OpAmps
Overview
At it's most basic level, an operational amplifier (opamp) is a discrete analogue integrated circuit which acts as a voltageamplifier with very high gain.
An opamp's gain can be adjusted with the appropriate external circuitry (the opamp's internal gain does not change, but the gain of the entire circuit does). This is almost always done for any opamp that is used as an amplifier. External circuitry can also manipulate the opamp to act as as a current amplifier or perform "mathematical" operations on signals.
Schematic Symbol
An opamp is commonly drawn on schematics as:
You may see this symbol with or without the voltage supply pins $+V_{S}$ and $V_{S}$. If they are not present, it is assumed that they are connected up to a power source which should be obvious from the design intent.
Always take note of the position of the inverting () and noninverting (+) input terminals. Sometimes they can be drawn swapped around relative to the symbol shown above.
Uses
 Buffers (a.k.a. voltage followers)
 Linear amplifiers, nonlinear amplifiers
 Preamplifiers (amplifier's with a input voltage of 10mV or less)
 Analogue mathematical operations (differentiators, integrators, summer, subtracters, e.t.c)
 Voltage to current (transconductance) and current to voltage conversions (transimpedance)
OpAmp Topologies
Voltage Followers (a.k.a. Buffers)
A voltage follower (also known as a buffer) is one of the most basic circuits you can make with an opamp.
The output voltage for a voltage follower is just simply:
$\begin{align} \label{eq:buffervouteqvin} v_{out} &= v_{in} \end{align}$As shown in Eq $\ref{eq:buffervouteqvin}$, the output voltage is the same as the input voltage. Well isn't this pointless? No, the key point to a voltage follower/buffer is that it can convert a highimpedance input into a low impedance output. Practically, this means that you can now sink/source more current from the output without the voltage changing. Buffers are great for boosting signals that travel across long distances, or for signals which get split and go to many devices (this is called fanout, and is common with digital clock signals).
A simulation schematic for a voltagefollower opamp is shown below.
The results of the simulation:
NonInverting Amplifiers
A opamp in the noninverting amplifier configuration is shown below.
The equation for the output voltage of the noninverting amplifier is:
$\begin{align} v_{out} = \left(1 + \frac{R_f}{R_i}\right) v_{in} \end{align}$Notice the $1$ in the gain equation? This means that no matter what you set the resistors $R_f$ and $R_i$ to, you can never get a gain which is less than one. This is one of the disadvantages of the noninverting amplifier (you can have a gain of less than one with an inverting amplifier).
There is no one common convention for labelling these feedback resistors. Sometimes they are called $R_i$ and $R_f$, other times $R_1$ and $R_2$ (with either resistor being $R_1$)
Simulation example for a noninverting amplifier.
Here is a simulation schematic (circuit) for a noninverting opamp amplifier running from a singleended supply. Because R1 ($R_f$) and R2 ($R_i$) are both $1k\Omega$, the opamp has a voltage gain $A_V$ of:
$\begin{align} A_V &= 1 + \frac{R_f}{R_i} \\ &= 1 + \frac{1k\Omega}{1k\Omega} \\ &= 2 \\ \end{align}$The results of the simulation are shown below. As you can see, the output voltage $v_{out}$ is exactly twice the input voltage $v_{in}$.
Inverting Amplifiers
A opamp amplifier in the inverting configuration is shown below.
The equation for the output voltage of an inverting amplifier is:
$\begin{align} v_{out} =  \frac{R_f}{R_i} v_{in} \end{align}$The negative sign is to show that the output is the inverse polarity of the input. Notice that, unlike the noninverting amplifier, an inverting amplifier lets you obtain a gain of less than 1.
Simulation example for a an inverting amplifier.
Below is the schematic used for simulating the behaviour of an inverting opamp. Note how is requires a negative voltage power supply.
And below are the simulation results for the above schematic:
Difference Amplifiers
A difference (a.k.a differential or subtractive) amplifier is an opamp circuit which amplifies the voltage difference between two electrical signals. It is the mathematical equivalent of subtraction with some additional gain. The schematic is shown below:
The output voltage is given by the equation^{1}:
$\begin{align} v_{out} = \left(\frac{R_2}{R_1 + R_2}\right)\left(\frac{R_3 + R_4}{R_3}\right)v_1  \frac{R_4}{R_3}v_2 \end{align}$It's easy to confuse a differential amplifier (what we are discussing here) with a differentiator amplifier, which performs the mathematical function of differentiation (opposite of integration).
To make a standard difference amplifier, $R_1$ is picked to equal $R_3$ and $R_2$ is picked to equal $R_4$. In this case, the above equation easily simplifies to:
$\begin{align} v_{out} = \left(\frac{R_2}{R_1}\right)(v_1  v_2) \end{align}$Now you can clearly see how it behaves as a subtractor ($v_1  v_2$) followed by some gain ($\frac{R_2}{R_1}$).
Simulation example for a differential amplifier.
Below is a schematic for simulating the behaviour of a differential opamp:
This schematic produces the following results:
Integration Amplifiers
An integration amplifier performs mathematical integration on the input signal  It's output voltage is proportional to the integral of the input voltage w.r.t. time. The below schematic shows an ideal opamp based integrator.
Eq. $\ref{eq:voutvinidealintegrator}$ shows the relationship between input and output voltage.
$\begin{align} \label{eq:voutvinidealintegrator} v_{out} = \frac{1}{R_i C} \int_0^t v_{in}\ dt \end{align}$However, this circuit is normally not practical in real world situations. Any errors such as the output offset voltage and input bias current (which all opamps invariably have), as well as a nonperfect input signal with small amounts of DC bias, will cause the output to drift, until it reaches saturation.
A way to fix this problem is to insert a highvalued feedback resistor, $R_f$, to limit the DC gain, as well as a resistor, $R_{bias}$, on the noninverting input terminal to compensate for the input bias current.
