Ideal op-amp model

The ideal op-amp model is the set of five simplifying assumptions that make op-amp circuits solvable by hand. A real op-amp is close enough to this idealisation that the model predicts circuit behaviour to a fraction of a percent, so we use it for essentially all design and only reach for the corrections (see Closed-loop gain, Real op-amp imperfections) when precision demands it.

The five idealisations:

• Infinite input impedance. No current flows into either input terminal. The inputs sense voltage but draw nothing.
• Zero output impedance. The output behaves as an ideal voltage source: whatever load is hung on $v_O$, the voltage does not sag.
• Infinite open-loop gain $A$. The differential gain is so large that, in any circuit using Negative feedback, the input difference $v_2-v_1$ must be essentially zero — otherwise $v_O=A(v_2-v_1)$ would be impossibly huge.
• Zero common-mode gain, equivalently infinite Common-mode rejection ratio. Any signal common to both inputs is rejected; the device responds only to the difference.
• Infinite bandwidth. $A$ does not depend on frequency. (In reality $A$ rolls off above a few hertz; that is Open-loop gain frequency dependence, treated under Real op-amp imperfections.)

The five ideal-op-amp characteristics.

The two golden rules

The whole model collapses, for analysis purposes, into two statements. These are the only tools you need to solve any op-amp circuit that uses negative feedback.

Golden rule 1 — no current into the inputs. This is just “infinite input impedance” restated for circuit analysis: $i_{+}=i_{-}=0$. Any current arriving at an input node must therefore go somewhere else (into a feedback resistor, typically). This is what lets you write KCL at the input node and ignore the op-amp as a current sink.

Golden rule 2 — the inputs are equal. In a negative-feedback configuration, $v_{+}=v_{-}$. The reasoning: the output is $v_O=A(v_{+}-v_{-})$ with $A\to \infty$. For $v_O$ to be a finite, sensible voltage (it must be — it is pinned between the rails), the difference $v_{+}-v_{-}$ must be $v_O/A\to 0$. The op-amp, through the feedback path, adjusts its own output until its two inputs are equal. It will do whatever it takes. This forced equality, with no physical wire between the inputs, is the Virtual short and virtual ground — important enough to get its own note.

Every circuit in the inverting/non-inverting family, the Summing amplifier, the Difference amplifier, the Op-amp integrator and Op-amp differentiator is derived by applying these two rules together with KCL at the inverting node. Memorise them; the rest is algebra.