What is a Lock-in Amplifier?
TN 1000
In its most basic form a lock-in amplifier is an instrument with dual capability. It can recover signals in the presence of an
overwhelming noise background or, alternatively, it can provide high resolution measurements of relatively clean signals over
several orders of magnitude and frequency. However, modern instruments offer far more than these two basic functions and this
increased capability has led to their acceptance, in many scientific disciplines, as units which can provide the optimum solution to a
large range of measurement problems. For example, the modern lock-in amplifier will function as: an AC Signal Recovery Instrument
a Vector Voltmeter
a Phase Meter
a Spectrum Analyzer
a Noise Measurement Unit
…..and much more.
It is this versatility, available in a single compact unit, which makes the lock-in amplifier an invaluable addition to any laboratory.
This Technical Note describes the basic “building blocks” of the lock-in amplifier so that the user and potential user may better
understand how the instruments work and how the choices made in their design affect their performance.
A lock-in amplifier, in common with most AC indicating
instruments, provides a DC output proportional to the AC signal
under investigation. In modern units the DC output may be
presented as a reading on a digital panel meter or as a digital
value communicated over a computer interface, rather than a
voltage at an output connector, but the principle remains the
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 traditional rectifier, which is found in a typical AC
voltmeter, makes no distinction between signal and noise and
produces errors due to rectified noise components. The noise at
the input to a lock-in amplifier, however, is not rectified but
appears at the output as an AC fluctuation. This means that the
desired signal response, now a DC level, can be separated from
the noise accompanying it in the output by means of a simple
low-pass filter. Hence in a lock-in amplifier the final output is
not affected by the presence of noise in the applied signal.
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 most 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, since the reference circuit is
“locked” to it. It is from this characteristic that the instrument
derives its name.
This inherent tracking ability allows extremely small bandwidths
to be defined for the purpose of signal-to-noise ratio
improvement since there is no frequency “drift”, as is the case
with analog “tuned filter/rectifier” systems. Because of the
automatic tracking, lock-in amplifiers can give effective “Q”
values (a measure of filter selectivity) in excess of 100,000,
whereas a normal bandpass filter becomes difficult to use with
Q’s greater than 50.
Phase-Sensitive Detection
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.
Figure 1 shows the situation where the lock-in amplifier is
detecting a noise-free sinusoid, identified in the diagram as
“Signal In”. The instrument is also fed with a reference signal,
from which it generates an internal sinusoidal reference which is
also shown in the diagram.
Figure 1
The demodulator operates by multiplying these two signals
together to yield the signal identified in the diagram as
“Demodulator Output”. Since there is no relative phase-shift
between the signal and reference phases, the demodulator output
takes the form of a sinusoid at twice the reference frequency, but
with a mean, or average, level which is positive.
What is a Lock-in Amplifier?
Figure 2 shows the same situation, except that the signal phase is
now delayed by 90° with respect to the reference. It can been seen
that although the output still contains a signal at twice the
reference frequency, the mean level is now zero.
Figure 2
From this it can be seen that the mean level is: proportional to the product of the signal and reference
frequency amplitudes
related to the phase angle between the signal and reference.
It will be appreciated that if the reference signal amplitude is
maintained at a fixed value, and the reference phase is adjusted to
ensure a relative phase-shift of zero degrees, then by measuring the
mean level the input signal amplitude can be determined.
The mean level is, of course, the DC component of the
demodulator output, so it is a relatively simple task to isolate it by
using a low-pass filter. The filtered output is then measured using
conventional DC voltmeter techniques.
The above discussion is based on the case of noise-free input
signals, but in real applications the signal will be accompanied by
noise. This noise, which by definition has no fixed frequency or
phase relationship to the reference, is also multiplied by the
reference signal in the demodulator, but does not result in any
change to the mean DC level. Noise components at frequencies
very close to that of the reference do result in demodulator outputs
at very low frequencies, but by setting the low-pass filter to a
sufficiently low cut-off frequency these can be rejected. Hence the
combination of a demodulator and low-pass output filter allows
signals to be measured even when accompanied by significant
Those readers who are interested in a mathematical derivation of
the same conclusions should refer to the Appendix at the end of
this Technical Note.
