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 reference 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 reference
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:
Vsig  V0 sig cos(t   sig )
,
Vref  V0 ref cos(t   ref )
1
PSD output  Vsig  Vref  V0 sig V0 ref (cos( sig   ref )  cos( 2 t   sig   ref ))
2
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 reference. There are two ways to get V0 sig :
1. Change φref until the output DC signal becomes maximal
cos(φsig - φref) =1 , Vmeasured = 1/2 V0 ref V0 sig , φsig = φref
V0 ref is known that make us able to obtain V0 sig.
2. Carry out two measurements (Vmeasured 1 , Vmeasured 2) with phase difference of
90 degree for reference signal, then
(Vmeasured 12 + Vmeasured 22)1/2 = 1/2 V0 ref V0 sig ,
φsig = atan(Vmeasured 2/Vmeasured 1).
The measurement of φsig as function of φref 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 reference sine wave
which is phase-locked to the reference 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
(~4oK).
connector A
R0
R
connector B
R1
common
ground
R0 – 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.
R1 – 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 reference
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 reference sine wave. This means
that the signal is multiplied by a single reference sine wave (instead of a reference and its
many harmonics) and only the signal at this single reference 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 reference 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 reference reconstruction by its frequency, phase and
amplitude detection.

Write Matlab function for signal-reference 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.
Vreference
computer
ADC
5V
1kΩ
1kΩ
Vcc
LM 741
1µF
+
-
A
+
100Ω
Rg2
1kΩ
1kΩ
10kΩ
LM 741
AD 620
100kΩ
+
ref
Rg1
-
B
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.