Inverting Amplifier Analysis

College Depth 164 in the knowledge graph ☐ I know this ☆ Set as goal

Core Idea

The inverting amplifier connects the input signal through a resistor R_in to the op-amp's inverting terminal, with a feedback resistor R_f from output to inverting input, while the non-inverting terminal is grounded. Negative feedback forces the inverting input to virtual ground (0 V), so the input current I_in = V_in / R_in flows entirely through R_f (since no current enters the op-amp input), producing V_out = -I_in * R_f = -(R_f / R_in) * V_in. The closed-loop gain is A_v = -R_f / R_in, with the negative sign indicating phase inversion. Input impedance equals R_in (not infinite, because the inverting terminal is at virtual ground, not floating), which is a key practical consideration — the source must drive current into R_in. This configuration is the basis for the summing amplifier (multiple input resistors to the inverting node) and the transimpedance amplifier (current input, voltage output). Practical limitations include finite open-loop gain (causing gain error), finite bandwidth (gain-bandwidth product limits usable frequency range), and output voltage swing limited by supply rails.

How It's Best Learned

Derive the gain formula from first principles using the virtual ground and virtual open rules rather than memorizing it. Then re-derive it including finite open-loop gain A_OL to see how the ideal formula emerges as A_OL approaches infinity and to quantify the gain error for realistic op-amps. Compare input impedance to the non-inverting configuration to understand why the choice between topologies matters for high-impedance sources.

Common Misconceptions

Assuming the inverting amplifier has infinite input impedance like the non-inverting configuration — because the inverting input is at virtual ground, the source sees R_in as the load, which can be problematically low for high-gain designs (small R_in).
Believing virtual ground means the inverting node is physically connected to ground — it is held at ground potential by the feedback loop, but current flows through R_f to the output, not to ground.
Ignoring the gain-bandwidth product constraint — an inverting amplifier with gain of -100 using an op-amp with GBW of 1 MHz has a usable bandwidth of only 10 kHz.

Explainer

From your work on op-amp fundamentals, you know the two golden rules that make ideal op-amp analysis tractable: no current enters the input terminals (virtual open), and negative feedback drives the differential input voltage to zero (virtual short, or in this circuit, virtual ground). The inverting amplifier is the circuit where these rules produce a result that initially seems paradoxical: the inverting input is held at 0 V even though it is not physically connected to ground.

Here is how the feedback loop creates virtual ground. Suppose the input V_in is positive. Current flows through R_in toward the inverting terminal. If that terminal were truly floating, voltage would build up there and the output would swing negative. But the op-amp's output swings negative in response — and the feedback resistor R_f connects that negative output back to the inverting terminal, pulling its voltage back toward zero. The feedback loop continuously corrects any departure from 0 V at the inverting node. In the ideal case (infinite open-loop gain), the correction is perfect: the inverting input is held exactly at 0 V. This is virtual ground — not a physical connection, but an enforced potential.

Once you accept virtual ground, the gain derivation follows directly from Ohm's Law. The current through R_in is I = V_in / R_in (since the inverting node is at 0 V). Because no current enters the op-amp input terminal itself (virtual open), this entire current must flow through R_f. The voltage at the output is V_out = 0 − I × R_f = −(R_f / R_in) × V_in. The gain magnitude is R_f / R_in, and the negative sign reflects the phase inversion. Notice what sets the gain: not the op-amp itself, but the external resistor ratio. The op-amp's job is to provide enough open-loop gain that the feedback loop enforces virtual ground; the precision of the closed-loop gain depends on the resistors, not on the exact value of the op-amp's open-loop gain (as long as it is large).

The practical tradeoff to internalize is the input impedance penalty. In a non-inverting configuration, the signal feeds directly into the op-amp's input (near-infinite impedance). In the inverting configuration, the source sees R_in as its load, because the inverting terminal is held at virtual ground — any current into the node flows through R_in to the virtual ground, so the source must supply it. High-gain inverting designs require small R_in (to keep R_f manageable), which loads down high-impedance sources. This is why you'll encounter the non-inverting configuration for sensors and low-impedance-sensitive applications, while the inverting configuration is preferred when gain accuracy, signal summation (summing amplifier), or current-to-voltage conversion (transimpedance amplifier) are the priority.

Practice Questions 5 questions

Prerequisite Chain

Counting to 10 → Counting to 20 → Understanding Zero → The Number Zero → Counting to Five → One-to-One Correspondence → Combining Small Groups Within 5 → Addition Within 10 → Addition Within 20 → Two-Digit Addition Without Regrouping → Two-Digit Addition with Regrouping → Addition Within 100 → Repeated Addition as Multiplication → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Two-Digit by One-Digit Division → Division with Remainders → Remainders and Quotients in Division → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Reading and Writing Decimals → Comparing and Ordering Decimals → Adding and Subtracting Decimals → Multiplying Decimals → Dividing Decimals → Dividing Fractions → Mixed Number Arithmetic → Order of Operations → Integer Order of Operations → Variable Expressions → Combining Like Terms → One-Step Equations → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Iterated Integrals and Fubini's Theorem → Double Integrals in Cartesian Coordinates → Double Integrals over Rectangular Regions → Double Integrals in Polar Coordinates → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Green's Theorem → Surface Integrals and Flux of Vector Fields → Surface Integrals and Flux of Vector Fields → Divergence Theorem: Flux and Outflow → Divergence Theorem → Electric Flux → Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → The Electromagnetic Spectrum → Blackbody Radiation and Planck's Law → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → Heisenberg Uncertainty Principle → Wavefunction and the Born Rule → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → Quantum Entanglement → Bell Theorem and Bell Inequalities → Postulates of Quantum Mechanics → Scattering Theory → Introduction to Scattering Theory → Partial Wave Analysis in Scattering → Spin Angular Momentum → Electron Spin and Intrinsic Magnetic Moment → Stern-Gerlach Experiment: Spin Quantization and Measurement → Electron Diffraction and Matter Wave Properties → Davisson-Germer Experiment: Crystal Diffraction of Electrons → Electron Diffraction and Matter Wave Interference → Wavefunctions and Probability Density Interpretation → Quantum Superposition and Linear Combinations of States → Quantum Operators and Observables → Canonical Commutation Relations and Uncertainty → Heisenberg Uncertainty Principle and Measurement Limits → Time-Independent Schrödinger Equation and Eigenvalues → Hydrogen Atom in Quantum Mechanics → Spectral Lines and Energy Transitions → Selection Rules for Atomic Transitions → LS and jj Coupling Schemes in Multi-Electron Atoms → Pauli Exclusion Principle and Antisymmetric Wavefunctions → Electron Configuration and the Aufbau Principle → The Periodic Table and Atomic Electronic Structure → The Periodic Table → Electron Configuration → Periodic Trends → Ionization Energy → Ionic Bonding → Lewis Structures → Polar Covalent Bonds and Dipole Moments → Classification of Bonds: Ionic, Covalent, and Metallic → Metallic Bonding and Properties of Metals → Crystal Structures and Solid Properties → Crystal Structure and Unit Cells → Electrical Properties of Materials → Diode Characteristics and Models → Diode Circuit Applications → Bipolar Junction Transistor (BJT) Fundamentals → MOSFET Fundamentals → Operational Amplifier Fundamentals → Op-Amp Circuit Applications → Inverting Amplifier Analysis

Longest path: 165 steps · 911 total prerequisite topics

Prerequisites (2)

Operational Amplifier Fundamentalshard Op-Amp Circuit Applicationssoft

Leads To (0)

No topics depend on this one yet.