Electronics Guide

Multiplication and Averaging

Consider an input signal of the form V_s sin(ωt + φ) and a reference of the form sin(ωt). Their product can be rewritten, using a standard trigonometric identity, as the sum of two terms:

V_s sin(ωt + φ) × sin(ωt) = (V_s/2) cos(φ) − (V_s/2) cos(2ωt + φ)

The first term sits at the difference frequency, which is zero hertz (direct current), and the second at the sum frequency, which is twice the reference frequency. The DC term, (V_s/2) cos(φ), depends on φ, the phase difference between the signal and the reference. A low-pass filter following the multiplier removes the term at twice the reference frequency and passes the DC term, producing a steady output proportional to the signal amplitude and to the cosine of the phase difference.

The key consequence is selectivity. If the input contains a second sinusoid at a different frequency, its product with the reference yields only sum and difference frequencies, both nonzero, both removed by the low-pass filter. Only input components at the reference frequency, or extremely close to it, produce a surviving DC contribution. Noise, which spreads its energy across a wide band of frequencies, contributes almost nothing to the averaged output because the filter rejects all but a narrow slice around the reference frequency.

Equivalent Noise Bandwidth

The width of the surviving frequency slice is set by the low-pass filter and is described by its equivalent noise bandwidth (ENBW), defined as the width of an ideal rectangular filter that would pass the same total noise power as the real filter. For the single-pole resistor-capacitor filter that forms the simplest lock-in output stage, the equivalent noise bandwidth equals one divided by four times the time constant:

ENBW = 1/(4τ)

Here τ is the filter time constant in seconds, and ENBW is in hertz. A 100-millisecond time constant therefore corresponds to an equivalent noise bandwidth of 2.5 hertz, and a 10-second time constant to 25 millihertz. Because the multiplier translates the reference frequency to zero hertz, this narrow band appears centered on the reference frequency, no matter where in the spectrum that frequency lies.

A lock-in amplifier can therefore be configured for an equivalent noise bandwidth well below one hertz, far narrower than any practical analog band-pass filter centered at the same frequency, and it holds that width without the center-frequency drift that plagues high-quality-factor filters. Because the random-noise contribution to a measurement is proportional to the square root of the bandwidth, narrowing the detection band from kilohertz to a fraction of a hertz improves the signal-to-noise ratio by a factor of tens or hundreds. This concentration of measurement sensitivity into a vanishingly narrow band is the central reason the technique is so effective.

A Worked Example

The advantage is easiest to see in numbers. Consider a 10-nanovolt sine wave at 10 kilohertz. A good low-noise preamplifier contributes about 5 nanovolts per root hertz of input noise; with a gain of 1,000 and a 100-kilohertz bandwidth, it raises the signal to 10 microvolts but also delivers roughly 1.6 millivolts of broadband noise, more than a hundred times the signal. Adding an excellent band-pass filter with a quality factor of 100 narrows the band to 100 hertz and cuts the noise to about 50 microvolts, which still buries the 10-microvolt signal. Further gain changes nothing, because it amplifies signal and noise alike.

A phase-sensitive detector set for a 0.01-hertz detection bandwidth reduces the noise reaching the output to about 0.5 microvolts while the signal remains at 10 microvolts. The signal-to-noise ratio becomes roughly 20, and an accurate measurement is possible. Nothing about the amplifier improved. What changed is the bandwidth over which its noise accumulates.

Generating and Synchronizing the Reference

A lock-in measurement requires that the quantity of interest be modulated at a known frequency. In an optical experiment, a mechanical chopper interrupts a light beam at a chosen rate; in an electrical measurement, a sinusoidal excitation drives the device under test. The same source that imposes the modulation also supplies the reference, ensuring that the signal and the reference remain locked in frequency. The instrument typically accepts an external reference input, such as the synchronizing output of a chopper controller or function generator, or it generates the excitation internally so that the reference is inherently available.

Within the instrument, a phase-locked loop regenerates a clean reference that follows the external source even if that source drifts slowly or carries its own noise. The phase-locked loop tracks the reference frequency and produces the internal sinusoids used for multiplication, suppressing jitter that would otherwise degrade the measurement. Because the detector responds only to signals locked to this reference, drift in the reference frequency is followed automatically rather than appearing as an error.

