Operational Amplifiers

An operational amplifier (op-amp) is a direct-coupled, very-high-gain differential voltage amplifier with two inputs and one single-ended output:

$$v_o=A_{OL}(v_+-v_-)=A_{OL}v_d.$$

It was named for its original use in analog computers to perform addition, subtraction, integration and differentiation.

Internal Arrangement and Ideal Model

The signal passes through a differential input stage (high input resistance, first gain and common-mode rejection), a high-gain and level-shift stage (most of $A_{OL}$ and dominant-pole compensation), and a class-AB push-pull output stage (low output resistance and load-current drive). Direct coupling permits amplification down to DC, but also allows offset and temperature drift to reach the output.

Parameter	Ideal	Typical $\mu$A741
Open-loop gain $A_{OL}$	$\infty$	$2\times10^5$ ($106\,\mathrm{dB}$)
Input impedance $Z_i$	$\infty$	$2\,\mathrm{M\Omega}$
Output impedance $Z_o$	$0$	$75\,\Omega$
Unity-gain bandwidth $f_T$	$\infty$	$1\,\mathrm{MHz}$
CMRR	$\infty$	about $90\,\mathrm{dB}$
Slew rate	$\infty$	$0.5\,\mathrm{V/\mu s}$
Input offset voltage $V_{OS}$	$0$	about $1\,\mathrm{mV}$
Input bias/offset currents	$0$	about $80/20\,\mathrm{nA}$

Ideal and representative practical op-amp characteristics.

The low-frequency practical model has a large differential resistance $R_i$, a controlled source $A_{OL}v_d$, and a non-zero series output resistance $R_o$. Its gain falls with frequency, output voltage and current are bounded by the supplies and output stage, and large-signal speed is limited by slew rate. The ideal model is the limiting case

$$A_{OL}\to\infty,\qquad R_i\to\infty,\qquad R_o\to0.$$

Note

Key Point — Virtual short and virtual ground

For negative feedback, linear operation and an unsaturated output,

$$v_+-v_-=\frac{v_o}{A_{OL}}\to0 \quad\Rightarrow\quad v_+\approx v_-, \qquad i_+\approx i_-\approx0.$$

The inputs have equal voltage but are not physically shorted. If $v_+$ is grounded, $v_-\approx0$ is a virtual ground: it has ground potential but cannot independently source or sink current. These rules fail in open-loop comparator operation, with positive feedback, or during saturation.

Practical Performance Parameters

Open-loop gain $A_{OL}$:
ratio $v_o/v_d$ without feedback. Large loop gain makes a feedback-set closed-loop gain accurate, but $A_{OL}$ falls with frequency.

Input and output impedance:
high $Z_i$ avoids source loading and low $Z_o$ makes output voltage insensitive to load. Voltage feedback generally raises the non-inverting closed-loop input impedance and lowers output impedance; both improvements diminish as loop gain falls at high frequency.

Common-mode rejection ratio:
if $v_{cm}=(v_++v_-)/2$, then

$$\boxed{\mathrm{CMRR}=\left|\frac{A_d}{A_{cm}}\right|},\qquad \mathrm{CMRR}_{\mathrm{dB}}=20\log_{10}(\mathrm{CMRR}).$$

High CMRR rejects hum or interference appearing equally at both inputs.

Power-supply rejection ratio:
under the common input-referred convention,

$$\boxed{\mathrm{PSRR}=\left|\frac{\Delta V_S}{\Delta V_{OS}}\right|}, \qquad \mathrm{PSRR}_{\mathrm{dB}}=20\log_{10}(\mathrm{PSRR}).$$

Higher is better. A data sheet may instead quote supply sensitivity $\Delta V_{OS}/\Delta V_S$ in $\mathrm{\mu V/V}$, for which smaller is better.

Input offset and currents:
$V_{OS}$ is the differential DC voltage required to force zero output. The input-current quantities are

$$\boxed{I_B=\frac{|I_{B+}|+|I_{B-}|}{2}},\qquad \boxed{I_{OS}=\bigl||I_{B+}|-|I_{B-}|\bigr|}.$$

Offset is multiplied approximately by noise gain, while currents create errors in source resistances. Making the DC resistance seen by both inputs equal cancels much of the average-bias-current error, not the mismatch $I_{OS}$.

Drift and noise:
offset and current change with temperature and time; voltage/current noise limit the smallest usable signal. These are decisive in precision DC and sensor interfaces.

Input/output ranges:
the common-mode input range, output swing, output-current limit and supply rails must all be respected. Rail-to-rail labels do not imply zero headroom at every load.

