Noninverting OPAMP
 Boris Poupet
 bpoupet@hotmail.fr
 15 min

4422Views
 0 Comments
Introduction
The voltage signal applied to an opamp can either be supplied to its noninverting input (+) or the inverting input (). These different configurations are simply known as a noninverting opamp, and inverting opamp. In this tutorial, we focus on the noninverting configuration and present its details.
An overview of the noninverting opamp will be given in the first section through the concept of the ideal amplifier.
In the second section, real noninverting configurations are discussed, we demonstrate the equations describing the gain and the input/output impedances.
Finally, examples of circuits based on the noninverting configurations are given in the last section.
Ideal noninverting opamp
The goal of this section is to properly demonstrate and explain the ideal characteristics of the noninverting configuration such as its input/output impedance and gain. The circuit representation of an ideal noninverting opamp is given in Figure 1 below.
Note that the symbol “∞” highlights the fact that the opamp is here to be considered ideal. We highly recommend the reader to refer to the tutorial Opamp basics for this section.
In this ideal model, the input impedance defined by the contribution of the resistance linking the inverting and noninverting inputs (R_{i} in Figure 3) and the resistors R_{1} and R_{2}, is infinite. Moreover, for an ideal circuit, R_{i} is supposed to be infinite, as a consequence, no currents can enter the opamp 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 R_{1} and R_{2} is a virtual short. For this same reason, all the feedback current across R_{1 }(I) is also found across R_{2}.
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 openloop 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 closedloop A_{CL} of an ideal noninverting 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 noninverting opamp
In a real opamp 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 nonzero value.
The noninverting configuration still remains the same as the one presented in Figure 1.
Closedloop gain
For a noninverting configuration, Equation 1 still applies for V_{– }, moreover, we have V_{+}=V_{in}. However, since a low current can flow from the noninverting 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 openloop 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 closedloop gain A_{CL} for a real noninverting configuration is given by Equation 4:
For a real configuration, the gain not only depends on the resistor values but also on the openloop gain
It is interesting to note that if we consider the opamp to be ideal (A_{OL}→+∞), the denominator is simplified to one term: A_{OL}R_{2}/(R_{1}+R_{2}). 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_{R}_{2}. Even if for real opamps, a small leaking current enters the inverting input, it is several orders of magnitude smaller than the feedback current.
The current I_{0} across R_{0} (see Figure 3) can be expressed as a function of the voltage drop across R_{0} and the same value of the impedance R_{0}:
Since V_{–} is described by Equation 1, the output current I_{out} can be expressed as the sum of I_{0} and the current flowing in the feedback branch given by V_{out}/(R_{1}+R_{2}):
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 noninverting configuration:
We can note that in the case of an ideal opamp, 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 R_{1}/(R_{1}+R_{2}). With that simplified version, we can still see that Z_{out}→0 for an ideal opamp situation.
Input impedance
The input impedance of a noninverting 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_{R2}R_{2}=0 with I_{R2} being the current across the resistor R_{2}.
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.
Noninverting opamp examples
Buffer circuits
The most simple designs for noninverting configurations are buffers, which have been described in the previous tutorial Opamp Building Blocks. In this configuration, R_{1}=0 and R_{2}→+∞ 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 noninverting configuration circuit given in Figure 5:
The resistors, input value, and gain in openloop are given such as:
 R_{1}=10 kΩ
 R_{2}=2 kΩ
 R_{L}=1 kΩ
 V_{in}=1 V
 A_{OL}=10^{5}
First of all, we can compute the value of the closedloop 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 opamps range between 2×10^{4 }and 2×10^{5}, 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 R_{1}) and I_{R}_{2} (across R_{2}) 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}/R_{2}=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 opamp is said to be in a noninverting 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 noninverting opamps are presented. Due to the parasitic phenomena that are intrinsic to their design, their properties change, the expression of the closedloop 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 openloop 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.