Electronics Guide

Circuit Operation

In the inverting configuration, the op-amp maintains its inverting input at virtual ground potential. Because the non-inverting input connects to actual ground and the op-amp has extremely high gain, the voltage difference between the two inputs approaches zero. Current flowing through the input resistor must equal the current through the feedback resistor (the high input impedance of the op-amp prevents any significant current from entering its inputs), creating a precise relationship between input and output voltages.

The voltage gain of an inverting amplifier is:

A_v = −R_f/R_in

where R_f is the feedback resistor and R_in is the input resistor. The negative sign indicates the phase inversion inherent to this configuration.

Design Considerations

The input impedance of an inverting amplifier equals the input resistor value, which may be insufficient for high-impedance sources. Selecting larger resistor values increases input impedance but also increases thermal noise and may introduce errors due to input bias currents.

For gains less than unity, the inverting configuration excels because it can easily achieve attenuation while maintaining a defined input impedance. Unity gain occurs when R_f equals R_in, while gains below one require R_f smaller than R_in. Note that the feedback fraction, and therefore the bandwidth, depends on R_f/R_in through the quantity 1 + R_f/R_in rather than through the closed-loop gain itself, so an inverting attenuator still loads the amplifier with a noise gain slightly above unity.

The virtual ground at the inverting input makes this configuration ideal for summing multiple signals, as each input sees only its own input resistor rather than interacting with other input sources.

Common Applications

• Signal amplification with controlled gain
• Signal attenuation and scaling
• Phase inversion in audio and signal processing
• Building block for summing amplifiers
• Active filter implementations

Circuit Operation

With the input applied to the non-inverting terminal, the op-amp adjusts its output to maintain virtually zero voltage difference between its inputs. This forces the voltage at the inverting input to equal the input signal voltage, and the resistive divider between output and ground must produce this voltage at its midpoint.

The voltage gain of a non-inverting amplifier is:

A_v = 1 + R_f/R₁

where R_f is the feedback resistor and R₁ is the resistor from the inverting input to ground. The minimum gain is unity, reached when R_f is zero or R₁ is omitted, which makes the non-inverting configuration unsuitable for attenuation.

Design Considerations

The differential input impedance of a non-inverting amplifier is raised by a factor of one plus the loop gain, which makes it very high at low frequencies. The benefit fades as loop gain falls with frequency, and the common-mode input impedance, which feedback does not raise, sets the eventual ceiling. Even so, the configuration is well suited to buffering high-impedance sources without loading them.

Input bias current flows through the source impedance, potentially creating offset errors. To minimize this effect, some designs add a resistor between the non-inverting input and ground equal to the parallel combination of R_f and R₁, balancing the bias current effects at both inputs.

Common-mode voltage handling becomes a consideration because the entire input signal appears at the op-amp's inputs. The selected op-amp must have adequate common-mode input range for the expected signal levels.

Common Applications

• High-impedance buffer amplification
• Sensor signal conditioning
• Voltage scaling without inversion
• Input stages for precision instruments
• Audio preamplifiers

Circuit Characteristics

With 100% negative feedback, the voltage follower exhibits unity gain (A_v = 1) with extremely high input impedance and low output impedance. The output precisely follows the input voltage, limited only by the op-amp's ability to drive the load and its finite bandwidth and slew rate.

The voltage follower isolates the source from the load, preventing loading effects that would otherwise reduce signal amplitude. A high-impedance sensor can drive the follower's input while the op-amp's output stage drives low-impedance loads without affecting the sensor.

Performance Limitations

While conceptually simple, voltage followers present stability challenges in certain situations. The direct feedback connection provides no frequency-dependent roll-off, potentially causing oscillation with capacitive loads or long cable runs. Many op-amps require compensation or isolation resistors when driving significant capacitive loads.

Input offset voltage and input bias current directly affect output accuracy since there are no resistors to introduce additional errors but also no gain to amplify them. For precision applications, selecting op-amps with low offset and bias specifications proves essential.

