Op-Amp Ideal Model & Configurations

Analog Electronics · Topic 6 · ~15 min

1 — Why the Op-Amp Matters

The operational amplifier is the most-used analog IC in history. A single 8-pin chip replaces dozens of discrete transistors and lets you build amplifiers, filters, oscillators, and voltage references with just a handful of resistors and capacitors.

1.1 What makes it special

Gain is set by external parts — two resistors decide the voltage gain, not the transistor's β or temperature.
Near-zero design effort — the four ideal rules (Section 2) let you analyse any circuit with KCL alone.
Universality — the same chip becomes an inverting amp, a filter, an integrator, or a comparator just by changing the feedback network.

1.2 The classic: µA741

Fairchild's µA741 (1968) defined the op-amp category. Its 8-pin DIP package became the industry standard pinout that almost every op-amp still follows.

Figure 2. µA741 8-pin DIP pinout — the industry-standard op-amp package.

1.3 Where you will see op-amps

Audio — microphone pre-amps, headphone drivers, tone controls, mixers.
Sensors — bridge amplifiers for strain gauges, thermocouples, load cells.
Signal conditioning — active filters, level shifters, voltage followers before ADCs.
Control — PID error amplifiers, comparators, integrators in feedback loops.

Every configuration in this topic solves one of these real problems. The nine circuits below are the building blocks.

2 — The Ideal Op-Amp Model

Figure 3. Ideal op-amp symbol (IEEE Std 315) with the four golden rules annotated.

2.1 The four rules

Infinite open-loop gain — $A_{OL} \to \infty$ . Even the tiniest differential input drives the output to a rail.
Infinite input impedance — $R_{\text{in}} \to \infty$ . Zero current flows into either input terminal.
Zero output impedance — $R_{\text{out}} = 0$ . The output is a perfect voltage source.
Zero input offset voltage — when $V^+ = V^-$ , the output is exactly zero (no built-in error).

2.2 Virtual short and virtual ground

Because the gain is infinite and the output is finite, negative feedback forces $V^+ = V^-$ . This constraint is the virtual short.

When the non-inverting input is tied to ground, the inverting input sits at 0 V even though it is not physically grounded. This special case is the virtual ground.

2.3 Why “ideal” works

Real op-amps like the LM741 have $A_{OL} \approx 2 \times 10^5$ and $R_{\text{in}} \approx 2\,\text{M}\Omega$ . For most audio and instrumentation circuits, these numbers are close enough to infinity that the ideal model predicts closed-loop gain within 1%.

👉 Think of the ideal model as training wheels: you design with it, then check bandwidth and slew-rate limits afterward.

3 — Nine Canonical Configurations

1. Inverting

A_v = -\frac{R_f}{R_{in}}

2. Non-Inverting

A_v = 1 + \frac{R_f}{R_{in}}

3. Follower (Buffer)

A_v = 1

4. Summing

V_{out} = -R_f \sum_k \frac{V_k}{R_k}

5. Difference

V_{out} = \frac{R_f}{R_1}(V_2 - V_1)

6. Integrator

V_{out} = -\frac{1}{RC}\int V_{in}\,dt

7. Differentiator

V_{out} = -RC\,\frac{dV_{in}}{dt}

8. Instrumentation

A_v = 1 + \frac{2R}{R_{gain}}

9. Comparator

V_o = A(V^+ - V^-)

Figure 4. The nine canonical op-amp circuits. Each cell shows the feedback network and gain formula.

3.1 Inverting amplifier

A_v = -\frac{R_f}{R_{\text{in}}}

The input signal enters through $R_{\text{in}}$ to the virtual-ground node. The output is inverted and scaled by the ratio of the two resistors.

DC input: Apply +1 V DC → output is −10 V DC (with $R_f/R_{\text{in}} = 10$ ). The output is a steady inverted, scaled copy.

AC input: Apply a 1 kHz sine → output is an inverted sine, 10× larger. The phase is flipped 180°.

3.2 Non-inverting amplifier

A_v = 1 + \frac{R_f}{R_{\text{in}}}

The input drives $V^+$ directly, giving near-infinite input impedance. Gain is always ≥1.

DC input: Apply +1 V DC → output is +11 V DC (gain 11). Non-inverting, so sign is preserved.

AC input: Apply a 1 kHz sine → same-phase sine, 11× larger amplitude.