The Typical Lock-In Amplifier
The block diagram of a typical lock-in amplifier is shown in
figure 3. Readers should be aware that the following discussion
makes no assumptions as to the technology used to implement each
of the circuit elements and that analog, mixed technology and
digital methods may be used.
Signal Channel
In the signal channel the input signal, including noise, is amplified
by an adjustable-gain, AC-coupled amplifier, in order to match it
more closely to the optimum input signal range of the PSD.
Instruments are usually fitted with high impedance inputs for
voltage measurements. Many also incorporate low impedance
inputs for better noise matching to current sources, although in
some cases the best results are obtained through the use of a
separate external preamplifier.
The performance of the PSD is usually improved if the bandwidth
of the noise voltages reaching it is reduced from that of the full
frequency range of the instrument. To achieve this, the signal is
passed through some form of filter, which may be simply a band
rejection filter centered at the power line frequency and/or its
second harmonic to reject line frequency pick-up, or alternatively a
more sophisticated tracking bandpass filter centered at the
reference frequency.
Reference Channel
It has been shown that proper operation of the PSD requires the
generation of a precision reference signal within the instrument.
When a high-level, stable and noise-free reference input is
provided, this is a relatively simple task. However there are many
instances where the available reference is far from perfect or
symmetrical, and in these cases a well designed reference channel
circuit is very important. Such circuits can be expensive and often
account for a significant proportion of the total cost of the
The internally generated reference is passed through a phaseshifter, which is used to compensate for phase differences that may
have been introduced between the signal and reference inputs by
the experiment, before being applied to the PSD.
Phase-sensitive Detector
There are currently three common methods of implementing the
PSD, these being the use of an Analog Multiplier, a Digital Switch
or a Digital Multiplier.
Analog Multiplier
In an instrument with an analog multiplier, the PSD comprises an
electronic circuit which multiplies the applied signal with a
sinewave at the same frequency as the applied reference signal.
Although the technique is very simple in principle, in practice it is
difficult to manufacture an analog multiplier which is capable of
operating linearly in the presence of large noise, or other
interfering, signals. Non-linear operation results in poor noise
rejection and thereby limits the signal recovery capability of the
Digital Switching Multiplier
The switching multiplier uses the simplest form of demodulator
consisting of an analog polarity-reversing switch driven at the
applied reference frequency. The great advantage of this approach
is that it is very much easier to make such a demodulator operate
linearly over a very wide range of input signals.
However, the switching multiplier not only detects signals at the
applied reference frequency, but also at its odd harmonics, where
the response at each harmonic relative to the fundamental is
defined by the Fourier analysis of a squarewave. Such a response
may well be of use if the signal being detected is also a
squarewave, but can give problems if, for example, the unit is
being used at 1 kHz and there happens to be strong interfering
signal at 7 kHz.
As discussed earlier, the use of a tuned low-pass or bandpass filter
in the signal channel prior to the multiplier modifies the response
of the unit so that it primarily detects signals at the reference
frequency. However, in order to fully reject the 3F response, while
still offering good performance at the reference frequency, very
complex and expensive filters would be required. These are
impractical for commercial instruments, so units fitted with filters
tend to show some response to signals and noise at the third and
fifth harmonics of the reference frequency and relatively poor
amplitude and phase stability as a function of operating frequency.
What is a Lock-in Amplifier?
Figure 3
SIGNAL RECOVERY analog lock-in amplifiers use an
alternative and more sophisticated type of switching demodulator
which replaces the single analog switch with an assembly of
several switches driven by a Walsh function. This may be thought
of as a stepped approximation to a sinewave. Careful selection of
components allows such a demodulator to offer all of the
advantages of the switching demodulator with one additional
benefit, which is the complete rejection of the responses at the
third and fifth harmonics and reduced responses for higher orders.
Such a demodulator, when used with a relatively slow roll-off, 4thorder, low-pass filter in the signal channel, produces an overall
response very near to the ideal. In this case the demodulator rejects
the third and fifth harmonic responses and the higher orders are
removed by the signal channel filter.
