PL230329B1 - Method for measuring the voltage period and the system for measuring the voltage period - Google Patents

Description

Description of the invention

The subject of the invention is a voltage period measuring system for testing the electrical properties of periodically alternating current circuits.

The solution for determining the signal period presented in the Zieliński TP textbook "Digital Signal Processing from Theory to Applications" by Wydawnictwa Komunikacji i Łączności, Warsaw 2005, 2009, is known, based on the value of the own correlation function R _x (z) in the form of the formula:

+ □ 0

Ą (r) = j -T ^ dt -00 where x (t) is the tested signal, and x (tr) is the tested signal, shifted by the time τ. The autocorrelation function takes maximum values for time shifts τ equal to the multiple of the signal period x (t), hence the signal period is equal to the smallest distance between the maxima of the autocorrelation function. In the case of real measurement systems, only a limited number of samples is available and the autocorrelation function is replaced by the unbiased estimator R _x (k) in the form of the formula:

w-i-H where N is the number of samples, -Λ / + 1 <k <O is the shift value, and x (n) are the signal samples for O <η <Λ / -1. For these systems, the period is determined in the same way as for the continuous signal and is approximately equal to the minimum distance between the maxima of the autocorrelation function.

The known solution is characterized above all by the fact that the resolution of the measurement result is equal to the sampling period, hence the error of the measurement result is strongly dependent on the synchronicity of the measured signal and the sampling period. An additional disadvantage is that the autocorrelation function estimator adopts approximate maximum values that differ slightly from each other and are not normalized, i.e. they assume different values for different signals.

For example, devices and systems for measuring the period of signals based on autocorrelation functions have been disclosed in US4463425, US5584295, US6477476.

The essence of the voltage period measuring system according to the invention is that this system measures the instantaneous voltage values at equal time intervals called the sampling period T _s less than half the time of the measured voltage period T, and the time of the entire measurement covers at least two full periods T of the measured voltage, i.e. t _n > 2T, and then, on the basis of the values of the measurement results, it analytically estimates the value of the voltage period.

The circuit for measuring the period T of the voltage u (f) comprises an amplifier whose input terminals are connected to points of the electrical circuit between which the measured voltage u (f) is present and whose output is connected to the input of an analog-to-digital converter. _{A digital clock signal generated by a clock signal generator with period T s} <0.57, triggering sampling at times 6, fc, ti, ... t _n is connected to the digital input that triggers the sampling of the converter (measurement of the instantaneous value of the voltage at the analog input of the converter) , where t, = (/ - 1) 7, and the time of the last measurement meets the condition t _n = (n- ^ Ts> 2T. The output of the analog-to-digital converter, to which the result signals with the values t / i = are sent for individual samples u (t}, ui = uz = u (t3), ... u _n = u (t _n ), is connected to the measuring line controller which is controlled by the interface controller used for communication with the user via the interface. The measuring data memory block is connected to the measuring line controller, which is used to store the data read by the measuring track controller from the analog-to-digital converter. The task is to determine the value of the objective function f _c for the value of the argument of the objective function T _m and store the value of the objective function f _c in the memory unit of the value of the objective function and the argument of the objective function T _m in the memory unit of the value of the argument of the objective function. Attached to the objective function value memory unit and the objective function argument value memory unit is the objective function minimum calculator, which determines the value of the objective function argument T _m , for which the objective function f _c takes the minimum value, and stores this value in the period memory unit. voltage. The system of determining the minimum of the objective function activates the system of calculating the value of the objective function to determine the value of the objective function f _c for successive values of the arguments of the objective function T _m from the range T _m in to T _max with a resolution of T _s tep, while the controller

PL 230 329 Β1 of the interface shows the measured value of the voltage period T u (f) read from the voltage period memory unit after performing calculations for the frequency search range from Τ ™ _η to T _max with resolution Tstep The system for determining the value of the objective function consists of the following systems: time modification, sequence sorting system, data averaging system, interpolation function determination system and final objective function determination system.

The time modification system reads the sequence of measurement data pairs from the data block in the form of the measurement time and the instantaneous voltage value at a given measurement time according to the formula:

(^, ^), ((^ 1 ^), ((^), ... ((,,, ^,) and modifies the measurement data string by modifying the values of measurement times according to the formula:

T _m where LJ is the rounding down operation, obtaining a string of the form:

(^), ((^), ((^), ... ((^, ^ which writes to the modified data memory block that is attached to the string sort system.

The sequence sort system sorts the modified sequence of measurement data pairs read from the modified data memory block according to the modified measurement time t ', obtaining a sequence in the form:

(, nn \ (n \ in \ in \ ^ 1> ^W 1) z V2 ' ^{Z /} 2)') '·> ^u ') with the property t <t _{( n} which writes to the sorted data memory block that is attached to to the data averaging system.

