Operational amplifier applications

Main article: Operational amplifier

This article illustrates some typical operational amplifier applications. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal features as shown in the diagram^{[ clarification needed ]}, but here a simplified schematic notation is used, many details such as device selection and power supply connections are not shown. Operational amplifiers are optimised for use with negative feedback, and this article discusses only negative-feedback applications. When positive feedback is required, a comparator is usually more appropriate. See Comparator applications for further information.

Practical considerations

Operational amplifiers parameter requirements

In order for a particular device to be used in an application, it must satisfy certain requirements. The operational amplifier must

• have large open-loop signal gain (voltage gain of 200,000 is obtained in early integrated circuit exemplars), and
• have input impedance large with respect to values present in the feedback network.

With these requirements satisfied, the op-amp is considered ideal, and one can use the method of virtual ground to quickly and intuitively grasp the 'behavior' of any of the op-amp circuits below.

Component specification

Resistors used in practical solid-state op-amp circuits are typically in the kΩ range. Resistors much greater than 1 MΩ cause excessive thermal noise and make the circuit operation susceptible to significant errors due to bias or leakage currents.

Input bias currents and input offset

Practical operational amplifiers draw a small current from each of their inputs due to bias requirements (in the case of bipolar junction transistor-based inputs) or leakage (in the case of MOSFET-based inputs).

These currents flow through the resistances connected to the inputs and produce small voltage drops across those resistances. Appropriate design of the feedback network can alleviate problems associated with input bias currents and common-mode gain, as explained below. The heuristic rule is to ensure that the impedance "looking out" of each input terminal is identical.

To the extent that the input bias currents do not match, there will be an effective input offset voltage present, which can lead to problems in circuit performance. Many commercial op-amp offerings provide a method for tuning the operational amplifier to balance the inputs (e.g., "offset null" or "balance" pins that can interact with an external voltage source attached to a potentiometer). Alternatively, a tunable external voltage can be added to one of the inputs in order to balance out the offset effect. In cases where a design calls for one input to be short-circuited to ground, that short circuit can be replaced with a variable resistance that can be tuned to mitigate the offset problem.

Operational amplifiers using MOSFET-based input stages have input leakage currents that will be, in many designs, negligible.

Power supply effects

Although power supplies are not indicated in the (simplified) operational amplifier designs below, they are nonetheless present and can be critical in operational amplifier circuit design.

Supply noise

Power supply imperfections (e.g., power signal ripple, non-zero source impedance) may lead to noticeable deviations from ideal operational amplifier behavior. For example, operational amplifiers have a specified power supply rejection ratio that indicates how well the output can reject signals that appear on the power supply inputs. Power supply inputs are often noisy in large designs because the power supply is used by nearly every component in the design, and inductance effects prevent current from being instantaneously delivered to every component at once. As a consequence, when a component requires large injections of current (e.g., a digital component that is frequently switching from one state to another), nearby components can experience sagging at their connection to the power supply. This problem can be mitigated with appropriate use of bypass capacitors connected across each power supply pin and ground. When bursts of current are required by a component, the component can bypass the power supply by receiving the current directly from the nearby capacitor (which is then slowly recharged by the power supply).

Using power supply currents in the signal path

Additionally, current drawn into the operational amplifier from the power supply can be used as inputs to external circuitry that augment the capabilities of the operational amplifier. For example, an operational amplifier may not be fit for a particular high-gain application because its output would be required to generate signals outside of the safe range generated by the amplifier. In this case, an external push–pull amplifier can be controlled by the current into and out of the operational amplifier. Thus, the operational amplifier may itself operate within its factory specified bounds while still allowing the negative feedback path to include a large output signal well outside of those bounds. ^[1]

Amplifiers

The first example is the differential amplifier, from which many of the other applications can be derived, including the inverting, non-inverting, and summing amplifier, the voltage follower, integrator, differentiator, and gyrator.

Differential amplifier (difference amplifier)

Main article: Differential amplifier

Amplifies the difference in voltage between its inputs.

The name "differential amplifier" must not be confused with the "differentiator", which is also shown on this page.

The "instrumentation amplifier", which is also shown on this page, is a modification of the differential amplifier that also provides high input impedance.

