Compensation of Accuracy Error for Time Interval Measurements

1. Introduction

Time measurements (time-to-digital conversion) [1,2,3,4] have always belonged to the class of the most demanded measurements since it is reasonable to reduce measurements of many heterogeneous physical quantities to time measurements. As modern methods and instruments for measuring time provide very high accuracy [5], these technologies are in demand in a wide range of fields and industries [6,7]. The most commonly used time measurements are measurements of time intervals (TI) between the start and stop input signals [8,9]. For a significant part of the experiments, the object of investigation is the time relations between initialising and secondary events [10,11].

In the analysis of measurement errors of time intervals, we assume that the result A_i of the i-th measurement is, in general, a random variable and can be represented as a sum of the measured quantity T_i and the absolute error of its measurement a_i

$${\mathit{A}}_{\mathit{i}}={\mathit{T}}_{\mathit{i}}+{\mathit{a}}_{\mathit{i}}$$

The absolute measurement error ai can, in turn, be represented as the sum of the random Δ_ir, accuracy Δ_is and offset Δ_c components of the total error [12], therefore:

$$A_i=T_i+a_i=T_i+{{\mathsf{\Delta }}_{ir}+\mathsf{\Delta }}_{is}+{\mathsf{\Delta }}_c$$

(1)

The random error (or precision error) Δ_ir has an unpredictable value and represents the scatter of the measurement results due to the influence of noises of different origins.

The accuracy error is determined by the repeatability of its value from measurement to measurement. However, the value of this error Δ_is itself depends on several parameters. The peculiarity of the accuracy error in TI measurement is its multiplicative character, i.e. the value of Δ_is is proportional to the duration of the measured TI.

The offset error is determined by the difference between the delays of the start and stop channels and is additive. Its value Δ_c is offset for a particular meter. The offset error can be determined and used to correct the measurement results accordingly.

Suppose the task is to test a meter. In that case, the value of T_i in (1) is generally unknown since the source (generator) of input signals possesses its errors (Accuracy, Offset, Jitter). Ti can be represented as the sum of the generator set value T_g, generator accuracy error Δ_isg, generator constant error Δ_cg and jitter of the TI generator Δ_irg. Therefore, the general equation of the measurement result is as follows:

$${\mathit{A}}_{\mathit{i}}={\mathit{T}}_{\mathit{i}}+{\mathit{a}}_{\mathit{i}}={\mathit{T}}_{\mathit{g}}+{{\mathsf{\Delta }}_{\mathit{i}\mathit{r}\mathit{g}}+\mathsf{\Delta }}_{\mathit{i}\mathit{s}\mathit{g}}+{\mathsf{\Delta }}_{\mathit{c}\mathit{g}}+{{\mathsf{\Delta }}_{\mathit{i}\mathit{r}}+\mathsf{\Delta }}_{\mathit{i}\mathit{s}}+{\mathsf{\Delta }}_{\mathit{c}}$$

(2)

A graphical interpretation of the generation and measurement errors is illustrated in Figure 1.

The accuracy error Δ_is in the TI measurement is determined by the stability level of the clock generator (CG) of the meter and is caused by the fact that the period F of the CG differs from the nominal value F₀ by τ. The value of τ is not constant and depends on four factors: changes in ambient temperature, the value of the initial deviation from F₀, variations in the generator supply voltage and ageing of its components.

In the specifications of a CG, the maximum value of its relative stability is K_max, defined as ratio K_max=±τ_max/F₀, where τ_max takes into account all destabilising factors. According to the methods of minimising the influence of destabilising factors, CGs are divided into stability classes [13,14]: standard XO (Crystal Oscillator), termo-compensated TCXO (Temperature Compensated Crystal Oscillators), thermo-stated OCXO (Oven Controlled Crystal Oscillators), microprocessor-corrected MCXO (Microprocessor Corrected Crystal Oscillator), and atomic standards.

Reducing the accuracy error is relatively easy to achieve by using high-end CG in the meter. However, for specific applications (e.g., space electronics), such a solution is not always feasible due to the high cost of high-class CGs, as the main cost component of space-level CGs are the testing procedures. In this paper, the problem of significant reduction of the accuracy error without increasing the CG class of the TI meter is solved.

2. Estimation of Accuracy Error

When measuring time intervals, the result of measurement A_i is proportional to the period of CG F, which determines the multiplicative nature of the accuracy error, and its value Δ_s increases proportionally to the duration of measured TI. The maximum value of accuracy error Δ_smax is usually the parameter of the meter, which can be estimated based on the certificate value K_max for the applied CG and A_max value, corresponding to the upper limit of the range of measured TI durations: Δ_smax= K_max·A_max.