The Role of Phase Adjustment

Since the detected output is proportional to the cosine of the phase difference between signal and reference, the phase of the reference must be adjusted so that the signal of interest produces the desired response. When the reference phase is set so that the phase difference is zero, the output reaches its maximum and reports the full signal amplitude. When the phase difference is ninety degrees, the cosine is zero and the detector reports nothing, even though the signal is present. Setting the reference phase correctly is therefore an essential step, and the phase at which the output is maximized itself conveys information about the signal path, including the cumulative delay introduced by the experiment.

Choosing the Modulation Frequency

The choice of modulation frequency shapes the whole measurement. The frequency should be high enough to clear the region where flicker noise and thermal drift dominate, yet low enough that the transducer, the excitation source, and any intervening amplifier still respond faithfully. A mechanical chopper sets a practical ceiling of several kilohertz for a beam a few millimeters across; finely slotted blades reach 20 kilohertz in Stanford Research Systems' SR542, but only for a beam about 1.4 millimeters wide. An electronic modulator imposes no such ceiling until the bandwidth of the device under test intervenes.

Within that window, the frequency should avoid the power-line frequency and its harmonics, which carry the most common laboratory interference, and it should avoid simple ratios with other periodic disturbances such as chopper-motor rotation, display refresh, and switching-converter rates. An odd, non-round frequency, 3.7 kilohertz rather than 4 kilohertz for example, reduces the chance of coincidence with a discrete interferer. Many instruments also provide switchable notch filters at the line frequency and its second harmonic, which relieve the detector of interference it would otherwise have to tolerate as dynamic reserve.

Modulation Moves the Signal Away from Low-Frequency Noise

Many measurements are limited by noise that is strongest at low frequencies, including flicker noise, often called 1/f noise, and slow drift from temperature changes, mechanical settling, and component aging. A direct measurement of a small steady quantity sits squarely within this low-frequency noise. By modulating the quantity at a higher frequency, the experimenter shifts the signal to a part of the spectrum where the dominant disturbance is the comparatively flat thermal noise floor rather than the rising 1/f contribution. The lock-in detector then recovers the signal at the modulation frequency, away from the worst of the noise. The gain can be dramatic: flicker noise in a semiconductor front end may exceed the thermal floor by an order of magnitude at 1 hertz while contributing almost nothing at 1 kilohertz.

Coherence as the Basis for Rejection

Noise rejection in a lock-in amplifier is not merely filtering; it is correlation. The multiplier compares the input against the reference point by point, and only those parts of the input that march in step with the reference accumulate a nonzero average. Random noise, having no fixed relationship to the reference, contributes positive and negative products in equal measure and averages away. This is why a lock-in amplifier can extract a signal whose amplitude is a small fraction of the root-mean-square noise: the measurement does not depend on the signal rising above the noise at any instant, only on its persistent coherence with the reference over the averaging time.

The Noise Floor at the Input

Narrowing the detection band cannot reduce noise that arrives with the signal or is generated by the instrument's own front end, so the input stage sets the ultimate floor. A typical laboratory lock-in amplifier contributes about 5 nanovolts per root hertz of input voltage noise, and Stanford Research Systems gives a typical 2.5 at 1 kilohertz for its current SR860. The source contributes as well, through the Johnson noise of its resistance R_s:

e_n = √(4kTR_s) ≈ 0.13√R_s nV/√Hz

Here e_n is the noise voltage density, k is Boltzmann's constant, and T is the absolute temperature; the approximation holds at room temperature, about 300 kelvins, with R_s in ohms. A 50-ohm source therefore adds about 1 nanovolt per root hertz, while a 2-kilohm source adds roughly 5.8, already exceeding the amplifier's own contribution.

Uncorrelated noise sources add in quadrature, so an amplifier at 5 nanovolts per root hertz driven from a 2-kilohm source presents a combined floor near 7.7 nanovolts per root hertz. Multiplying a noise density by the square root of the equivalent noise bandwidth gives the noise expected at the output. With a 100-millisecond time constant and its 2.5-hertz equivalent noise bandwidth, the same amplifier yields about 7.9 nanovolts root mean square when driven from a low-impedance source, and about 12 from the 2-kilohm source. Converting to peak-to-peak noise takes a convention. Stanford Research Systems multiplies the root-mean-square value of Gaussian noise by about five, a band the noise leaves about 1 percent of the time, which puts the first figure at roughly 40 nanovolts peak to peak; the stricter factor of 6, a band the noise leaves only about 0.3 percent of the time, gives about 47. Lowering the source impedance, cooling the source, or coupling through a matching transformer improves this floor; adding gain does not.