The circuit shown computes the difference of two voltages, multiplied by some gain factor. The output voltage

${\displaystyle V_{\text{out}}={\frac {\left(R_{\text{f}}+R_{1}\right)R_{\text{g}}}{\left(R_{\text{g}}+R_{2}\right)R_{1}}}V_{2}-{\frac {R_{\text{f}}}{R_{1}}}V_{1}=\left({\frac {R_{1}+R_{\text{f}}}{R_{1}}}\right)\cdot \left({\frac {R_{\text{g}}}{R_{\text{g}}+R_{2}}}\right)V_{2}-{\frac {R_{\text{f}}}{R_{1}}}V_{1}.}$

Or, expressed as a function of the common-mode input V_com and difference input V_dif:

${\displaystyle V_{\text{com}}=(V_{1}+V_{2})/2;V_{\text{dif}}=V_{2}-V_{1},}$

the output voltage is

${\displaystyle V_{\text{out}}{\frac {R_{1}}{R_{\text{f}}}}=V_{\text{com}}{\frac {R_{1}/R_{\text{f}}-R_{2}/R_{\text{g}}}{1+R_{2}/R_{\text{g}}}}+V_{\text{dif}}{\frac {1+(R_{2}/R_{\text{g}}+R_{1}/R_{\text{f}})/2}{1+R_{2}/R_{\text{g}}}}.}$

In order for this circuit to produce a signal proportional to the voltage difference of the input terminals, the coefficient of the V_com term (the common-mode gain) must be zero, or

${\displaystyle R_{1}/R_{\text{f}}=R_{2}/R_{\text{g}}.}$

With this constraint ^{[nb 1]} in place, the common-mode rejection ratio of this circuit is infinitely large, and the output

${\displaystyle V_{\text{out}}={\frac {R_{\text{f}}}{R_{1}}}V_{\text{dif}}={\frac {R_{\text{f}}}{R_{1}}}\left(V_{2}-V_{1}\right),}$

where the simple expression R_f / R₁ represents the closed-loop gain of the differential amplifier.

The special case when the closed-loop gain is unity is a differential follower, with

${\displaystyle V_{\text{out}}=V_{2}-V_{1}.}$

Inverting amplifier

An inverting amplifier is a special case of the differential amplifier in which that circuit's non-inverting input V₂ is grounded, and inverting input V₁ is identified with V_in above. The closed-loop gain is R_f / R_in, hence

${\displaystyle V_{\text{out}}=-{\frac {R_{\text{f}}}{R_{\text{in}}}}V_{\text{in}}\!\,}$.

The simplified circuit above is like the differential amplifier in the limit of R₂ and R_g very small. In this case, though, the circuit will be susceptible to input bias current drift because of the mismatch between R_f and R_in.

To intuitively see the gain equation above, calculate the current in R_in:

${\displaystyle i_{\text{in}}={\frac {V_{\text{in}}}{R_{\text{in}}}}}$

then recall that this same current must be passing through R_f, therefore (because V₋ = V₊ = 0):

${\displaystyle V_{\text{out}}=-i_{\text{in}}R_{\text{f}}=-V_{\text{in}}{\frac {R_{\text{f}}}{R_{\text{in}}}}}$

A mechanical analogy is a seesaw, with the V₋ node (between R_in and R_f) as the fulcrum, at ground potential. V_in is at a length R_in from the fulcrum; V_out is at a length R_f. When V_in descends "below ground", the output V_out rises proportionately to balance the seesaw, and vice versa. ^[2]

As the negative input of the op-amp acts as a virtual ground, the input impedance of this circuit is equal to R_in.

Non-inverting amplifier

A non-inverting amplifier is a special case of the differential amplifier in which that circuit's inverting input V₁ is grounded, and non-inverting input V₂ is identified with V_in above, with R₁ ≫ R₂. Referring to the circuit immediately above,

${\displaystyle V_{\text{out}}=\left(1+{\frac {R_{\text{2}}}{R_{\text{1}}}}\right)V_{\text{in}}\!\,}$.

To intuitively see this gain equation, use the virtual ground technique to calculate the current in resistor R₁:

${\displaystyle i_{1}={\frac {V_{\text{in}}}{R_{1}}}\,,}$

then recall that this same current must be passing through R₂, therefore:

${\displaystyle V_{\text{out}}=V_{\text{in}}+i_{1}R_{2}=V_{\text{in}}\left(1+{\frac {R_{2}}{R_{1}}}\right)}$

Unlike the inverting amplifier, a non-inverting amplifier cannot have a gain of less than 1.

