Lecture 27
Op- Amp
Op-Amp Differentiator
R
0
to
C
t1
t2
V

i
V
0
o
+
to
t1
t2
 dV 
vo   i  RC
 dt 
Ref:080114HKN
Operational Amplifier
2
Non-ideal case (Inverting Amplifier)
Rf
Ra
Vin ~

V
in

V
Vin
Zout
~
Vout
 AVin
+
Equivalent Circuit
3 categories are considering
Rf
Ra

R
 R
V
+ +

Ref:080114HKN
o
Zin
+
Practical op-amp
V
o
 Close-Loop Voltage Gain
 Input impedance
 Output impedance
-AV
Operational Amplifier
3
Close-Loop Gain
Applied KCL at V– terminal,
Rf
Vin  V  V Vo  V


0
Ra
R
Rf
V
Ra

in
V
By using the open loop gain,
R R 
+ +

Vo   AV


Vin Vo
V
V
V

 o  o  o 0
Ra ARa AR R f AR f
V
R R f  Ra R f  Ra R  ARa R
Vin
 Vo
Ra
ARa R R f
Ra
in
V
o
-AV
Rf
V
o
V  R
The Close-Loop Gain, Av
 AR R f
Vo
Av 

Vin R R f  Ra R f  Ra R  ARa R
Ref:080114HKN
Operational Amplifier
4
Close-Loop Gain
When the open loop gain is very large, the above equation become,
Av ~
 Rf
Ra
Note : The close-loop gain now reduce to the same form
as an ideal case
Ref:080114HKN
Operational Amplifier
5
Input Impedance
Rf
Input Impedance can be regarded as,
Rin  Ra  R // R
R
V R
+ +

V
if
V
o
-AV
R'
However, with the below circuit,
V  ( AV )  i f ( R f  Ro )
V R f  Ro
 R 

if
1 A
Ref:080114HKN

in
where R is the equivalent impedance
of the red box circuit, that is
R 
V
Ra
if
V
Operational Amplifier
Rf
R
+

-AV
6
Input Impedance
Finally, we find the input impedance as,
1
1 A 
Rin  Ra   

R
R

R

f
o
 
Since,
1

Rin  Ra 
R ( R f  Ro )
R f  Ro  (1  A) R
R f  Ro  (1  A) R , Rin become,
Rin ~ Ra 
Again with
( R f  Ro )
(1  A)
R f  Ro  (1  A)
Rin ~ Ra
Note: The op-amp can provide an impedance isolated from
input to output
Ref:080114HKN
Operational Amplifier
7
Output Impedance
Only source-free output impedance would be considered,
i.e. Vi is assumed to be 0
Firstly, with figure (a),
Ra
Ra // R
Ra R
Vo  V 
Vo
R f  Ra // R
Ra R f  Ra R  R f R
By using KCL, io = i1+ i2
V 
io 
Rf
R
V
Vo
V  ( AV )
 o
R f  Ra // R f
Ro
io
R
+
 -AV
V
By substitute the equation from Fig. (a),
The output impedance, Rout is
Ro ( Ra R f  Ra R  R f R )
Vo

io (1  Ro )( Ra R f  Ra R  R f R )  (1  A) Ra R
R and A comparably large,
Ro ( Ra  R f )
Rout ~
ARa
Ref:080114HKN
Operational Amplifier
Rf
V
Ra
i2
R
V
R
(a)
i1
V
+

-AV
(b)
8
o