9.2.3.2. Amplifiers

Table of Contents

Basic idea of amplification

An amplifier is a circuit that takes a small input signal and produces a larger output signal. In operational amplifier circuits, the op amp itself is the active device, but the amount of amplification is usually controlled by external components, most often resistors.

The important point is that an op amp by itself has a very large internal gain, but useful amplifier circuits are made by using feedback. Feedback lets us set a predictable gain and makes the circuit more stable and practical.

If the input signal is a voltage, then the amplifier usually aims to produce an output voltage proportional to that input. This proportionality is called the voltage gain.

For a linear amplifier,
$$
A_v = \frac{V_{\text{out}}}{V_{\text{in}}}
$$
where $A_v$ is the voltage gain.
If $A_v$ is positive, output and input have the same sign.
If $A_v$ is negative, the output is inverted.

Why op amp amplifiers need feedback

An ideal op amp has extremely large open loop gain. That means even a tiny difference between its two input terminals would try to drive the output to a very large value. In practice, without feedback, the output usually saturates near the power supply limits. That is not useful for ordinary signal amplification.

By connecting part of the output back to the input, we create negative feedback. This causes the op amp to adjust its output so that the voltage difference between the two input terminals becomes very small.

In many ideal op amp amplifier circuits, we use two key rules. These belong to ideal op amp behavior, but they are especially important here because they make gain calculations simple.

For an ideal op amp operating with negative feedback,
$$
I_+ = I_- = 0
$$
and
$$
V_+ \approx V_-
$$
This second relation is often called the virtual short condition.

Inverting amplifier

One of the most common op amp amplifier circuits is the inverting amplifier. The input signal goes through a resistor to the inverting input, and another resistor connects the output back to that same input. The noninverting input is connected to ground.

Because the noninverting input is at $0 \, \text{V}$, the inverting input is also held very close to $0 \, \text{V}$ by negative feedback. This point is often called a virtual ground. It is not physically connected to ground, but its voltage is approximately zero.

Using current conservation at the inverting input node gives the gain formula.

$$
\frac{V_{\text{in}} - 0}{R_{\text{in}}} = \frac{0 - V_{\text{out}}}{R_f}
$$

So,

$$
V_{\text{out}} = -\frac{R_f}{R_{\text{in}}} V_{\text{in}}
$$

For the inverting amplifier,
$$
A_v = \frac{V_{\text{out}}}{V_{\text{in}}} = -\frac{R_f}{R_{\text{in}}}
$$
The negative sign means the output is inverted.

Rendering scientific sketch…

Inverting op amp amplifier

Noninverting amplifier

Another very important circuit is the noninverting amplifier. Here the input signal is applied directly to the noninverting input. The inverting input receives a fraction of the output through a resistor network.

The op amp adjusts the output so that the inverting input voltage matches the noninverting input voltage. If the noninverting input is at $V_{\text{in}}$, then the inverting input is also approximately at $V_{\text{in}}$.

If the resistor from output to the inverting input is $R_f$, and the resistor from the inverting input to ground is $R_1$, then the inverting input sees a divided version of the output:

$$
V_- = V_{\text{out}} \frac{R_1}{R_1 + R_f}
$$

Since $V_- \approx V_+ = V_{\text{in}}$,

$$
V_{\text{in}} = V_{\text{out}} \frac{R_1}{R_1 + R_f}
$$

So,

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

For the noninverting amplifier,
$$
A_v = 1 + \frac{R_f}{R_1}
$$
This gain is positive, so the output is not inverted.

Rendering scientific sketch…

Noninverting op amp amplifier

Voltage follower

A special case of the noninverting amplifier is the voltage follower, also called a buffer. In this circuit, the output is connected directly to the inverting input, with no resistor divider. The input signal goes to the noninverting input.

This gives a gain of 1:

$$
V_{\text{out}} = V_{\text{in}}
$$

Why is this useful if it does not increase voltage? Because it can isolate one part of a circuit from another. An ideal op amp has very high input resistance and low output resistance, so the buffer can pass a voltage from one stage to the next without significantly loading the source.

