Lock-in amplifier

A lock-in amplifier (also phase-sensitive rectifier , carrier frequency amplifier ( TFV) or phase sensitive detector (PSD)) is an amplifier for measuring a weak electrical alternating signal that is modulated with a reference signal known in frequency and phase . The device is an extremely narrow bandpass filter and thereby improves the signal-to-noise ratio (SNR, Signal to Noise Ratio). The advantage is that direct voltages , alternating voltages of other frequencies and noise are filtered efficiently. ${\displaystyle f_{\rm {ref}}}$

Structure and functionality

Schematic structure of a simple lock-in amplifier

Measurement setup with lock-in amplifier

A lock-in amplifier requires the following functional elements:

• Signal input for the modulated measurement signal.
• Signal input for the sinusoidal (sometimes also square) reference signal.
• Input amplifier for the signal input, possibly with an input filter.
• Phase shifter for the adjustment between reference and measurement signal.
• Mixer (multiplier) that multiplies the input signal with the reference signal.
• Low pass to perform a time averaging over several signal periods.
• Optional: A built-in oscillator for modulating the measurement signal.

The two input signals are multiplied together in the mixer and then integrated in a low-pass filter . The lock-in amplifier thus calculates the cross-correlation between the measurement signal and the reference signal for a fixed phase shift . The cross-correlation for signals of different frequencies is zero. If the frequency of the measurement signal is different from that of the reference signal, the lock-in does not provide an output signal. The cross-correlation only provides a value other than zero for the same frequencies and thus a contribution to the output signal of the lock-in amplifier. By selecting the appropriate frequency for the reference, the corresponding component in the measurement signal can be filtered out. The reference signal is locked onto the measurement signal . ${\displaystyle \Delta \phi }$

Another interpretation is based on the mixer: When the measurement signal is multiplied by the reference signal, the difference and sum frequencies are created as a mixed signal . The difference frequency for the useful signal is ideally zero. A low-pass filter is sufficient for filtering this DC voltage signal, and it only needs to be designed to be wide enough to allow desired changes in the signal to pass. All other frequencies, especially high-frequency noise , mains hum or other interference signals, are filtered out by the low-pass filter.

Ideally, the lock-in amplifier provides a DC voltage as the output signal. It is proportional to:

• Input voltage;
• Cosine of the phase shift between the input signal and the reference signal. ${\displaystyle \Delta \phi }$

The output signal results as follows: ${\displaystyle U_{\mathrm {out} }(t)}$ ${\displaystyle U_{\mathrm {out} }(t)=1/T\int \limits _{t-T}^{t}\;{\sin \left[2\pi f_{\mathrm {ref} }\cdot s+\Delta \varphi \right]U_{\mathrm {in} }(s)}\mathrm {d} s}$

If the input signal is also modulated sinusoidally, the result for the output signal is a sufficiently long integration time : ${\displaystyle U_{\mathrm {in} }}$ ${\displaystyle T}$

${\displaystyle U_{\mathrm {out} }(t)\propto U_{\mathrm {in} }\cdot \cos \left(\Delta \varphi \right)}$

If the reference signal and the measurement signal are in phase ( ), the output signal generated by the lock-in amplifier becomes maximum. If the phase shift is 90 °, the output signal is zero. ${\displaystyle {\Delta }{\varphi }=0}$

If you consider the lock-in amplifier in the frequency range, it corresponds to a bandpass around the reference frequency , the bandwidth of which is inversely proportional to the integration time . Interference signals in the measurement channel with frequencies that are within this bandwidth lead to a beat at the output.

This formulation applies to a sinusoidal reference signal. In practical application (see optical modulators ) one often has to deal with rectangular reference signals, where the output signal then looks different. Rectangular reference signals mean that the odd harmonics of the signal also contribute to the output signal, as do interference signals in the corresponding bands.

The phase between the measurement signal and the reference signal is therefore extremely important and, as a measurement result, is equivalent to the amplitude of the measurement signal. It can provide valuable information for some measurements. If, for example, on / off amplitude-modulated light is used, which causes photoconductivity on a sample , the measured current will lag behind the excitation somewhat, since various effects within the sample cause time delays, which is reflected in a phase shift. So one can draw conclusions about the type and extent of these effects in the sample from the degree of the phase shift.

There are single phase lock-in and dual phase lock-in amplifiers. The latter determine the output signal for two different phase shifts that differ by 90 °. The Pythagorean addition of the two resulting output signals makes the final result of the measurement independent of the phase, which enables both simpler and more precise measurements (for more information see web links).

Digital lock-in amplifier

DSP-based lock-in amplifier

A PC-controlled digital lock-in amplifier

FPGA-based lock-in amplifier with integrated oscilloscope and frequency spectrum

The best signal sensitivities can be achieved with the help of digital lock-in amplifiers based on digital signal processors (DSP). First the input signal and the reference signal are digitized ( ADC ). The further steps such as preparation of the reference signal, phase shifting, multiplication and low-pass filtering are then carried out purely digitally. If necessary, the result is converted back into an analog signal (DAC, digital-to-analog converter, digital-analog converter ). Lock-ins based on DSP also enable a more precise determination of the phase position between the input signal and the reference signal. The purely digital data processing makes it possible to use more than just one demodulator per channel. This expands the possibilities of evaluation. FPGA-based lock-in amplifiers can also feed multiple reference frequencies (e.g. f1 and f2) from one source. Multiple or mixed frequencies (e.g. f1-f2) can thus be evaluated in a phase-stable manner with additional demodulators.

See also

• Chopper amplifiers internally chop up an electrical DC voltage signal to be measured, in contrast to the lock-in amplifier, which always receives an AC voltage signal that is generated in the measurement arrangement via chopper devices or other modulators (e.g. optical ).
• The principle of the lock-in amplifier is used in radio technology in direct mix receivers .
• Homodyne detection

Web links

• Lock-in tutorial (English, PDF, 457 kB)
• Manual of the Stanford Research Systems SR530 (English, PDF: 1.05 MB) - manual of a dual lock-in, contains an introduction to the dual lock-in principle on p. 28f. and very detailed information on the measurement with a real lock-in (dynamic reserve etc.)
• Explanation of the correct time constant setting for noise measurements with lock-in amplifiers (English, PDF: 230 kB) - Explanations for setting the low-pass filter for comparative noise measurements with lock-in amplifiers.
• [1] Explanation and structure of a lock-in amplifier
• Basics of lock-in amplifier detection and the state of the art , review article 2016.