Dynamic Reserve

The ability of a lock-in amplifier to tolerate large interfering signals without overload is described by its dynamic reserve, the ratio of the largest tolerable noise or interference to the full-scale signal, expressed in decibels. A dynamic reserve of 60 decibels with a full-scale sensitivity of 1 microvolt means that interference as large as 1 millivolt can be present at the input without spoiling the measurement, provided that interference lies outside the narrow detection band.

The limits differ sharply between architectures. Analog phase-sensitive detectors are held to roughly 60 decibels of reserve, and the ceiling is set not by simple overload but by multiplier nonlinearity and by direct-current offsets that appear as gain and zero errors. Running an analog detector at high reserve also forces the output amplifier to high gain, where zero drift on the order of 1,000 parts per million per degree Celsius is common. Digital detectors, in which the reserve is limited chiefly by the linearity and resolution of the analog-to-digital conversion, routinely exceed 100 decibels without measurable error, at the cost of somewhat higher output noise.

Dynamic reserve is also frequency dependent. It falls to zero at the reference frequency itself, where interference is indistinguishable from signal, and it rises as the interfering frequency moves away, at a rate set by the roll-off of the output filter. This is one reason a filter of twenty-four decibels per octave tolerates nearby interference better than one of six decibels per octave. Because high reserve costs stability, the sound practice is to use the least reserve that avoids overload rather than the most the instrument offers.

The Two Components

The in-phase output, conventionally labeled X, is produced by multiplying the input by a reference at the nominal phase. The quadrature output, labeled Y, is produced by multiplying the same input by a reference shifted ninety degrees. These two outputs are proportional to V_s cos(φ) and V_s sin(φ) respectively, where φ is the phase between the signal and the in-phase reference. Together they specify both the magnitude and the phase of the signal at the reference frequency, the two numbers that fully characterize a sinusoid of known frequency.

Magnitude and Phase

From the in-phase and quadrature outputs, the instrument computes the magnitude R and the phase θ:

R = √(X² + Y²)

θ = arctan(Y/X)

The arctangent is resolved into the correct quadrant from the signs of X and Y, so θ covers the full circle from −180 to 180 degrees. The magnitude R is independent of the reference phase setting, which is a significant practical advantage: an experimenter can measure the true signal amplitude without having to first null the phase. The phase θ reports the delay between the signal and the reference and is itself a useful measurement in impedance and time-of-flight work. Reporting results as X and Y, or equivalently as R and θ, depends on whether the application calls for vector components or for amplitude and phase.

The two representations are not equally well behaved at low signal-to-noise ratio. X and Y are linear in the signal and average toward the correct value even when noise dominates, whereas R, formed from squares, is strictly positive and therefore reads high when the signal is small: pure noise produces a nonzero magnitude, averaging about 1.25 times the root-mean-square noise in each of X and Y. When the signal approaches the noise floor, the disciplined practice is to null the phase and report X, reserving R for measurements comfortably above the floor.

What the Displayed Value Means

A lock-in amplifier reports the root-mean-square amplitude of the Fourier component at the reference frequency, not the peak value and not the amplitude of the whole waveform. A displayed magnitude of 1 volt therefore corresponds to a sine wave of about 1.414 volts peak, or 2.8 volts peak to peak. Everything outside that one Fourier component is discarded, which is exactly the intent of the technique but a frequent source of confusion when the input is not sinusoidal.

Chopped-light experiments make the point concretely. A mechanical chopper produces a square wave, and the Fourier series of a square wave of peak amplitude V_p about its mean contains only odd harmonics:

v(t) = (4V_p/π)[sin(ωt) + (1/3) sin(3ωt) + (1/5) sin(5ωt) + …]

The fundamental has amplitude 4V_p/π, about 1.273V_p, larger than the square wave's own peak. A square wave that swings ±1 volt, which is what a detector output switching between 0 and 2 volts becomes once its direct-current level is set aside, therefore reads 0.90 volts root mean square on the instrument, not 1 volt. Converting a lock-in reading into the physical amplitude at the detector requires this factor, and neglecting it introduces an error of roughly 10 percent.

Setting the Time Constant

The output low-pass filter is characterized by a time constant, the interval over which the detector averages the multiplier output. A long time constant produces a narrow equivalent noise bandwidth and strong noise rejection, at the cost of a sluggish response that takes several time constants to settle after a change. A short time constant responds quickly but admits more noise. The operator chooses the time constant to match the rate at which the measured quantity changes: slow or static measurements tolerate long time constants and benefit from the resulting noise reduction, while measurements that track a changing quantity require a time constant short enough to follow it.

