# 2. The Inverting Amplifier

*The Inverting Amplifier*

Figure 3 (a) illustrates an inverting amplifier with the feedback, and Figure 3 (b) shows the equivalent circuit for this ideal inverting op-amp circuit. We have used the properties of the ideal op-amp to model the op-amp input as an open circuit. The controlled source is *Gv _{d}*, but under the given assumptions, we will not have to use this information explicitly. We wish to solve for the output voltage,

*v*, in terms of the input voltage,

_{out}*v*. We write equations for

_{a}*v*and

_{+}*v*and then set these expressions equal to each other. Since the current through

_{–}*R*is zero,

Also Kirchhoff’s node equation at *v _{–}* yields,

Since *v _{+} = v_{–}* and

*v*0, then

_{+}=*v*is also zero. Therefore, we have one equation in two unknowns,

_{–}*v*and

_{a}*v*, so we can solve for the closed-loop gain as,

_{out}Notice that the closed-loop gain, *v _{out }/v_{a}*, is negative (inverted) and is dependent only on the ratio of two resistors,

*R*. It is independent of the very high open-loop gain,

_{F }/R_{a}*G*. This desirable result is caused by the use of feedback of a portion of the output voltage to subtract from the input voltage. The feedback from output to input through

*R*serves to drive the differential voltage,

_{F}*v*, close to zero. Since the non-inverting input voltage,

_{d}= v_{+}– v_{–}*v*, is zero, the feedback has the effect of driving

_{+}*v*to zero. Hence, at the input of the op-amp,

_{–}No matter how complex the ideal op-amp circuit, by following this simple procedure the engineer can quickly analyze (and soon design) op-amp systems.

###### We may now expand this result to the case of multiple inputs.

The amplifier shown in Figure (4) produces an output which is a negative weighted summation of several input voltages.

Since the current through *R* is zero, *v _{+} = *0. The node equation at the inverting input terminal is given by Equation (16):

Since *v _{+} = v_{–}*, then

*v*0

_{+}=*= v*

_{–}and we find

*v*in terms of the inputs as follows:

_{out}The extension to *n* inputs is straightforward.

__APPLICATIONS__

Analyze the following circuits using with the TINACloud circuit simulator to determine V* _{out}* in terms of the input voltages by clicking the links below.