Frequency and Large-Signal Response

For a dominant-pole compensated op-amp,

$$A_{OL}(s)\simeq\frac{A_0}{1+s/\omega_p}.$$

At $f_p$ the open-loop magnitude is $3\,\mathrm{dB}$ below $A_0$; beyond it the response falls at about $20\,\mathrm{dB/decade}$. The unity-gain or transition frequency $f_T$ satisfies $|A_{OL}|=1$. Thus

$$\boxed{\mathrm{GBW}\simeq A_0f_p\simeq f_T},\qquad \boxed{BW\simeq\frac{f_T}{A_N}},$$

where $A_N$ is the closed-loop noise gain. The familiar $|A_{CL}|BW\simeq f_T$ is valid only when signal gain and noise gain coincide.

Dominant-pole open-loop response and the gain–bandwidth trade-off.

The slew rate is the maximum large-signal output slope:

$$\boxed{SR=\max\left|\frac{\mathrm{d}v_o}{\mathrm{d}t}\right|}.$$

For $v_o=V_p\sin(2\pi ft)$, the maximum required slope is $2\pi fV_p$; hence the full-power bandwidth is

$$\boxed{f_{FP}=\frac{SR}{2\pi V_p}}.$$

This is distinct from small-signal bandwidth. The usable frequency is bounded by the stricter of GBW, slew rate, swing and output-current requirements.

A large step cannot make the output change faster than the slew rate.

Note

Example — Full-power bandwidth of a $\mu$A741

For $SR=0.5\,\mathrm{V/\mu s}$ and required peak output $V_p=10\,\mathrm{V}$,

$$f_{FP}=\frac{0.5\times10^6}{2\pi(10)}=7.96\,\mathrm{kHz}.$$

Even though the device has about $1\,\mathrm{MHz}$ unity-gain bandwidth, a $10\,\mathrm{V}$-peak sine above $7.96\,\mathrm{kHz}$ becomes slew-distorted.

After a step, settling time is the time required to enter and remain inside a stated error band such as $0.1\%$, $0.01\%$ or one-half LSB. It may contain a slew interval, linear settling and ringing due to limited phase margin; therefore settling time is not interchangeable with either bandwidth or slew rate.

Basic Closed-Loop Configurations

Inverting and non-inverting feedback circuits used in virtual-short derivations.

Inverting amplifier

Since $v_+=0$, the inverting node is at virtual ground. KCL there gives

$$\frac{v_i-v_-}{R_1}=\frac{v_--v_o}{R_f}, \qquad v_-=0,$$

so

$$\boxed{\frac{v_o}{v_i}=-\frac{R_f}{R_1}},\qquad \boxed{Z_{in}=R_1}.$$

The minus sign denotes $180^\circ$ inversion. A bias-current compensation resistor $R_B\simeq R_1\parallel R_f$ is often placed from $v_+$ to ground.

Non-inverting amplifier and follower

Here $v_-=v_+=v_i$. With negligible input current, the feedback network is an unloaded divider:

$$v_i=v_o\frac{R_1}{R_1+R_f} \quad\Rightarrow\quad \boxed{\frac{v_o}{v_i}=1+\frac{R_f}{R_1}}.$$

The signal is not inverted and input impedance is ideally infinite. Directly connecting output to the inverting input gives the voltage follower, $A_v=1$, used to buffer a high-resistance source from a low-resistance load.

Note

Example — Closed-loop gain and output check

For $R_1=10\,\mathrm{k\Omega}$, $R_f=40\,\mathrm{k\Omega}$ and $v_i=0.2\,\mathrm{V}$, the inverting circuit gives $A_v=-4$, $v_o=-0.8\,\mathrm{V}$; the non-inverting circuit gives $A_v=5$, $v_o=1.0\,\mathrm{V}$. These results are valid only if input common-mode range, output swing/current, bandwidth and slew rate are satisfied.

Linear Computing Applications

Summing, integrating and differentiating applications.

Summing amplifier

Applying KCL to the virtual-ground summing node gives

$$\sum_{k=1}^{m}\frac{v_k}{R_k}=-\frac{v_o}{R_f} \quad\Rightarrow\quad \boxed{v_o=-R_f\sum_{k=1}^{m}\frac{v_k}{R_k}}.$$

Equal $R_k=R$ produce a scaled sum; unequal values produce a weighted sum. Applications include audio mixing, level shifting and DACs.

Difference amplifier

For the standard four-resistor circuit, exact ratio matching gives

$$\frac{R_2}{R_1}=\frac{R_4}{R_3}=k \quad\Rightarrow\quad \boxed{v_o=k(v_2-v_1)}.$$

Equal common-mode components cancel ideally. Ratio mismatch converts common-mode voltage to differential error and therefore limits circuit CMRR.