Filter Order and Roll-Off

Lock-in amplifiers typically offer a selectable filter order, providing roll-off rates such as six, twelve, eighteen, or twenty-four decibels per octave. Steeper roll-off rejects nearby interference more sharply and yields a narrower equivalent noise bandwidth for the same time constant. With identical poles of time constant τ in cascade, the equivalent noise bandwidth falls from 1/(4τ) for one pole to 1/(8τ) for two, 3/(32τ) for three, and 5/(64τ) for four, so a 100-millisecond time constant gives an equivalent noise bandwidth of 2.5 hertz at six decibels per octave but about 0.78 hertz at twenty-four. The price is additional phase lag and a longer settling time. A higher-order filter is useful when interference lies close to the signal frequency, whereas a lower-order filter settles more quickly when the spectrum is comparatively clean.

Settling time follows from the time constant and the filter order together. A single-pole filter comes within 1 percent of its final value after about 4.6 time constants, since the residual error decays exponentially; each additional pole lengthens that interval, to about 6.6 time constants for two poles, 8.4 for three, and 10 for the four poles of a twenty-four-decibel-per-octave filter. In a swept measurement, the dwell time at each point must exceed this settling time, or the recorded curve will lag the sweep and distort peak positions and line shapes.

The Speed-Versus-Noise Trade-Off

No setting escapes the underlying relationship between measurement speed and noise. Because the noise admitted by the detector scales with the square root of its equivalent noise bandwidth, and that bandwidth is inversely related to the time constant, halving the noise requires roughly quadrupling the averaging time. Designing a lock-in measurement therefore means deciding how much time can be spent at each data point and accepting the noise floor that the available time allows. Where the experiment permits, increasing the modulation amplitude or the integration time improves the result more reliably than any change to the detector alone.

Dual-Phase Detection

A dual-phase lock-in amplifier contains two phase-sensitive detectors driven by references ninety degrees apart, producing the in-phase and quadrature outputs simultaneously. This arrangement allows direct computation of magnitude and phase and frees the measurement from manual phase nulling. Dual-phase detection became the standard configuration precisely because the phase-independent magnitude it provides removes a recurring source of operator error and makes the instrument far easier to use for general measurements.

Analog Versus Digital Implementation

Early lock-in amplifiers performed multiplication with analog mixers, including switching demodulators that multiply the input by a square wave derived from the reference. Because a square wave is, by its Fourier series, the sum of a fundamental and odd harmonics, a square-wave reference responds not only to the fundamental but also to its odd harmonics, so analog detectors of this type can register signals at three times, five times, and higher odd multiples of the reference frequency unless a band-pass filter precedes the detector. The sensitivity to each odd harmonic falls off in inverse proportion to its order, so the third harmonic is admitted at one-third the gain of the fundamental and the fifth at one-fifth, but interference near these multiples can still fold down to the output. Analog low-pass filters built from resistors and capacitors then perform the averaging.

A digital lock-in amplifier converts the input to numbers with a high-resolution analog-to-digital converter and performs the multiplication and filtering in a digital signal processor or field-programmable gate array. The reference sinusoids are generated numerically to twenty bits or better, leaving harmonic content around 120 decibels below the fundamental, so the detector multiplies by what is effectively a pure sine and largely escapes the odd-harmonic response that affects switching demodulators; Stanford Research Systems, for example, specifies 80 decibels of harmonic rejection for its SR860. Digital multiplication introduces no direct-current offset and no gain error from reference-amplitude drift, which removes the two chief sources of analog detector inaccuracy. Digital filters realize precise, repeatable time constants and filter orders that are difficult to achieve with analog components, and the same hardware computes X, Y, R, and θ directly.

The digital architecture also makes capabilities practical that analog hardware could offer only awkwardly. A single instrument can run several demodulators in parallel, each locked to a different frequency, so that a device excited at two tones is characterized in one measurement. Detection at an integer multiple of the reference is a standard feature, exploited where the response of interest appears at a harmonic of the drive. In wavelength-modulation spectroscopy the second-harmonic signal traces the second derivative of the absorption line when the modulation is small; inelastic electron tunneling spectroscopy reads the second derivative of current with respect to voltage at the second harmonic of a small bias modulation; and the 3ω method measures thermal conductivity from the third harmonic of the voltage across a heater strip driven at ω. Instrument bandwidth has grown with converter performance, from about 100 kilohertz in classic designs to megahertz and, in specialized instruments, gigahertz operation: Zurich Instruments rates its SHFLI to 8.5 gigahertz. The underlying principle of phase-sensitive detection is unchanged throughout.