$\begin{align} v_{out} &= \frac{1}{R_i C} \int_0^t v_{in}\ dt \\ R_{bias} &= R_i\ \ R_f \ R_L \\ V_E &= \left(\frac{R_f}{R_i} + 1\right) V_{OS} \end{align}$Differentiator Amplifiers
TODO: Add content here.
Transconductance Amplifiers
A transconductance amplifier is an opamp topology which is used to convert a voltage into a current. Coincidentally, it is also known as a voltagetocurrent converter.
A basic transconductance amplifier can be built with an opamp in a noninverting configuration.
A transconductance amplifier is useful creating an industry standard 420mA (or 020mA) currentloop signal. The input voltage can come from something like a potentiometer or microcontroller (coupled with either using a VDAC peripheral or PWM/RCfilter technique to create a variable voltage).
One disadvantage with this design is that the current output is not ground referenced, that is, ground is not used as the return path for the current. This complicates the wiring.
Transimpedance Amplifiers
See the Photodiodes page for information on how to design a transimpedance amplifier to convert a photodiodes output current signal into a voltage.
Current Sources
An opamp can be easily wired up with a MOSFET and sense resistor to make a voltage controlled current source. See Current Sources And Sinks: Opamp Current Sink for info on how to make an opamp current source, including schematics, equations and worked examples.
Summing Amplifiers
Opamps can be used to sum (add) voltages together. There are called summing amplifiers.
Summing amplifiers are used for things such as:
 Mixing audio signals: Audio mixers can be made from summing amplifiers, which combine multiple audio tracks together (e.g. singing with guitar, piano and drums).
 Creating a DAC: Many inputs with resistances following a $2^n$ pattern are used. These days however you should probably just buy a dedicated DAC IC!
 Levelshifting: For example, shifting a +5V signal to 03.3V for input into a DAC. A two input summing amplifier can do this, with one input being the AC signal you want to levelshift and the other input being a fixed DC offset. Combined with the gain of the opamp you can both offset and scale the signal.
Inverting Summing Amplifier
The below schematic shows a inverting summing amplifier with three inputs.
How Does The Inverting Summing Amplifier Work?
The current $i_f$ going through the resistor $R_f$ is just the sum of the currents going through resistors $R_i$.
$\begin{align} i_f &=  (i_1 + i_2 + i_3) \\ \end{align}$The inverting input of the amplifier is at "virtual ground".
$\begin{align} \label{eq:ifeqv1riv2riv3ri} i_f &=  \left(\frac{v_1}{R_i} + \frac{v_2}{R_i} + \frac{v_3}{R_i}\right) \\ \end{align}$Again, because the inverting input is at $0V$, the voltage $v_{out}$ is just:
$\begin{align} \label{eq:vouteqifrf} v_{out} &= i_f R_f \end{align}$Substitute Eq. $\ref{eq:ifeqv1riv2riv3ri}$ into Eq. $\ref{eq:vouteqifrf}$, and then simplify a little:
$\begin{align} v_{out} &=  \left(\frac{v_1}{R_i} + \frac{v_2}{R_i} + \frac{v_3}{R_i}\right) R_f \nonumber \\ \label{eq:invsumampgaineq} &=  \frac{R_f}{R_i} (v_1 + v_2 + v_3) \\ \end{align}$And there we have it! Eq. $\ref{eq:invsumampgaineq}$ shows you [^summingamplifierinverting] works as an inverting summing amplifier for three input voltages $v_1$, $v_2$, and $v_3$. Note that aside from summing, the inverting summing amplifier also applies the negative gain $ \frac{R_f}{R_i}$, just like a standard [^_inverting_amplifiers, inverting amplifier]. You can make $R_f$ equal to $R_i$ if you don't want any gain, but you can't avoid the inverting part. If you don't want it to invert (perhaps your using a singlesided supply?), then take a look at the [^_non_inverting_summing_amplifier, noninverting summing amplifier].
Whilst common, the resistors $R_i$ do not need to be equal. If they are not equal, the circuit in [^summingamplifierinverting] is called a scaling summing amplifier. Each input voltage is scaled by a value determined by it's input resistance before being summed together.
$\begin{align} \label{eq:invscalingsumampgaineq} v_{out} &=  R_f \left(\frac{v_1}{R_1} + \frac{v_2}{R_2} + \frac{v_3}{R_3}\right) \\ \end{align}$Noninverting Summing Amplifier
The noninverting summing amplifier is similar to the inverting summing amplifier except that it does not invert the input voltages. The advantages and disadvantages of the noninverting vs the inverting summing amplifier are highlighted below:
Advantages
 Does not invert the input signals (this might actually be a disadvantage, depending on the application!)
 May only need a singleended voltage supply (depending on the voltage ranges of the inputs).
 Higher input impedance
Disadvantages
 More complicated gain equation/resistor values (because there is no virtual ground at the inputs).
 Unused inputs cannot be left floating, they must be grounded (more on this below).
The below schematic shows a noninverting summing amplifier with three inputs.
If all the input resistors are equal ($R_1 = R_2 = R_3 = R$), the gain equation is:
$\begin{align} v_{out} = \left(1 + \frac{R_f}{R_i}\right) \frac{v_1 + v_2 + v_3}{3} \end{align}$How Does The NonInverting Summing Amplifier Work?
Because we have no nice virtual ground at the noninverting input, we have to analyze the noninverting summing amplifier a little differently to the inverting version. First, use Kirchhoff's current law:
$\begin{align} i_1 + i_2 + i_3 = 0 \\ \end{align}$Let's call the voltage at the noninverting pin $v_+$. We can then write replace the currents above with the voltage difference divided by the resistance:
$\begin{align} \frac{v_1  v_+}{R_1} + \frac{v_2  v_+}{R_2} + \frac{v_3  v_+}{R_3} = 0 \\ \end{align}$Let's assume the input resistors are equal ($R_1 = R_2 = R_3 = R$), then simplify and solve for $v_+$:
$\begin{align} \frac{1}{R} ( v_1  v_+ + v_2  v_+ + v_3  v_+ ) &= 0 \nonumber \\ v_1 + v_2 + v_3  3v_+ &= 0 \nonumber \\ v_+ = \frac{v_1 + v_2 + v_3}{3} \\ \end{align}$So now we have an equation for $v_+$ in terms on the input voltages. As described in the [^_non_inverting_amplifiers], the equation for a noninverting amplifier is:
$\begin{align} v_{out} = \left(1 + \frac{R_f}{R_i}\right) v_{+} \end{align}$Substituting in $v_+$ we get the final equation for a 3input noninverting summing amplifier:
