Understanding Op Amp Equation Derivation: A Detailed Guide for You

Operational amplifiers, or op-amps, are fundamental components in electronic circuits. They are versatile and widely used in various applications due to their high input impedance, low output impedance, and high gain. One of the key aspects of understanding op-amps is mastering the derivation of their equations. In this article, we will delve into the derivation of op-amp equations, providing you with a comprehensive guide to help you grasp this concept better.

What is an Operational Amplifier?

An operational amplifier is an electronic device that amplifies the difference between two input voltages. It has two inputs, a non-inverting input (+) and an inverting input (-), and one output. The op-amp is designed to have a very high gain, which means that even a small difference between the input voltages can result in a large output voltage.

The basic symbol of an op-amp is shown below:

Op-Amp Equation Derivation: The Basics

The derivation of op-amp equations starts with the basic principles of circuit analysis. Let’s consider a simple inverting amplifier circuit, which is one of the most common configurations of an op-amp.

In an inverting amplifier, the input voltage is applied to the inverting input terminal, and the output is taken from the output terminal. The non-inverting input terminal is connected to ground. The basic circuit diagram is shown below:

Now, let’s derive the equation for this circuit. We will use the virtual ground concept, which states that the voltage at the inverting input terminal is equal to the voltage at the non-inverting input terminal when the op-amp is in a linear region.

Let’s assume that the input voltage is Vi and the output voltage is Vo. According to the virtual ground concept, the voltage at the inverting input terminal (V-) is equal to the voltage at the non-inverting input terminal (V+), which is connected to ground. Therefore, V- = 0V.

Now, let’s apply Kirchhoff’s voltage law (KVL) to the feedback loop. The voltage across the feedback resistor (Rf) is equal to the voltage across the input resistor (Ri) multiplied by the gain of the op-amp (A). Mathematically, this can be expressed as:

Vo = A (Vi – V-)

Since V- = 0V, the equation simplifies to:

Vo = A Vi

Now, let’s express the gain of the op-amp in terms of the feedback resistor and the input resistor. The gain (A) is defined as the ratio of the output voltage (Vo) to the input voltage (Vi). Mathematically, this can be expressed as:

A = Vo / Vi

Substituting the value of Vo from the previous equation, we get:

A = (A Vi) / Vi

Canceling out the Vi terms, we get:

A = A

This equation tells us that the gain of the op-amp is equal to the ratio of the feedback resistor (Rf) to the input resistor (Ri). Mathematically, this can be expressed as:

A = Rf / Ri

Therefore, the equation for the inverting amplifier is:

Vo = -A Vi = -Rf / Ri Vi

This equation shows that the output voltage is the negative of the input voltage multiplied by the ratio of the feedback resistor to the input resistor.

Op-Amp Equation Derivation: Non-Inverting Amplifier

Now, let’s derive the equation for a non-inverting amplifier, which is another common configuration