A mechanical analogy is a class-2 lever, with one terminal of R₁ as the fulcrum, at ground potential. V_in is at a length R₁ from the fulcrum; V_out is at a length R₂ further along. When V_in ascends "above ground", the output V_out rises proportionately with the lever.

The input impedance of the simplified non-inverting amplifier is high:

${\displaystyle Z_{\text{in}}=(1+A_{\text{OL}}B)Z_{\text{dif}}}$

where Z_dif is the op-amp's input impedance to differential signals, and A_OL is the open-loop voltage gain of the op-amp (which varies with frequency), and B is the feedback factor (the fraction of the output signal that returns to the input). ^{[3] [4]} In the case of the ideal op-amp, with A_OL infinite and Z_dif infinite, the input impedance is also infinite. In this case, though, the circuit will be susceptible to input bias current drift because of the mismatch between the impedances driving the V₊ and V₋ op-amp inputs.

The feedback loop similarly decreases the output impedance:

${\displaystyle Z_{\text{out}}={\frac {Z_{\text{OL}}}{1+A_{\text{OL}}B}}}$

where Z_out is the output impedance with feedback, and Z_OL is the open-loop output impedance. ^[4]

Voltage follower (unity buffer amplifier)

Used as a buffer amplifier to eliminate loading effects (e.g., connecting a device with a high source impedance to a device with a low input impedance).

${\displaystyle V_{\text{out}}=V_{\text{in}}\!}$

${\displaystyle Z_{\text{in}}=\infty }$ (realistically, the differential input impedance of the op-amp itself (1 MΩ to 1 TΩ), multiplied by the open-loop gain of the op-amp)

Due to the strong (i.e., unity gain) feedback and certain non-ideal characteristics of real operational amplifiers, this feedback system is prone to have poor stability margins. Consequently, the system may be unstable when connected to sufficiently capacitive loads. In these cases, a lag compensation network (e.g., connecting the load to the voltage follower through a resistor) can be used to restore stability. The manufacturer data sheet for the operational amplifier may provide guidance for the selection of components in external compensation networks. Alternatively, another operational amplifier can be chosen that has more appropriate internal compensation.

The input and output impedance are affected by the feedback loop in the same way as the non-inverting amplifier, with B=1. ^{[3] [4]}

Summing amplifier

A summing amplifier produces the negative of the sum of several (weighted) voltages:

${\displaystyle V_{\text{out}}=-R_{\text{f}}\left({\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}\right)}$

• When ${\displaystyle R_{1}=R_{2}=\cdots =R_{n}}$, and ${\displaystyle R_{\text{f}}}$ independent

${\displaystyle V_{\text{out}}=-{\frac {R_{\text{f}}}{R_{1}}}(V_{1}+V_{2}+\cdots +V_{n})\!}$

• When ${\displaystyle R_{1}=R_{2}=\cdots =R_{n}=R_{\text{f}}}$

${\displaystyle V_{\text{out}}=-(V_{1}+V_{2}+\cdots +V_{n})\!}$

• Input impedance of the nth input is ${\displaystyle Z_{n}=R_{n}}$ (${\displaystyle V_{-}}$ is a virtual ground)

Instrumentation amplifier

Main article: Instrumentation amplifier

Combines very high input impedance, high common-mode rejection, low DC offset, and other properties used in making very accurate, low-noise measurements

• Is made by adding a non-inverting buffer to each input of the differential amplifier to increase the input impedance.

Oscillators

Wien bridge oscillator

Main article: Wien bridge oscillator

Produces a very low distortion sine wave. Uses negative temperature compensation in the form of a light bulb or diode.

Filters

Main article: Active filter

Operational amplifiers can be used in construction of active filters, providing high-pass, low-pass, band-pass, reject and delay functions. The high input impedance and gain of an op-amp allow straightforward calculation of element values, allowing accurate implementation of any desired filter topology with little concern for the loading effects of stages in the filter or of subsequent stages. However, the frequencies at which active filters can be implemented is limited; when the behavior of the amplifiers departs significantly from the ideal behavior assumed in elementary design of the filters, filter performance is degraded.

