Non-inverting OPAMP

Introduction

The voltage signal applied to an op-amp can either be supplied to its non-inverting input (+) or the inverting input (-). These different configurations are simply known as a non-inverting op-amp, and inverting op-amp. In this tutorial, we focus on the non-inverting configuration and present its details.

An overview of the non-inverting op-amp will be given in the first section through the concept of the ideal amplifier.

In the second section, real non-inverting configurations are discussed, we demonstrate the equations describing the gain and the input/output impedances.

Finally, examples of circuits based on the non-inverting configurations are given in the last section.

Ideal non-inverting op-amp

The goal of this section is to properly demonstrate and explain the ideal characteristics of the non-inverting configuration such as its input/output impedance and gain. The circuit representation of an ideal non-inverting op-amp is given in Figure 1 below.

Note that the symbol “∞” highlights the fact that the op-amp is here to be considered ideal. We highly recommend the reader to refer to the tutorial Op-amp basics for this section.

In this ideal model, the input impedance defined by the contribution of the resistance linking the inverting and non-inverting inputs (R_i in Figure 3) and the resistors R₁ and R₂, is infinite. Moreover, for an ideal circuit, R_i is supposed to be infinite, as a consequence, no currents can enter the op-amp through any input because of the presence of an open circuit.

This observation can also be summarized by saying that the node interconnecting the inverting input and resistances R₁ and R₂ is a virtual short. For this same reason, all the feedback current across R₁ (I) is also found across R₂.

For the ideal model, the equality V₊=V_–=V_in is assured by the fact that the differential signal V₊-V_– can only be equal to 0 in order to produce a finite output V_out when multiplied by an infinite open-loop gain.

We can see the branches connected to the inverting input acting as a voltage divider circuit:

According to the voltage divider formula, we can express the inverting voltage V_– as a function of the output voltage and the resistances:

Since V_–=V_in, after some simplification, we prove the expression of the gain in closed-loop A_CL of an ideal non-inverting configuration:

We can note that the ideal gain presented in Equation 2 is strictly positive and higher than 1, meaning that the output signal is amplified and in phase with the input signal.

Real non-inverting op-amp

In a real op-amp circuit, the input (Z_in) and output (Z_out) impedances are not idealized to be equal to respectively +∞ and 0 Ω. Instead, the input impedance has a high but finite value, the output impedance has a low but non-zero value.

The non-inverting configuration still remains the same as the one presented in Figure 1.

Closed-loop gain

For a non-inverting configuration, Equation 1 still applies for V_– , moreover, we have V₊=V_in. However, since a low current can flow from the non-inverting input to the inverting input, the voltages are not equal anymore: V₊≠V_–.

We also need to remind that the inputs V₊ and V_– are linked with the output through the open-loop gain formula:

The equations for V₊ and V_– can be injected in Equation 3. After regrouping the terms “V_out” on one side of the equation and the terms “V_in” on the other, we get:

Finally, the closed-loop gain A_CL for a real non-inverting configuration is given by Equation 4:

For a real configuration, the gain not only depends on the resistor values but also on the open-loop gain

It is interesting to note that if we consider the op-amp to be ideal (A_OL→+∞), the denominator is simplified to one term: A_OLR₂/(R₁+R₂). As a consequence, Equation 4 is simplified back to Equation 2.

Output impedance

We start by assuming the equality of the currents across the resistances: I_R1=I_R2. Even if for real op-amps, a small leaking current enters the inverting input, it is several orders of magnitude smaller than the feedback current.

The current I₀ across R₀ (see Figure 3) can be expressed as a function of the voltage drop across R₀ and the same value of the impedance R₀:

Since V_– is described by Equation 1, the output current I_out can be expressed as the sum of I₀ and the current flowing in the feedback branch given by V_out/(R₁+R₂):

Finally, after rearranging the equation to obtain the ratio Z_out=V_out/I_out, we can write the expression of the output impedance for a real non-inverting configuration:

We can note that in the case of an ideal op-amp, that is to say when A_OL→+∞, we observe indeed Z_out→0.

A simplified version for the expression of Z_out is given by the following Equation 6:

The term β is known as the feedback factor and is given by the ratio R₁/(R₁+R₂). With that simplified version, we can still see that Z_out→0 for an ideal op-amp situation.

Input impedance

The input impedance of a non-inverting configuration can be defined by the ratio V₊/I_in (see Figure 3). For the input loop, we can write Kirchoff’s voltage law such as V₊-V_in+I_R2R₂=0 with I_R2 being the current across the resistor R₂.

It can be shown that the expression of the input impedance can also be written as a function of the feedback factor:

Again, when the ideal situation is satisfied (A_OL→+∞) we find that Z_in→+∞ such as specified in the first section.

Non-inverting op-amp examples

Buffer circuits

The most simple designs for non-inverting configurations are buffers, which have been described in the previous tutorial Op-amp Building Blocks. In this configuration, R₁=0 and R₂→+∞ as we can present in Figure 4 below:

This buffer (or voltage follower) has a unity gain and does not invert the output, meaning that V_out=V_in. Its high input impedance and low output impedance are very useful to establish a load match between circuits and make the buffer to act as an ideal voltage source.

Example

We consider a real non-inverting configuration circuit given in Figure 5:

The resistors, input value, and gain in open-loop are given such as:

• R₁=10 kΩ
• R₂=2 kΩ
• R_L=1 kΩ
• V_in=1 V
• A_OL=10⁵

First of all, we can compute the value of the closed-loop gain A_CL. By using Equation 4 we obtain A_CL=5.99 while Equation 2 gives A_CL=6. We can remark that both values are very similar since A_OL is high. Typical values of A_OL for real op-amps range between 2×10⁴ and 2×10⁵, which is high enough to always consider Equation 2 valid.

From this value, we can simply say that the output voltage is given by V_out=A_CL×V_in= 6 V.

The currents I_R1 (across R₁) and I_R2 (across R₂) are approximately equal if we consider the leaking current in the inverting input to be much lower than the feedback current. Due to the virtual short existing at the node N, V_N=V_in, and therefore we have I_R1=I_R2=V_in/R₂=0.5 mA.

Since the current I_L through the output load is given by V_out/R_L=6 mA, we can determine the output current thanks to Kirchoff’s current law: I_out=I_L+I_R1=6.5 mA.

Finally, we can also specify the output impedance to be Z_out=V_out/I_out=920 Ω.

Conclusion

When the input signal is supplied to the pin “+”, the op-amp is said to be in a non-inverting configuration. The design and main properties of this configuration are presented in the first section that presents its ideal model.

In the second section, the real non-inverting op-amps are presented. Due to the parasitic phenomena that are intrinsic to their design, their properties change, the expression of the closed-loop gain, input, and output impedances are different. However, the simplified version of these formulas that describe the ideal model can indeed be recovered when we set the open-loop gain to be infinite.

Examples of real configurations are shown in the last section, we present how to calculate the main characteristics of a configuration with the knowledge of the resistors value and input voltage.

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