The effect of crystal phase noise on the proton precession magnetometer magnetic field measurement accuracy

The proton precession magnetometer is a crucial piece of equipment for measuring magnetic fields because it can determine the value of the magnetic field by measuring the frequency of the Free Induction Decay (FID) signal. The frequency source, one of the crucial parts of the frequency measurement system, has a direct impact on how well the system works as a whole. The communication quality and detection accuracy of the frequency measuring system are directly influenced by phase noise, which is a crucial indicator of frequency stability. By acquiring various output frequencies of crystal oscillators using the Phase Locked Loop (PLL) frequency doubling technique, this study investigates the extent to which crystal phase noise impacts the accuracy of magnetometer frequency measurement.


Introduction
The detection of weak magnetic signals is widely used in earth resource exploration, archaeology, and earthquake monitoring.Since the frequency of the FID signal generated by the proton magnetometer sensor is proportional to the magnitude of the measured magnetic field, the volume of the measured magnetic field can be calculated by measuring the frequency of the FID signal generated by the sensor.The measurement accuracy will directly determine the accuracy of the obtained magnetic field data [1].Improving the frequency accuracy of the FID signal is the key to improving the measurement accuracy of the magnetometer.
High-precision time-frequency measurement and control technology play a crucial role in the development of measurement and control of various other physical quantities.Therefore, signals' generation, measurement, transformation, and control have been the focus of attention in countries worldwide [2].Especially in recent years, high-precision frequency measurement has been playing an increasingly important role in many fields.
Currently, known instruments in China also use hardware frequency measurement [3][4].Hardware frequency measurement is the primary measurement method, generally using direct frequency measurement, which generally uses three methods: the multi-period method, the frequency multiplication method, and the differential frequency method.Among them, the multi-period method of frequency measurement accuracy is the most effective in the entire frequency range required for better accuracy.This method, however, essentially only counts beyond zero and does not use all of the signal's amplitude information because the feed signal is an exponentially decaying sine wave that contains noise.The main disadvantage is that the auxiliary circuit noise spikes may lead to counting errors in the process of calculating the magnetic field.The error in one count during the calculation of the magnetic field will cause a substantial error in the overall measurement after the error accumulation.
The equal precision frequency measurement method and the direct frequency measurement method are used to achieve the measurement of the frequency of the signal to be measured by the method of gate counting.This measurement method, on the other hand, improves test accuracy and can achieve equal precision measurement in this frequency range.The measurement results of the equal precision frequency measurement method are not related to the size of the measured signal frequency but only to the size and stability of the reference frequency.For the equal precision measurement method, increasing the standard clock frequency can improve frequency measurement accuracy.If the gate time is 1 μs, the frequency measurement must be accurate to ±1×10 −3 orders of magnitude, which corresponds to a standard clock frequency of at least 1000 MHz.However, the current maximum counting frequency of the counter is about 500 MHz, which cannot meet the measurement requirements, and for close to 1000 MHz standard signal, it is pretty challenging to achieve.
In this study, we present a design method that multiplies the crystal output frequency using the ALTPLL IP core in Altera's Quartus II software, based on the crystal frequency phase-locked frequency multiplication from the crystal phase noise.The relationship between the crystal oscillator phase noise and the frequency measurement accuracy of the magnetometer is deduced from the experimental results after an experimental platform is constructed, circuit debugging and test experiments are carried out, and experimental results are obtained.

Crystal oscillator
Crystal oscillators are widely used as stable frequency reference sources in modern electronic systems requiring frequency control and management and in fields such as precision and accurate frequency metrology [5].The application of crystal oscillators in various communication systems needs to focus on their short-term frequency stability, also known as phase noise.Good phase noise performance means a more desirable oscillation signal with shallow phase distortion, which can provide a more accurate signal.

