Electronics

C. Operational amplifiers

Operational Amplifiers

Use the two golden rules of the ideal op-amp to solve inverting, noninverting, summing, difference, integrator, and differentiator circuits, then bound them with gain-bandwidth and slew rate.

The operational amplifier is the highest-yield analog topic on the FE because almost every problem reduces to two assumptions applied with discipline. An ideal op-amp has infinite open-loop gain, infinite input impedance, and zero output impedance; the FE Reference Handbook states it directly under Operational Amplifiers as

v_o=A(v_1-v_2)

with

A

large and, in the linear region,

v_2-v_1=0

. From that single line every closed-loop formula follows. Points are lost not on the algebra but on sign and on which input the feedback returns to — so this concept drills the method, not just the results.

The two golden rules

When an ideal op-amp is in linear operation with negative feedback, two rules hold and solve nearly everything. First, no current flows into either input (infinite input impedance). Second, the two input voltages are equal — the virtual short — because infinite gain drives the differential input to zero. When the noninverting input is grounded, the inverting input is held at

0\,\text{V}

: a virtual ground. Apply KCL at the inverting node using these two rules and the circuit falls out in one equation.

i_+ = i_- = 0,\qquad v_+ = v_-\quad(\text{virtual short})

Inverting amplifier

Ground the noninverting input, drive through

R_1

into the inverting node, and feed back through

R_f

. The inverting node is a virtual ground, so the input current

v_{in}/R_1

Noninverting amplifier

Drive the signal into the noninverting input and feed

R_f

R_1

from the output back to the inverting input, with

R_1

Summing and difference amplifiers

An inverting summer ties several inputs through their own resistors into the virtual-ground node; by KCL each contributes independently, producing a weighted, inverted sum. A difference (subtractor) amplifier feeds one signal to the inverting side and one to the noninverting side; with the matched-resistor condition

R_f/R_1

on both sides it outputs a scaled difference — the basis of the instrumentation amplifier and the practical realization of common-mode rejection.

v_o=-R_f\!\left(\frac{v_1}{R_1}+\frac{v_2}{R_2}+\cdots\right),\qquad v_o=\frac{R_f}{R_1}\,(v_b-v_a)

Integrator and differentiator

Replace the inverting amplifier's feedback resistor with a capacitor and you get an integrator: the virtual-ground current

v_{in}/R

charges

C

, so the output is the (inverted) time integral of the input. Swap the roles — capacitor in, resistor feedback — and you get a differentiator, whose output is proportional to the rate of change of the input. Integrators are common in filters and analog computers; differentiators are noise-sensitive and usually tamed with a series resistor.

v_o(t)=-\frac{1}{RC}\int v_{in}\,dt\quad(\text{integrator}),\qquad v_o(t)=-RC\,\frac{dv_{in}}{dt}\quad(\text{differentiator})

Gain-bandwidth product

A real op-amp's open-loop gain rolls off at

-20\,\text{dB/decade}

above a low dominant pole, so the product of closed-loop gain and bandwidth is a constant — the gain-bandwidth product (GBW), equal to the unity-gain frequency

f_t

. A circuit with a higher closed-loop gain therefore has proportionally less bandwidth. This is the first thing to check when an op-amp circuit fails to pass a high frequency.

\text{GBW}=A_{CL}\cdot f_{-3\text{dB}}=f_t\ \Rightarrow\ f_{-3\text{dB}}=\frac{f_t}{A_{CL}}

Slew rate and full-power bandwidth

GBW is a small-signal limit; slew rate is a large-signal one. The output cannot change faster than the slew rate

SR

(in

\text{V}/\mu\text{s}

), set by how fast the internal compensation capacitor can charge. For a sinusoid

v_o=V_p\sin(2\pi f t)

Exam strategy

For any op-amp problem: (1) note whether the input enters the inverting or noninverting terminal — that fixes the sign and base formula; (2) if you are unsure, write KCL at the inverting node using

i=0

and

v_+=v_-

from scratch — it always works and protects against memorized-formula slips; (3) for a difference amp, check the matched-resistor condition before using the simple form; (4) decide whether the limit is gain-bandwidth (small-signal) or slew rate (large-signal swing) and apply the right one. Inverting input impedance is

Key equations

Ideal op-amp golden rules

i_+=i_-=0,\qquad v_+=v_-

Worked examples

Inverting amplifier output

Problem. An inverting amplifier has $R_f=100\,\text{k}\Omega$ and $R_1=10\,\text{k}\Omega$

Common pitfalls

•Sign errors: the inverting amp is $-R_f/R_1$ , the noninverting is $+(1+R_f/R_1)$

References

NCEES FE Reference Handbook — Electrical and Computer Engineering: Operational Amplifiers — Ideal op-amp definition, two-source output, inverting/noninverting forms, CMRR.
NCEES FE Reference Handbook — Electrical and Computer Engineering: Electronics — Surrounding device and amplifier context.
Sedra & Smith, Microelectronic Circuits — Virtual-short method, integrator/differentiator, GBW and slew-rate derivations.

A. Models, biasing, and performance of discrete devices

Discrete Device Models and Biasing

B. Amplifiers

Transistor Amplifiers