The experience shows that the K_max value of a particular CG for the specific temperature conditions of its application can be much less than the value taken from its specifications data. Respectively, the real value of Δ_smax can also be several times less. Therefore, the first step in reducing the accuracy error is to estimate the Δ_smax value of a particular CG meter.

Evaluation of Δ_smax of a particular TI meter is based on measurement and accumulation of an array [A] of n results of the TI measurement (Figure 2) for the case of measuring input TI of maximum duration T_gmax when the ambient temperature t° changes in the operating temperature range of the meter. Based on (2), it can be assumed that the average value of the array of measurement results at temperature t° will be:

$${\overline{A}}_{t^{°}max}=T_{gmax}+{\mathsf{\Delta }}_{cg}+{{\mathsf{\Delta }}_{cg}+\mathsf{\Delta }}_{t^{°}s}+{\mathsf{\Delta }}_c$$

(3)

In this case, we assume zero values of the meter’s random error and the TI oscillator’s jitter. The influence of the accuracy error of the generator Δ_sg included in this expression can be neglected if the stability of its CG is 1-2 classes higher than the CG stability of the TI meter under study. Thus Figure 2 shows the use of an external OCXO (reference oscillator). In this case, expression (3) is simplified:

$${\overline{A}}_{t^{°}max}=T_{gmax}+{{\mathsf{\Delta }}_{cg}+\mathsf{\Delta }}_{t^{°}s}+{\mathsf{\Delta }}_c$$

(4)

Equation (4) used to evaluate the accuracy error Δ_t°c is complicated due to the presence of unknown error constants Δ_cg and Δ_c in the expression. The value of Δ_c is determined by the difference in delays for the start and stop inputs of the TI meter. In turn, Δ_cg is determined by the difference in delays in the TI meter start and stop signal generation circuitry and the difference in the lengths of the connection wires.

It is possible to estimate the values of Δ_cg and Δ_c when two conditions are met. Firstly, the TI must be measured with a minimum duration of T_gmin, which allows for omitting the accuracy errors of the generator Δ_sg and the measurer Δ_s, then:

$${\overline{A}}_{min}=T_{gmin}+{\mathsf{\Delta }}_{cg}+{\mathsf{\Delta }}_c$$

Secondly, additional measurements are required when the wires are cross-connected to the meter inputs (shown in dashed in Figure 2), and the sign of inter-channel delay of the TI generator is changed. In this case, a system of equations can be formulated:

$${\overline{A}}_{1min}=2T_{gmin}+{\mathsf{\Delta }}_{cg}+{\mathsf{\Delta }}_c$$

$${\overline{A}}_{2min}=2T_{gmin}-{\mathsf{\Delta }}_{cg}+{\mathsf{\Delta }}_c$$

This implies that:

$${\mathsf{\Delta }}_{cg}=\frac{{\overline{A}}_{1max}-{\overline{A}}_{2max}}{2}$$

$${\mathsf{\Delta }}_c=\frac{{\overline{A}}_{1max}+{\overline{A}}_{2max}-2T_{gmin}}{2}$$

Estimation of the error constant of the meter relative to its inputs Δ_c is an independent value, but at the same time, it provides the capability for estimation of Δ_t°s:

$${\mathsf{\Delta }}_{t°s}=\overline{A}-\left(T_{gmax}+{\mathsf{\Delta }}_{cg}+{\mathsf{\Delta }}_c\right)=\frac{1}{n}\sum _{i=1}^n{A}_{t^{°}i}-\left(T_{gmax}+{\mathsf{\Delta }}_{cg}+{\mathsf{\Delta }}_c\right)$$

(5)

While testing the meter in a climate chamber within the meter’s operating temperature range (Figure 2), a dependence of Δ_t°s on ambient temperature changes t° can be derived by TI measurement of maximum duration T_gmax. The graphical representation provides a capability to estimate the value of Δ_smax for a particular TI meter and the actual stability of its clock generator K_max=±Δ_smax/T_gmax.

Figure 3 shows an example of the plot of change of Δ_s as a function of temperature for a TI meter with a standard XO class clock generator (according to its specification K_max=±50 ppm). The dependence character in the operating temperature range of the meter (-40°C ÷ +60°C) is of a pronounced non-linear nature and corresponds to the frequency-temperature characteristics of quartz AT-slice plates [5].

