog Circuits Formulas for GATE ECE Exam | Analog Circuits - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


• Energy gap
???? G/si
=1.21- 3.6 × 10
-4
.T   ev
???? G/Ge
=0.785- 2.23 × 10
-4
.T   ev
? Energy gap depending on temperature 
• E
F
 = E
C
 - KT ln?
???? ???? ???? ???? ? = E
v
 + KT ln ?
???? ???? ???? ???? ? 
• No. of electrons  n = N
c
 e
-(E
c
-E
f
)/RT
         (KT in ev) 
• No. of holes  p = N
v
 e
-(E
f
-E
v
)/RT
• Mass action law  n
p
 = n
i
2
 = N
c
N
v
 e
-EG/KT
• Drift velocity  ???? d
 = µE  (for si ???? d
 = 10
7
 cm/sec)
• Hall voltage ???? H
 = 
B.I
w
e
 . Hall coefficient  R
H
 = 1/? .            ? ? charge density = qN
0
 = ne … 
• Conductivity  s = ?µ ;  µ = sR
H
 .
• Max value of electric field @ junction E
0
 =  -
q
?
si
N
d
. n
n0
 =  -
q
?
si
 N
A
. n
p0
 . 
• Charge storage @ junction  Q
+
 = - Q
-
 = qA x
n0
N
D
 =  qA x
p0
N
A
• Diffusion current densities  J
p
 = - q D
p
 
dp
dx
J
n
 = - q D
n
 
dn
dx
• Drift current Densities  = q(p µ
p
+ nµ
n
)E
• µ
p
 , µ
n
 decrease with increasing doping concentration .
•
D
n
µ
n
= 
D
p
µ
p
 = KT/q ˜ 25 mv @ 300 K 
• Carrier concentration in N-type silicon n
n0
 = N
D
 ;  p
n0
 = n
i
2
 / N
D
• Carrier concentration in P-type silicon p
p0
 = N
A
 ;  n
p0
 = n
i
2
 / N
A
• Junction built in voltage  V
0
 = V
T
 ln ?
???? ???? ???? ???? ???? ???? 2
? 
• Width of Depletion region  W
dep
 =  x
p
 + x
n
 = ?
2e
s
q
?
1
N
A
+
1
N
D
?(V
0
+ V
R
) 
* ?
2???? ???????? ???? = 12.93???? ???????????? ???????? ?  
•
x
n
x
p
 = 
N
A
N
D
 
• Charge stored in depletion region  q
J
 = 
q.N
A
 N
D
N
A
+N
D
 . A . W
dep
  
• Depletion capacitance  C
j
 = 
e
s
A
W
dep
 ;   C
j0
 = 
e
s
A
W
dep
/ V
R
=0
C
j
 = C
j0
/? 1 +
V
R
V
0
?
m
  C
j
 = 2C
j0
  (for forward Bias) 
• Forward current  I = I
p
+ I
n
 ;       I
p
 = Aq n
i
2
D
p
L
p
N
D
 ? ???? ???? /???? ???? - 1? 
I
n
 = Aq n
i
2
 
D
n
L
n
N
A
 ? ???? ???? /???? ???? - 1?  
• Saturation Current  I
s
 = Aq n
i
2
  ?
D
p
L
p
N
D
+
D
n
L
n
N
A
? 
• Minority carrier life time  t
p
 = L
p
2
 / D
p
  ;    t
n
 = L
n
2
 / D
n
Analog Circuits (Formula Notes/Short Notes)
Page 2


• Energy gap
???? G/si
=1.21- 3.6 × 10
-4
.T   ev
???? G/Ge
=0.785- 2.23 × 10
-4
.T   ev
? Energy gap depending on temperature 
• E
F
 = E
C
 - KT ln?
???? ???? ???? ???? ? = E
v
 + KT ln ?
???? ???? ???? ???? ? 
• No. of electrons  n = N
c
 e
-(E
c
-E
f
)/RT
         (KT in ev) 
• No. of holes  p = N
v
 e
-(E
f
-E
v
)/RT
• Mass action law  n
p
 = n
i
2
 = N
c
N
v
 e
-EG/KT
• Drift velocity  ???? d
 = µE  (for si ???? d
 = 10
7
 cm/sec)
• Hall voltage ???? H
 = 
B.I
w
e
 . Hall coefficient  R
H
 = 1/? .            ? ? charge density = qN
0
 = ne … 
• Conductivity  s = ?µ ;  µ = sR
H
 .
• Max value of electric field @ junction E
0
 =  -
q
?
si
N
d
. n
n0
 =  -
q
?
si
 N
A
. n
p0
 . 
• Charge storage @ junction  Q
+
 = - Q
-
 = qA x
n0
N
D
 =  qA x
p0
N
A
• Diffusion current densities  J
p
 = - q D
p
 