$\begin{align} v_{out} = \left(1 + \frac{R_f}{R_i}\right) \frac{v_1 + v_2 + v_3}{3} \end{align}$Notice the $3$. This scaling factor is equal to the number of inputs the noninverting summing amplifier has, so remember that the scaling factor changes if you add/remove inputs!
WARNING: Unused inputs into a inverting summing amplifier can be left floating. Unused inputs into a noninverting amplifier CANNOT be left floating, they need to be tied to ground (0V).
Level Shifters
Opamps can be used to perform levelshifting. The concept of levelshifting is very similar to what is achieved by [^_summing_amplifiers, summing amplifiers].
Simulation example for an opamp based level shifter.
The below schematic shows a opamp based level shifter which converts a $\pm 5.0V$ signal into a $03.3V$ signal, suitable for passing into a microcontrollers DAC for digital conversion and processing (microcontrollers often run of $+3.3V$, hence that is also a common choice for the reference voltage of the DAC). If you used railtorail opamps, you might be able to get away with lowering the voltage rails for the opamp to $\pm 5.0V$.
Ideal Diodes/Rectifiers
Basic Ideal Diode/Rectifier
An opamp can be combined with a regular diode to make an "ideal diode" (also called a superdiode^{2}, precision rectifier, or half/fullwave rectifier), one that rectifies and has no voltage drop. Ideal diode circuits are useful when:
 You have to rectify a signal with precision (the forward voltage drop across the diode is unacceptable).
 You have to rectify a small signal, whose amplitude is less than a forward diode drop.
The below schematic shows the most basic form of an ideal diode (a.k.a. a halfwave rectifier). Because feedback to the inverting pin is taken from the load ($v_{out}$), the opamp will drive it's output to whatever is needed to make the output voltage the same as the input voltage (when $v_{in}$ is positive), thereby compensating for the forward voltage drop of the diode. When $v_{in}$ is negative, the opamp will try as hard as it can to lower the output voltage, but this will be blocked from the output by the diode, providing the rectifying action.
The below graph shows the response of the basic ideal diode circuit.
The ideal diode is only ideal up to a point. It is not limited by the bandwidth and max. slew rates of the opamp, which is typically much less than of a basic diode! There is also an issue with slew rates with the above circuit. As soon as $v_{in}$ goes negative, the opamp will "saturate" and swing it's output to the maximum negative voltage. This will cause some delay on the output (limited by the slew rate) when the input once again goes positive.
TIP: The standard diode in the ideal diode circuit can be replaced by a Schottky diode for a quickandeasy speed improvement (Schottky diodes have faster switching speeds).
Improved Halfwave Rectifier
The below schematic shows an improved halfwave rectifier with additional circuitry to prevent the opamp from saturating when in the blocked part of the cycle^{3}. Note that this also is an inverting halfwave rectifier  it rectifies the negative half of the input and outputs it as a positive wave on the output.
The gain of the halfwave rectifier above is:
$\begin{align} A = \begin{cases} \frac{R_2}{R_1} & v_{in} < 0 \\ 0 & v_{in} \ge 0 \\ \end{cases} \end{align}$Sample And Hold Circuits
The basic opamp based sample and hold circuit is an extension of the ideal diode circuit, but with an added capacitor on the output to maintain with the voltage when the input signal is removed.
Important Variables
Sorted by function.
CommonMode Input Voltage Range
The commonmode input voltage range is the range of voltages that can appear at the input to the opamp and it still work correctly. For standard singlesupply opamps, the typical range is approximately $0V$ to $V_+  1.5V$. Note how it includes the most negative rail $V_$ (which is 0V for a singlesupply opamp) but only gets within 1.5V
of the most positive rail, $V_+$
Input Offset Voltage (Vos/Vio)
The input offset voltage $V_{OS}$ (or sometimes called $V_{IO}$) is the voltage difference required between the two input pins to force the output to 0. It is a DC measurement parameter. In an ideal opamp, the opamp only amplifies a difference between the inputs, and so the output is 0V when the difference is 0V, hence the input offset voltage is 0V. However, realworld opamps always have some unavoidable differences in the internal components that make up the opamps (specifically, in the input differential stage of the internal circuitry), and thus the inputs are not perfectly identical. The below circuit shows how the input offset voltage is modelled as a voltage source in series with one of the inputs of an ideal opamp.
A nonzero input offset voltage results in gain errors between the input and output of a opamp. The input offset voltage is typically in the following ranges:
 15mV for good general purpose opamps, 515mV for really bad ones.
 200uV1mV for specialized low input offset voltage opamps
 10uV200uV for the best "ultra" low input offset voltage opamps.
 < 1uV for chopperstabilized (autozero) opamps.
For example, the general purpose LM324 has a typical input offset voltage of 2mV and a maximum of 3mV, at $T_A = 25°C$^{4}. "Low" input offset voltage opamps will have a $V_{OS}$ in the range of 50200uV. For example, the OPAx196 family of opamps has a max. $V_{IO}: 100uV$^{5}.
Input offset voltages vary by opamp transistor technology. Bipolar opamps typically have the lowest input offset voltage, followed by CMOS and the BiFET opamps^{6}.
Input Offset Voltage Drift
The input offset voltage varies with both temperature and time (drift). The variation with temperature is usually represented by $T_C V_{OS}$ (I've also seen $\Delta V_{IO}/\Delta T$ used^{7}). Typical temperature drift for precision opamps is in the range of $110uV^{\circ}C$^{8}. The venerable LM324 has a $T_C V_{OS} = 7uV^{\circ}C$ (max)^{7}.
The change of input offset voltage with time is called aging. Aging is normally specified in $uV/1000hours$. But since aging is a physical process that follows the "random walk pattern" (Brownian motion), it is more accurate to describe it proportional to the square root of elapsed time.