For a voltage follower,
$$
A_v = 1
$$
It does not amplify voltage, but it improves signal transfer between circuits.

Comparing common op amp amplifier types

The three circuits above are the foundation of many analog electronics applications.

Amplifier type	Input terminal used	Gain formula	Inverts signal
Inverting	Inverting input through resistor	$-\dfrac{R_f}{R_{\text{in}}}$	Yes
Noninverting	Noninverting input directly	$1 + \dfrac{R_f}{R_1}$	No
Voltage follower	Noninverting input directly	$1$	No

Input and output limitations

Real op amp amplifiers cannot produce any output voltage whatsoever. The output is limited by the supply voltages. If the circuit tries to produce a larger output than the power supply allows, the output saturates.

For example, if an op amp is powered by $\pm 12 \, \text{V}$, the output cannot usually go beyond those limits, and often not quite all the way to them. So even if the gain formula predicts a certain output, the actual output will flatten if that value is too large.

Suppose an amplifier has gain $10$ and the input is $2 \, \text{V}$. The predicted output is

$$
V_{\text{out}} = 10 \times 2 = 20 \, \text{V}
$$

If the supply does not allow $20 \, \text{V}$, the circuit saturates instead of remaining linear.

Amplifier formulas such as
$$
V_{\text{out}} = A_v V_{\text{in}}
$$
are valid only when the op amp remains in its linear operating region and does not saturate.

Gain in decibels

Amplifier strength is often expressed using decibels, especially when comparing signal levels.

For voltage gain, when the same resistance conditions apply at input and output, the gain in decibels is

$$
G_{\text{dB}} = 20 \log_{10} \left| A_v \right|
$$

If $A_v = 10$, then

$$
G_{\text{dB}} = 20 \log_{10}(10) = 20 \, \text{dB}
$$

If $A_v = 2$, then

$$
G_{\text{dB}} \approx 20 \log_{10}(2) \approx 6.0 \, \text{dB}
$$

This does not replace the ordinary gain formula, but it is a convenient way to describe amplification.

A simple example

Consider an inverting amplifier with $R_{\text{in}} = 2 \, \text{k}\Omega$ and $R_f = 10 \, \text{k}\Omega$.

Its gain is

$$
A_v = -\frac{R_f}{R_{\text{in}}} = -\frac{10}{2} = -5
$$

If the input is

$$
V_{\text{in}} = 0.3 \, \text{V}
$$

then the output is

$$
V_{\text{out}} = -5(0.3) = -1.5 \, \text{V}
$$

The output is five times larger in magnitude and opposite in sign.

Now consider a noninverting amplifier with $R_1 = 1 \, \text{k}\Omega$ and $R_f = 4 \, \text{k}\Omega$.

Its gain is

$$
A_v = 1 + \frac{R_f}{R_1} = 1 + \frac{4}{1} = 5
$$

With the same input, $V_{\text{in}} = 0.3 \, \text{V}$,

$$
V_{\text{out}} = 5(0.3) = 1.5 \, \text{V}
$$

This time the output keeps the same sign as the input.

Practical meaning

In real electronics, amplifier circuits are used to boost weak sensor signals, prepare audio signals, and connect circuit stages without unwanted loading effects. The op amp makes this possible with simple resistor networks that set the desired gain.

The most important skill at this stage is to recognize the circuit form and apply the correct gain relation. Later circuits are often built from these same amplifier ideas.

Key amplifier formulas for ideal op amp circuits:
Inverting amplifier:
$$
V_{\text{out}} = -\frac{R_f}{R_{\text{in}}} V_{\text{in}}
$$
Noninverting amplifier:
$$
V_{\text{out}} = \left(1 + \frac{R_f}{R_1}\right) V_{\text{in}}
$$
Voltage follower:
$$
V_{\text{out}} = V_{\text{in}}
$$

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