Introduction A lock-in amplifier, in common with most AC indicating instruments, provides a DC output proportional to the AC signal under investigation. The special rectifier, called a phase-sensitive detector (PSD), which performs this AC to DC conversion forms the heart of the instrument. It is special in that it rectifies only the signal of interest while suppressing the effect of noise or interfering components which may accompany that signal. The noise at the input to a lock-in amplifier is not rectified but appears at the output as an AC fluctuation. This means that the desired signal response, a DC level, can be separated from the noise accompanying it in the output by means of a simple low-pass filter. In order to function correctly the detector must be programmed to recognize the signal of interest. This is achieved by supplying it with a erence voltage of the same frequency and with a fixed phase relationship to that of the signal. This is commonly done by ensuring that they are derived from the same source. The use of such a erence signal ensures that the instrument will track any changes in the frequency of the signal of interest. This inherent tracking ability allows extremely small bandwidths to be defined for the purpose of signal-to-noise ratio improvement. As mentioned above, the heart of the lock-in amplifier is the phase-sensitive detector (PSD), which is also known as a demodulator or mixer. The detector operates by multiplying two signals together, and the following analysis indicates how this gives the required outputs: V sig = V cos( ω t + ϕ sig ), V = V0 cos( ωt + ϕ 0 sig ) PSD output = V sig V 1 = V 2 V (cos( ϕ ϕ ) + cos(2ωt + ϕ + ϕ 0 sig 0 sig sig )) The output from the PSD passes to a low-pass filter which removes the 2ωt component, leaving the output of the lock-in amplifier as the DC signal that is proportional to the signal amplitude and dependent on phase difference between the signal and the erence. There are two ways to get V 0 sig :
1. Change φ until the output DC signal becomes maximal cos(φ sig - φ ) =1, V measured = 1/2 V 0 V 0 sig, φ sig = φ V 0 is known that make us able to obtain V 0 sig. 2. Carry out two measurements (V measured 1, V measured 2 ) with phase difference of 90 degree for erence signal, then (V 2 measured 1 + V 2 measured 2 ) 1/2 = 1/2 V 0 V 0 sig, φ sig = atan(v measured 2 /V measured 1 ). The measurement of φ sig as function of φ makes a Lock-in amplifier very sensitive to phase differences that the signal accumulates passing the measured system. A block diagram of Lock-in Amplifier is shown below. The input signal is amplified by a low-noise differential amplifier, and selectively filtered to remove line frequency related interference and other unwanted signals. The signal which results is amplified by a high-gain AC amplifier, and is then multiplied by a erence sine wave which is phase-locked to the erence input. The output of the multiplier contains the sum and difference frequency components. Two stages of low pass filtering provide the lock-in's time constants. The purpose of the filtering is twofold. First, the filters remove the 2f components which are introduced by the multipliers. Secondly, the filters provide noise reduction by narrowing the lock-in's detection bandwidth. The output of the filter stages is amplified by a chopper stabilized DC amplifier and becomes the lock-in's output. The tradeoff between AC gain at the front end of the lock-in, and post-filter DC gain determines the dynamic reserve of the lock-in amplifier - the ratio of the largest noise signal to the full scale input that the lock-in can tolerate before overload. If very little AC gain is used, large interfering signals can be present without overloading the front end. However, high DC gains must then be used which make the output more unstable. If the DC gain is lowered for more stability, higher AC gains must be used making the unit more susceptible to overloads. This tradeoff between dynamic reserve and stability is inherent to all analog lock-in amplifiers.
