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: one at the difference frequency, which is zero hertz (direct current), and one at the sum frequency, which is twice the reference frequency. The DC term has amplitude proportional to V_s cos(φ), where φ is 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. A lock-in amplifier can be configured for an equivalent noise bandwidth of a fraction of a hertz, far narrower than any practical analog band-pass filter centered at the same frequency. Because the random-noise contribution to a measurement is proportional to the square root of the bandwidth, narrowing the bandwidth from kilohertz to a fraction of a hertz can improve 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.

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.

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. Choosing a modulation frequency that avoids power-line harmonics and other discrete interference further improves the result.

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.

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, often expressed in decibels. A high dynamic reserve allows accurate measurement of a small signal in the presence of interference many times larger, provided that interference lies outside the narrow detection band. Practical instruments trade dynamic reserve against output stability and noise, and the operator selects a setting appropriate to the measurement conditions.

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 as the square root of the sum of the squares of X and Y, and the phase θ as the arctangent of Y divided by X. 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.

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 for a given time constant but introduces 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. The settling time of the measurement is determined jointly by the time constant and the filter order, and an adequate number of time constants must elapse before a reading is trusted.

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 with high spectral purity, so the detector multiplies by a true sine rather than a square wave and is insensitive to the odd-harmonic response that affects switching demodulators. Digital filters realize precise, repeatable time constants and filter orders that are difficult to achieve with analog components, and the same hardware can compute X, Y, R, and θ directly. Modern instruments extend this approach to multiple simultaneous reference frequencies and to internal sweep and data-logging functions, while the underlying principle of phase-sensitive detection remains unchanged.

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.