Some opamps are trimmed by the manufacturer after the opamp is packaged. Performing trimming after packaging prevents any productionline effects from effecting the input offset voltage. One such example is the family of AD8601, AD8602 and AD8604 opamps from Analog Devices (they call it DigiTrim). The offset is trimmed with a special digital code using no extra pins (i.e. reuses existing opamp pins). Once programmed, polysilicon fuses are blown to permanently set the trim values^{8}.
Trimming Input Offset Voltage
If your opamp lacks a dedicated trim pin, you can make your own trimming circuit as shown below. This is for an opamp in the inverting configuration. $VR1$ is a potentiometer, manually adjusted until it cancels out the opamps input offset voltage.
The maximum input offset voltage you can compensate for with this circuit is given by Eq $\ref{eq:triminvertingoffset}$^{8}.
$\begin{align} \label{eq:triminvertingoffset} V_{offset} = \pm\left(1 + \frac{R2}{R1}\right) \left(\frac{R_P}{R_P + R3}\right) V_R \end{align}$Input Bias Current (Ib+ and Ib)
The input bias current $I_{B+}$ and $I_{B}$ describe the currents that flow in and out of the opamps input pins. In an ideal opamp, no current flows into/out of the input pins (the opamp has infinite input impedance). In reality, always some small amount of current will flow. Typical input bias currents range from 110nA. Special "low input bias current" opamps can have much lower values. For example, the Texas Instruments OPA396 has a typ. $I_B = \pm 10fA$ at $25^{\circ}C$! This increases to approx. $\pm 4pF$ across it's entire temperature range (typical)^{9}.
The amount and behaviour of input bias current depends on the opamp transistor technology. A FETbased opamp's input bias current will approximately double with every 10°C rise in temperature^{10} ^{11}.
Input bias currents are a problem because these currents will flow through external circuitry connected to the opamps inputs. This current when flowing through resistors and other impedances will create unwanted voltages which will increase the systematic errors. Finding an opamp with a low input bias current can be important for a high sensitivity transimpedance amplifier, such as one connected to a photodiode.
Do not confuse the input bias current with the input offset current (although they are related). The input offset current $I_{OS}$ is the difference between the input bias current at the +
pin and the 
pin.
Input Impedance
The input impedance is the internal resistance to ground from the two input pins. In an ideal opamp, this value is infinite. For most opamps, this value is somewhere between 110MΩ.
GainBandwidth (GBW) Product
The gainbandwidth product (initialised as GBWP, GBW, GBP or GB) is an important parameter which puts a limit on the maximum gain and frequency an opamp can operate at. An opamp's maximum possible gain reduces as the frequency of the signal increases. The multiplication of the gain with the frequency gives the gainbandwidth product, which is relatively constant for a particular opamp.
Hence if the gain bandwidth of a particular opamp is 1Mhz, and the gain is 10, the maximum frequency that the opamp can operate linearly at (still provide a gain of 10) is at 100kHz. Or if the gain was set to 100, then the maximum frequency is 10kHz. This also means that an opamp acts as a lowpass filter, as the gain drops for very high frequencies. Like wise for a GBW of 1MHz, is the gain was just 1 (i.e. acting as a buffer/follower), then it should be able to operate up to frequencies of 1MHz.
An example of an ultrahigh gain bandwidth is 1700MHz, which are present in 'Wideband CFB" opamps, designed for applications such as RGB line drivers (such as the OPA695). A 'normal' GBW can be anywhere between 100kHz and 10MHz. A low gainbandwidth is around 1kHz (reminiscent of less advanced, older opamps). Remember gain is unitless (V/V), so gain bandwidth is expressed as a frequency only. Not realising this can be confusing! The GBW product is closely related to the [^_slew_rate, slew rate].
High Level Output Voltage
The high level output voltage ($V_{OH}$) defines the highest voltage which the opamp can drive the output to (with respect to the power supply $V_+$).
Low Level Output Voltage
The low level output voltage ($V_{OL}$) defines the lowest voltage which the opamp can drive the output to. The LM324 is rumoured to only be able to drive the output near ground if it is sourcing current, but only to 0.5V minimum if sinking (see this EDA Forum post, LM324 Opamp Gain Instability).
Slew Rate
The slew rate of an opamp defines the maximum rate the output voltage can change with respect to time. In an ideal opamp, this would be infinite. It has the SI units $V/s$, and is commonly expressed in $uV/s$. It can be thought of as the slope of the output waveform if one of the inputs of the input was subjected to a step voltage change.
Opamps have a limited output slew rate due to internal compensation capacitor combined with a finite output drive current. Charing a capacitive output with a constant current (a good approximation) gives a linear increase in voltage (recall that the equation relating voltage to current for a capacitor is $i = C \frac{dV}{dt}$).
The plot below shows how the noninfinite slew rate of an opamp distorts the shape of an input squarewave "pulse". The slew rate of $0.4V/us$ (which is not particularly good by any standard) of the "jellybean" LM324 opamp was used in this plot^{12}.
The max. slew rate of an opamp limits the amplitude of output waveforms it can produce at high frequencies without distortion. This parameter usually increases as the GBW of the opamp increases. Higher slew rate opamps also tend to have higher quiescent currents.
Slew Rate Equation
The maximum frequency of a signal that an opamp can output before it gets distorted because of it's maximum slew rate limit is given by the following equation:
$\begin{align} SR = 2\pi f_{max} V_P \end{align}$where:
$SR$ is the slew rate, in $Vs^{1}$
$f_{max}$ is the maximum frequency (of a sine wave) that is allowed before distortion, in $Hz$
$V_P$ is the peak voltage of the sine wave (i.e. amplitude, or half the peaktopeak voltage), in $V$