Digital Multiplier
In an instrument employing this type of multiplier the input signal
is amplified and then immediately digitized. This digital
representation is then multiplied by a digital representation of a
sinewave at the reference frequency. A digital signal processor
(DSP) is used for this task and the output is therefore no longer an
analog voltage but rather a series of digital values.
The technique offers the advantages of a perfect multiplication
with no inherent errors and minimizes the DC coupled electronics
that are needed with other techniques, thereby reducing output
drift. It has been used for a number of years in such applications as
swept-frequency spectrum analyzers.
There are, however, a number of major problems with this method
when applied to recovering signals buried in noise. The most
important of these is dynamic range. Consider the case of an input
signal in the presence of 100 dB (100,000 times larger) of noise. If
the signal is to be digitized to an accuracy of “n” bits then the
input converter must handle a dynamic range of 2n × 100,000 to
fully accommodate the signal and noise amplitudes. With a typical
value for n of 15, this equates to a range of 3.2 × 109:1,
corresponding to 32 bits. An analog to digital converter (ADC) can
be built with such an accuracy, but would be extremely expensive
and quite incapable of the sampling rates needed in a lock-in
amplifier operating to 100 kHz.
Practical digital lock-in amplifiers use a 16 or 18-bit ADC.
Consequently, in the presence of strong interfering signals, the
required signal may only be changing the least significant bits of
the converter, and indeed may actually be so small that there is no
change at all in the ADC output. Hence the measurement
resolution of an individual output sample is very coarse.
Resolution may be improved however by averaging many such
samples. For example 256 samples of 1-bit resolution can average
to 1 sample of 8-bit resolution, but this is at the expense of reduced
response time. This averaging only operates predictably if the
spectral power distribution of the interfering noise is known. If it is
not, then noise has to be added by the instrument from its own
internal noise source to ensure that it dominates. The addition of
this noise, which is only needed in demanding signal recovery
situations, tends to lengthen the response time for a given
measurement accuracy compared to an analog type of instrument.
Low-pass Filter and Output Amplifier
As mentioned earlier, the purpose of the output filter is to remove
the AC components from the desired DC output. Practical
instruments employ a wide range of output filter types,
implemented either as analog circuits or in digital signal
processors. Most usually, however, these are equivalent to one or
more stages of simple single-pole “RC” type filters, which exhibit
the classic 6 dB/octave roll-off with increasing frequency.
There is usually also some form of output amplifier, which may be
either a DC-coupled analog circuit or a digital multiplier. The use
of this amplifier, in conjunction with the input amplifier, allows
the unit to handle a range of signal inputs. When there is little
accompanying noise, the input amplifier can be operated at high
gain without overloading the PSD, in which case little, if any, gain
is needed at the output. In the case of signals buried in very large
noise voltages, the reverse is the case.
The output from a lock-in amplifier was traditionally a DC voltage
which was usually displayed on an analog panel meter. Nowadays,
especially when the instruments are used under computer control,
the output is more commonly a digital number although the analog
DC voltage signal is usually provided as well. Units using an
analog form of phase-sensitive detector use an ADC to generate
their digital output, whereas digital multiplying lock-in amplifiers
use a digital to analog converter (DAC) to generate the analog
Single Phase and Dual Phase
The discussion above is based around a single-phase instrument. A
development of this is the dual-phase lock-in amplifier, which is
not, as some people think, a dual channel unit. Rather it
incorporates a second phase-sensitive detector, which is fed with
the same signal input as the first but which is driven by a reference
signal that is phase-shifted by 90 degrees. This second detector is
followed by a second output filter and amplifier, and is usually
referred to as the “Y” output channel. The original output being
referred to as the “X” channel.
An advantage of the dual-phase unit is that if the signal channel
phase changes (but not its amplitude) then although the output
from one detector will decrease, that from the second increases. It
can be shown, however, that the vector magnitude, R, remains
constant, where:- R = √(X2 + Y2)
Hence if the lock-in amplifier is set to display R, changes in the
signal phase will not affect the reading and the instrument does not
require the adjustment of the reference phase-shifter circuit. This
capability has led to the dual-phase instrument becoming by far the
most common type of unit.