Crystal performance evaluation
As crystal oscillators are susceptible to their own and external environment, resulting in their unstable working performance, in order to better use crystal oscillators, their working performance needs to be evaluated [6].At present, there are two main parameters to evaluate the performance of a crystal oscillator: one is frequency accuracy, that is, whether its output vibration frequency is in line with its traditional value and whether it meets the factory standard; the other is frequency stability, that is, whether its output frequency signal sequence is smooth, whether there is frequency jump, and whether the output oscillation signal is within the safe range after long-term work.A study shows that the following formula can express the frequency accuracy of the crystal oscillator: In Equation (1), ft is the actual operating output frequency of the crystal; f0 is the expected frequency value; the unit to the left of the equal sign is ppm.

Phase noise of crystal oscillator
As the performance of radio communication systems improves, the performance of the signal source phase noise becomes a significant limiting factor in the overall system performance.Often referred to as phase noise, it refers to the short-term stability of the frequency, sometimes referred to as instantaneous stability.Short-term frequency stability is the relative frequency stability within one hour, usually the unwanted signal or noise and the interaction between the desired signal caused by the change in oscillator frequency, mainly caused by changes in circuit parameters, and does not include changes in frequency due to component aging or changes in the surrounding temperature.Frequency stability measures the oscillator's ability to produce the same frequency over the specified time range.If there is a transient change in signal frequency that cannot be kept constant.Then, there is instability in the signal source.Its cause is phase noise.Due to the presence of phase noise, it causes the expansion of the carrier spectrum, which can range from deviations from the carrier of less than 1 Hz to several megahertz.The oscillator's phase noise consists of various components, such as white noise and 1/f noise [7].
In practice, all signal sources suffer from instabilities, i.e., there are useless signal amplitudes, frequencies, or phase undulations that never produce a pure sine signal.Since frequency is the derivative of phase concerning time, these instabilities can be equivalently viewed as useless frequency or phase undulations.The characterization of these phase undulations is often called phase noise.Phase noise is difficult to see in the time domain, and it exhibits zero crossover variation relative to a pure sinusoidal signal without phase noise.Because phase noise is generated by mixing low-frequency signals at much lower frequencies than the carrier frequency, the zero-crossing variation is not observed in the time domain until after many weeks of carrier waves.
Phase noise is an essential indicator of the stability of the frequency reference source, and in general, phase noise refers to the short-term stability of the signal frequency [8].As shown in Figure 1, a represents a standard sine wave signal without any noise, and b represents a sine wave signal with phase noise.Relative to the standard sine wave signal, the phase of waveform b is arbitrarily disturbed at the set time grid, i.e., over or lagged.From the frequency domain, an ideal crystal oscillator outputs a signal that behaves as a single spectrum.However, due to phase undulation or other factors, an actual crystal oscillator can never produce a single pure sine or cosine signal that behaves as an impure signal frequency.Figure 2 shows the spectrum of a pure sine signal and an impure sine signal.The output of an ideal oscillator is the ideal waveform, but the actual oscillator's output could be better.As shown in Figure 3, the ideal degree of the waveform can be measured from three perspectives: amplitude, frequency, and phase.For the oscillator output signal, its amplitude instability is shown in the fluctuation of peak value; its phase instability is shown in the fluctuation of voltage at zero point; its voltage frequency instability is shown in the period change.Due to the limiting characteristics of the oscillation circuit, the amplitude noise will be removed, and the non-idealities in the amplitude of the oscillation waveform will become less critical.However, the phase noise caused by the phase jitter of the crystal is introduced into the communication system referenced by it as the frequency reference source, significantly reducing the performance of the communication system.The phase noise at the low-frequency bias of the frequency synthesizer affects the modulated signal and reduces its signal-to-noise ratio; the phase noise at the high-frequency bias interferes with other signal channels.As shown in Figure 4, the frequency synthesizer's phase noise affects the transmitter performance process.Furthermore, the crystal oscillator is the primary source of low-frequency bias phase noise in the frequency synthesizer.Therefore, the low phase noise performance of the crystal oscillator is significant for the whole radio frequency (RF) transceiver system.