Comparator

Main articles: Comparator and Comparator applications

An operational amplifier can, if necessary, be forced to act as a comparator. The smallest difference between the input voltages will be amplified enormously, causing the output to swing to nearly the supply voltage. However, it is usually better to use a dedicated comparator for this purpose, as its output has a higher slew rate and can reach either power supply rail. Some op-amps have clamping diodes on the input that prevent use as a comparator. ^[5]

Integration and differentiation

Inverting integrator

Main article: Op amp integrator

The integrator is mostly used in analog computers, analog-to-digital converters and wave-shaping circuits. A simple version is:

Assuming ideal elements, it integrates the input signal (multiplied by ${\displaystyle -{\tfrac {1}{RC}}}$) over a time interval from t₀ to t₁, yielding an output voltage at time t = t₁ of:

${\displaystyle V_{\text{out}}(t_{1})=V_{\text{out}}(t_{0})-{\frac {1}{RC}}\int _{t_{0}}^{t_{1}}V_{\text{in}}(t)\,dt,}$

where V_out(t₀) is the capacitor's initial voltage at time t = t₀. In other words, the circuit's output voltage changes over the time interval by an amount proportional to the time integral of the input voltage:

${\displaystyle -{\frac {1}{RC}}\int _{t_{0}}^{t_{1}}V_{\text{in}}(t)\,dt.}$

This circuit can be viewed as an active low-pass electronic filter with a single pole at DC (i.e., where ${\displaystyle \omega =0}$).

Its practicality is limited by a significant problem: unless the capacitor is periodically discharged, the output voltage will eventually drift outside of the operational amplifier's operating range. This can be due to any combination of:

• a non-zero DC component in the input V_in,
• a non-zero opamp input bias current,
• a non-zero opamp input offset voltage. ^[6]

The following slightly more complex circuit can ameliorate the second two problems, and in some cases, the first as well, but has a limited bandwidth of integration:

Here, the feedback resistor R_f provides a discharge path for capacitor C_f. The series resistor R_n at the non-inverting input alleviates input bias current and common-mode problems, provided it is set to the parallel resistance of R_i || R_f:

${\displaystyle R_{\text{n}}=R_{\text{i}}||R_{\text{f}}={\frac {1}{{\frac {1}{R_{\text{i}}}}+{\frac {1}{R_{\text{f}}}}}}\,.}$

Op amp integrator § Practical circuit explains the output drift adds a small finite DC error voltage:

${\displaystyle V_{\text{error}}=\left({\frac {R_{\text{f}}}{R_{\text{i}}}}+1\right)\left(V_{\text{OS}}+I_{\text{B-}}\left(R_{\text{f}}\parallel R_{\text{i}}\right)\right).}$

Because the circuit is a first-order low-pass filter with a flat response up to its cutoff frequency ${\displaystyle {\tfrac {1}{2\pi C_{f}R_{f}}}}$, it only functions as an integrator for frequencies significantly higher than that cutoff.

Inverting differentiator

Main article: Differentiator § Active differentiator

Assuming ideal elements, this circuit differentiates the signal (multiplied by ${\displaystyle -RC}$) over time:

${\displaystyle V_{\text{out}}=-RC{\frac {dV_{\text{in}}}{dt}},}$

where ${\displaystyle V_{\text{in}}}$ and ${\displaystyle V_{\text{out}}}$ are functions of time.

The transfer function of the inverting differentiator has a single zero in the origin (i.e., where angular frequency ${\displaystyle \omega =0}$). The high-pass characteristics of a differentiating amplifier can lead to stability challenges when the circuit is used in an analog servo loop (e.g., in a PID controller with a significant derivative gain). In particular, as a root locus analysis would show, increasing feedback gain will drive a closed-loop pole toward marginal stability at the DC zero introduced by the differentiator.

Synthetic elements

Inductance gyrator

Main article: Gyrator

Simulates an inductor (i.e., provides inductance without the use of a possibly costly inductor). The circuit exploits the fact that the current flowing through a capacitor behaves through time as the voltage across an inductor. The capacitor used in this circuit is geometrically smaller than the inductor it simulates, and its capacitance is less subject to changes in value due to environmental changes. Applications where this circuit may be superior to a physical inductor are simulating a variable inductance or simulating a very large inductance.

