IN13 · Keep the rhythm

Lock-In Amplifier

How can a periodic signal smaller than the noise still be measured? A lock-in multiplies the input by a known rhythm, then averages. A signal in step leaves a steady component; rapidly changing noise tends to cancel. Longer averaging narrows the noise band but delays changes. Interference at the same rhythm remains.

Turn the reference dial and follow two filtered channels. Compare waiting times on the same noisy capture, then try beats and coherent interference.

Turn the reference knob on the instrument. The two channels keep signals in step with that reference.

Reference frequency

32\,\mathrm{Hz}

Reference phase

0\,\mathrm{deg}

Time constant per pole

0.2\,\mathrm{s}

Signal detuning

0\,\mathrm{Hz}

A default acquisition is loaded.

Source, noise, input range and filter

Cascaded RC polesSignal RMS voltage

0.003

Signal starting phase / degrees

30

White noise density

0.0006

Coherent interferer RMS voltage

0

Interferer starting phase / degrees

90

ADC bipolar range / V

0.2

Random seed

186

Voltages are RMS for sinusoids. Noise density is in volts per square root hertz, white up to the sampled Nyquist frequency. Changing frequency or range needs a new known-standard calibration.

u_X=\sqrt{2}v\cos\theta,\quad u_Y=-\sqrt{2}v\sin\theta

Model assumptions

RMS-scaled dual-phase detection with recorded cosine references, an 18-bit symmetric ADC at 1024 samples/s and cascaded held-input RC filters. A separate known sine calibrates gain, offset and phase at the saved reference frequency and range. The fixed input delay leaves an additional phase on detuned tones. No PLL dynamics, synchronous notch filter, source loading, analog anti-alias circuit or commercial performance is modeled. White noise is defined below sampled Nyquist; deterministic tones stay below it. Magnitude is positively biased near the noise floor. Same-frequency coherent interference is inseparable from the target. Saved outputs come only from saved ADC codes, reference samples and calibration. Pulse speed is schematic; the apparatus uses planar wiring.

Simulated ADC observations

Two averages from the same noisy samples

Saved acquisition

In-phase output

2.60\,\mathrm{mV}

Quadrature output

1.50\,\mathrm{mV}

Output magnitude

3.00\,\mathrm{mV}

Output phase at record end

30.0^{\circ}

Bandwidth, calibration and saved metadata

The current controls preview a new setup. Saved traces retain their original phase, clock, range, filter, seed and independently acquired standard. X and Y are instantaneous filtered RMS-scaled components at the last sample. A detuned component keeps rotating; this phase is not a fixed recovered signal phase. The average magnitude of noise is positive. Last-second scatter includes drift and beats and is not a confidence interval.

H(z)=\left(\frac{1-a}{1-az^{-1}}\right)^p,\quad a=e^{-1/(f_s\tau)}

Model and primary source

A teaching instrument with an 18-bit symmetric ADC at 1024 samples/s, a fixed gain and delay, and ideal recorded quadrature references. Each low-pass pole uses the exact held-input RC recursion from zero initial state. This is not an SR850 firmware emulation. All plotted samples are retained observations; no synchronous notch filter, PLL dynamics, source loading, analog anti-alias circuit or commercial accuracy is modeled. Deterministic tones stay below Nyquist; sampled white noise is defined within that band. A scalar phase calibration is valid at its saved reference frequency; detuned tones retain the additional delay at their offset. Coherent same-frequency interference adds to the target and cannot be separated by this reference alone. Wire pulses encode instantaneous voltage; their speed is schematic.

SRS SR850 manual, Rev. 2.0 (01/2009), pp. 3-1–3-3, 3-11–3-12, 3-19–3-20, 3-27–3-28

Start with the essentials

Focus question: How can a periodic signal smaller than the noise still be measured?
One-sentence intuition: A component in step with the reference leaves a steady product that survives averaging.

Core mathematical model

RMS-scaled quadratures

X=V_{\rm rms}\cos(\phi_s-\phi_r),\qquad Y=V_{\rm rms}\sin(\phi_s-\phi_r)

For a settled same-frequency sinusoid, the cosine reference and negative-sine reference produce two signed components. A leading signal gives positive quadrature.

Each sampled low-pass stage

y_n=ay_{n-1}+(1-a)u_n,\qquad a=e^{-1/(f_s\tau)}

The sample rate and time constant determine the pole. Several identical stages reduce noise further and take longer to respond.

Discrete noise bandwidth

B_{\rm ENBW}=\frac{f_s}{2}\sum_{n=0}^{\infty}h_n^2

The impulse response of the actual sampled filter determines its one-sided noise bandwidth. This differs from its half-power frequency.

Common difficulties

Averaging limits

Typical misconceptionLonger averaging removes every unwanted signal.

Better mental modelIt attenuates components away from the reference. A coherent component at the same frequency remains, and genuine changes are delayed.

Magnitude bias

Typical misconceptionA positive magnitude proves a periodic signal.

Better mental modelMagnitude is nonnegative, so noise gives a positive average too. Compare signed quadratures and their stability.

Run the experiment

01
Turn the physical reference knob
Drag the green phase knob, then save a capture.
What to observe: The components rotate between the two channels. Their combined magnitude stays nearly fixed for a clean same-frequency input.
02
Compare the same noisy input
Choose Buried signal and inspect the solid and dashed outputs.
What to observe: The dashed curves use a shorter time constant on the identical raw samples. They react faster and fluctuate more.
03
Detune the signal
Choose A slow beat.
What to observe: The output rotates continuously. Its final phase describes one recorded instant, not a fixed source phase.
04
Check the limits
Compare Coherent impostor and Noise alone.
What to observe: A coherent interferer survives averaging. Noise alone also has a positive average magnitude.
05
Recalibrate a new range
Change input range, acquire the known standard, then save a new capture.
What to observe: The new acquisition freezes that separate standard. Older records and exports retain their original calibration.

Lock-In Amplifier

Two averages from the same noisy samples

A reference turns an oscillation into a slow voltage

Core mathematical model

RMS-scaled quadratures

Each sampled low-pass stage

Discrete noise bandwidth

Common difficulties

Averaging limits

Magnitude bias

Run the experiment

Turn the physical reference knob

Compare the same noisy input

Detune the signal

Check the limits

Recalibrate a new range

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