3.3 Voltage follower (buffer)

A_v = 1

A special case of the non-inverting amp with $R_f = 0$ and $R_{\text{in}} \to \infty$ . It is the op-amp equivalent of an emitter follower: unity gain, massive impedance transformation.

DC input: Apply +3 V DC → output is +3 V DC. The point is impedance transformation, not gain.

AC input: Apply a sine → identical sine, same amplitude, same phase. Isolates a high-impedance source from a low-impedance load.

3.4 Summing amplifier

V_{\text{out}} = -R_f \sum_k \frac{V_k}{R_k}

Multiple inputs feed separate resistors into the virtual-ground node. Each input is independently weighted by its resistor ratio.

This is the circuit behind audio mixers: each channel fader is a variable $R_k$ .

DC input: $V_1 = 2$ V, $V_2 = 3$ V with equal R → output = −5 V (inverted sum).

AC input: Two sines at different frequencies → inverted mixed output (audio mixer behavior).

3.5 Difference amplifier

V_{\text{out}} = \frac{R_f}{R_1}\,(V_2 - V_1)

Amplifies only the difference between two signals and rejects any voltage common to both. Requires matched resistors for good CMRR.

DC input: $V_1 = 3$ V, $V_2 = 5$ V → output = gain × (5 − 3). Only the difference matters.

AC input: Common-mode noise is rejected; only the differential signal is amplified.

3.6 Integrator

V_{\text{out}} = -\frac{1}{R\,C}\int V_{\text{in}}\,dt

Replace $R_f$ with a capacitor. The output is the running time-integral of the input, scaled by $-1/(RC)$ .

Feed in a square wave and you get a triangle wave — the capacitor “accumulates” charge over each half-cycle.

DC input: A DC step causes the output to ramp linearly (runaway without a DC reset resistor).

AC input: A sine becomes a −cosine (90° lag), with amplitude scaling as $1/(RC\omega)$ .

3.7 Differentiator

V_{\text{out}} = -R\,C\,\frac{dV_{\text{in}}}{dt}

The capacitor moves to the input and $R_f$ returns to the feedback path. The output tracks the rate of change of the input.

DC input: A constant DC level produces 0 V output (derivative of a constant is zero).

AC input: A sine becomes a −cosine (90° lead), with amplitude scaling as $RC\omega$ .

3.8 Instrumentation amplifier

A_v = 1 + \frac{2R}{R_{\text{gain}}}

A three-op-amp topology: two input buffers plus a difference output stage. Gain is set by a single resistor $R_{\text{gain}}$ .

Preferred over a single difference amp because the buffered inputs eliminate source-loading errors and the symmetry delivers very high CMRR.

DC input: Measures tiny differential voltages (e.g. bridge sensor offset) with high gain while rejecting common-mode DC.

AC input: Same differential amplification, but gain rolls off at high frequencies due to each op-amp's bandwidth limit.

3.9 Comparator

V_o = A\,(V^+ - V^-)

With no feedback, the open-loop gain $A \approx 2 \times 10^5$ amplifies even a microvolt difference to the rail. The output is effectively $+V_{\text{sat}}$ or $-V_{\text{sat}}$ .

DC input: $V^+ > V^-$ → output = $+V_{\text{sat}}$ . $V^+ < V^-$ → output = $-V_{\text{sat}}$ .

AC input: A sine compared against a DC reference → square wave output at the crossing frequency.

4 — The Comparator Boundary

4.1 Why comparator is included

Strictly, a comparator is not an amplifier — it has no linear region. It is included here because it uses the same op-amp package and because understanding its limitations motivates the next topic.

4.2 The chatter problem

When the input hovers near the threshold, noise causes the output to toggle rapidly — this is chatter (or “contact bounce” in relay terms). The solution is positive feedback, which adds hysteresis.

👉 The fix is a Schmitt Trigger — positive feedback that creates two thresholds and a hysteresis band wider than the noise.

The ideal op-amp model has four rules: infinite gain, infinite

R_{\text{in}}

, zero

R_{\text{out}}

, and the virtual short (

V^+ = V^-

). Nine canonical configurations cover inverting, non-inverting, buffer, summing, difference, integrator, differentiator, instrumentation amp, and comparator. The comparator is the open-loop boundary case — its chatter problem is solved by adding positive feedback (Schmitt trigger / hysteresis).

Next: switch to the Simulate tab to build each configuration interactively — adjust resistor and capacitor values and watch the output waveform respond in real time.