The data averaging system averages the modified measurement data with the same modified measurement time read from the sorted data memory block by replacing several pairs of measurement data with the same modified time with one pair of measurement data with the same modified time and the voltage value equal to the arithmetic mean of the averaged measurement data voltages, yielding element string m:

wherein m <n and which stores in an average data memory block which is connected to a string evaluation circuit.

The string value determination system reads m pairs of averaged measurement data from the averaged data memory block and based on them performs the following operations:

- using the argument T _m determines the sequence of time differences Δ7 according to the formula:

At, - t _{i + l} -t, for i = 1,2,3 ... m -1

Δί = T -t m m m whose values are saved in the first block of memory,

- using the argument T _m determines the sequence of time differences according to the formula:

Δ ₂ ί, = ζ ₊₂ -i, for i = l, 2,3 ... w-2, the values of which are saved in the second block of memory,

PL 230 329 Β1

- determines the sequence a according to the formula:

- u.

a. = ¹

Δ /, u - u

a. _ £ + i ---- d | a j = 2 3 4 ... m -1

Δ ₂ <-ι ^ m + l

Δζ, the values of which are saved in the third memory block,

- determines the sequence b according to the formula:

ξ = (α _{/ +)} -aj Δ /. for i = 1,3,4 · - ^m, the values of which are stored in the fourth block of memory,

- determines the sequence c according to the formula:

c, = 0 _{c =} ^Μ Μ ^Δ ύ + ^ ιΔζ ι _d | _a / = 2,3,4 ... ^ - 1 δ ₂ <._, _ w ,,,, ΔΖ + μ.Δ /, ", __ jy__M ~ I IM________1J ^- A #

Δ ₂ ς-, the values of which are saved in the fifth block of memory,

- determines the mean μ according to the formula:

^m i A \ / = ik ² J the value of which is stored in the mean memory unit μ;

- determines the sequence d according to the formula:

d _i = c, - (-!) 'μ for i = 1,3,4. ..m, the values of which are saved in the sixth block of memory.

Connected to the first, fourth and sixth memory blocks is a terminator for the objective function.

The end evaluator of the objective function reads m elements of the sequences ΔΖ, b and c from the first, fourth and sixth memory blocks, respectively, from which it determines the final value of the objective function f _c for the argument T _m according to the formula;

\ ^m ffh) λ / Λ) νΣ ', + ς' k / = 1 \ A ³ / J which is stored in the memory unit of the objective function values.

An advantage of the inventive period measurement circuitry is that it has any period measurement resolution, since the voltage period T _m argument of the objective function f _c does not have to be a multiple of the sampling period Tsjak, as is the case with the autocorrelation method. The possibility of determining the objective function for any value of the argument T _m satisfying the condition 2T _S <T _m <± (n-1) T _S and determining the value of this function over the entire appropriately interpolated interval [0, Tm] results in a smaller impact of sampling asynchrony on the measurement error in compared to the estimator method

PL 230 329 Β1 of the autocorrelation function. Another advantage is the non-negative value of the objective function, belonging to the range [0, + oo], which allows for easier determination of the minimum value close to 0. An additional advantage is the uncomplicated way of determining the objective function, which allows for quick numerical calculations. Acceleration of the calculations allows you to reduce the clock frequency of the processor that would perform such calculations, and thus reduces the electricity consumption of the measuring device.

An embodiment of the measurement system according to the invention will be shown in the drawing, in which Fig. 1 shows an overall measurement arrangement and Fig. 2 shows a diagram of a UWWFC target function value evaluation arrangement.

The measuring system is adapted to be operated by the user via the INT interface. Through the INT interface, the user can set the operating parameters of the measuring system, start the voltage period measurement or read the measurement result. The INT interface is connected by a bi-directional digital bus with the KI interface controller. The KI interface controller is adapted to work in two modes: in communication mode (with the user) and in standby mode. In the communication mode, the KI controller supports the INT interface - it sends data for display, downloads setting parameters and gets a command to start the measurement. In addition, the KI controller saves the system operation parameters using digital buses in the appropriate memory units: sampling period T _s in the unit T _s , number of samples n in the unit n, lower voltage period search range T _mm in the Tmin unit, upper voltage period search range T _max in unit of Tmax, and the measurement resolution (i.e. step of the searched voltage periods) T _ste p in the unit Tstep. In addition, the controller KI reads via the digital bus the determined period of the electric voltage T _u stored in the memory unit T _u . When a command to initiate a measurement is received from the INT interface, the KI interface controller checks the correctness of the measurement parameter settings using five conditions:

