Past exam of the physics A course of the University of Cambridge / 2011 / Paper 2 / Section D / D13

An ideal operational amplifier draws no current at either input. With stable operational-amplifier negative feedback in its unsaturated linear range, infinite open-loop gain also gives a virtual short, $V_{+}=V_{-}$. The amplifier adjusts its output to enforce that equality; it is not a conducting connection between the inputs. Here $V_{+}=0$, so the ideal summing node is a virtual ground. Kirchhoff's current law gives $V_{\mathrm{in}}/R_1=-V_{\mathrm{out}}/R_2$, hence the ideal gain is $-R_2/R_1$.

For the finite-input-resistance inverting amplifier let the inverting-node voltage be $v$, and continue to neglect output impedance. The controlled output and node-current balance areEliminating $v$ givesDividing the denominator by $R_2$ gives the requested resistance-sum form. The ideal result follows whenever $|A|\gg 1+R_2/R_1+R_2/R_{\mathrm{in}}$; taking $R_{\mathrm{in}}\to \infty$ alone is insufficient if $A$ is small. At finite frequency $A$ and the gain are phasor ratios, subject to feedback stability.

With equal resistors and sufficiently large loop gain, $V_{\mathrm{out}}\simeq -V_{\mathrm{in}}$. This gives a controlled sign reversal or 180-degree phase inversion, useful for subtraction and opposite-phase signal paths, with amplitude set by a resistor ratio rather than uncertain open-loop gain. The source still sees approximately $R_1$, so this is not a high-input-impedance voltage follower. For the ideal-input, single-pole model, the noise gain of an operational amplifier is two and the closed-loop bandwidth is approximately half the gain-bandwidth product. Output swing, slew rate and load constraints must also permit linear operation.

A representative bipolar-input 741 amplifier has $R_{\mathrm{in}}$ of order $10^6\Omega$; the LM741 specification gives a typical $2\mathrm{M\Omega }$ at $25^{\circ }\mathrm{C}$. This is an illustrative device specification, not an identification of an unspecified practical-class part. Other input architectures can have much larger electrical resistance, and finite input capacitance makes the input impedance frequency dependent.^[ref]

The open-loop gain is large and roughly constant at low frequency, then falls as internal poles become important. A dominant-pole approximation isAbove $f_p$, but before higher poles dominate, $|A|\propto f^{-1}$, a slope of $-20\mathrm{dB}$ per decade, with increasing phase lag. The approximate unity-gain frequency is $f_T=A_0{f}_p$. A 741-class amplifier has typical low-frequency gain about $2\times 10^5$; this is the open-loop gain, distinct from the feedback-set signal gain. The finite-gain formula at the parent level explains why the closed-loop response eventually rolls off.^[ref]