# Lock - in Amplifier and Applications

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1 Lock - in Amplifier and Applications What is a Lock in Amplifier? In a nut shell, what a lock-in amplifier does is measure the amplitude V o of a sinusoidal voltage, V in (t) = V o cos(ω o t) where ω o = 2πf o and f o are the angular- and natural frequencies of the signal respectively. You supply this voltage to the signal input of the lock-in, and its meter tells you the amplitude Vo, typically calibrated in V-rms. What makes a lock-in different from a simple AC voltmeter, which is what I just described, is that you must also supply the lock-in with a reference input, that is, a decent size, say 1 volt p-p, sinusoidal voltage that is synchronized with the signal whose amplitude you are trying to measure. The lock-in uses this signal (like an external trigger for an oscilloscope) to "find" the signal to be measured, while ignoring anything that is not synchronized with the reference. In practice, the lock-in can measure voltage amplitudes as small as a few nano volts, while ignoring signals even thousands of times larger. In contrast, an AC voltmeter would measure the sum of all of the voltages at its input. Let's consider an example. Suppose the signal is a 10 nv sine wave at 10 khz. Clearly some amplification is required to bring the signal above the noise. A good low noise amplifier will have about 5 nv/ Hz of input noise. If the amplifier bandwidth is 100 khz and the gain is 1000, then we can expect our output to be 10μV of signal (10 nv x 1000) and 1.6 mv of broadband noise (5 nv/ Hz x 100 khz x 1000). We won't have much luck measuring the output signal unless we single out the frequency of interest. If we follow the amplifier with a bandpass filter with a Q=100 (a VERY good filter) entered at 10 khz, any signal in a 100 Hz bandwidth will be detected (10 khz/q). The noise in the filter pass band will be 50 μv (5 nv/ Hz x 100 Hz x 1000) and the signal will still be 10 μv. The output noise is much greater than the signal and an accurate measurement can not be made. Further gain will not help the signal to noise problem. Now try following the amplifier with a phase-sensitive detector (PSD). The PSD can detect the signal at 10 khz with a bandwidth as narrow as 0.01 Hz! In this case, the noise in the detection bandwidth will be only 0.5 μv (5 nv/ Hz x.01 Hz x 1000) while the signal is still 10 μv. The signal to noise ratio is now 20 and an accurate measurement of the signal is possible.

2 What is Phase Sensitive Detection? Lock-in measurements require a frequency reference. Typically an experiment is excited at a fixed frequency (from an oscillator or function generator) and the lock-in detects the response from the experiment at the reference frequency. In the diagram below, the reference signal is a square wave at frequency ω r. This might be the sync output from a function generator. If the sine output from the function generator is used to excite the experiment, the response might be the signal waveform shown below. The signal is V sig sin(ω r t + θ sig ) where Vsig is the signal amplitude, ω r is the signal frequency, and θ sig is the signal s phase. Lock-in amplifiers generate their own internal reference signal usually by a phase-locked-loop locked to the external reference. In the diagram below the external reference, the lock-in s reference and the signal are all shown. The internal reference is V L sin(ω o t + θ ref ). The lock in amplifies the signal and then multiplies it by the lock-in reference using a phase-sensitive detector or multiplier. The output of the PSD is simply the product of two sine waves. V psd = V sig V L sin(w r t + θ sig )sin(ω L t + θ ref ) = 1/2 V sig V L cos([ω r - ω L ]t + θ sig - θ ref ) - 1/2 V sig V L cos([ω r + ω L ]t + θ sig + θ ref ) The PSD output is two AC signals, one at the difference frequency (ω r - ω L ) and the other at the sum frequency (ω r + ω L ). If the PSD output is passed through a low pass filter, the AC signals are removed. What will be left? In the general case, nothing.

3 However, if ω r equals ω L, the difference frequency component will be a DC signal. In this case, the filtered PSD output will be V psd = 1/2 V sig V L cos(θ sig - θ ref ). This is a very nice signal it is a DC signal proportional to the signal amplitude. It s important to consider the physical nature of this multiplication and filtering process in different types of lock-ins. In traditional analog lock-ins the signal and reference are analog voltage signals. The signal and reference are multiplied in an analog multiplier, and the result is filtered with one or more stages of RC filters. In a digital lock-in such as the SR850, the signal and reference are represented by sequences of numbers. Multiplication and filtering are performed mathematically by a digital signal processing (DSP) chip. We ll discuss this in more detail later. Block Diagram While a lock-in amplifier is an extremely important and powerful measuring tool, it is also quite simple. The block diagram of a lock-in amplifier is shown in Figure 1. The lock-in consists of five stages: 1. AC amplifier, called the signal amplifier; 2. Voltage controlled oscillator (VCO); 3. Multiplier, called the phase sensitive detector (PSD); 4. Low-pass filter; and 5. DC amplifier. The signal to be measured is fed into the input of the AC Amplifier. The output of the DC amplifier is a DC voltage proportional to V 0. This voltage is displayed on the lock-in's own meter, and is also available at the output connector. I now elaborate on the functions of the five stages.

