Op-Amps: The Virtual Short and Essential Circuits

The operational amplifier is a near-magical IC: infinite input impedance, zero output impedance, and gain of a million. With one simple rule — the virtual short — you can analyze every op-amp circuit by algebra alone. Understanding the inverting amplifier, integrator, active filter, and comparator covers 90% of analog design.

1. The Ideal Op-Amp

An operational amplifier has two inputs (inverting (−) and non-inverting (+)) and one output. The ideal model has:

Infinite open-loop gain A_OL: V_out = A_OL × (V₊ − V₋). Real LM741: 200,000. Real OPA827: 20,000,000.
Infinite input impedance: No current flows into either input terminal. This allows the op-amp to measure voltage without loading the source.
Zero output impedance: Output can drive any load without voltage drop. Real ICs: 10–100 Ω; external buffers used for heavy loads.
Infinite bandwidth: Gain applies at all frequencies. Real: gain-bandwidth product (GBW) is constant — LM741 has GBW = 1 MHz; TL071: 3 MHz; OPA657: 1.6 GHz.

Open-loop, even a 1 µV input difference produces a saturated output. Op-amps are almost never used open-loop except as comparators — they're used with negative feedback, which is what makes them useful and predictable.

2. The Virtual Short Principle

When an op-amp is connected with negative feedback (output connected back to the inverting input through a network), it will do whatever necessary to make the voltage difference between its inputs zero:

V+ = V- (virtual short) I+ = I- = 0 (no input current)

These two rules — along with Kirchhoff's current law — are all you need to analyze any linear op-amp circuit:

Identify V+ from the circuit (usually a voltage divider or direct connection)
Set V− = V+ (virtual short)
Apply KCL at the V− node to find V_out

The op-amp adjusts V_out through the feedback network to enforce V+ = V−. The output is the op-amp "working to make both inputs equal." This is feedback control — the same principle as a PID controller's integral action.

Why "virtual": The inputs are not electrically connected — there's no real short. The op-amp's high gain and feedback create the effect of equal voltages without current flow between inputs.

3. Inverting Amplifier

Circuit: V_in connects to V− through R₁. V_out connects to V− through R_f. V+ is grounded (0 V).

By virtual short: V− = V+ = 0 V (V− is a "virtual ground").

KCL at V−: current through R₁ must equal current through R_f (no current into op-amp input):

I_in = (V_in - 0) / R_1 = V_in / R_1 I_f = (0 - V_out) / R_f I_in = I_f \to V_in/R_1 = -V_out/R_f Gain: A_v = V_out/V_in = -R_f/R_1

The gain is set by the ratio of two resistors. Negative sign means output is inverted. Input impedance = R₁. Precision requires low-drift metal-film resistors.

4. Non-Inverting Amplifier & Voltage Follower

Circuit: V_in connects to V+. R₁ connects V− to ground. R_f connects V_out to V−.

Virtual short: V− = V+ = V_in. Voltage divider from V_out to ground through R_f and R₁:

V- = V_out \times R_1/(R_1 + R_f) = V_in Gain: A_v = V_out/V_in = 1 + R_f/R_1

Gain ≥ 1 (non-inverting). Input impedance is very high (~10⁷ Ω for real IC), ideal for loading sensitive sources.

Voltage follower (unity-gain buffer): R_f = 0, R₁ = ∞. Gain = 1. V_out = V_in. Used to isolate high-impedance sources from low-impedance loads. Every sensor output should be buffered this way before driving cables or ADCs.

5. Summing and Difference Amplifiers

Summing Amplifier

V_out = −R_f(V₁/R₁ + V₂/R₂ + ...)

Multiple inputs each through their own resistor to V−. Virtual ground means no interaction between inputs. Used in DAC circuits (R-2R ladder) and audio mixers.

Difference Amplifier

V_out = R_f/R_1 × (V₂ − V₁)

Amplifies the difference between two signals while rejecting common-mode noise. Fundamental for measuring sensor output in noisy environments. CMRR (Common-Mode Rejection Ratio) >80 dB typical.

Instrumentation Amplifier

Gain = 1 + 2R/R_G

Three op-amps. Very high CMRR (>110 dB), adjustable gain with a single resistor. Standard for medical electrodes, Wheatstone bridge sensors (strain gauges, load cells). INA128, AD620.

6. Integrator and Differentiator

Replace the feedback resistor with a capacitor to get circuits that perform calculus:

Integrator (Miller integrator): R₁ input resistor, capacitor C_f in feedback:

V_out(t) = -(1/R₁C_f) \int V_in(t) dt Square wave input \to triangle wave output Sine wave f \to sine wave with -90° phase shift, gain = 1/(2πf\cdotR₁C_f)

Used in waveform generators, ADC sample-and-hold, control system integrators. Add a large feedback resistor in parallel with C_f to limit DC gain and prevent saturation.

Differentiator: C input, R_f feedback:

V_out(t) = -R_f\cdotC \cdot dV_in/dt Triangle wave \to square wave output

Differentiators amplify noise (noise has high frequency components with large derivatives) — add a small resistor R_s in series with the input capacitor to limit high-frequency gain.

7. Real-World Limitations & Common ICs

Gain-Bandwidth Product (GBW): Closed-loop gain × bandwidth = GBW. A TL071 (GBW = 3 MHz) configured for gain 100 has bandwidth of only 30 kHz. Use faster op-amps (OPA657, AD8099) for video or audio.
Slew rate: Maximum rate of output voltage change (V/µs). LM741: 0.5 V/µs. OPA657: 700 V/µs. Limits output swing at high frequencies.
Input offset voltage (V_OS): Small voltage (1 µV – 10 mV) causing output offset. Critical in precision instrumentation. Chopper-stabilized op-amps (LTC1050) achieve V_OS < 5 µV.
Rail-to-rail: Traditional op-amps can't output within ~1V of supply rails. Rail-to-rail (RRO) designs maintain output swing to within mV of rails — needed in single-supply 3.3V or 5V systems.

Common IC families:

LM741: 1968 original monolithic op-amp. ±15V supply, GBW 1 MHz. Obsolete but still sold.
TL071/072: JFET-input, low noise, general purpose. Still standard in audio circuits.
OPA827: Ultra-low noise (4 nV/√Hz), JFET, precision. Used in ADC front-ends, scientific instruments.
MCP6002: 2.7–5.5V single supply, rail-to-rail, 1 MHz. Standard for microcontroller projects.