Integrator

With $Z_f=1/(sC)$,

$$\boxed{\frac{V_o(s)}{V_i(s)}=-\frac{1}{sRC}},\qquad \boxed{v_o(t)=v_o(0)-\frac{1}{RC}\int_0^t v_i(\tau)\,\mathrm{d}\tau}.$$

A constant input gives a ramp of slope $\mathrm{d}v_o/\mathrm{d}t=-v_i/(RC)$; a square wave gives a triangular wave. Because the ideal circuit has infinite DC gain, offsets drive it into saturation. A large resistor parallel to $C$ gives a finite low-frequency gain in a practical integrator.

Differentiator

The capacitor current $i=C\,\mathrm{d}v_i/\mathrm{d}t$ flows through $R$, so

$$\boxed{\frac{V_o(s)}{V_i(s)}=-sRC},\qquad \boxed{v_o(t)=-RC\frac{\mathrm{d}v_i(t)}{\mathrm{d}t}}.$$

A triangular input gives a square output and a step gives a narrow pulse. Since $|H(j\omega)|=\omega RC$ rises without bound, the ideal circuit amplifies high-frequency noise and can become unstable. A practical differentiator adds a series input resistor and a small capacitor across $R$ to confine operation to a finite band.

Note

Example — Weighted summer

Let $R_f=20\,\mathrm{k\Omega}$, $R_1=10\,\mathrm{k\Omega}$, $R_2=20\,\mathrm{k\Omega}$, $v_1=0.5\,\mathrm{V}$ and $v_2=1\,\mathrm{V}$. Then

$$v_o=-\left(2v_1+v_2\right)=-2.0\,\mathrm{V}.$$

Circuit	Mathematical action/use	Main limitation
Summer	Weighted addition; mixers and DACs	Resistor accuracy and output range
Difference amplifier	Subtraction/common-mode rejection	Ratio matching determines CMRR
Integrator	$1/s$; ramps, waveform generation, filters	Offset-driven DC saturation
Differentiator	$s$; edge detection and pulse shaping	Noise gain and high-frequency stability

Linear op-amp applications and their dominant practical limits.

Types of Op-Amp

Type	Defining features	Trade-off	Typical use
General-purpose	Balanced gain, speed, offset and cost	No extreme specification	Basic amplifiers and filters
High-speed	High GBW/SR, short settling	Power, noise, layout sensitivity	Video, pulse and fast ADC drive
Precision	Low offset, drift, bias and noise; high CMRR/PSRR	Often lower speed or higher cost	Bridges and DC measurement
Power	High output current/voltage and protection	Heat and large package	Actuators, speakers, supplies
Comparator	Fast open-loop switching, logic output	Poor linear-feedback behavior	Threshold/zero crossing
Norton	Responds to $i_+-i_-$; often single supply	Nonstandard input behavior	Current-mode filters/oscillators
Instrumentation	High $Z_i$, accurate differential gain/CMRR	Matching and extra amplifiers	Sensor and biomedical signals
Isolation	Galvanic input–output barrier	Bandwidth, error and cost	High-side and safety isolation

Classification by the property optimized.

A comparator may share the triangle symbol but is optimized to switch open loop; an ordinary op-amp used this way may have slow saturation recovery, invalid common-mode inputs or unsuitable output levels. A Norton amplifier forms an input-current difference. An isolation amplifier transfers the signal optically, capacitively or magnetically while maintaining separate grounds; isolation rating, leakage and barrier safety are specifications in addition to gain and bandwidth.

Three-Op-Amp Instrumentation Amplifier

Three-op-amp instrumentation amplifier; one resistor sets first-stage gain.

The two non-inverting input amplifiers buffer both sources. Their inverting nodes are joined by $R_G$, so the differential current is $(v_2-v_1)/R_G$ and each equal feedback resistor $R_1$ contributes gain. The matched final subtractor rejects the buffered common-mode voltage. For the shown ratios,

$$\boxed{v_o=\left(1+\frac{2R_1}{R_G}\right) \frac{R_3}{R_2}(v_2-v_1)}.$$

Note

Key Point — Why instrumentation amplifiers are preferred

Both sources see very high input impedance; one resistor adjusts differential gain without disturbing the subtractor ratios; accurate ratio matching gives high CMRR. Practical accuracy is limited by resistor-ratio mismatch, source imbalance and bias currents, offset/drift/noise, common-mode range, output swing, GBW and slew rate. Applications include strain gauges, RTDs, thermocouples, ECG/EEG, shunt-current sensing and data acquisition.

[!NOTE]

Example — Instrumentation-amplifier gain

With $R_1=10\,\mathrm{k\Omega}$, $R_G=1\,\mathrm{k\Omega}$ and a unity-gain matched difference stage ($R_3=R_2$), the differential gain is $1+2R_1/R_G=21$. An input difference of $v_2-v_1=8\,\mathrm{mV}$ therefore gives $v_o=168\,\mathrm{mV}$ while an ideal common-mode component is rejected.