Op-Amp Gain: Inverting & Non-Inverting Configurations

Hello! Welcome back to your course on Radioelectronics.

Introduction

Approximate time to complete: 1 hour

In our last lesson, we established the characteristics of an ideal operational amplifier and the two "golden rules" for analyzing circuits that use them. Today, we'll put that theory into practice. Our goal is to analyze the two most fundamental op-amp circuits: the inverting amplifier and the non-inverting amplifier. We will derive the equation for the voltage gain for each configuration.

Recap from the previous lesson:

When analyzing an ideal op-amp circuit with a negative feedback connection, we apply two simple but powerful rules:

The inputs draw no current: $I_{+} = I_{-} = 0$ . This is due to the infinite input resistance.
The voltage difference between the inputs is zero: $V_{+} = V_{-}$ . This is a consequence of the infinite open-loop gain working to counteract any difference.

This lesson will first give you the intuition behind why negative feedback leads to these rules and then use them to derive the gain for our two main configurations. Your background in Radiophysics will make the underlying circuit theory (like KVL, KCL, and voltage dividers) very familiar.

The Power of Negative Feedback

Before we jump into specific circuits, it's incredibly valuable to understand how negative feedback tames the op-amp's enormous open-loop gain to produce a stable, controllable output. The video below provides an excellent conceptual and graphical explanation that justifies the "golden rules" we rely on.

Please watch this detailed segment. It will connect the ideal theory from our last lesson to the practical analysis we're about to perform.

Watch from 00:13:21 to 00:51:19. Pay close attention to these key ideas:

The Problem (29:39): Why an op-amp's huge open-loop gain makes it saturate and act like a comparator without feedback.
The Solution (33:14): How a feedback loop connects a fraction of the output back to the inverting input.
The Mechanism (36:50): The graphical analysis showing how the op-amp automatically adjusts its output voltage to force the differential input voltage ( $V_{+} - V_{-}$ ) to become vanishingly small. This is the core intuition behind the second golden rule.
The Rules Formalized (44:07): How the characteristics of an ideal op-amp (infinite gain, infinite input resistance) lead directly to the two golden rules when negative feedback is applied.

This deep dive solidifies why our simplified analysis model works so well. Now, let's apply it.

The Non-Inverting Amplifier

The non-inverting amplifier is a configuration where the output signal is in phase with the input signal. The input signal ( $V_{S}$ ) is applied directly to the non-inverting (+) input.

Please watch from 00:51:19 to 00:58:18. This segment walks through the derivation of the circuit's gain.

Here is a breakdown of the analysis steps, which apply our golden rules:

Analyze the non-inverting input (+): The input voltage source $V_{S}$ is connected directly to the non-inverting terminal. Therefore: $V_{+} = V_{S}$
Apply Golden Rule #2: The op-amp forces the voltage at the inverting terminal to be equal to the voltage at the non-inverting terminal. $V_{-} = V_{+} = V_{S}$
Analyze the feedback network: The resistors $R_{F}$ (feedback) and $R_{G}$ (to ground) form a simple voltage divider. The voltage at the inverting terminal, $V_{-}$ , is the output of this divider, with $V_{o u t}$ as its input. Using the voltage divider formula: $V_{-} = V_{o u t} (\frac{R _{G}}{R _{G} + R _{F}})$
Combine and solve for gain: We now have two expressions for $V_{-}$ . Let's set them equal to each other: $V_{S} = V_{o u t} (\frac{R _{G}}{R _{G} + R _{F}})$ Rearranging to find the closed-loop gain, $A_{C L} = \frac{V _{o u t}}{V _{S}}$ , we get: $A_{C L} = \frac{V _{o u t}}{V _{S}} = \frac{R _{G} + R _{F}}{R _{G}} = 1 + \frac{R _{F}}{R _{G}}$

Key Characteristics of the Non-Inverting Amplifier:

Gain: The voltage gain is determined by the ratio of the two external resistors and is always greater than or equal to 1.
Phase: The output voltage is in phase with the input voltage.
Input Impedance: The input signal is connected directly to the op-amp's non-inverting terminal, which has infinite input impedance. This makes it an excellent buffer, as it doesn't load the source circuit.

The Inverting Amplifier

In the inverting configuration, the output signal is 180° out of phase with the input signal. Here, the input signal is applied through a resistor ( $R_{i}$ ) to the inverting (-) input, and the non-inverting (+) input is connected to ground.

Now watch from 01:02:20 to 01:08:09 for the derivation of the inverting amplifier's gain.

The analysis for this circuit uses Kirchhoff's Current Law (KCL), leveraging both golden rules.

Analyze the non-inverting input (+): The non-inverting terminal is connected to ground. $V_{+} = 0$
Apply Golden Rule #2: This creates a crucial condition at the inverting terminal. $V_{-} = V_{+} = 0$ This point is known as a virtual ground. It's at 0V potential, but it's not directly connected to ground. The op-amp actively holds it at this voltage.
Apply KCL at the inverting node ( $V_{-}$ ): According to Golden Rule #1, no current flows into the op-amp. Therefore, the current flowing through $R_{i}$ must be equal to the current flowing through $R_{F}$ . $I_{i} = - I_{F} or I_{i} + I_{F} = 0$ (The video assumes both currents flow into the node, leading to the second form.)
Express currents using Ohm's Law:
- The current through $R_{i}$ is $I_{i} = \frac{V _{S} - V _{-}}{R _{i}} = \frac{V _{S} - 0}{R _{i}} = \frac{V _{S}}{R _{i}}$ .
- The current through $R_{F}$ is $I_{F} = \frac{V _{o u t} - V _{-}}{R _{F}} = \frac{V _{o u t} - 0}{R _{F}} = \frac{V _{o u t}}{R _{F}}$ .
Combine and solve for gain: Substitute these into the KCL equation: $\frac{V _{S}}{R _{i}} + \frac{V _{o u t}}{R _{F}} = 0$ Solving for the closed-loop gain, $A_{C L} = \frac{V _{o u t}}{V _{S}}$ : $A_{C L} = \frac{V _{o u t}}{V _{S}} = - \frac{R _{F}}{R _{i}}$

Key Characteristics of the Inverting Amplifier:

Gain: The magnitude of the voltage gain is set by the ratio $\frac{R _{F}}{R _{i}}$ . It can be less than, equal to, or greater than 1.
Phase: The negative sign indicates a 180° phase inversion between the output and input.
Input Impedance: The input source "sees" the resistor $R_{i}$ as the input impedance, because the $V_{-}$ node is a virtual ground.

Conclusion

In this lesson, you've seen how to apply the two golden rules of ideal op-amp analysis to derive the gain for the two most essential amplifier configurations.

Key Takeaways:

Negative feedback is the mechanism that makes op-amps behave as stable, linear amplifiers by forcing the differential input voltage to zero.
The non-inverting amplifier provides a gain of $A_{C L} = 1 + \frac{R _{F}}{R _{G}}$ , has a very high input impedance, and does not invert the signal.
The inverting amplifier provides a gain of $A_{C L} = - \frac{R _{F}}{R _{i}}$ , creates a virtual ground at its inverting input, and inverts the signal by 180°. Its input impedance is equal to $R_{i}$ .

These two configurations are the fundamental building blocks for many other op-amp circuits. In our next lesson, "Op-Amp Applications: Summing and Difference Amplifiers," we will see how the principles of the inverting amplifier can be extended to create circuits that perform mathematical addition and subtraction.