# A = ∞ ```NON-INVERTING AMPLIFIER GAIN DERIVATION
with FINITE OPEN LOOP GAIN ANALYSIS
ASSUMPTIONS: INFINITE INPUT IMPEDANCE: ∴ i+ = 0;
ZERO VOLTAGE DROP BETWEEN INPUTS, and A = ∞ .
i− = 0
ZERO AC INPUT CURRENT.
ASSUMPTIONS HOLD FOR A &gt;&gt; AV = 1 +
Rs
3
+15
+
7
6
A
+
_
v+
vin
2
4
-
Let Rs = 0; Therefore v+ = vin
R1
v− =
but v+ = v−
&times; vout ;
R 1+ R 2
+
-15
v-
R2
R1
vout
R2
_
R1
so vin =
R1
&times; vout ;
R1 + R 2
vout R1 + R 2
=
vin
R1
or Av = 1 +
vout
R2
R1
FINITE OPEN-LOOP GAIN ANALYSIS:
R1
R1
= Avid = A( v+ − v− ) = A( vin − v− );
v− =
vout = βvout where β =
.
R1 + R2
R1 + R2
β is called the feedback transfer function and represents the fraction of the
output voltage that is fed back from the output to the input. Combining the
equations above gives:
vout
A
vout = A[ vin − βvout ];
vout + Aβvout = Avin ;
= Av =
vin
1 + Aβ
This gives the classic negative feedback amplifier gain expression. The product
Aβ is called the loop gain or loop transmission. For Aβ&gt;&gt;1, Av approaches the
ideal gain expression found above [=1/β]. In reality, A is a function whose value
decreases with increasing frequency, until, at some point when Aβ is no longer
&gt;&gt;1, the ideal gain equation, a function of only two resistor values, no longer
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Cite as: Ron Roscoe, course materials for 6.101 Introductory Analog Electronics Laboratory, Spring 2007.