# Op-Amp Non-inverting Amplifier Circuit

In this article, “Op-Amp Non-nverting Amplifier Circuit” will be explained in detail.

The non-inverting amplifier circuit, like the inverting amplifier circuit, is one of the basic circuits in an op-amp.

## Op-Amp Non-inverting Amplifier Characteristics

**Non-inverting Amplifier Circuit**

$$V_{out}=\left(1+\frac{R_2}{R_1}\right)V_{in}$$

As shown in the equation above, the non-inverting amplifier circuit amplifies $V_{in}$ by a factor of $1+\frac{R_2}{R_1}$ and outputs it to $V_{out}$.

For example, if $R_1=1kΩ, R_2=10kΩ, V_{in}=1V$;

$$V_{out}=\left(1+\frac{10k}{1k}\right) \times 1=11[V]$$

- Output signal is input signal non-inverted
- Amplification factor is more than 1x
- Electrically unstable in operation
- Very high input impedance
- Output impedance is almost 0

### Output signal is input signal non-inverted

In the op-amp non-inverting amplifier circuit, the phase of the input and output signals are the same because the output signal is non-inverted of the input signal.

For example, if the input voltage is +, the output voltage will be +, and if the input voltage is -, the output voltage will be -.

### Amplification factor is more than 1x

Since the ratio of the op-amp non-inverting amplification circuit is $1+\frac{R_2}{R_1}$, the amplification factor is always more than one times.

However, by setting the resistor $R_1$ to open (∞Ω) and $R_2$ to short (0Ω), a voltage follower (a buffer with an amplification factor of 1x) can be designed.

### Electrically unstable in operation

The op-amp non-inverting amplifier circuit is electrically unstable in operation when compared to the op-amp inverting amplifier circuit.

In an op-amp, the $+$ and $-$ pins are virtually shorted inside the op-amp.

In the case of the op-amp non-inverting amplifier circuit, the operating point is swung by the voltage input to the $+$ pin, and the $-$ pin connected through a virtual short circuit is similarly affected.

Therefore, the operating point is swung by the input voltage, which may cause unstable operation when the input signal is high frequency.

### Very high input impedance

The op-amp non-inverting amplifier circuit has a very high input impedance because the input voltage is directly input to the $+$ pin.

(In general, both the $+$ and $-$ input pins of an op-amp have very high input impedance.)

Therefore, even if the impedance of the circuit connected to the input of the op-amp non-inverting amplifier circuit is somewhat high, the voltage drop will not occur with little effect. It is suitable for high-precision measurements.

### Output impedance is almost 0

The output impedance of the op-amp non-inverting amplifier circuit is kept at a voltage that satisfies the equation $V_{out}=\left(1+\frac{R_2}{R_1}\right)V_{in}$ by the negative feedback, so it is almost 0.

Therefore, it is almost unaffected by the input impedance of the circuit connected to the output pin of the op-amp, and the required signal level can be extracted without causing a voltage drop.

## Op-Amp Non-inverting Amplifier Equations(Formulas)

To obtain the equation for the op-amp non-inverting amplifier circuit, a calculation is made from the equation for the voltage of each part of the circuit and the ideal op-amp is replaced by the nullor model.

### Equation Example 1

**Op-Amp Schematic Symbol**

The op-amp amplifies the potential difference between the two input voltages, $V_+$ and $V_-$, with an open-loop gain $A_{OL}$.

$$V_{out}=A_{OL}(V_+-V_-)$$

**Non-inverting Amplifier Circuit**

In the case of the op-amp non-inverting amplifier circuit, $V_+$ and $V_{in}$ are the same.

$$V_+=V_{in}$$

As a result of the above, $V_{out}$ is:

$$V_{out}=A_{OL}(V_{in}-V_-)\cdots(1)$$

Also, since the $-$ pin of the inverting amplifier circuit has high input impedance and no current can flow through it, the circuit can be represented as shown below:

Furthermore, the voltage divider circuit can be changed for clarity as shown below:

