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Inverting Amplifier
We can use the concept of the virtual short to easily determine the input resistance of the inverting amplifier. Recall that the input resistance of an amplifier is:
Ri and Ro of the
Ri =
vi ii
For the inverting amplifier, it is evident that the input current ii is i1 :
R2
i2
vi R1 v1
-
ii = i1
v2
+
ideal
vo
From Ohm's Law, we know that this current is:
ii = i1 =
vi - v 1 R1
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The non-inverting terminal is "connected" to virtual ground:
v1 = 0
and thus the input current is:
ii = i1 =
vi R1
We now can determine the input resistance:
Ri =
vi ii
R = vi 1 vi = R1
The input resistance of this amplifier is therefore Ri = R1 .
Now, let's attempt to determine the output resistance Ro. Recall that we need to determine two values: the short-circuit output current (ios ) and the open-circuit output voltage (voo ) . To accomplish this, we must replace the op-amp in the circuit with its linear circuit model:
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R2 i2 vi R1 ii = i1
+ v 2
+ -
-
v1
Roop ioop
Aop (v2 - v1 )
vo = 0
ios
From KCL, we find that:
ios =
where:
i2 =
and:
v 1 - vo = R2
-Aop v1
i
op o
=
-Aop v1 - vo
Roop
=
Roop
Therefore, the short-circuit output current is:
ios
v1 Aop v1 = - R2 Roop
Roop - R2 Aop = R R op 2 o v 1
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The open-circuit output voltage can likewise be determined in terms of Aop and v1:
R2 i2 vi R1 ii = i1
+ v 2
+ -
-
v1
Roop ioop Aop (v2 - v1 ) io =0 + voo
-
Here, it is evident that since io=0:
i2 = -ioop
where we find from Ohm's Law:
i2 =
v1 - ( - Aopv1 ) R2 + Roop
v1
1 + Aop = op R2 + Ro
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and thus voo is:
voo = v1 - R2 i2
1 + Aop = v1 - R2 v R2 + Roop 1 R + R op R2 1 + Aop o = 2 - R2 + Roop R2 + Roop Roop - R2 Aop v = R + R op 1 o 2
(
)v
1
Now, we can find the output resistance of this amplifier:
Ro =
voo ios
Roop - R2 Aop R R op 2 o
-1
Roop - R2 Aop = R + R op o 2
R2 Roop = R2 + Roop
= R2 Roop In other words, the inverting amplifier output resistance is simply equal to the value of the feedback resistor R2...