This circuit is of limited use in applications relying on the back EMF property of an inductor, as this effect will be limited in a gyrator circuit to the voltage supplies of the op-amp.

Negative impedance converter (NIC)

Main article: Negative impedance converter

Creates a resistor having a negative value for any signal generator.

In this case, the ratio between the input voltage and the input current (thus the input resistance) is given by

${\displaystyle R_{\text{in}}=-R_{3}{\frac {R_{1}}{R_{2}}}.}$

In general, the components ${\displaystyle R_{1}}$, ${\displaystyle R_{2}}$, and ${\displaystyle R_{3}}$ need not be resistors; they can be any component that can be described with an impedance.

Non-linear

Precision rectifier

Main article: Precision rectifier

The voltage drop V_F across the forward-biased diode in the circuit of a passive rectifier is undesired. In this active version, the problem is solved by connecting the diode in the negative feedback loop. The op-amp compares the output voltage across the load with the input voltage and increases its own output voltage with the value of V_F. As a result, the voltage drop V_F is compensated, and the circuit behaves very nearly as an ideal (super) diode with V_F = 0 V.

The circuit has speed limitations at high frequency because of the slow negative feedback and due to the low slew rate of many non-ideal op-amps.

Exponential output

The Shockley diode equation gives the current–voltage relationship for an ideal semiconductor diode:

${\displaystyle I_{\text{D}}=I_{\text{S}}\left(e^{\frac {V_{\text{D}}}{V_{\text{T}}}}-1\right),}$

where ${\displaystyle I_{\text{S}}}$ is the saturation current, ${\displaystyle V_{\text{D}}}$ is the forward voltage across the diode, and ${\displaystyle V_{\text{T}}}$ is the thermal voltage (approximately 26 mV at room temperature). When ${\displaystyle V_{\text{D}}\gg V_{\text{T}},}$ the diode's current is approximately proportional to an exponential function:

${\displaystyle I_{\text{D}}\simeq I_{\text{S}}e^{\frac {V_{\text{D}}}{V_{\text{T}}}}.}$

The opamp's inverting input is virtually grounded and ideally draws no current. Thus, the output voltage will be:

${\displaystyle V_{\text{out}}=-RI_{\text{D}}.}$

The output voltage ${\displaystyle V_{\text{out}}}$ is thus approximately an exponential function of the input voltage ${\displaystyle V_{\text{in}}}$:

${\displaystyle V_{\text{out}}\simeq -RI_{\text{S}}e^{\frac {V_{\text{in}}}{V_{\text{T}}}}\,.}$

This implementation does not consider temperature stability and other non-ideal effects.

Logarithmic output

See also: Log amplifier

Since the logarithm is the inverse function of exponentiation, the exponential output circuit described above can be rearranged by swapping the diode into the feedback path of the opamp to form a log amplifier:

Since the opamp's inverting input is virtually grounded and ideally draws no current, ${\displaystyle V_{\text{out}}{=}V_{\text{D}}}$ and the current flowing from the source through the resistor and diode is:

${\displaystyle {\frac {V_{\text{in}}}{R}}=I_{\text{R}}=I_{\text{D}},}$

where ${\displaystyle I_{\text{D}}}$ is the current through the diode, which as described earlier is approximately:

${\displaystyle I_{\text{D}}\simeq I_{\text{S}}e^{\frac {V_{\text{D}}}{V_{\text{T}}}}.}$

Solving for ${\displaystyle V_{\text{out}}}$ gives an approximately logarithmic relationship between the input voltage ${\displaystyle V_{\text{in}}}$ and the output voltage ${\displaystyle V_{\text{out}}}$:

${\displaystyle V_{\text{out}}\simeq -V_{\text{T}}\ln \left({\frac {V_{\text{in}}}{I_{\text{S}}R}}\right)\,.}$

This implementation does not consider temperature stability and other non-ideal effects.