dp
dx
J
n
 = - q D
n
 
dn
dx
• Drift current Densities  = q(p µ
p
+ nµ
n
)E
• µ
p
 , µ
n
 decrease with increasing doping concentration .
•
D
n
µ
n
= 
D
p
µ
p
 = KT/q ˜ 25 mv @ 300 K 
• Carrier concentration in N-type silicon n
n0
 = N
D
 ;  p
n0
 = n
i
2
 / N
D
• Carrier concentration in P-type silicon p
p0
 = N
A
 ;  n
p0
 = n
i
2
 / N
A
• Junction built in voltage  V
0
 = V
T
 ln ?
???? ???? ???? ???? ???? ???? 2
? 
• Width of Depletion region  W
dep
 =  x
p
 + x
n
 = ?
2e
s
q
?
1
N
A
+
1
N
D
?(V
0
+ V
R
) 
* ?
2???? ???????? ???? = 12.93???? ???????????? ???????? ?  
•
x
n
x
p
 = 
N
A
N
D
 
• Charge stored in depletion region  q
J
 = 
q.N
A
 N
D
N
A
+N
D
 . A . W
dep
  
• Depletion capacitance  C
j
 = 
e
s
A
W
dep
 ;   C
j0
 = 
e
s
A
W
dep
/ V
R
=0
C
j
 = C
j0
/? 1 +
V
R
V
0
?
m
  C
j
 = 2C
j0
  (for forward Bias) 
• Forward current  I = I
p
+ I
n
 ;       I
p
 = Aq n
i
2
D
p
L
p
N
D
 ? ???? ???? /???? ???? - 1? 
I
n
 = Aq n
i
2
 
D
n
L
n
N
A
 ? ???? ???? /???? ???? - 1?  
• Saturation Current  I
s
 = Aq n
i
2
  ?
D
p
L
p
N
D
+
D
n
L
n
N
A
? 
• Minority carrier life time  t
p
 = L
p
2
 / D
p
  ;    t
n
 = L
n
2
 / D
n
Analog Circuits (Formula Notes/Short Notes)
• Minority carrier charge storage  Q
p
 = t
p
I
p
 ,  Q
n
 = t
p
I
n
     Q = Q
p
 + Q
n
 = t
T
I              t
T
 = mean transist time 
• Diffusion capacitance  C
d
 =  ?
???? ???? ???????? ???? ? I = t.g  ? C
d
 ? I. 
t? carrier life time ,  g = conductance = I /  ???????? ????  
• I
02
 = 2
(T
2
-T
1
)/10
 I
01
• Junction Barrier Voltage  V
j
 = V
B
 = V
r
 (open condition)
           = V
r
 - V (forward Bias) 
           =  V
r
 + V (Reverse Bias) 
• Probability of filled states above ‘E’  f(E) = 
1
1+e
(E-E
f
)/KT
  
• Drift velocity of  e
-
        ???? d
 = 10
7
 cm/sec 
• Poisson equation  
d
2
V
dx
2
 = 
-?
v
?
  = 
-nq
?
 ? 
dv
dx
 = E = 
-nqx
?
  
Transistor :- 
• I
E
 = I
DE
 + I
nE
• I
C
 = I
Co
 – a I
E
  ? Active region
• I
C
 = – a I
E
 + I
Co
 (1- e
V
C
/V
T
 )
Common Emitter :- 
• I
C
 = (1+ ß) I
Co
 + ßI
B
    ß = 
a
1-a
• I
CEO
 = 
I
Co
1-a
  ? Collector current when base open 
• I
CBO
 ? Collector current when I
E
 = 0        I
CBO
 > I
Co
 .
• V
BE,sat
  or  V
BC,sat
 ?  - 2.5 mv /
0
 C ;    V
CE,sat
 ? 
V
BE,sat
10
 = - 0.25 mv /
0
C 
• Large signal Current gain  ß =  
I
C
- I
CBo
I
B
+ I
CBo
• D.C current gain  ß
dc
 = 
I
C
I
B
 = h
FE
  
• (ß
dc
 = h
FE
 ) ˜ ß   when  I
B
 > I
CBo
• Small signal current gain ß
'
 = 
?I
C
?I
R
?
V
CE
 =  h
fe
 =  
h
FE
1-(I
CBo
+ I
B
)
?h
FE
?I
C
  
• Over drive factor =
ß
active
ß
forced
?under saturation
 ? I
C sat
 = ß
forced
 I
B sat
   