Proof
A sine wave at frequency $f$ and has peak voltage $V_P$ is described with the equation:
$\begin{align} f(t) = V_P \sin (2\pi f t) \end{align}$The slope at any point in time $t$ is found by taking the derivative. As per standard derivative rules, the derivative of $\sin$ is $\cos$ and you multiply the function by what is inside the $\sin$:
$\begin{align} \frac{df}{dt} = 2\pi f V_P \cos (2\pi f t) \end{align}$We want to find the maximum slope, $max\left(\frac{df}{dt}\right)$. We know that the maximum value of any $\cos$ function is $1$, so we can simplify to:
$\begin{align} max\left(\frac{df}{dt}\right) = 2\pi f V_P \end{align}$This maximum slope $max\left(\frac{df}{dt}\right)$ is the maximum slew rate of the opamp $SR$. Hence:
$\begin{align} SR = 2\pi f_{max} V_P \end{align}$Finding the max. frequency of the LM324 due to slew rate limitations.
So just as an example, the jellybean LM324 with a slew rate of $0.4V/us$ can generate a $10V$ peak ($20V$ peaktopeak) waveform of the following frequency before distortion:
$\begin{align} f_{max} &= \frac{SR}{2\pi V_P} \nonumber \\ &= \frac{0.4V/us}{2\pi 10V} \nonumber \\ &= 6.37kHz \\ \end{align}$Slew Rate Effects On Sine Waves
The below plot shows different levels of distortion for a 10V (peak) 20kHz sine wave when it is generated by an opamps with different slew rates. The jellybean LM324 slew rate of $0.4V/us$ is shown (which outputs a triangle wave!) along with a slightly better $1.0V/us$ (which just clips the sine wave). The minimum slew rate required to prevent any distortion of the sine wave is $1.26V/us$.
Quiescent Current
The quiescent current (current with no load, device in steadystate) is generally constant over the total rated supply voltage range. Obviously, if there is a load on the opamp, the current drawn through the power pins (the supply current) will be the sum of the quiescent current and the current going through the load.
Quiescent currents for standard opamps are typically between 1.54mA. A 'lowpower' opamp has a typical quiescent current between 0.51.5mA (such as the LM258N
). Then there are ultralow power opamps that only draw 520pA (such as the LMC6464
). You normally sacrifice slewrate and gainbandwidth for ultralow power. Likewise, higher gainbandwidth and higher slew rate opamps typically have larger quiescent currents.
Cascading OpAmps
Cascading opamps is concept when the output of one opamp is connected to the input of another. There can be an arbitrary number of opamps in the cascade, but usual limits are 34.
For a fixedgain, cascading opamps can also be used to increase the bandwidth, as each individual opamp now can operate at a lower gain and therefore has a larger bandwidth as defined by the gainbandwidth product. Note though that each additional opamp added to increase the bandwidth gives diminishing returns. Also important to note that opamp bandwidth is defined as the 3dB gain points. Hence the bandwidth does not stay the same (total bandwidth gets smaller) when two identical opamps are cascaded, as these will now the 6dB points. A practical limit for fixedtotalgain increasedbandwidth cascading is about 34 opamps.
The Gain
When cascading opamps, the total gain is the product of all of the individual opamps gains, i.e.:
$\begin{align} A_{total} = A_0 A_1 A_2 ... A_n \end{align}$The Bandwidth
The bandwidth of cascaded opamps is not as simple to calculate as the gain.
If all of the opamps are identical, then the following equation can be used:
$\begin{align} BW_{tot} = BW \times \sqrt{2^{\frac{1}{N}}  1} \end{align}$where:
$BW_{tot}$ = the total bandwidth of the cascaded opamp system
$BW$ = the bandwidth of the individual opamps (remember, they have to be identical)
$N$ = the number of opamps in the cascaded system
The above equation gives diminishing returns with every additional opamp added.
Feedback Resistor Values
As a ruleofthumb, you should use the lowest acceptable resistances in opamp feedback paths to reduce instabilities.
Types Of OpAmps
General Purpose
General purpose opamps typically have parameters in the following ranges:
 Gain Bandwidth Product: 1MHz
 Input Bias Current: 15pA
 Input Voltage Offset: 1mV
 Output Current: 2050mA
 Icc: 1mA
RailtoRail OpAmps
A railtorail opamp is an opamp which supports input voltages near the power rails, and can drive the output close to the one or more of the power rails. We must stress the word NEAR, as the opamp's output voltage will never get exactly to the rail, due to the finite voltage drop across the outputstage transistors. Railtorail opamps just support wider ranged input voltages and can drive closer to the rails than general purpose opamps can. Look for the low level output voltage ($V_{OL}$) parameter in the opamp's datasheet. For "railtorail" opamps, this will usually be about 100200mV about ground at normal load currents.
"Railtorail" opamps cannot really output either rail voltage, just closer to it that general purpose opamps.
This also means that a railtoral singlesupply opamp cannot output 0V. To achieve a true ground output, you need a negative voltage supply. There are dedicated ICs designed to provide a small negative power supply to opamps so that they can output true ground. One such example is the Texas Instruments LM7705, a "Low Noise Negative Bias Generator". This IC only generates 230mV, which allows the designer to use CMOSbased opamps which usually have a maximum supply voltage of 5.5V.
Micropower OpAmps
Micropower is a termed used for extremely low quiescent current opamps that are designed for battery or energy recoverybased power supplies. The supply current of micropower opamps is typically within the range of 50100uA at a supply voltage of 210V. Because they are designed for batterybased systems, they are also commonly singlesupply opamps.
Instrumentation Amplifiers
Instrumentation amplifiers are analog voltage amplifier circuits that, although are drawn the the same symbol as an opamp, are typically made up internally from three opamps (and passives). You can either make an instrumentation amplifier out of discrete opamps or purchase a instrumentation amplifier IC which contains all the opamps within the same chip.
Manufacturer Part Number Families