1.The lower limit of the measuring range must be smaller than the upper limit of the measuring range, i.e. Tmin <Tmax

2. The measurement resolution must not be less than 0.1 of the measurement range, i.e. 10T _step <T _max -T _mn

3.sample period T _s must not be less than the minimum data processing period Tη, e.g. resulting from the device parameters, i.e. T _s > T _m , _n p

4. the measurement time must cover at least two full periods of the measured voltage, i.e. (n-AjTs> 2T _max

5. The sampling period must be less than half the period of the measured electrical voltage, C yli 2T _S <Tmin

If the settings are incorrect, the KI controller sends the appropriate error information to the INT interface and proceeds to setting the data. If, however, the measurement settings are correct, the KI interface controller sends to the INT interface information about the start of the measurement and sends a digital signal to the KTP measurement line controller in order to start the measurement. Then the KI controller suspends its operation in the communication mode, going to the standby mode until the appearance of a digital signal informing about the end of the measurement coming from the system for determining the minimum value of the target function UWMINFC. When the signal of measuring completion from the UWMINFC system appears, the KI interface controller downloads the result from the T _u unit, sends it to the INT interface and then returns to the communication mode with the user.

The KTP measurement path controller reads the data from the measurement path and places the results in the BD measurement data memory block. The measuring path includes: the W amplifier, the ADC analog-digital converter and the CLK clock signal generator. The generator G, connected to the input terminals of the test circuit, symbolically represents the system generating the alternating voltage, the period of which is being tested. The terminals of the periodic voltage generator G are connected with the input of the linear amplifier W. The output of the amplifier W is connected with the analog-to-digital converter ADC. The task of the W amplifier is to condition the input voltage to the ADC range. This means that the voltage at the output of the amplifier in relation to the mass is directly proportional to the input voltage of the measuring system and the amplifier gain is selected so that at a given measuring range, the voltage at its output does not exceed the operating range of the ADC converter. The measurement result of the ADC converter is transmitted via the digital bus to the KTP measurement track controller. The measurement in the ADC converter is triggered by an external signal coming from the controlled clock CLK. The signal from the CLK clock is also fed to the digital input of the KTP controller in order to provide information about the performed measurement. The time between successive signals of the measurement excitation is set in the CLK clock by a signal

PL 230 329 Β1 digital from the KTP measurement circuit controller. The CLK clock settings are constant throughout the measurement and are set so that the ADC excitation level lasts for the entire transducer processing period, and then the clock output is set to the opposite and re-changed to the transducer excited state after T _s .

The KTP controller is connected to the memory units T _s i, n and the BD memory block. The controller, by means of digital buses, reads the sampling period T _s from the memory unit T _s and the number of samples n from the memory unit n, writes the measurement data to the measurement data memory block BD and writes and reads the value of the variable of the measurement number and from the memory unit i. a measurement initiation digital signal from the KI interface controller; and sends a data processing initiation digital signal to the UWMINFC target function minimum calculator.

The KTP measurement path controller, after receiving the digital measurement initiation signal from the KI controller, sets the value of the measurement number i = 1 in the unit i and reads the sampling period T _s from the unit T _s and sets the CLK clock, Then the controller starts the measurement cycle and performs the following sequence of actions :

1.reads the data i from the unit i, on the basis of which it determines the measurement time t, according to the formula ti = (/ -1) Ts

2. waits for a digital signal from the ADC excitation clock CLK

3. waits for a digital signal from the CLK clock with the opposite state, which informs that the ADC converter has finished the measurement

4. Reads data from ADC

5. processes data from the ADC converter taking into account the amplification of the amplifier W and obtains the voltage value at the input of the measurement circuit

6.saves the measurement result in the form of a pair (u ,, t) in the BD block, where ui = u (tt) is the voltage at the input of the measuring line at time t,

7.increases the value of the measurement number / in the i unit by one (i = i + 1)

8. compares the values in the units i and n, if i <n, it goes to the next measurement - the beginning of the cycle, and if i> n, the controller ends the measurement cycle.