Conversion formula :- 
   CC  ? CE 
• h
ic
 = h
ie
 ;     h
rc
 = 1 ;         h
fc
 = - (1+ h
fe
) ;      h
oc
 = h
oe
 
 CB  ? CE 
• h
ib
 = 
h
ie
1+h
fe
;  h
ib
 = 
h
ie
 h
oe
1+h
fe
- h
re
 ; h
fb
 = 
-h
fe
1+h
fe
;    h
ob
 = 
h
oe
1+h
fe
CE parameters in terms of CB can be obtained by interchanging B & E . 
Page 3


• Energy gap
???? G/si
=1.21- 3.6 × 10
-4
.T   ev
???? G/Ge
=0.785- 2.23 × 10
-4
.T   ev
? Energy gap depending on temperature 
• E
F
 = E
C
 - KT ln?
???? ???? ???? ???? ? = E
v
 + KT ln ?
???? ???? ???? ???? ? 
• No. of electrons  n = N
c
 e
-(E
c
-E
f
)/RT
         (KT in ev) 
• No. of holes  p = N
v
 e
-(E
f
-E
v
)/RT
• Mass action law  n
p
 = n
i
2
 = N
c
N
v
 e
-EG/KT
• Drift velocity  ???? d
 = µE  (for si ???? d
 = 10
7
 cm/sec)
• Hall voltage ???? H
 = 
B.I
w
e
 . Hall coefficient  R
H
 = 1/? .            ? ? charge density = qN
0
 = ne … 
• Conductivity  s = ?µ ;  µ = sR
H
 .
• Max value of electric field @ junction E
0
 =  -
q
?
si
N
d
. n
n0
 =  -
q
?
si
 N
A
. n
p0
 . 
• Charge storage @ junction  Q
+
 = - Q
-
 = qA x
n0
N
D
 =  qA x
p0
N
A
• Diffusion current densities  J
p
 = - q D
p
 
dp
dx
J
n
 = - q D
n
 
dn
dx
• Drift current Densities  = q(p µ
p
+ nµ
n
)E
• µ
p
 , µ
n
 decrease with increasing doping concentration .
•
D
n
µ
n
= 
D
p
µ
p
 = KT/q ˜ 25 mv @ 300 K 
• Carrier concentration in N-type silicon n
n0
 = N
D
 ;  p
n0
 = n
i
2
 / N
D
• Carrier concentration in P-type silicon p
p0
 = N
A
 ;  n
p0
 = n
i
2
 / N
A
• Junction built in voltage  V
0
 = V
T
 ln ?
???? ???? ???? ???? ???? ???? 2
? 
• Width of Depletion region  W
dep
 =  x
p
 + x
n
 = ?
2e
s
q
?
1
N
A
+
1
N
D
?(V
0
+ V
R
) 
* ?
2???? ???????? ???? = 12.93???? ???????????? ???????? ?  
•
x
n
x
p
 = 
N
A
N
D
 
• Charge stored in depletion region  q
J
 = 
q.N
A
 N
D
N
A
+N
D
 . A . W
dep
  
• Depletion capacitance  C
j
 = 
e
s
A
W
dep
 ;   C
j0
 = 
e
s
A
W
dep
/ V
R
=0
C
j
 = C
j0
/? 1 +
V
R
V
0
?
m
  C
j
 = 2C
j0
  (for forward Bias) 
• Forward current  I = I
p
+ I
n
 ;       I
p
 = Aq n
i
2
D
p
L
p
N
D
 ? ???? ???? /???? ???? - 1? 
I
n
 = Aq n
i
2
 
D
n
L
n
N
A
 ? ???? ???? /???? ???? - 1?  
• Saturation Current  I
s
 = Aq n
i
2
  ?
D
p
L
p
N
D
+
D
n
L
n
N
A
? 
• Minority carrier life time  t
p
 = L
p
2
 / D
p
  ;    t
n
 = L
n
2
 / D
n
Analog Circuits (Formula Notes/Short Notes)
• Minority carrier charge storage  Q
p
 = t
p
I
p
 ,  Q
n
 = t
p
I
n
     Q = Q
p
 + Q
n
 = t
T
I              t
T
 = mean transist time 
• Diffusion capacitance  C
d
 =  ?
???? ???? ???????? ???? ? I = t.g  ? C
d
 ? I. 
t? carrier life time ,  g = conductance = I /  ???????? ????  
• I
02
 = 2
(T
2
-T
1
)/10
 I
01
• Junction Barrier Voltage  V
j
 = V
B
 = V
r
 (open condition)
           = V
r
 - V (forward Bias) 
           =  V
r
 + V (Reverse Bias) 
• Probability of filled states above ‘E’  f(E) = 
1
1+e
(E-E
f
)/KT
  
• Drift velocity of  e
-
        ???? d
 = 10
7
 cm/sec 
• Poisson equation  
d
2
V
dx
2
 = 
-?
v
?
  = 
-nq
?
 ? 
dv
dx
 = E = 
-nqx
?
  