INA ** INAx126: Precision instrumentation amplifiers by Texas Instruments. The INA126 has one amplifier per package, the INA2126 has two. ** INA290: Precision currentsense amplifier.

LM741: Very popular and old "741" style opamp produced by Texas Instruments, ON Semiconductor and Rochester Electronics.

LT: The prefix Linear Technology (now Analog Devices) uses for their range of opamps. ** LT1006: Precision, singlesupply opamp. ** LT1077: Micropower, singlesupply opamp. ** LT1167: Instrumentation amplifier.

MAX: Opamps by Maxim. ** MAX4194: Instrumentation amplifier.

OP07: Analog Devices/Texas Instruments ranges of low input offset voltage opamps. ** OP07C: ±318V VCC, 40 to +85°C industrial temp. range *** OP07CP: DIP8 package *** OP07CS: SOIC8 package ** OP07D: ±418V VCC ** OP07E: 0 to 70°C commercial temp. range

OPA: Texas Instruments (previously BurrBrown) family of opamps. ** OPAx187: Zerodrift 36V railtorail opamps. Includes the OPA187 (1 opamp), OPA2187 (2 opamps) and OPA4187 (4 opamps). ** OPA241: Singlesupply ** OPA251: Dualsupply ** OPA27: Texas Instruments (previously BurrBrown) family of ultralow noise, precision opamps. Internally compensated for unitygain stability. ** OPA37: Uncompensated version of the OPA27.

TLE202: Texas Instruments family of "highspeed lowpower" precision operational amplifiers. Belong to the Excalibur family of TI opamps which uses "isolated vertical PNP transistors" to give unitygain bandwidth and slew rate improvements.

TLV: Texas Instruments family of opamps.

