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Electrical Circuits and Sensors · Module 7 of 10

Operational Amplifiers

A sensor's output is often a few millivolts: too small to read. The operational amplifier scales, adds, and subtracts voltages with a handful of resistors, and two simple rules let you analyse almost any of them.

01

Readiness check

This module applies your circuit laws to an active element. Tick only what you can do closed-notes.

  • Write KCL at a node.
  • Use Ohm's law on a resistor between two known voltages.
  • Recognise negative feedback in a circuit.
  • Form a ratio of two resistances.
  • Recall that a sensor signal may be only millivolts.
0 or 1 weak itemsContinue with this module.
2 weak itemsRevisit KCL at a node in Module 2.
3 or more weak itemsRebuild nodal analysis from Module 3.
02

The core idea

An ideal op-amp with negative feedback draws no input current and forces its two inputs to the same voltage. Those two rules, applied with KCL, give the gain of any standard amplifier from its resistors alone.

no input current: i+ = i− = 0virtual short: V+ = V−inverting gain = −Rf/Rin

The operational amplifier is a high-gain difference amplifier. On its own its gain is enormous, but wrapped in negative feedback it becomes precise and predictable. Two idealisations carry almost all the analysis. First, the input terminals draw no current, because the input resistance is effectively infinite. Second, the feedback drives the difference between the inputs to zero, so the two input voltages are equal, a virtual short. When the non-inverting input is grounded, the inverting input sits at zero too, a virtual ground. Applying KCL at that node, with no current entering the op-amp, the output voltage is fixed entirely by the external resistors. The same two rules give the inverting, non-inverting, summing, and difference amplifiers.

The skill works when: you set the input currents to zero and the inputs equal, then write KCL at the inverting node.
The skill breaks down when: current is assumed to flow into the op-amp, or the virtual short is used without negative feedback.
The concept. The inverting input is held at a virtual ground by feedback. With no current into the op-amp, the input current Rin carries flows on through Rf, setting Vout = −(Rf/Rin)Vin.
03

The skills, taught in order

Five skills build from the ideal model to the four amplifiers a sensor circuit needs most.

7.1 The ideal op-amp

The ideal op-amp has infinite open-loop gain, infinite input resistance, and zero output resistance. The first means a tiny input difference drives a large output; the second means no current enters the inputs. With negative feedback these give the two working rules.

7.2 The virtual short and virtual ground

Negative feedback forces V+ = V−, the virtual short. If the non-inverting input is grounded, the inverting input is held at 0 V, a virtual ground, even though no wire connects it to ground. This node is where KCL is written.

7.3 The inverting amplifier

With the input through Rin to the virtual ground and feedback through Rf, KCL gives Vout = −(Rf/Rin)Vin. The gain is negative (the output is inverted) and set purely by the resistor ratio.

ConfigurationGainSignUse
Inverting−Rf/Rininvertedscaling, summing
Non-inverting1 + Rf/R1samebuffering, gain ≥ 1
Summing−Σ Rf/Rkinvertedmixing signals
DifferenceRf/R1 (matched)differentialbridge, sensor

The four standard amplifiers, each derived from the same two rules applied with KCL.

7.4 The non-inverting amplifier

Feeding the signal to the non-inverting input gives Vout = (1 + Rf/R1)Vin: a positive gain, always at least one, with a very high input resistance that barely loads the source, ideal for reading a sensor.

7.5 Summing and difference amplifiers

The summing amplifier adds several inputs, each scaled by its own resistor, at one virtual ground. The difference amplifier, with matched resistor pairs, outputs a gain times the difference of two inputs, the heart of reading a Wheatstone bridge.

Engineering connection: the instrumentation amplifier that reads a strain-gauge bridge, in Module 9, is a difference amplifier built from these rules.

04

Worked example 1: an inverting amplifier

An inverting amplifier has Rin = 10 kΩ and Rf = 50 kΩ. The input is 2 V. Find the gain and the output voltage.

Figure 1. The 5-to-1 resistor ratio sets a gain of −5. A 2 V input produces a −10 V output: amplified and inverted.
  1. ProblemFind the gain and output of the inverting amplifier in Figure 1.
  2. Given / findRin = 10 kΩ, Rf = 50 kΩ, Vin = 2 V. Find the gain and Vout.
  3. AssumptionsIdeal op-amp; negative feedback; non-inverting input grounded.
  4. ModelThe inverting node is a virtual ground; KCL gives Vout = −(Rf/Rin)Vin.
  5. Equationsgain = −Rf/RinVout = gain × Vin
  6. Solvegain = −50/10 = −5. Vout = −5 × 2 = −10 V.
  7. CheckThe input current is 2/10 kΩ = 0.2 mA; it flows through Rf, dropping 0.2 mA × 50 kΩ = 10 V below the virtual ground, so Vout = −10 V.
  8. ConclusionThe output is five times the input and inverted, set entirely by the resistor ratio, not by the op-amp's own gain.
Result. gain = −5 and Vout = −10 V.
05

Worked example 2: a non-inverting amplifier

A non-inverting amplifier has R1 = 4 kΩ from the inverting input to ground and Rf = 20 kΩ as feedback. The input is 1.2 V. Find the gain and the output voltage.