4 The AC amplifier is simply a voltage amplifier combined with variable filters. Some lock-in amplifiers let you change the filters as you wish, others do not. Some lock-in amplifiers have the output of the AC amplifier stage available at the signal monitor output. Many do not. The voltage controlled oscillator is just an oscillator, except that it can synchronize with an external reference signal (i.e., trigger) both in phase and frequency. Some lock-in amplifiers contain a complete oscillator and need no external reference. In this case they operate at the frequency and amplitude that you set, and you must use their oscillator output in your experiment to derive the signal that you ultimately wish to measure. Virtually all lock-in amplifiers are able to synchronize with an external reference signal. The VCO also contains a phase-shifting circuit that allows the user to shift its signal from degrees with respect to the reference. The phase sensitive detector is a circuit which takes in two voltages as inputs V1 and V2 and produces an output which is the product V 1 *V 2. That is, the PSD is just a multiplier circuit. The low pass filter is an RC filter whose time constant may be selected. In many cases you may choose to have one RC filter stage (single pole filter) or two RC filter stages in series (2-pole filter). In newer lock-in amplifiers, this might be a digital filter with the attenuation of a "many pole" filter. The DC amplifier is just a low-frequency amplifier similar to those frequently assembled with op-amps. It differs from the AC amplifier in that it works all the way down to zero frequency (DC) and is not intended to work well at very high frequencies, say above 10 KHz. Time Constant and DC Gain The output of the PSD contains many signals. Most of the output signals have frequencies which are either the sum or difference between an input signal frequency and the reference frequency. Only the component of the input signal whose frequency is exactly equal to the reference frequency will result in a DC output. The low pass filter at the PSD output removes all of the unwanted AC signals, both the 2F (sum of the signal and the reference) and the noise components. This filter is what makes the lock-in such a narrow band detector. Time Constants Lock-in amplifiers have traditionally set the low pass filter bandwidth by setting the time constant. The time constant is simply 1/2pf where f is the -3 db frequency of the filter. The low pass filters are simple 6 db/oct roll off, RC type filters. A 1 second time constant referred to a filter whose -3 db point occurred at 0.16 Hz and rolled off at 6 db/oct beyond 0.16 Hz. Typically, there are two successive filters so that the overall filter can roll off at either 6 db or 12 db per octave. The time constant referred to the -3

5 db point of each filter alone (not the combined filter). The notion of time constant arises from the fact that the actual output is supposed to be a DC signal. In fact, when there is noise at the input, there is noise on the output. By increasing the time constant, the output becomes more steady and easier to measure reliably. The trade off comes when real changes in the input signal take many time constants to be reflected at the output. This is because a single RC filter requires about 5 time constants to settle to its final value. The time constant reflects how slowly the output responds, and thus the degree of output smoothing. The time constant also determines the equivalent noise bandwidth (ENBW) for noise measurements. The ENBW is NOT the filter -3 db pole, it is the effective bandwidth for Gaussian noise. DC Output Gain How big is the DC output from the PSD? It depends on the dynamic reserve. With 60 db of dynamic reserve, a noise signal can be 1000 times (60 db) greater than a full scale signal. At the PSD, the noise can not exceed the PSD's input range. In an analog lock-in, the PSD input range might be 5V. With 60 db of dynamic reserve, the signal will be only 5 mv at the PSD input. The PSD typically has no gain so the DC output from the PSD will only be a few mill volts! Even if the PSD had no DC output errors, amplifying this mill volt signal up to 10 V is error prone. The DC output gain needs to be about the same as the dynamic reserve (1000 in this case) to provide a 10 V output for a full scale input signal. An offset as small as 1 mv will appear as 1 V at the output! In fact, the PSD output offset plus the input offset of the DC amplifier needs to be on the order of 10 μv in order to not affect the measurement. If the dynamic reserve is increased to 80dB, then this offset needs to be 10 times smaller still. This is one of the reasons why analog lockins do not perform well at very high dynamic reserve. The digital lock-in does not have an analog DC amplifier. The output gain is yet another function handled by the digital signal processor. We already know that the digital PSD has no DC output offset. Likewise, the digital DC amplifier has no input offset. Amplification is simply taking input numbers and multiplying by the gain. This allows the SR830 to operate with 100 db of dynamic reserve without any output offset or zero drift.

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