Buffer Variations

Several variations on the basic voltage follower address specific application requirements:

• AC-coupled follower: Capacitor in series with the input blocks DC while passing AC signals
• Bootstrapped buffer: Increased effective input impedance by driving the bias network with a buffered copy of the input, so little voltage develops across it
• Composite buffer: Output transistor stage adds current drive capability
• Differential buffer: Converts differential signals to single-ended output

Operating Principles

Because the inverting input maintains virtual ground potential, each input resistor operates independently: the current from each source depends only on its voltage and resistor value, unaffected by other inputs. All input currents combine at the summing junction and flow through the feedback resistor, producing an output voltage proportional to their sum.

For a summing amplifier with n inputs:

V_out = −R_f(V₁/R₁ + V₂/R₂ + V₃/R₃ + ... + V_n/R_n)

Equal input resistors produce a simple sum (or average, if scaled appropriately), while different resistor values weight the inputs differently.

Design Considerations

The summing amplifier's virtual ground isolates inputs from each other, preventing interaction between sources. This makes it ideal for audio mixing applications where multiple signals must combine without crosstalk.

Each input sees only its own input resistor as impedance, regardless of other inputs or their impedances. This predictable loading simplifies interface design between multiple signal sources.

The total input current capability limits the number of inputs and their gains. Too many inputs with low resistor values may exceed the output current capability of the feedback network or cause excessive noise.

Weighted Summation

Varying the input resistor values creates weighted sums useful for scaling signals before combining. For example, in an audio mixer, different channels may require different gain factors to balance levels from various sources like microphones, instruments, and playback devices.

Digital-to-analog converters often use summing amplifier principles with binary-weighted resistors, where each input resistor doubles in value for each bit position. This converts a digital binary code into a proportional analog voltage.

The Inverting Integrator

An input resistor R feeding the virtual ground, with a capacitor C in the feedback path, produces an output that is the running integral of the input:

V_out(t) = −(1/RC) ∫ V_in(t) dt

In the frequency domain the gain falls at 20 decibels per decade and passes through unity at a frequency of 1/(2πRC). A 100 kΩ resistor with a 10 nF capacitor gives a unity-gain frequency near 159 Hz and integrates cleanly well above it.

The practical difficulty lies at the other end of the spectrum. The capacitor presents an open circuit at direct current, so the circuit has no feedback there at all, and the amplifier's own input offset voltage and input bias current integrate without limit until the output sits against a supply rail. A resistor R_f placed across the capacitor fixes this by setting a finite direct-current gain of −R_f/R and a corner at 1/(2πR_fC), below which the circuit stops integrating and behaves as an ordinary inverting amplifier. Choosing R_f at least ten times R keeps that corner a decade below the lowest frequency of interest. Where a true integrator is required, as in a ramp generator or a dual-slope converter, a switch across the capacitor resets the accumulated charge instead.

The Inverting Differentiator

Exchanging the two elements, so that a capacitor C feeds the virtual ground and a resistor R_f provides feedback, produces the derivative:

V_out(t) = −R_fC dV_in(t)/dt

The differentiator is the harder of the two to build, for a reason that is structural rather than incidental. Its noise gain, the factor by which the amplifier's own input noise and offset reach the output, is 1 + jωR_fC, which rises at 20 decibels per decade. The amplifier's open-loop gain falls at 20 decibels per decade over the same range, so the two curves close on each other at 40 decibels per decade. That rate of closure leaves almost no phase margin, and the circuit peaks sharply or oscillates outright.

The standard remedy adds a resistor R_in in series with the input capacitor. Above 1/(2πR_inC) the noise gain flattens at 1 + R_f/R_in instead of continuing to climb, the rate of closure returns to 20 decibels per decade, and the circuit is stable. The price is that differentiation holds only below that corner, which must therefore sit above the highest signal frequency the circuit is meant to differentiate. A small capacitor across R_f is often added as well, rolling the response off at high frequency and suppressing the noise a differentiator otherwise amplifies. Because of these limits, designers frequently avoid the differentiator altogether and obtain a derivative by other means, such as a high-pass filter followed by a flat gain stage.