Frequency domain characterization of phase noise
The mathematical formula represents an ideal oscillator whose output is an ideal sinusoidal signal: In Equation ( 2), V(t) is the output waveform amplitude, A, ω0, and φ are the vibration amplitude, angular frequency, and initial phase, respectively.
In practice, the crystal oscillator, due to the influence of other factors, its amplitude and phase are generally time-varying, which the mathematical formula can express: The vibration amplitude A(t) and the phase φ(t) are functions that change with time.Generally, amplitude stabilization circuits are designed in high-precision crystal oscillators to suppress fluctuations in vibration amplitude A(t), so the effect of phase φ(t) variation (i.e., phase noise) is much greater than the effect of amplitude A(t) fluctuations.
When using a spectrum analyzer to observe the signal noise sidebands, phase noise is symmetrically distributed on both sides of the center frequency.The actual engineering applications usually have single sideband phase noise (SSB).Single sideband phase noise is: at a particular frequency offset from the center corner frequency ω0, the ratio of the power per unit bandwidth to the total power of the output signal, the unit is dBc/Hz.

Time domain characterization of phase noise
In addition to the frequency domain method to represent the phase noise, the time domain representation has also been proposed.With the ongoing in-depth study of the oscillator noise characteristics, the defining formula for the time domain representation of the phase noise has evolved from the classical variance form to the Arlen variance.In the case of a limited number of measurements, the best expression of the Arlen variance is given in [9] and [10]: In Equation ( 4), t is the sampling time, and m is the total number of groups of measurements.
___ k y is the measured average frequency value, and every two measurements are taken as a group.It is important to ensure that there is no time interval between each two measurements.Sometimes, to reduce the time required for the measurement, it is possible to use a sampling number of m + 1.This gives another expression for the Allan variance: Allan variance applies to most of the current frequency sources and scales.It has gained the most widespread application due to its easy calculation and is an essential quantity for measuring frequency fluctuations.However, Allan variance also has some drawbacks and limitations.For example, the difference between adjacent frequency measurements in practice includes the random fluctuations caused by noise but also the frequency shift of the system itself.

Phase noise analysis based on the Leeson model
Leeson [11], [12] proposed the feedback oscillator model in the early 1960s.In the following years, Sauwvage [13] proved the feasibility of Leeson's model mathematically and theoretically.Figure 5 shows the schematic diagram of the feedback oscillator model proposed by Leeson.As seen in Figure 5, the Leeson model consists of two main parts, the amplifier and the filter resonant loop, and it is a phase-positive feedback system.Since the equivalent circuit model of the quartz crystal is a resistor-inductor-capacitor (RLC) circuit, the resonant loop in Figure 5 can be viewed as an RLC circuit, i.e., a resonant loop that acts as a bandpass filter.We relate the phase noise spectral density of both at the oscillator's output and inside the amplifier so that the total power spectral density of the oscillator can be found.
  in the equation, S∆θ (fm) is the phase undulation spectral density of the oscillator.S∆θ (fm) is the phase undulation spectral density at the input of the amplifier; P and QL are the input excitation power and the on-load quality factor, respectively.Fc, fm, f0 represent the corner frequency, offset carrier frequency, and oscillator's carrier frequency, respectively.T, k, F represents the absolute temperature, Boltzmann's constant, and the feedback coefficient.
Equation ( 6) represents the phase undulation spectral density of the output of the Leeson model oscillator, i.e., the phase noise.It can be seen that due to the presence of positive phase feedback caused by the phase noise S∆θ (fm) at the input of the amplifier, the oscillator output phase noise increases by a factor of 1 + (f0/2QLfm) 2 .We can derive the single-sideband phase noise of the oscillator from the relationship between the corner frequency fc of the flicker noise and the half bandwidth f0/2QL of the resonant loop and the relationship between the single-sideband phase noise and the output power spectral density as: From Equation ( 8), we can know that the single sideband phase noise spectral density of the oscillator is related to many circuit factors, only a comprehensive weighing of the choice of various parameters of the oscillator circuit, especially the choice of parameters such as fc, f0, P, F, and QL.For example, for the carrier frequency f0 of the oscillator, the higher the frequency is, the higher the output phase noise of the oscillator will be.

