The basic function of the lock-in amplifier is to extract the target signal from the noisy environment. In all electronic systems, noise is ubiquitous. According to the cause of the noise, it can be roughly divided into two types: the internal noise of the system (intrinsic noise) and the external environmental noise. From the microscopic point of view, the internal noise is generated due to the randomness of the instantaneous motion of the carrier. This randomness leads to the fact that we can not accurately predict the instantaneous value at any future moment, that is, the system noise is introduced. To the external noise, some unpredictable interference outside the system will cause the system's internal response, causing system error. From a subjective point of view, any signal that is not wanted or that affects the measurement can be considered as noise. Although the noise has the unpredictable characteristics with instantaneous value, but such randomness is not completely chaotic, they have a certain statistical law.
1. Common noise source of electronic system
1.1. Thermal noise
No matter whether the system is working or not, the thermal noise exists in any electronic system. Even if the system is not powered, the electronic device will exhibit noise characteristics. Thermal noise was first discovered by J.B. Johnson in 1928, so thermal noise is also known as Johnson noise. The voltage of thermal noise can be calculated by the following formula:
In the above formula, k is the Boltzmann constant with a value of 1.38 × 10-23 J / K. T is the Kelvin temperature. R is the resistance of the resistor in ohms Ω and Δf is the measured bandwidth. As can be seen from the above equation, the power spectral density function of thermal noise has nothing to do with the frequency. When the temperature and the resistance value is constant, the power spectral density function is a straight line, indicating that the thermal noise has the property of white noise.
1.2. Shot noise
Shot noise usually exists in the PN junction. The micro-mechanism is that the random emission and annihilation of carriers in the PN junction cause the current flowing through the barrier to randomly fluctuate. Shot noise was first discovered by W. Schottky in 1918 in a study of hot cathode tubes and it was theoretically proved a kind of white noise. Shot noise can occur as a noise current in the current measurement and is given by the following formula:
In the above formula, q is the electronic element charge and its value is 1.62 × 10-19C. I is the effective value of AC current or average value of DC current of the PN junction. △ f is the measured bandwidth. To system or environment with small bandwith, shot noise generally has a negligible effect and it can be ignored.
1 / f noise was first discovered by Johnson in 1925 in tube current. For resistive elements with the same resistance and different materials, they have the same thermal noise. When the two conductor contact is not ideal, their contact resistance will be random fluctuations, which caused the noise, called 1 / f noise. The salient features of the noise is its power spectral density inversely proportional to the operating frequency f. So that, the lower the frequency, the more serious the influence of noise. The power spectral density function is:
In the above formula, K is a fixed value which is related to the material type, contact surface, geometry and others. Id is the average DC current flowing through the conductor and the unit is A; f is the operating frequency. The 1 / f noise of carbonaceous resistors is typically 0.1-0.3μVrms. The noiseof metal film resistors or winding resistors is approximately 1 order of magnitude smaller than the carbonaceous resistance.
1.4. Total noise of the electronic system
The internal noise of the electronic system described above is independent of each other. If the total internal noise of the electronic system is described statistically, we should square the RMS values of the independent noise sources and then accumulate and carry out the operation. At last we get the total noise.
2. External noise source
In addition to the above-mentioned internal noise of the electronic system, various interferences from the external environment may appear during the experiment. Most external noise sources are asynchronous with the system in the time domain. In the frequency domain, these noises are not related to the reference signal frequency and the harmonics of the reference source in the system. These noises may come from lighting Equipment, refrigeration equipment, motors, radios, computer screens and so on. Eliminating the above noise will increase the requirements for dynamic reserve and time constant of the measurement system.
Some external noise sources may have synchronization and correlation with the source signal in the system. When these external noise is introduced into the system, it will cause serious distortion of the target signal, such as fluctuation in the target signal amplitude. This distortion can not be ruled out by frequency-dependent methods, that is, the lock-in amplifier will consider this part of the noise as the source signal, resulting in a measurement error. A typical source of synchronous noise comes from the in-ground current design of the experimental environment and instrumentation. External noise sources can couple into the source signal path in a variety of ways.