Basic Configuration

The single-op-amp difference amplifier uses four resistors: two form an inverting amplifier from one input, while two form a voltage divider that sets the reference voltage at the non-inverting input from the other signal. Take V₁ to reach the inverting input through R₁, with R₂ as the feedback resistor, and V₂ to reach the non-inverting input through R₃, with R₄ from that input to ground. Superposition then gives the general result:

V_out = V₂[R₄/(R₃ + R₄)](1 + R₂/R₁) − V₁(R₂/R₁)

The two coefficients become equal in magnitude, and the common-mode response therefore vanishes, only when the divider ratio matches the inverting ratio:

R₄/R₃ = R₂/R₁

Under that condition the expression collapses to a clean differential gain:

V_out = (R₂/R₁)(V₂ − V₁)

Equal resistors throughout make R₂/R₁ unity and reduce the output to the plain difference V₂ − V₁. The balance condition, not the individual resistor values, is what the circuit depends on, and it is the condition that manufacturing tolerance violates.

Common-Mode Rejection

The difference amplifier's ability to reject common-mode signals depends critically on resistor matching, typically far more than on the op-amp's own CMRR. Any mismatch breaks the balance condition, the common-mode voltage appears partially in the output, and the common-mode rejection ratio (CMRR) falls. Writing G for the differential gain R₂/R₁ and t for the fractional resistor tolerance, the worst case follows:

CMRR_min ≈ (1 + G)/(4t)

This is a worst-case bound rather than a typical figure. It assumes all four resistors sit at their tolerance limits in the least favorable directions at once, which places R₁ and R₄ low while R₂ and R₃ run high. Four independent parts rarely conspire that way, so a measured unit built from the same resistors commonly does 5 to 10 decibels better. The bound is nevertheless what a design must be guaranteed against.

With standard 1% resistors at unity gain, the bound is a CMRR of only about 34 decibels, a ratio near 50. Stepping to 0.1% resistors improves the worst case to roughly 54 decibels, and 0.01% parts reach about 74 decibels: each factor of ten in tolerance buys exactly 20 decibels. Raising the gain helps as well, since the numerator carries 1 + G, so the same 1% resistors at a gain of 100 give a worst case near 68 decibels. Reaching high rejection at low gain therefore calls for integrated difference amplifiers whose resistors are laser-trimmed and thermally tracked on a single die, because matched temperature behavior matters as much as initial tolerance. Discrete resistors trimmed to 0.01% at room temperature drift apart if their temperature coefficients differ, and a difference of 10 parts per million per degree Celsius across a 30 degree Celsius rise is enough to undo the trim.

Input Impedance Considerations

Unlike the instrumentation amplifier, the simple difference amplifier presents different impedances to its two inputs, and these impedances vary with the common-mode voltage. The inverting input sees the input resistor impedance, while the non-inverting input sees the divider resistance.

This impedance imbalance can cause errors when source impedances are significant or unequal. Buffering both inputs with voltage followers before the difference amplifier addresses this limitation while maintaining the basic topology's simplicity. The buffers fix only the impedance, however. They add no gain, so the resistor mismatch downstream is still referred to the input undiminished and the CMRR is unchanged. Improving that requires gain ahead of the subtraction, which is exactly what the instrumentation amplifier adds.

Three-Op-Amp Topology

The classic instrumentation amplifier uses two input buffer stages followed by a difference amplifier. The input op-amps connect in a unique configuration that provides gain while maintaining high input impedance: a single resistor bridges their inverting inputs, with feedback resistors connecting each op-amp's output back to its inverting input.

The differential gain of the input stage follows from a single observation: the voltage across R_g equals the input difference, so the current through it is (V₂ − V₁)/R_g, and because no current enters the amplifier inputs, that same current flows through both feedback resistors R. Each output therefore moves away from its own input by that current times R, and the two move in opposite directions, which is where the factor of two originates. With G₂ for the gain of the output difference amplifier, most often unity, the complete transfer function is:

V_out = (V₂ − V₁)(1 + 2R/R_g)G₂

The common-mode behavior is the more interesting half. When both inputs carry the same voltage, no voltage appears across R_g, so no current flows through it or through either feedback resistor, and both outputs sit at exactly the input voltage. The common-mode gain of the input stage is therefore exactly unity, and, notably, it stays exactly unity even if the two resistors R are badly mismatched, because the current that mismatch would act upon is zero. Only the two amplifiers' own finite CMRR and their offset voltages disturb this.

