# 7. Non-inverting Amplifier

*Non-inverting Amplifier*

Figure 29(a) illustrates the *non-inverting amplifier*, and Figure 29(b) shows the equivalent circuit.

The input voltage is applied through *R _{1}* into the non-inverting terminal.

**7.1 Input and Output Resistances**

**7.1 Input and Output Resistances**

The *input resistance *of this amplifier is found by determining the Thevenin equivalent of the input circuit. The load resistance is normally such that *R _{load}* >>

*R*. If this were not true, the effective gain would be reduced and the effective value of

_{o}*R*would be the parallel combination of

_{o}*R*with

_{o}*R*. Let us again define and

_{load}*R’*=

_{F}*R*+

_{F}*R*. We shall neglect

_{o}*R*, since it is so much less than

_{1}*R*. Now since

_{in}*R*>>

_{load}*R*, we can reduce Figure 29(a) to the simplified form of Figure 30(a).

_{o}We find the Thevenin equivalent of the circuit surrounded by the elliptical curve, resulting in Figure 30(b). In Figure 30(c), the resistance to the right of 2*R _{cm}* is given by

*v*/

*i’*. In order to evaluate this, we write a loop equation to obtain

Therefore,

The input resistance is the parallel combination of this quantity with 2*R _{cm}*.

Recall that , *R’ _{F}* =

*R*+

_{F}*R*, and

_{o}*R*>>

_{load}*R*. If we retain only the most significant terms and note that

_{o}*R*is large, Equation (55) reduces to

_{cm}where we again use the zero-frequency voltage gain, *G _{o}*.

Equation (56) can be used to find the input resistance of the 741 op-amp. If we substitute the parameter values as given in Table 1, Equation (56) becomes

We again use the assumptions that *R _{cm}* is large, that is

*R’*»

_{F}*R*and

_{F}*R’*»

_{A}*R*. Then the output resistance of a 741 op-amp is given by

_{A}__EXAMPLE__

Calculate the input resistance for the unity-gain follower shown in Figure 31(a).

*Solution:* The equivalent circuit is shown in Figure 31(b). Since we assume the zero-frequency gain, *G _{o}*, and the common-mode resistance,

*R*, are high, we can neglect the term compared to (1+

_{cm}*G*)

_{o}*R*. Equation (57) cannot be used since

_{i}*R*= 0. The input resistance is then given by

_{A}This is typically equal to 400 MΩ or more, so we can neglect *R _{1}* (i.e., set

*R*= 0).

_{1 }**7.2 Voltage Gain**

**7.2 Voltage Gain**

We wish to determine the voltage gain, *A _{+}* for the non-inverting amplifier of Figure 32(a).

This gain is defined by

The equivalent circuit is shown in Figure 32(b). If we assume *R _{F}*>>

*R*,

_{o}*R*>>

_{load}*R*and , the circuit can be reduced to that shown in Figure 32(c). If we further define , then Figure 32(d) results.

_{o}The assumed conditions are desirable in order to prevent reduction of the effective gain. The operation of taking Thevenin equivalents modifies the dependent voltage source and the driving voltage source as in Figure 32(d). Note that

The output voltage is given by

We can find *i* by applying KVL to the circuit of Figure 32(d) to obtain

where

and implying .

Solving for the current,* i*, we obtain

The voltage gain is given by the ratio of output to input voltage.

As a check of this result, we can reduce the model to that of the ideal op-amp. We use the zero-frequency gain, *G _{o}*, in place of

*G*in Equation (64) and also the following equalities.

When we let , Equation (64) becomes

which agrees with the result for the idealized model.

__Example__

__Example__

Find the gain of the unity-gain follower shown in Figure 33.

Figure 33 – Unity gain follower*Solution:* In this circuit, , *R’ _{A}* = 2

*R*, and

_{cm}*R*<<

_{F}*R’*. We assume that

_{A}*G*is large, , and we set

_{o}*R*=

_{1}*R*. Equation (64) then reduces to

_{F}(67)

so *v _{out}* =

*v*as expected.

_{in}

**7.3 Multiple-Input Amplifiers**

**7.3 Multiple-Input Amplifiers**

We extend the previous results to the case of the non-inverting amplifier with multiple voltage inputs. Figure 34 shows a multiple-input non-inverting amplifier.

If inputs *v _{1}*,

*v*,

_{2}*v*, …,

_{3}*v*are applied through input resistances

_{n}*R*,

_{1}*R*,

_{2}*R*, …,

_{3}*R*, we obtain a special case of the general result derived in Chapter “Ideal Operational Amplifiers”, as follows:

_{n}We choose

to achieve bias balance. The output resistance is found from Equation (52).

As a specific example, let us determine the output voltage of the two-input summer of Figure 35.

The output voltage is found from Equation (68) , as follows:

We choose to achieve bias balance. If we assume *R _{F}* =

*R*=

_{1}*R*=

_{2}*R*, then Equation (70) reduces to

_{A}*v*=

_{out}*v*+

_{1}*v*, which is a unity-gain two-input summer.

_{2}