# vout v ```Phy335
Unit 5 Handout
Op Amp Feedback and Effective Gain
For the Op Amp circuit with no feedback, like that one shown in Fig. 1, the output voltage vout
is given by the expression
vout = g OL (v+ − v− ).
(1)
Over a wide frequency range, the open loop gain is g OL = O(106 ). For typical rail voltages
VCC+ and VCC− of 15 V, the maximum usable input difference (v+ − v− ) is therefore limited to
15 V/gOL = 15&micro;V.
RF
v-
-
v+
+
vin
RS
vout
vout
+
Figure 1: Op Amp circuit without feedback.
Figure 2: Op Amp circuit with feedback.
However, adding negative feedback makes the (closed loop) voltage gain g ≡ vout /vin controllable
by external components. For the circuit shown in Fig. 2, and using the high input impedance of
an Op Amp so that i+ = i− = 0, one finds the equations governing the currents and voltages in
the resistors to be
vout = g OL (0 − v− )
vin − v−
is =
RS
vout − v−
if =
RF
where the first line comes from the Op Amp basic definition, the second and third are simply from
Ohm’s Law, and v+ = 0 because it is connected to ground. Because is = −if and vout = g OL v− ,
v− can be eliminated from the above equations:
1 &sup3;
RS
is = −if
&acute;
vout &acute;
−1 &sup3;
vin + OL
=
vout g OL
g
RF
&sup3; 1
1
1
1 &acute;
vout
vin = − OL
+ OL
+
RS
g RS
g RF
RF
(continued on back)
1
and since g OL RS &gt;&gt; RF and g OL RF &gt;&gt; RF , the last line becomes
1
1
vin = −
vout
RS
RF
which is trivially rearranged to become
g≡
vout
RF
=−
.
vin
RS
(2)
Back substitution then shows that v− = 0 which gives rise to the second golden rule v+ = v−
(which we did not use, but derived).
2
```