That asymmetry is the whole reason the topology outperforms the single-op-amp difference amplifier. The front end multiplies the difference by 1 + 2R/R_g while leaving the common-mode signal at unity, so the output stage receives a signal in which the wanted component has already been raised relative to the unwanted one. The output stage's resistor-matching error is still present, but referred back to the input it is divided by the front-end gain. The consequence is that an instrumentation amplifier's CMRR improves with gain, and that at a gain of one it holds no advantage at all, since the front end contributes nothing and the specification reduces to that of the output difference amplifier.

Advantages Over Simple Difference Amplifiers

• High input impedance: Both inputs see the input impedance of their respective buffer op-amps
• Balanced inputs: Both input paths are symmetric, so source impedance affects them equally
• Single-resistor gain control: R_g matches nothing, so trimming gain cannot disturb the resistor ratios that set CMRR; raising the gain in fact improves it
• CMRR that scales with gain: Integrated parts commonly guarantee around 90 decibels at unity gain, rising past 110 decibels at a gain of 10 and 130 decibels at a gain of 100
• Low drift: Matched input stages minimize offset voltage temperature drift

Integrated Instrumentation Amplifiers

Modern integrated instrumentation amplifiers contain all three op-amps plus precisely matched internal resistors on a single chip. These devices offer superior performance compared to discrete implementations due to tight resistor matching and thermal tracking.

Features often include programmable gain through external resistor or pin-strapping, reference voltage inputs for level shifting, and specifications optimized for specific applications like medical instrumentation, strain gauge interfaces, or thermocouple amplifiers.

Basic Transimpedance Amplifier

The simplest current-to-voltage converter connects the current source to the inverting input with only a feedback resistor to the output. The non-inverting input connects to ground (or a reference voltage). The op-amp maintains virtual ground at its inverting input, so the input current flows entirely through the feedback resistor.

V_out = −I_inR_f

The transimpedance (transfer impedance) equals the feedback resistor value, expressed in volts per ampere (or ohms).

Photodiode Amplifiers

Photodiode current-to-voltage conversion presents unique challenges due to the photodiode's junction capacitance and the high gains typically required. The feedback resistor and photodiode capacitance form a pole in the feedback network that can cause instability.

Compensation typically involves adding a small capacitor in parallel with the feedback resistor. This capacitor value must be carefully chosen to maintain stability while preserving adequate bandwidth for the application. The optimal value depends on the feedback resistor, photodiode capacitance, and op-amp gain-bandwidth product.

Design Trade-offs

Higher feedback resistance increases sensitivity but also increases noise and reduces bandwidth. The Johnson noise of the feedback resistor often dominates in high-gain configurations, requiring careful optimization between sensitivity and noise performance.

For very low current levels (picoampere range), junction FET-input or CMOS op-amps with femtoampere-level input bias currents become necessary to prevent the op-amp's bias current from overwhelming the signal current.

Grounded-Load Converter

Many practical loads have one terminal tied to ground. The Howland current pump drives such a load with a single op-amp and four resistors that combine negative and positive feedback. The load hangs from the non-inverting input node to ground, one input signal reaches that node through a resistor R₁, and the other reaches the inverting input through R₃, with R₂ returning from the output to the load node and R₄ closing the negative feedback loop. The circuit works when the positive and negative feedback are equally strong:

R₁/R₂ = R₃/R₄

Under that condition the output current depends only on the differential input and the single resistor R₁:

I_load = (V_in+ − V_in−)/R₁

The inputs are genuinely differential, and the current may flow in either direction, so one circuit both sources and sinks. Grounding one input and driving the other reduces the expression to I_load = V_in/R₁.

Output impedance is the whole point of the topology, and resistor matching alone sets it. Writing ε for the fractional error in the ratio balance above, the output resistance is approximately R₁/ε. With four 10 kΩ resistors, a 1% error in one of them permits an output resistance as low as 1 MΩ, and all four at their 1% limits in the worst directions bring it down to about 250 kΩ. Moving to 0.1% parts recovers roughly 2.5 MΩ. The sign matters as much as the magnitude: the ratio error can go either way, and when it does the output resistance turns negative. A negative output resistance driving a capacitive load does not droop toward zero but runs away from it, accelerating toward a supply rail. Precision versions therefore trim the balance with a potentiometer or buy a matched resistor network, and the improved Howland variant moves the current-setting element to a separate sense resistor so that compliance and matching sensitivity both improve.