Transistor :- 
• I
E
 = I
DE
 + I
nE
• I
C
 = I
Co
 – a I
E
  ? Active region
• I
C
 = – a I
E
 + I
Co
 (1- e
V
C
/V
T
 )
Common Emitter :- 
• I
C
 = (1+ ß) I
Co
 + ßI
B
    ß = 
a
1-a
• I
CEO
 = 
I
Co
1-a
  ? Collector current when base open 
• I
CBO
 ? Collector current when I
E
 = 0        I
CBO
 > I
Co
 .
• V
BE,sat
  or  V
BC,sat
 ?  - 2.5 mv /
0
 C ;    V
CE,sat
 ? 
V
BE,sat
10
 = - 0.25 mv /
0
C 
• Large signal Current gain  ß =  
I
C
- I
CBo
I
B
+ I
CBo
• D.C current gain  ß
dc
 = 
I
C
I
B
 = h
FE
  
• (ß
dc
 = h
FE
 ) ˜ ß   when  I
B
 > I
CBo
• Small signal current gain ß
'
 = 
?I
C
?I
R
?
V
CE
 =  h
fe
 =  
h
FE
1-(I
CBo
+ I
B
)
?h
FE
?I
C
  
• Over drive factor =
ß
active
ß
forced
?under saturation
 ? I
C sat
 = ß
forced
 I
B sat
   
Conversion formula :- 
   CC  ? CE 
• h
ic
 = h
ie
 ;     h
rc
 = 1 ;         h
fc
 = - (1+ h
fe
) ;      h
oc
 = h
oe
 
 CB  ? CE 
• h
ib
 = 
h
ie
1+h
fe
;  h
ib
 = 
h
ie
 h
oe
1+h
fe
- h
re
 ; h
fb
 = 
-h
fe
1+h
fe
;    h
ob
 = 
h
oe
1+h
fe
CE parameters in terms of CB can be obtained by interchanging B & E . 
• A
I
 =
-h
f
1+h
0
Z
L
 Z
i
 = h
i
 + h
r
 A
I
Z
L
          A
vs
 = 
A
v
.Z
i
Z
i
+R
s
Z
i
+R
s
 = 
A
I
.Z
L
 = 
A
I
s
.Z
L
R
s
A
V
 = 
A
I
  Z
L
Z
i
Y
0
 = h
o
 - 
h
f
 h
r
h
i
+ R
s
A
Is
 = 
A
v
.R
s
Z
i
+R
s
  = 
A
vs
.R
s
Z
L
  
Choice of Transistor Configuration :- 
• For intermediate stages CC can’t be used as  A
V
 < 1
• CE can be used as intermediate stage
• CC can be used as o/p stage as it has low o/p impedance
• CC/CB can be used as i/p stage because of i/p considerations.
Stability & Biasing :- ( Should be as min as possible) 
• For  S = 
?I
C
?I
Co
?
V
B0,ß
    S
'
 = 
?I
C
?V
BE
?
I
C0,ß
          S
''
 = 
?I
C
?ß
?
V
BE,I
Co
 ?I
C
 = S. ?I
Co
  + S
'
 ?V
BE
 + S
''
 ?ß  
• For fixed bias  S =
1+ß
1-ß
dI
B
dI
C
  = 1 + ß  
• Collector to Base bias  S = 
1+ß
1+ß
R
C
R
C
+R
B
           0 < s < 1+ ß  = 
1+ß
1+ß?
R
C
+ R
E
R
C
+ R
E
+ R
B
?
• Self bias  S =  
1+ß
1+ß
R
E
R
E
+R
th
  ˜ 1+ 
R
th
R
e
          ßR
E
  > 10 R
2
 
• R
1
 = 
V
cc
 R
th
V
th
;  R
2
 = 
V
cc
 R
th
V
cc
-V
th
• For thermal stability [ V
cc
 - 2I
c
 (R
C
 + R
E
)] [ 0.07 I
co
 . S]  <  1/?  ; V
CE
 < 
V
CC
2
Hybrid –pi(p)- Model :- 
  g
m
 = |I
C
 | / V
T
   
 r
b
'
e
  =  h
fe
 / g
m
         
r
b
'
b
 = h
ie
 - r
b
'
e
r
b
'
c
 = r
b
'
e
 / h
re
        
g
ce
 = h
oe
 - (1+ h
fe
 ) g
b
'
c
    
Specifications of An amplifier :-