Piecewise linear output

Piecewise linear functions can approximate non-linear functions as a series of connected line segments. Gain compression circuits (like sine or square root) use diodes or transistors to switch between line segments with slopes determined by resistive voltage divider networks. Expansion circuits may be built using a compression circuit as feedback of an opamp. ^[7]

Temperature-compensated compression

Temperature-compensated three-segment compression function

The schematic shown for a "temperature-compensated three-segment compression function" ^{[8] [9]} produces a gain compression transfer function where each subsequent line segment reduces the steepness of the transfer function. For small signals, transistors Q2 and Q3 produce very little base current, and so the circuit's gain is determined just by the feedback resistance of R2 divided by the input resistance of R1. Once the output voltage exceeds around 2 V (whose exact voltage depends on R3 and R4 and the -15 V supply), then Q3 saturates, so the circuit's feedback resistance is determined by R4 in parallel with R2, reducing the gain. As the output voltage increases more, Q2 will saturate, so the circuit's gain is again reduced by the additional inclusion of R6 into the parallel feedback resistance. Temperature-compensation transistors Q4 and Q1 cancel out the temperature-dependent p–n junction base-emitter forward voltage drop of Q3 and Q2. Additional linear segments can be added using additional copies of the resistor-transistor-resistor chains (like the chain R5, Q2, R6 or the chain R3, Q3, R4 but with different values) in a similar manner to further compress the input. This circuit's compression function only works for negative inputs. Diode D1 forces the output to zero if a positive input is applied.

Other applications

• audio and video preamplifiers and buffers
• filters
• voltage regulator and current regulator
• analog-to-digital converter
• digital-to-analog converter
• voltage clamp
• oscillators and waveform generators
• Analog computer
• Capacitance multiplier
• Charge amplifier

See also

• Current-feedback operational amplifier
• Frequency compensation
• Operational amplifier
• Operational transconductance amplifier
• Transimpedance amplifier

Notes

1. If you think of the left-hand side of the relation as the closed-loop gain of the inverting input, and the right-hand side as the gain of the non-inverting input, then matching these two quantities provides an output insensitive to the common-mode voltage of ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$.

References

1. Paul Horowitz and Winfield Hill, The Art of Electronics . 2nd ed. Cambridge University Press, Cambridge, 1989 ISBN   0-521-37095-7
2. Basic Electronics Theory, Delton T. Horn, 4th ed. McGraw-Hill Professional, 1994, p. 342–343.
3. "Benefits of Negative Feedback". HyperPhysics. Retrieved 2018-05-07.
4. Simpson, Robert E. (1987). "7.2 Negative Voltage Feedback". Introductory electronics for scientists and engineers (2nd ed.). Boston: Allyn and Bacon. p. 291. ISBN   0205083773. OCLC   13821010. input impedance of an amplifier without negative feedback is increased by adding negative feedback. .. ${\displaystyle Z_{\mathrm {inf} }=(1+A_{0}B)Z_{\mathrm {ino} }}$ .. output impedance .. is decreased .. ${\displaystyle Z_{\mathrm {outf} }=Z_{\mathrm {out} }/(1+A_{0}B)}$
5. "Op Amps used as Comparators—is it okay? - the Signal - Archives - TI E2E support forums".
6. "AN1177 Op-Amp Precision Design: DC Errors" (PDF). Microchip. 2 January 2008. Archived (PDF) from the original on 2019-07-09. Retrieved 26 December 2012.
7. Kuhn, Kenneth A. (2004-03-24). "Piecewise Linear Circuits" (PDF).
8. "AN-4 Monolithic Op Amp—the Universal Linear Component" (PDF). Texas Instruments . April 1968.
9. "National Semiconductor Application Note 31" (PDF). September 2002.

Further reading

Main article: Operational amplifier further reading

External links

• "Single supply op-amp circuit collection" (PDF). (163  KiB)
• "Op-amp circuit collection" (PDF). (2980  KiB)
• "A Collection of Amp Applications" (PDF). (1.06  MiB) – Analog Devices Application note
• "Basic OpAmp Applications" (PDF). (173  KiB)
• "Handbook of operational amplifier applications" (PDF). (2.00  MiB) – Texas Instruments Application note
• Low Side Current Sensing Using Operational Amplifiers Archived 2009-04-08 at the Wayback Machine
• "Log/anti-log generators, cube generator, multiply/divide amp" (PDF). Archived from the original (PDF) on 2008-05-09. (165  KiB)
• Logarithmically variable gain from a linear variable component
• Impedance and admittance transformations using operational amplifiers by D. H. Sheingold
• High Speed Amplifier Techniques very practical and readable –with photos and real waveforms
• Properly terminating an unused op-amp