For larger currents, a common variant abandons the Howland entirely and adds a pass transistor. The op-amp servos the voltage across a current-sense resistor until it equals the input, forcing a defined current through the transistor and the grounded load:

I_load = V_in/R_sense

This arrangement is unidirectional, since the transistor conducts one way only, but it handles currents far beyond an op-amp's own output rating and needs no resistor matching at all.

Floating-Load Converter

When the load can float, with neither terminal grounded, the simplest voltage-to-current converter places the load directly in the op-amp's feedback path. The op-amp holds the input voltage across a reference resistor, and because negligible current enters its inputs, that same current flows through the load:

I_load = V_in/R_ref

The output current depends only on the input voltage and the reference resistor, not on the load resistance, giving excellent compliance as long as the voltage the op-amp must develop stays within its output range. Because both load terminals float, this topology cannot drive a load that has one end grounded.

Current Loop Applications

Industrial 4 to 20 mA current loop transmitters commonly use voltage-to-current converters. The input voltage from a sensor signal chain converts to a current that transmits over long cable runs immune to voltage drops and noise pickup. Because the range starts at 4 mA rather than zero, a reading of no current is unambiguous evidence of a broken wire rather than a legitimate zero-scale measurement.

Logarithmic Amplifier Principles

The logarithmic relationship between current and voltage in a forward-biased diode or transistor junction provides the basis for log amplifiers. Placing a transistor in the feedback path of an inverting amplifier creates an output proportional to the logarithm of the input current:

V_out = −(kT/q)ln(I_in/I_s)

where kT/q is the thermal voltage, about 25.7 mV at 25 degrees Celsius, and I_s is the transistor saturation current. One decade of input current therefore moves the output by (kT/q)ln(10), close to 59 mV, which is the figure a log amplifier's scale is usually quoted against.

Because the logarithm of voltage requires first converting voltage to current (through an input resistor), the complete log amplifier equation becomes:

V_out = −(kT/q)ln(V_in/(I_sR_in))

Temperature Compensation

Two separate temperature effects act on this circuit, and they are not of the same size or the same kind. The thermal voltage kT/q is proportional to absolute temperature, so its fractional change per kelvin is simply 1/T. Near room temperature that is 1/298, or about 0.34% per degree Celsius, and it appears as a scale-factor error: the volts-per-decade slope stretches with temperature, moving from roughly 54 mV per decade at 0 degrees Celsius to 68 mV per decade at 70 degrees Celsius.

The saturation current I_s is the larger problem. It rises steeply with temperature, roughly doubling every 10 degrees Celsius for a silicon junction, and it enters the expression inside the logarithm. Doubling I_s shifts the output by (kT/q)ln(2), near 18 mV, which against a 59 mV decade is three tenths of a decade. Referred back to the input, the circuit reports a current error of roughly 7% per degree Celsius, around twenty times the scale-factor error, and it appears as an offset in the logarithmic domain rather than as a gain error. The two effects do not cancel, and no adjustment of the slope removes the I_s term.

This is precisely why a practical log amplifier uses a matched transistor pair rather than a single device. The second transistor carries a stable reference current, and subtracting the two base-emitter voltages yields an output proportional to ln(I_in/I_ref). The I_s terms appear in both halves and cancel outright, provided the two devices share a die and a temperature. That leaves only the kT/q scale-factor drift, a much smaller and far better behaved error, which designers correct with a gain-setting divider containing a resistor whose resistance rises with temperature at about 3300 parts per million per degree Celsius, cancelling the 1/T dependence. Monolithic log amplifiers integrate both the matched pair and that compensating element.

Antilog Amplifiers

The antilogarithmic amplifier places the diode or transistor junction at the input rather than in the feedback path. The exponential relationship between voltage and current in the junction produces an output current (and through a resistor, an output voltage) that is the exponential (antilog) of the input voltage.

Combined with log amplifiers, antilog circuits enable analog computation of multiplication, division, powers, and roots. Two input voltages converted to logarithms can be added (for multiplication) or subtracted (for division), then converted back through an antilog stage.