PLL phase-locked frequency doubling
The frequency multiplication circuit is the core of the atomic clock signal source.The input crystal signal can be multiplied by the frequency multiplication circuit to the required frequency of the atomic clock.The traditional frequency multiplication method uses analog frequency multiplication, and discrete devices make the circuit large in size.In order to meet the demand for the miniaturization of atomic clocks, digital phase-locked frequency multiplication is now used.Phase Locked Loop (PLL) is integrated, which reduces the size and has better performance in terms of phase noise and spurious.The loop filter is an essential part of the phase-locked loop and consists mainly of capacitors, resistors, or amplifiers.The loop filter filters out the high-frequency components of the discriminator output signal.
It dramatically impacts important loop parameters such as spurious rejection, phase noise, loop stability, and lock time of the phase-locked loop.
The PLL consists of four main modules: phase discriminator, loop filter, voltage-controlled oscillator, and frequency divider, as shown in Figure 6.The input signal introduced through the oscillator enters the phase discriminator.The output signal after the frequency divider is multiplied to discriminate the phase.The discriminator outputs a voltage signal related to its phase difference, which is processed by the loop filter and enters the voltage-controlled oscillator (VCO).The VCO voltage control terminal controls the VCO output signal.Finally, the feedback loop stabilizes the output frequency, and the VCO outputs the target frequency.

Experiments and analysis of results
The experimental platform is shown in Figure 7.The experimental platform includes a computer, a signal generator, a magnetometer, and a sensor.The crystal uses an SG-8018 series quartz programmable oscillator with a fixed output frequency of 50 MHz and the phase jitter (Root-Mean-Square, RMS), as shown in Figure 8.The frequency multiplication link mainly uses the IP core ALTPLL in Altera's Quartus II software to multiply the crystal frequency.The IP core ALTPLL in Altera's Quartus II software is mostly used by the frequency multiplication link to multiply the crystal frequency.This IP core can establish or alter the scale factor to produce the desired frequency value as soon as the procedure is finished, as indicated in Figure 9.The frequency multiplication to 100 MHz and 200 MHz is selected as the output in this experiment.The signal generator has a fixed output of 2 kHz and 200 mV frequency.The signal amplitude of 100,000 times is attenuated through the attenuator into the magnetometer.Finally, the measured frequency value of the magnetometer is obtained to study the effect of the crystal phase noise on the measurement accuracy after frequency doubling.
The test results are shown in Figure 10 and Figure 11.In summary, after multiplying the crystal frequency, it is known from the Leeson model that the oscillator output phase noise increases, resulting in a decrease in frequency measurement accuracy.

Conclusion
Phase noise is one of the main parameters of crystal oscillators, and it has an important impact on the performance of various electronic devices.This paper details the concept of phase noise and the characterization methods of phase noise, primary domain characterization method, and frequency domain characterization method.The phase noise model based on the Leeson model is analyzed, and the impact of the phase noise of the crystal oscillator on frequency measurement accuracy is analyzed theoretically.Finally, the frequency doubling experiment proves that increasing the crystal frequency will increase the phase noise and reduce the measurement accuracy.Since the signal frequency value is closely related to the magnetic field strength, the magnetic measurement accuracy of the proton precession magnetometer will also decrease when the measurement frequency accuracy decreases.

Figure 2 .
Figure 2. Spectrum of pure and impure sine signals.

Figure 4 .
Figure 4.The process by which phase noise affects transmitter performance.

Figure 9 .
Figure 9. PLL frequency doubling.The interface lets you directly set or adjust the scale factor to output the desired frequency value.For this experiment, the frequency multiplier to 100 MHz and 200 MHz is selected as the output.The signal generator has a fixed output of 2 kHz and 200 mV frequency.The signal amplitude of 100,000 times is attenuated through the attenuator into the magnetometer.Finally, the measured frequency value of the magnetometer is obtained to study the effect of the crystal phase noise on the measurement accuracy after frequency doubling.The test results are shown in Figure10and Figure11.

Figure 11 .
Figure 11.Root mean square error for different frequencies.