Vsupply is voltage from Vdd/Vcc to Vss/Vee.
The Different Types Of Gain, Explained
Openloop gain $A_V$ (sometimes written as $A_{OL}$) is the gain of the opamp without any feedback.
Closed loop gain $G_V$ is the overall gain of the opamp with feedback.
We can generalize the circuit of an opamp with negative feedback to the block diagram shown below.
From the above block diagram we can write an equation for $v_{out}$:
$\begin{align} v_{out} &= A\times v_{sum} \\ &= A(v_{in}  Bv_{out}) \\ \end{align}$What we are really interested is an equation for the closedloop gain $G_V$ which is equal to $\frac{v_{out}}{v_{in}}$...all we need to do is to rearrange the equation as shown below:
$\begin{align} v_{out} &= Av_{in}  ABv_{out} & \text{Expanding} \\ v_{out}(1  AB) &= Av_{in} & \text{Shift $v_{out}$ onto left side and factor.} \\ \frac{v_{out}}{v_{in}} &= \frac{A}{1  AB} & \text{Jiggle things to get $\frac{v_{out}}{v_{in}}$} \\ \end{align}$And so we come to an equation for the closed loop gain $G_V$ as:
$\begin{align} G_V &= \frac{A}{1  AB} \end{align}$We can take this one step further, since normally the openloop gain $A$ for an opamp is very large, in the range of $100,000$ to $1,000,000$. With that, we can simply the closed loop gain to be:
$\begin{align} G_V &\approx \frac{1}{B} \end{align}$In summary:
 Openloop gain: $A$
 Loop gain: $AB$
 Closedloop gain: $\approx \frac{1}{B}$
Examples
Below are some examples of opamps that stand out from the crowd for some reason, be it popularity, years in service, or functionality wise.
AD860x
US$3.50 (1) Good for high precision stuff! Awesome for photodiode amplification (both currenttovoltage and voltagetovoltage configurations).
LM32x
A common family of opamps that has been around for along time, they can operate of a single supply and can swing right to ground, but cannot swing to the rail voltage. The LM321 has one opamp, the LM328 has two (dual), and the LM324 has 4 (quad).
LM833
One of the cheapest 'audio' opamps available (about US$0.20 as of 2011). Features a high GBW for it's price.
OPA695
US$3.50 (1) This is a ultrawideband, currentfeedback opamp. If you need an opamp with a ridiculously high gainbandwidth product, this is along the lines of what you want to use. It has a GBW of 1700Mhz and a maximum slewrate of 4300V/us.
OP07
A opamp with a "ultra" low input offset voltage (resistors are trimmed at production time to achieve this), guaranteed to be no more than 75uV. This opamp also features offset nulling pins to further reduce the input offset voltage by performing trimming once the opamp is installed in a circuit.
Negative Voltage Rails
Dedicated chargepump topology power supply ICs are available that supply a small negative voltage to the opamps V
pin.
Isolation Amplifiers
Isolation amplifiers provide galvanic isolation between the input (sensor) and output (measurement circuitry). They are used to protect the sensor measurement and recording circuitry (e.g. a microcontroller with onboard ADC) from dangerously high voltages at the sensor, and also the opposite, to protect the sensor environment from potentially dangerous voltages on the rest of the system.
A common application would be to isolate and amplify the voltage across a currentsense resistor on a highpower motor, or to protect humans with medical sensors connected to them from the measurement system.
Basic isolation amplifiers require two power supplies (one for each side of isolation), while others incorporate builtin transformers so that you only have to provide one power source.
Input Resistors
One of the first things you learn about an opamp is that the input impedance on the input pins are very large (ideally infinite). So naturally you would start to question why resistors would be connected to the input pins of an opamp like shown below.
These input resistors serve to limit the input current if the voltage on the input pin goes above $V_{CC}$. Most opamps have protection/clamping diodes from the input pins to $V_{CC}$ (typically you can determines this if in the datasheet the input pins max voltage is rated to $V_{CC} + 0.3V$, which is one diode voltage drop). If there was no resistor there, the builtin diode would conduct and sink a large current from the input pin to $V_{CC}$, possibly damaging the opamp. The resistor limits this current to a safe value.
Offset Nulling Circuits
Some opamps which are designed to have very low input offset voltages also come with offset nulling pins to further trim the input offset voltage once the opamp is installed in circuit. The OP07 is one opamp which has these pins. Typically, a $1050k\Omega$ potentiometer is connected across these pins with the wiper going to $V_{CC}$, as shown below.
If not using the trim pins, leave them not connected (open circuit). Do NOT connect them to ground.
Negative Impedance Converters (NICs)
What Is A NIC?
A negative impedance converter (NIC) is a clever opamp circuit which creates negative impedance (you might be wondering what negative impedance actually is, more on this later). A NIC can be constructed from an opamp and a few passive components as shown in the following schematic:
Typically the two resistances are the same ($R1 = R2$), and then the input impedance $Z_{IN}$ is:
$\begin{align} Z_{IN} = Z \end{align}$where:
$Z_{IN}$ is the input impedance, in Ohms ($\Omega$)
$Z$ is the impedance of the component connected between the inverting terminal and ground, as shown on the diagram, in Ohms ($\Omega$)
What Does Negative Impedance Actually Mean?
What does negative impedance actually mean? If $Z$ is just a simple resistor with resistance $R$ (the most basic kind of NIC), then the impedance is $Z_{IN} = R$. This means the circuit behaves just like a simple resistor connected to ground, except that the current comes out of the resistor, not into it.
Things get more interesting when you replace this resistor with a capacitor.
TODO: Add explanation of what happens when capacitor is added.
NIC Input Impedance Proof
Skip this section if you are not interested in the maths.
To prove $Z_{IN} = Z$, we need to find the input current at inverting terminal, and then use $Z_{IN} = \frac{V}{I_{IN}}$. The input current can be found by application of Ohm's law and the golden rules of opamps. Using the rule that the voltage at the two input terminals will be the same, we know the voltage across the impedance $Z$ is going to be:
$\begin{align} \label{eqn:vzeqv} V_Z = V \end{align}$Using Ohm's law, the current through the impedance $Z$ is therefore:
$\begin{align} I_Z = \frac{V}{Z} \end{align}$Because there is no current going into the inverting terminal of the opamp, this current $I_Z$ must also be flowing through $R2$:
$\begin{align} \label{eqn:i_r2} I_{R2} = \frac{V}{Z} \end{align}$Now knowing the voltage at the inverting terminal and the current through $R2$ we can write an equation for the voltage at the output of the opamp:
\begin{align} \label{eqn:vout_eq} V_{OUT} &= V_Z + V_{R2} \\ &= V_Z + I_{R2} \cdot R \\ &= V + \frac{V}{Z} \cdot R & &\text{Subs. in \ref{eqn:vzeqv} and \ref{eqn:i_r2}} \end{align}Now that we know the voltage on both sides of $R1$ we can find the voltage across it:
\begin{align} \label{eqn:v_r1} V_{R1} &= V_{OUT}  V &\\ &= V + \frac{V}{Z} \cdot R  V & &\text{Subs. in \ref{eqn:vout_eq}} \\ &= \frac{V}{Z} \cdot R & &\text{Simplifying} \end{align}Now we know the voltage across $R1$ we can find the current going through it using Ohm's law:
\begin{align} \label{eqn:i_r1} I_{R1} &= \frac{V_{R1}}{R} &\\ &= \frac{\frac{V}{Z} \cdot R}{R} & &\text{Subs. in \ref{eqn:v_r1}} \\ &= \frac{V}{Z} & &\text{Simplifying} \end{align}" Because the voltage on the righthand side of $R1$ is higher, this current is flowing righttoleft. Since no current flows into the inverting terminal of the opamp, this also must be current flowing "out" of the input terminal. Thus:
$\begin{align} \label{eqn:i_in} I_{IN} = \frac{V}{Z} \end{align}$Knowing the input current and voltage, we can finally write an equation for $Z_{IN}$:
\begin{align} \label{eqn:z_in} Z_{IN} &= \frac{V_{IN}}{I_{IN}} & &\text{Ohms law} \\ &= \frac{V}{\frac{V}{Z}} & &\text{Subs. in \ref{eqn:i_in}} \\ &= Z & &\text{Simplifying} \\ \end{align}Proof complete!
ChopperStabilised OpAmps
TODO: Add info.
Capacitive Loading
TODO: Add info.
Industry Standard Package Pinouts For OpAmps
Opamps are usually packaged in industry standard throughhole and surface mount packages. For many of these packages, there are industry standard pinouts which means you can easily find pincompatible alternatives for any given opamp. This section aims to illustrate some of these industry standard pinouts.
For 8pin packages:
Footnotes