Applications

• Dynamic range compression: Compressing signals spanning many decades into a manageable voltage range
• Analog computation: Multiplication, division, and power functions
• Signal level measurement: dB-scaled outputs for audio and RF applications
• Linearization: Compensating for exponential sensor characteristics
• True RMS conversion: As part of implicit RMS computation circuits

Half-Wave Precision Rectifier

The simplest precision rectifier uses one op-amp and one or two diodes in the feedback path. During positive input half-cycles, the diode conducts, and the circuit operates as a unity-gain (or amplifying) follower. During negative half-cycles, the feedback loop opens, and the output remains at zero (or the reverse diode clamps it).

The op-amp compensates for the diode's forward voltage by raising its output by roughly that drop, so the voltage delivered to the load tracks the input. The effective threshold falls by a factor of the op-amp's open-loop gain, allowing accurate rectification of signals well below the diode's 0.6 V drop.

Full-Wave Precision Rectifier

Full-wave precision rectification requires combining half-wave rectification with inversion and summing. One common approach uses two op-amps: a half-wave precision rectifier followed by a summing amplifier that adds the original signal to twice the rectified signal with appropriate polarity.

Alternatively, an absolute value circuit using two op-amps in a different configuration provides full-wave rectification with improved high-frequency response by keeping both op-amps active at all times.

Speed Limitations

When the input crosses zero, the op-amp must transition from one feedback path to another. During this transition, the op-amp may saturate momentarily, causing crossover distortion and limiting maximum operating frequency.

Fast precision rectifiers use techniques to minimize these transitions: keeping the op-amp output from saturating through additional feedback paths, using faster op-amps, or employing synchronous rectification with external switching control.

Applications

• AC voltage measurement: Converting AC signals to DC for measurement
• Peak detection: Capturing signal peaks for envelope detection
• Signal demodulation: Extracting modulation from carrier signals
• Absolute value circuits: Generating |x| function for computation
• Precision AC-DC conversion: Power measurement and control applications

Basic Clamping Operation

A simple diode clamp uses a capacitor to block DC and a diode to establish the reference level. During one half-cycle, the diode conducts, charging the capacitor. During the opposite half-cycle, the diode is reverse-biased, and the charged capacitor adds its voltage to the input signal, shifting the entire waveform.

Passive diode clamps suffer from the diode's forward voltage drop and require multiple time constants to stabilize. The output level drifts as the capacitor slowly discharges through the load or leakage paths.

Precision Clamping with Op-Amps

Placing the diode within an op-amp feedback loop eliminates the forward voltage drop, enabling precision clamping of small signals. The op-amp actively drives the capacitor charging, reducing settling time and improving level accuracy.

The reference voltage for the clamp level can be any value within the op-amp's output range, providing flexible level shifting. Changing the diode orientation changes whether positive or negative peaks are clamped to the reference.

Applications

• DC restoration: Restoring DC level lost through AC coupling
• Video signal processing: Black level clamping in video circuits
• Radar and pulse systems: Baseline restoration for pulse detection
• Level shifting: Repositioning signals for subsequent processing stages
• Peak-to-peak detection: Combined with peak detectors for amplitude measurement

Component Selection

Resistor selection significantly impacts circuit performance. Metal film resistors offer better temperature coefficients and lower noise than carbon composition types. For precision applications, resistor tolerance of 0.1% or better may be necessary, particularly in difference amplifiers and instrumentation amplifiers where matching affects CMRR.

Capacitors in AC-coupled circuits and compensation networks should be film types rather than ceramic for critical applications. Ceramic capacitors (except C0G/NP0 types) exhibit voltage coefficients and microphonic effects that can introduce distortion.

Layout Considerations

Power supply bypassing close to the op-amp power pins is essential for high-frequency performance and stability. Typically, 0.1 microfarad ceramic capacitors should connect within a few millimeters of each power pin.

Signal routing should keep input traces short and away from output traces to minimize crosstalk and oscillation risks. Ground planes help shield sensitive signals but can add capacitance that affects high-frequency response.

Testing and Troubleshooting

When circuits malfunction, first verify power supply voltages at the op-amp pins. Then check that inputs are within common-mode range and that the output is not saturated against a supply rail. For oscillating circuits, check power supply bypassing, probe without adding excessive capacitive load, and consider whether the op-amp is stable with the feedback network and load present.