2.1. Effect of capacitive coupling
Imagine an AC voltage signal close to the measurement environment, which may be coupled to the path where the detector is located by a parasitic capacitance. Although the parasitic capacitance may be very small, the noise generated by coupling may be much larger than the weak signal under test, especially when there is synchronization and correlation between the coupling noise and the source signal in the system.
The lock-in amplifier will see all signals on the target frequency as the source signal. If the AC voltage signal is a high-frequency signal, the coupling noise gain will be more obvious.
We can calculate the value of the noise current by the following formula :
i=Cstray dV/dt=ωCstray Vnoise
ω is the angular velocity corresponding to the source signal frequency. Vnoise is the noise amplitude; Cstray is the value of parasitic capacitance.
To avoid the impact of capacitive coupling noise on the system, the following points should be considered in the design:
1) For a defined measurement environment, removing or turning off the noise source.
2) Ensure that the noise source is away from the measurement environment, reducing the value of parasitic capacitance. Signal channels should keep away from the channel that may bring noise.
3) When the noise source can not be eliminated, use the experiment of measuring the terminal voltage of the device with miniature impedance to replace the experiment of measuring current since the voltage generated by the noise current is generally small.
4) Place the measurement source and detector in a metal body for isolation.
2.2. Effect of inductive coupling
An AC current in a nearby piece of apparatus can couple to the experiment via a magnetic field. A changing current in a nearby circuit gives rise to a changing magnetic field which induces an emf (d?B/dt) in the loop connecting the detector to the experiment. This is like a transformer with the experiment-detector loop as the secondary winding.
In order to avoid the impact of inductive coupling noise on the system, the following points should be considered in the design:
1) For a defined measurement environment, Removing or turning off the interfering noise source.
2) Reduce the area of the pick-up loop by using twisted pairs or coaxial cables, or even twisting the 2 coaxial cables used in
3) Using magnetic shielding to prevent the magnetic field from crossing the area of the experiment.
4) Measuring currents, not voltages, from high impedance detectors.
2.3. Resistive coupling or ground loops
Currents flowing through the ground connections can give rise to noise voltages. In this illustration, the detector is measuring the
signal relative to a ground far from the rest of the experiment. The experiment senses the detector signal plus the voltage due to the noise source's ground return current passing through the finite resistance of the ground between the experiment and the detector. The detector and the experiment are grounded at different places which, in this case, are at different potentials.
Cures for ground loop problems include:
1） Grounding everything to the same physical point.
2） Using a heavy ground bus to reduce the resistance of ground connections.
3） Removing sources of large ground currents from the ground bus used for small signals.
Not all sources of noise are electrical in origin. Mechanical noise can be translated into electrical noise by microphonic effects. Physical changes in the experiment or cables (due to vibrations for example) can result in electrical noise over the entire frequency range of the lock-in.
For example, consider a coaxial cable connecting a detector to a lock-in. The capacitance of the cable is a function of its geometry. Mechanical vibrations in the cable translate into a capacitance that varies in time, typically at the vibration frequency. Since the cable is governed by:
I=dQ/dt=C dV/dt+V dC/dt
Mechanical vibrations in the cable which cause a dC/dt will give rise to a current in the cable. This current affects the detector and the measured signal.
Some ways to minimize microphonic signals are:
1) Eliminate mechanical vibrations near the experiment.
2) Tie down cables carrying sensitive signals so they do not move.
3) Use a low noise cable that is designed to reduce microphonic effects.
2.5. Thermocouple effects
The emf created by junctions between dissimilar metals can give rise to many microvolts of slowly varying potentials. This source of noise is typically at very low frequency since the temperature of the detector and experiment generally changes slowly. This effect is large on the scale of many detector outputs and can be a problem for low frequency measurements, especially in the mHz range.
Some ways to minimize thermocouple effects are:
1) Hold the temperature of the experiment or detector constant.
2) Use a compensation junction, i.e. a second junction in reverse polarity which generates an emf to cancel the thermal potential of the first junction. This second junction should be held at the same temperature as the first junction.