A frequency meter based on Costas FLL for resonant accelerometers

The resonant accelerometer (RA) is an important part for high-performance inertial devices. And high accuracy frequency measurement is one of the key technology for RA. This paper developed a frequency meter based on Costas frequency lock loop (FLL). This structure is flexible as the analog input sinusodial voltage wave is converted into digital domain by an ADC and the frequency measurement function dealing with that voltage could be implemented only with digital circuits. The transfer function of the frequency meter is determined by the loop filter of the FLL and the closed system is always stable for PID controller. The noise in the loop is reshaped by a first order ∑-Δ modulator formed by the frequency meter loop, which relaxes the requirement for the accuracy of ADC. A test board for the frequency meter is implemented with a 24-bit ADC and FPGA, and achieves a noise of 1mHz/Hz at low frequency band, 7µHz/Hz around 50 Hz with a 40 kHz sinusodial input voltage.


Introduction
RA has a great potential in high accuracy inertial applications as it is characterized by large dynamic range, strong stability and high accuracy [1].Silicon and quartz are two kinds of crystalline materials used to fabricate RA, and the fabrication processes are quite different.The oscillating beam of the RA is the inertial sensing element, which converts the acceleration of the RA frame to its resonant frequency shift [2].The resonant state of the beam is extracted by the oscillating control circuit of the RA.The time-varying voltage from the circuit carries the acceleration information, and measuring the frequency of the voltage wave with high accuracy is one of the key technologies of RA.

CSMNT-2023
Journal of Physics: Conference Series 2740 (2024) 012025 IOP Publishing doi:10.1088/1742-6596/2740/1/012025 2 Two kinds of methods have been reported to measure the frequency of RA.One is based on frequency counter [3], and another is based on Σ-Δ noise shaping technology [4,5].Frequency counter based methods require high speed clock to get a higher accuracy or a larger bandwidth.Σ-Δ noise shaping based methods are combined with analog-digital hybrid circuits, which is suitable to implement with ASIC technology.
This paper proposes a frequency meter based on Costas FLL [6], which is different from the aforementioned methods.This frequency meter circuit consists of two parts.One of them is an analog circuit dealing with input voltage signal, and another is a FPGA based digital circuit running the frequency measurement algorithm.The Costas FLL algorithm is implemented with Verilog, which is flexible to develop and has the feature of first order Σ-Δ noise shaping.

Modelling and Analysis
The system diagram of the frequency meter based on Costas FLL is shown in Figure 1.Sinusoidal wave or square wave could be fed into the input of the circuit.These signal would be filtered by a bandpass filter before they are sampled by a 24-bit ADC.The circuit after the ADC is a Costas FLL built in a FPGA.The FLL consists of multipliers, a phase-frequency detectors (PFD), a loop filter and a numerical controlled oscillator (NCO).The FLL would indicate the frequency of the input voltage wave.