Experimental setup 1. Acquaintance with Lock-in amplifier and first operation. We learn a small signal measurement by the Lock-in amplifier through electrical circuit that simulates resistance measurement of the thin film samples at low temperatures (~4 o K). R 0 connector A R connector B R 1 common ground R 0 large resistor ~1MΩ that is intended to limit circuit current for prevention of the system heating. R small (sample) resistor ~10Ω, his resistance will be measured. R 1 precise resistor ~10kΩ that is placed out of the low temperature system for the circuit current monitoring. We use internal oscillator (VCO type) for the circuit input and the erence source. Set the SINE OUT amplitude to 1 V level (on the rare panel) and connect it to the circuit input. Connect the REF OUT from the rare panel to the REFERENCE INPUT on the front panel. The oscillator frequency is controlled by the VCO IN (rare panel) voltage. A voltage from 0 to 10 V will adjust the frequency according to the VCO RANGE (rare panel)
selected. Three ranges are available: 1 Hz/V, 100 Hz/V and 10 khz/v. There are three ways to set the frequency: 1) If the VCO IN is left open, then the oscillator will run at the top of its range (i.e. 10 Hz, 1 KHz, or 100 khz). 2) A variable resistor may be connected to the VCO IN (in series with input resistance that is 10 kω) to create voltage divider. 3) Connect the VCO IN to an external voltage source which can provide 0 to 10 V. At first measurements round leave the VCO IN connector open and choose 100 Hz/V range the frequency should be about 1000 Hz. Perform a series of measurements of the voltage on the resistor R at different configurations of the Lock-in amplifier and check a precision and accuracy of measurement dependent on filters on or off, sensitivity, dynamic reserve and time constants. Pay attention what are the conditions that cause OVERLOAD? Compare the measured voltage with calculated one. To calculate the voltage fall on R you should perform accurate measurement of the current in the circuit and the resistance of R. Now connect the variable resistor to VCO IN connector to be able to set different frequencies. Make a sequence of measurements in more noisily part of spectrum - around frequency 100 Hz (70-130 Hz) with and without LINE 2 filter to receive Notch filter frequency response. Perform at least 30 measurements, reduce the intervals close to the Notch bottom. Compare the width and the attenuation of the Notch filter with instrument specifications from manual. 2. Analog to Digital Conversion The main difference between analog and digital Lock-in is in demodulation stage (Phase Sensitive Detection (PSD)). The digital PSD multiplies the digitized signal (after amplification and filtering) with a digitally computed erence sine wave. This means
that the signal is multiplied by a single erence sine wave (instead of a erence and its many harmonics) and only the signal at this single erence frequency is detected. In the digital lock-in, the dynamic reserve is limited by the quality of the A/D conversion. Once the input signal is digitized, no further errors are introduced. Certainly the accuracy of the multiplication does not depend on the size of the numbers. The A/D converter used in this experiment is NI USB-6008 with analog input resolution - 11 bit and maximal sample rate 10 ksam/sec. Connect signal generator to ADC and sample the signal by Matlab (see Data Acquisition Toolbox). Check sampling limitations: low sampling rate, quantization error. Write Matlab function for simultaneously sampling of the erence and the signal. Write Matlab function for precise frequency detection by FFT implementation: reduce DC level, apply appropriate window (optional), use zero padding and then find a frequency that corresponding a maximum value of FFT. Find optimal parameters for accurate frequency detection. Write Matlab functions for erence reconstruction by its frequency, phase and amplitude detection. Write Matlab function for signal-erence demodulation. Write Matlab function for low-pass filtering. 3. Amateurish digital Lock-in amplifier. Now, when you are familiar with Lock-in operation and basics of digital signal processing, build a simple digital Lock-in. For this purpose you need to construct amplification circuit and light emitting-detecting circuit with LED and Phototransistor for measurement trial. The scheme of the circuit is shown below.
V erence ADC computer 5V - V cc 100Ω + LM 741 10kΩ 1µF 100kΩ A B R g1 + AD 620 - R g2 - LM 741 + You should measure accurately the amplification gain of the circuit it will serve you in your program. Make a series of measurements of the voltage between points A and B: 1) by your Lock-in, 2) by real one that we have in laboratory for calibration. Change the AC amplitude on LED to find a maximal sensitivity of your Lock-in. Make the measurements for a variety of frequencies.