Internal Oscillator
All lock-in amplifiers use some form of oscillator within their
reference circuits. Many units however also have a separate
internal oscillator which can be used to generate an electrical
stimulus for the experiment, usually with user-adjustable
frequency and amplitude.
What is a Lock-in Amplifier?
Computer Control
Virtually all modern instruments include a microprocessor. This
can simplify and automate manual measurements as well as
supporting remote control of the instrument over common
computer interfaces, such as the GPIB (IEEE-488) and RS232
links. The ability of the microprocessor to perform mathematical
manipulations adds such useful functions as vector phase and noise
measurements to the basic signal recovery capabilities of the lockin amplifier.
Further Information
This Technical Note is intended as an introduction to the
techniques used in lock-in amplifiers. Additional information may
be found in other SIGNAL RECOVERY publications, which
may be obtained from your local SIGNAL RECOVERY office
or representative, or by download from our website at
TN 1001
TN 1002
TN 1003
TN 1004
TN 1005
TN 1006
TN 1007
AN 1000
AN 1001
AN 1002
AN 1003
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AN 1005
Consider the case where a noise-free sinusoidal signal voltage Vin
is being detected, where
Vin = A cos (ωt)
ω is the angular frequency of the signal which is related to the
frequency, F, in hertz by the equality:ω = 2 πF
The lock-in amplifier is supplied with a reference signal at
frequency F derived from the same source as the signal, and uses
this to generate an internal reference signal of:Vref = B cos (ωt + θ)
where θ is a user-adjustable phase-shift introduced within the lockin amplifier.
The detection process consists of multiplying these two
components together so that the PSD output voltage is given by:Vpsd = A cos (ωt) . B cos (ωt + θ)
= AB cos ωt (cos ωt cos θ - sin ωt sin θ)
= AB(cos2 ωt cos θ - cos ωt sin ωt sin θ)
= AB((½ + ½cos 2ωt)cos θ - ½sin 2ωt sin θ)
TN 37831-2011
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Specifying a Lock-in Amplifier
The Analog Lock-in Amplifier
The Digital Lock-in Amplifier
How to Use Noise Figure Contours
What is a Boxcar Averager?
Boxcar Averager Specification Comparison
The Incredible Story of Dr D.P. Freeze
Dual-Channel Absorption Measurement with Source
Intensity Compensation
Input Offset Reduction using the Model
7265/7260/7225/7220 Synchronous
Oscillator/Demodulator Monitor Output
Using the Model 7225 and 7265 Lock-in Amplifiers
with software written for the SR830
Low Level Optical Detection using Lock-in Amplifier
Multiplexed Measurements using the 7225, 7265 and
7280 Lock-in Amplifiers
Dual Beam Ratiometric Measurements using the
Model 198A Mixed Beam Light Chopper
= ½AB((1+ cos 2ωt)cos θ - sin 2ωt sin θ)
= ½AB(cos θ + cos 2ωt cos θ - sin 2ωt sin θ)
= ½ABcos θ + ½AB(cos 2ωt cos θ - sin 2ωt sin θ)
= ½AB cos θ + ½ABcos(2ωt + θ)
If the magnitude, B, of the reference frequency is kept constant,
then the output from the phase-sensitive detector is a DC signal
which is: proportional to the magnitude of the input signal A
proportional to the cosine of the angle, θ, between it and the
reference signal
modulated at 2ωt, i.e. it contains components at twice the
reference frequency.
The output from the PSD then passes to a low-pass filter which
removes the 2ωt component, leaving the output of the lock-in
amplifier as the required DC signal.
In a practical situation the signal will usually be accompanied by
noise, but it can be shown that as long as there is no consistent
phase (and therefore by implication frequency) relationship
between the noise and the signal, the output of the multiplier due
to the noise voltages will not be steady and can therefore be
removed by the output filter.
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