Transfer function
In Figure 1, the sampling rate of the ADC is f A (namely the sample period is T A = 1 f A ), so the sampling time could be t m = mT A , where m is an integer.The sinusoidal voltage got by the ADC at sampling time t m could be expressed with u * (t m )=A cos (θ(t m )), at the same sampling time the wave expressions from the NCO are Bcosφ(t m ) and Bsinφ(t m ).The cos and sin wave from the NCO multiply the samples from ADC separately, after which the products are filtered with the lowpass filters.Then the data are downsampled by a factor of K, so the data rate after being downsampled is f P = f A K (namely the sample period is T P = 1 f P ).We can get . δ(n) is the phase difference between the input voltage wave and the NCO wave at the sampling time t n .According to the two-quadrant arctangent method in [6], PFD output expression is is the frequency of the input voltage and   is the frequency of the output of NCO.And their expressions are , Following the PFD is the loop filter, which tunes the dynamic properties of the FLL and acts as a controller.The loop filter generally consists of a digital PI or PID block, whose output controls the frequency of the NCO's output waves.The two waves from the NCO have the save frequency, and their phase difference is always π 2 .Based on these analysis, the nonlinear frequency meter based on Costas FLL could be converted to a simple linear system, and its diagram is shown in Figure 2. In Figure 2, G(z) is the transfer function of the loop filter, θ(z) is the z-transform of the phase of the input voltage and u(z) is z-transform of the frequency output of the FLL.According to this system diagram we can get So the transfer function from the angular frequency of the input voltage to the FLL output is Clearly it can be seen that the transfer function of the loop filter G(z) dominates the dynamic properties of the frequency meter.
CSMNT-2023 Journal of Physics: Conference Series 2740 (2024) 012025 IOP Publishing doi:10.1088/1742-6596/2740/1/0120254 which could be calculated based on the principle of superposition of the linear system.And the noise spectrum of θ (t n ) could be expressed as where N ADC , N NCO , N PFD are the spectrums of ADC's noise, PFD's noise and NCO's noise, separately.It can be seen from equation ( 7) that the noises from the ADC and NCO have the save influence on the frequency measurement noise, and they are both reshaped by the FLL loop in the first order of Σ-Δ modulation, while the noise from the PFD will effect the measure noise directly as its has no coefficient related to the FLL's structure.Equation ( 7) also tells us that if used a longer phase sampling time T P we would get a lower frequency noise while other parameters in the system remain the same.
Figure 3 Noise model of the frequency meter

Experiment for function and noise test
According to the design and analysis above, we made a test circuit board as shown in Figure 4.This board has a 24-bit ADC with two separated channels and a FPGA acting as the processor.The frequency measurement system is implemented with Verilog language, and runs in the FPGA on the test board.The frequency data measured is transmitted to the host computer via the USB interface.
Figure 4 Circit board of the frequency meter based on Costas FLL A frequency modulation (FM) signal for the frequency meter test board is generated by a Keysight waveform generator.This FM signal has a 40 kHz sinusoidal carrier, and the shape of its time-varying frequency is a symmetrical triangular wave with a period of 10 Hz and a varying frequency range of ±10 Hz.The ADC on the test board samples this FM signal at the rate of 2 Msps, and the downsampling rate for frequency sampling is 512, which results a 1.9 kHz output data rate for the frequency measurement.The test board measured the frequency of the FM signal in time.Figure 5 shows the measured frequency wave of the FM signal.The wave is in agreement with the FM parameter, which indicates that the frequency meter system functions properly.
We evaluated the noise level of the frequency circuit board by feeding a pure 40 kHz sinusoidal wave to the board.The wave was generated by the Keysight waveform generator.Figure 6 shows the noise spectrum of the frequency meter.It is shown that the noise at the frequency range lower than 0.3 Hz is almost flat, about 1 mHz Hz .The noise spectrum rolls down at the slope of -20 dB dec in the band from 0.3 Hz to 30 Hz.The frequency noise level gets its local minimum around 50 Hz, about 7 μHz Hz.The noise in the lower band is higher than that in the higher band, and it may be affected by the jitter of the clock on the circuit board.

Conclusion
This paper proposes a frequency meter based on Costas FLL.Its linearized system model and noise model are also analysed.It's found that the transfer function of the loop filter determines the dynamic properties of the frequency meter, and the FLL loop has a first order Σ-Δ modulation effect to the noises from ADC and NCO.We built a frequency meter circuit board implementing the Costas FLL with a 24-bit ADC and a FPGA.The noise spectrum around 50 Hz is 7 μHz Hz, which is the local minimum of the noise level.The clock jitter of the frequency meter circuit might have effects on the noise level of the frequency meter and needs to be investigated.

Figure 1
Figure 1 Frequency meter structure based on Costas FLL

Figure 2
Figure 2 Simplifed linear system model of the frequency meter

Figure 5 Figure 6
Figure 5 Frequency of triangular FM signal measured by the test board