Figure 2. Feeding the signal to the non-inverting input gives a positive gain of 1 + 20/4 = 6. A 1.2 V input becomes 7.2 V, upright and amplified.
  1. ProblemFind the gain and output of the non-inverting amplifier in Figure 2.
  2. Given / findR1 = 4 kΩ, Rf = 20 kΩ, Vin = 1.2 V. Find the gain and Vout.
  3. AssumptionsIdeal op-amp; negative feedback; input drives the non-inverting terminal.
  4. ModelThe virtual short puts Vin at the R1-Rf divider node; gain = 1 + Rf/R1.
  5. Equationsgain = 1 + Rf/R1Vout = gain × Vin
  6. Solvegain = 1 + 20/4 = 1 + 5 = 6. Vout = 6 × 1.2 = 7.2 V.
  7. CheckThe divider from Vout across R1 must equal Vin: 7.2 × 4/(4 + 20) = 7.2 × 4/24 = 1.2 V, matching the input at the virtual short.
  8. ConclusionThe non-inverting amplifier gives a positive gain of 6 and presents a very high input resistance, so it reads a source without loading it.
Result. gain = 6 and Vout = 7.2 V.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Current into the op-ampKCL includes an input current"What is the input resistance?"Ideal inputs draw no current.
Virtual short without feedbackInputs set equal in a comparator"Is there negative feedback?"The virtual short needs negative feedback.
Sign dropped on inverting gainOutput reported positive"Is this inverting or non-inverting?"Inverting gain is −Rf/Rin.
Non-inverting gain below 1Gain given as Rf/R1"Did I add the 1?"Non-inverting gain is 1 + Rf/R1.
07

Practice ladder

Level 1 · Direct skill

An inverting amplifier has Rin = 2 kΩ and Rf = 8 kΩ. Find the gain.

Show answer

gain = −Rf/Rin = −8/2 = −4.

Level 2 · Mixed concept

A non-inverting amplifier needs a gain of exactly 10. With R1 = 1 kΩ, find Rf.

Show answer

gain = 1 + Rf/R1 = 10, so Rf/R1 = 9, Rf = 9 kΩ.

Level 3 · Independent problem

A summing amplifier with Rf = 12 kΩ has two inputs: 1 V through 6 kΩ and 2 V through 4 kΩ. Find the output.

Show answer

Vout = −(Rf/R1 · V1 + Rf/R2 · V2) = −(12/6 · 1 + 12/4 · 2) = −(2 + 6) = −8 V.

Transfer task | Real engineering

A thermocouple gives 6 mV at full scale, and your data logger needs about 3 V. Choose an amplifier and resistor ratio to bridge the gap.

What good work looks like

A gain of about 3/0.006 = 500 is needed. A non-inverting amplifier with, say, R1 = 1 kΩ and Rf = 499 kΩ gives 1 + 499 = 500, scaling 6 mV to 3 V while barely loading the thermocouple.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I applied the virtual ground and no input current correctly."
"Give me three op-amp circuits; I will identify each configuration and its gain."
"Find Vout for me." Writing KCL at the virtual ground is the skill.
"Which resistor sets the gain?" Deriving the gain yourself is the point.

Portfolio task

Design an amplifier to scale a real sensor's output to a target range, then verify the gain by computing Vout at two input values.

Must include: the configuration, the resistor ratio, the gain, and two checked output points.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. State the two ideal op-amp rules.

No current enters the inputs, and negative feedback makes V+ = V−.

2. What is a virtual ground?

The inverting input held at 0 V by feedback when the non-inverting input is grounded.

3. Give the inverting gain.

−Rf/Rin.

4. Give the non-inverting gain.

1 + Rf/R1, always at least one.

5. What does a difference amplifier output?

A gain times the difference of its two inputs, used to read a bridge.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-derive the inverting and non-inverting gains from a blank page.
+3 daysAnalyse a summing and a difference amplifier.
+7 daysCarry amplification into sensors and transducers, Module 8.
+30 daysReuse the difference amplifier to read a Wheatstone bridge.
10

Textbook mapping

This module follows Alexander and Sadiku, Fundamentals of Electric Circuits, 4th edition. Use these references to read further.

Topic in this moduleWhere to read more
The ideal op-amp and feedbackAlexander & Sadiku, Chapter 5
Inverting and non-inverting amplifiersAlexander & Sadiku, Chapter 5
Summing and difference amplifiersAlexander & Sadiku, Chapter 5

Chapter numbers refer to the 4th edition. The op-amp configurations are standard, so any recent edition will align closely.