Based on Figure 3, Δ_smax=740 ps and T_gmax=134 µs, hence the estimate of K_max=±740·10^-12/134·10^-6=5.5-10^-6=±5.5 ppm. The estimation of K_max obtained in this way is underestimated since it does not take into account the influence of instability of the generator supply voltage and the ageing of its components. Nevertheless, the difference between the specification listed and the actual component's parameter values is sufficient.

3. Compensation for Accuracy Error

The investigation of the dependence of the accuracy error value Δ_s on the temperature for a particular CG provides the capability to perform a procedure for offsetting the influence of the accuracy error on the measurement result.

The compensation procedure is based on a special calibration, during which a series of high-precision TI measurements are carried out under changing ambient temperature t°. This is done by determining the average values of the TI measurement Ā_t° and the accuracy error Δ_t°s for the current temperature t° according to (5). The correction factors K_t°=Δ_t°s/Ā_t° and the corresponding temperature calculated from this data are recorded as a table of K_t° and t° values in the memory of the meter or processing computer.

Considering the non-linear nature of the relationship between the accuracy error and temperature (Figure 3), the choice of a table to record the relationship between the correction factors and temperature is the simplest method, although it requires memory resources.

Several conditions have to be met during calibration: the CG class of the TI generator must be higher than the CG class of the measured (compensated) meter, the results of the calibration measurements must be averaged to increase the accuracy of the table, and the T_gmax value must correspond to the maximum working range of the TI durations to be measured.

In general, calibration should be carried out under the real conditions of the meter’s use or its equivalent. For example, if we are talking about space applications, then calibration must be carried out under thermo-vacuum conditions.

To implement the procedure for compensating for the accuracy error under actual measurement conditions, it is essential to monitor the current temperature t°. The algorithm for compensation of the i-th measurement result consists of extracting from the table the K_t° coefficient corresponding to the temperature t° at the time of measurement and calculating the compensated measurement result A_t°i: comA_t°i=A_t°i(1-K_t°).

In a general case, the measurement result A_t°i contains both an accuracy error and a constant error Δ_c, so comA_t°i°=(A_t°i-Δ_c)(1-K_t°). Figure 4 represents the dependence of the accuracy error Δ_s on the temperature after the compensation of the accuracy error shown in Figure 3. The value of Δ_smax decreased from 740 ps to 20 ps, and the equivalent stability of CG, in this case, reaches a value of ±0.15 ppm.

The compensation procedure also has its limitations since it does not consider the influence of instability of TG supply voltage and the ageing of its components.

As in the proposed solution, a variant of accuracy error minimisation implying a pre-calibration procedure is presented in [15]. This option is based not on the correction of measurement results but on microprocessor correction of frequency of VCXO class CG, the frequency of which is controlled by DAC voltage.

The microprocessor performs the calibration by the user. As a result of this procedure, the coefficients of the 4^th-degree polynomial are determined and stored. This coefficient is used to calculate the DAC codes, which compensate the VCXO frequency deviation for the current temperature value t°. The CGs based on this principle belong to the CCXO (Coefficient Corrected Oscillator) class and can ensure the stability of the CG at the level of 0.1÷0.3 ppb. In its application, there is no need to manually correct measurement results, but the cost of such CGs is high, and at this time, it is not manufactured for the space-level ready version.

4. Conclusions

The accuracy error at measuring time intervals (IT) is determined by the relative stability of the clock generator (CG) used in a meter and the duration of the measured time interval. The discussed method of accuracy error compensation eliminates the dependence of accuracy error on the duration of measured TI. It reduces to a minimum the influence of such destabilising factors as changes in ambient temperature and initial deviation of the clock frequency from its nominal value.

To implement the compensation method, three procedures have been developed: a procedure for registering an accuracy error, a calibration procedure in which a correction table is generated, and a procedure for correcting measurement results in real conditions of the meter’s application. The main expenditures of implementing the method consist primarily in correctly conducting the meter’s calibration, and hardware support for temperature monitoring and performing relatively straightforward computational operations is also required. The main costs of the method implementation primarily consider the correct calibration of the meter, as well as the required hardware support for temperature monitoring and relatively simple computational operations performance.

Experimental validation of the method demonstrated that an equivalent instability of ±0.15 ppm can be achieved for a standard class XO meter (specification instability of ±50 ppm), which reduces the accuracy error to the equivalent degree. The accuracy error compensation procedure has its limitations related to the replicability of the field conditions in which the calibration procedure has been performed.

5. Patent

The patent application of the Republic of Latvia LVP2023000010 has been filed (filing date: 02.02.2023) in perspective of the study conducted and the consequent conclusions.