Interference at the Reference Frequency

The one disturbance a lock-in amplifier cannot reject is interference coherent with its own reference, because such interference is by definition indistinguishable from signal. Excitation current leaking into the detector through stray capacitance, a ground loop carrying return current at the modulation frequency, stray light from a source that the chopper also modulates, and electrical pickup from the chopper motor itself all produce an output that behaves exactly like a measurement. The cures are physical rather than electronic: grounding the experiment at a single point, shielding the detector, keeping reference and excitation cables away from signal cables, and reducing the area of any loop that could couple magnetically.

The essential diagnostic is a blank measurement. Removing the physical stimulus while leaving the excitation and reference running reveals the residual output, which is the coherent pickup. Any measurement smaller than that residual is not a measurement at all. Reversing the sign of the stimulus, or shifting the modulation to a different frequency and confirming that the result does not change, provides a further check.

Connections, Grounding, and Mechanical Noise

A single-ended connection is convenient but exposes the cable shield to pickup, which the instrument cannot distinguish from signal on the center conductor. A differential connection rejects that pickup and is preferred whenever the source permits it, provided both cables follow the same path so that they enclose no significant loop area. Common-mode rejection is finite. Stanford Research Systems' application note on lock-in amplifiers quotes about 100 decibels at low frequencies, falling by about six decibels per octave from around 1 kilohertz, and the data sheet of its current SR860 specifies 90 decibels up to 1 kilohertz. Even 100 decibels leaves a 100-millivolt common-mode signal looking like a 1-microvolt differential one, so a large common-mode voltage at the reference frequency remains dangerous even with a differential input.

Mechanical effects matter at the levels a lock-in amplifier can resolve. Flexing a coaxial cable changes its capacitance and injects charge, a microphonic signal that appears at the vibration frequency; tying cables down and using low-noise cable suppresses it. Junctions between dissimilar metals generate thermoelectric voltages of several microvolts per degree, which dominate below about one hertz and argue for modulating rather than measuring slowly. None of these effects is removed by a longer time constant if it happens to fall inside the detection band.

Optical Measurements

Optical experiments are a classic application. A mechanical chopper modulates a light beam at a fixed rate and supplies the reference, so that a photodetector signal arising from the chopped light is recovered by the lock-in while steady background light and detector drift are rejected. This approach enables measurement of weak optical signals in spectroscopy, photoluminescence, and absorption studies, and it underlies modulation techniques in which a property of the sample or the illumination is varied periodically to isolate a small differential response.

Impedance and Resistance Measurement

Lock-in detection is well suited to impedance measurement because it reports both magnitude and phase, which together separate the resistive and reactive parts of an impedance. Driving a device under test with a small sinusoidal excitation and detecting the resulting current or voltage with a lock-in yields the in-phase and quadrature components, from which resistance and reactance follow directly. The narrow detection band allows the use of very small excitation levels, which is valuable when larger currents would heat the sample or disturb the quantity being measured, as in the characterization of delicate materials and low-value resistances.

Sensor Measurement

Many sensor interfaces benefit from synchronous detection. Resistance-bridge sensors, capacitive sensors, and other transducers can be excited with an alternating drive and read out with a lock-in, moving the measurement away from low-frequency noise and rejecting interference from power lines and the environment. This strategy improves resolution in strain, displacement, and temperature measurement and is widely embedded in precision instrumentation, where a synchronous detector recovers a small sensor signal that would otherwise be lost in drift and pickup. The same architecture appears in integrated form well beyond the laboratory bench: capacitive touch controllers, inductive position sensors, and optical pulse-oximetry front ends all drive a transducer at a known frequency and demodulate synchronously, on silicon, for exactly the reasons a bench lock-in amplifier exists.

Materials and Device Characterization

Materials work relies on lock-in detection wherever the quantity of interest is a small differential response. Four-terminal resistance measurements on thin films, contacts, and low-value standards use a small alternating current and a lock-in to reach nanovolt resolution while avoiding both self-heating and the thermoelectric voltages that limit direct-current methods. Differential conductance measurements superimpose a small alternating modulation on a swept bias and recover the derivative of current with respect to voltage directly, rather than differentiating a noisy curve after the fact. Hall-effect and magnetoresistance measurements modulate either the current or the magnetic field and detect at that frequency. In scanning-probe microscopy, the cantilever is driven near resonance and a lock-in recovers the amplitude and phase of its motion, with the phase carrying information about dissipation in the tip-sample interaction.