Thus, $V_-$ is:

$$V_-=\frac{R_1}{R_1+R_2}V_{out}\cdots(2)$$

By substituting Eq.(2) into Eq.(1), $V_{out}$ is:

$$V_{out}=A_{OL}\left(V_{in}-\frac{R_1}{R_1+R_2}V_{out}\right)$$

$$V_{out}=A_{OL}V_{in}-\frac{R_1}{R_1+R_2}A_{OL}V_{out}$$

$$V_{out}+\frac{R_1}{R_1+R_2}A_{OL}V_{out}=A_{OL}V_{in}$$

$$\frac{R_1+R_2}{R_1+R_2}V_{out}+\frac{A_{OL}R_1}{R_1+R_2}V_{out}=A_{OL}V_{in}$$

$$\frac{R_1+R_2+A_{OL}R_1}{R_1+R_2}V_{out}=A_{OL}V_{in}$$

$$V_{out}=\frac{R_1+R_2}{R_1+R_2+A_{OL}R_1}A_{OL}V_{in}$$

$$V_{out}=\left(\frac{R_1+R_2}{\frac{R_1+R_2+A_{OL}R_1}{A_{OL}}}\right)V_{in}$$

$$V_{out}=\left(\frac{1}{\frac{R_1+R_2+A_{OL}R_1}{A_{OL}(R_1+R_2)}}\right)V_{in}$$

$$V_{out}=\left(\frac{1}{\frac{A_{OL}R_1}{A_{OL}(R_1+R_2)}+\frac{R_1+R_2}{A_{OL}(R_1+R_2)}}\right)V_{in}$$

$$V_{out}=\left(\frac{1}{\frac{R_1}{R_1+R_2}+\frac{1}{A_{OL}}}\right)V_{in}$$

Given that $A_{OL}$ is an extremely large value (∞), $\frac{1}{A_{OL}}⇒0$:

$$V_{out}=\left(\frac{1}{\frac{R_1}{R_1+R_2}}\right)V_{in}$$

$$V_{out}=\frac{R_1+R_2}{R_1}V_{in}$$

$$V_{out}=\left(1+\frac{R_2}{R_1}\right)V_{in}$$

### Equation Example 2

**Non-inverting Amplifier Circuit**

In the case of the op-amp non-inverting amplifier circuit, $V_+$ and $V_{in}$ are the same.

$$V_+=V_{in}$$

Furthermore, the $+$ and $-$ pins can be considered to be connected through a virtual short.

$$V_-=V_+=V_{in}\cdots(1)$$

Also, since the $-$ pin of the inverting amplifier circuit has high input impedance and no current can flow through it, the circuit can be represented as shown below:

Furthermore, the voltage divider circuit can be changed for clarity as shown below:

Thus, $V_-$ is:

$$V_-=\frac{R_1}{R_1+R_2}V_{out}\cdots(2)$$

By substituting Eq.(2) into Eq.(1), $V_{out}$ is:

$$V_{in}=\frac{R_1}{R_1+R_2}V_{out}$$

$$\frac{R_1}{R_1+R_2}V_{out}=V_{in}$$

$$V_{out}=\frac{R_1+R_2}{R_1}V_{in}$$

$$V_{out}=\left(1+\frac{R_2}{R_1}\right)V_{in}$$

### Equation Example 3(Using Nullor Model)

**Non-inverting Amplifier Circuit**

We replace the ideal op-amp with the nullor model and calculate the non-inverting amplifier circuit as shown below:

The nullor model can represent a virtual short in an op amp, which simplifies the calculation.

For a more detailed explanation of the nullor model, please refer to the following article.

**Non-inverting Amplifier Circuit**(Nullor Model)

We will try to change the above schematic for clarity.

From Kirchhoff’s Voltage Law(KVL), the relation between $V_{out}$, $V_1$ and $V_2$ is:

$$V_{out}=V_1+V_2$$

Therefore, we will obtain $V_1$ and $V_2$, respectively. First, $V_1$ is:

$$V_1=V_{in}\cdots(1)$$

Next, from Kirchhoff’s Current Law(KCL), $I_2=I_1$, so $V_2$ is:

$$V_2=R_2I_2=R_2I_1$$

Since $I_1$ is obtained as shown below:

$$I_1=\frac{V_1}{R_1}=\frac{V_{in}}{R_1}$$

Therefore, $V_2$ is:

$$V_2=R_2I_2=R_2I_1=\frac{R_2}{R_1}V_{in}\cdots(2)$$

By substituting Eq.(1) and Eq.(2) into $V_{out}=V_1+V_2$, the equation for the non-inverting amplifier circuit is:

$$V_{out}=V_1+V_2=V_{in}+\frac{R_2}{R_1}V_{in}$$

$$V_{out}=\left(1+\frac{R_2}{R_1}\right)V_{in}$$

## Other Op-Amp Circuit Examples

In this article, the “Op-Amp Non-inverting Amplifier Circuit” has been explained in detail, but there are various other circuits for op-amps as well.

Please refer to the following article for an introduction to the commonly used op-amp circuits.