Electronics Tutorials. Operational Amplifiers  The Differential Amplifier. Retrieved 20230829, from https://www.electronicstutorials.ws/opamp/opamp_5.html. ↩

Keim, Robert (2018, Feb 8). When a Diode Simply Isn’t Good Enough: The Superdiode. AllAboutCircuits. Retrieved 20210927, from https://www.allaboutcircuits.com/technicalarticles/whenadiodesimplyisntgoodenoughthesuperdiode/. ↩

Ducu, Dragos. (2011). AN1353: Op Amp Rectifiers, Peak Detectors and Clamps. Microchip. Retrieved 20211001, from http://ww1.microchip.com/downloads/en/AppNotes/01353A.pdf. ↩

Texas Instruments (2015). LMx24N, LM2902N LowPower, QuadOperational Amplifiers. Retrieved 20201020, from https://www.ti.com/lit/ds/snosc16d/snosc16d.pdf. ↩

Texas Instruments (2017, Jul). OPAx196 36V, LowPower, Low Offset Voltage, RailtoRail Operational Amplifier. Retrieved 20210903, from https://www.ti.com/lit/ds/symlink/opa196.pdf. ↩

Palmer, Richard (2001). DC Parameters: Input Offset Voltage (Vio). Texas Instruments. Retrieved 20201020, from https://www.ti.com/lit/an/sloa059/sloa059.pdf. ↩

OnSemiconductor (2021, Aug). Single Supply Quad Operational Amplifiers (Datasheet). Retrieved 20210903, from https://www.onsemi.com/pdf/datasheet/lm324d.pdf. ↩ ↩^{2}

Analog Devices (2008, Oct). MT037: Op Amp Input Offset Voltage. Retrieved 20210903, from https://www.analog.com/media/en/trainingseminars/tutorials/MT037.pdf. ↩ ↩^{2} ↩^{3}

Texas Instruments (formally BurrBrown) (2021, Jul). OPA396 Precision, Low IQ, Low Input Bias Current Op Amp (datasheet). Retrieved 20221010, from https://www.ti.com/lit/ds/symlink/opa396.pdf. ↩

Analog Devices (2008, Oct). Op Amp Input Bias Current. Retrieved 20201020, from https://www.analog.com/media/en/trainingseminars/tutorials/MT038.pdf. ↩

Luis Orozco (2014). MS2624: Optimizing Precision Photodiode Sensor Circuit Design. Analog Devices. Retrieved 20220224, from https://www.analog.com/en/technicalarticles/optimizingprecisionphotodiodesensorcircuitdesign.html. ↩

Texas Instruments (2014). LM321 Low Power Single Operational Amplifier. Retrieved 20211002, from https://www.ti.com/lit/ds/symlink/lm321.pdf. ↩ ↩^{2}