When all measurements are made, the KTP measurement path controller stops the CLK clock, sends a digital signal of excitation of the system for determining the minimum value of the UWMINFC target function, and then waits for the next digital signal of measurement initiation from the KI interface controller. As a result of the KTP controller operation, the measurement data in the BD block are stored in the form of a sequence of pairs:

The system for determining the minimum of the objective function UWMINFC determines the value of the period of the measured voltage on the basis of the argument for which the value of the objective function takes the minimum value. Objective function values are determined in a separate UWWFC objective function value evaluation system. The UWMINFC circuit uses digital buses to read the values of the lower search range of the voltage period Tmin from the memory unit Tmin, the upper range of the voltage period search T _ma xz from the memory unit Tmax, measurement resolution - the step of the searched voltage periods T _s tep from the memory unit Tstep and the values of the objective function f _c from memory unit f _c . In addition, the UWMINFC chip reads and writes the argument values of the objective function T _m in the memory unit Tm, the minimum value of the objective function f _m in the memory unit fm, and the value of the voltage period T _u for which the objective function takes the minimum value, in the memory unit T _u . At the same time, the UWMINFC system receives the digital signal of initiation of measurement data processing from the KTP measurement path controller, sends the digital signal of initiation of the determination of the value of the objective function to the UWWFC system for determining the value of the objective function, receives the digital signal of the completion of determining the value of the objective function from the UWWFC system and sends the digital signal of the end of the measurement to KI interface controller.

The circuit for determining the minimum of the objective function UWMINFC, after receiving the digital data processing initiation signal from the measurement path controller KTP, reads the lower voltage period search range Tmin from the memory unit Tmin and stores it, as an argument of the objective function T _m , in the memory unit Tm. Then the UWMINFC system starts the argument search cycle, for which the objective function takes the minimum value and performs the following sequence of actions:

1.sends the digital signal initiating the determination of the value of the objective function to the UWWFC system,

2.waits for the digital signal of the end of determining the value of the objective function from the UWWFC system,

PL 230 329 Β1

3.Reads the data: T _{m from the unit of} Tm, T _m j _{n from the unit of} Tmin, / unit from f _c , f _{m from the unit} f _m and if

T _m = Tmin or simultaneously T _m > Tmin and fc <fm it assigns the value of f _{c to the} variable f _m writing it to the fm unit and assigns the value T _{m to the} variable T _u writing it to the unit T _u

4.reads the data T _{m from the} Tm unit and T _has x from the Tmax unit and if T _m = T _has x, it assigns the variable T _m from the T _m unit to 0, and if the condition T _m = T _max is not satisfied, it reads the Tstep value of the unit Tstep i assigns the variable T _{m from the unit} Tm a value according to the formula T _m = min (T _m + T _slep , T _ma x), where the function min returns the minimum value of two variables Tm + Tsiep 0Γ3Ζ Tm _a x

5.reads the value of T _{m from the} T m unit and if T _m = 0 it ends the argument search cycle for which the objective function takes the minimum value, and if the condition T _m = 0 is not satisfied, it goes to the beginning of the cycle

At the moment of the end of the cycle of searching for the argument for which the objective function takes the minimum value, the UWMINFC system sends the digital signal of the end of the measurement to the controller of the KI interface, and then waits for the next digital signal of initiating data processing from the controller of the measurement path KTP to appear.

The system for determining the value of the objective function UWWFC waits for the digital signal initiating the determination of the value of the objective function from the UWMINFC chip, and then reads the value of the argument T _{m of} the objective function f _c from the memory unit Tm, the number of samples n from the memory unit n and the measurement data values from the of the measurement data memory block BD and on the basis thereof determines the value of the objective function f _c , which it stores in the memory unit f _c , and then sends the digital signal of the completion of the evaluation of the objective function to UWMINFC and waits for another digital signal of the determination initiation of the evaluation of the objective function. A detailed structure diagram of the UWWFC target function value determination system together with digital signals and external buses, and the measurement data memory block BD and memory units Tm, n and f _c is shown in Fig. 2.

The time modification circuit UMC, after receiving the objective function evaluation initiation digital signal from the UWMINFC circuit, reads the number of samples n _{from the memory unit n and the argument T m} from the memory unit Tm. Then the UMC system reads successively n measurement results in the form of pairs (Ui, ti) from the BD measurement data memory block and performs time modification according to the formula:

where L J is a rounding down operation and then writes the time-modified data obtained to a BDM-modified data memory block. As a result of modifying the times in the BDM block, the data is saved in the form of a sequence:

with property = 0. After modifying all the data, the UMC sends an initial digital signal to the USC sequence sort system and waits for another target function determination digital initiation signal from UWMINFC.

The USC string sorter, upon receipt of the initial digital signal from the UMC, reads the number of samples n from memory unit n, and n data pairs from the BDM modified data memory block. Then it sorts the data pairs modified by time / 'to obtain the string:

V1 5 ^W l) / (^ 2 ' ^W 2)' (A ')' ·· ' ^U n) where t <and t "= 0. The sorted sequence USC writes to the BDP sorted data block. The USC then sends the digital initiation signal to the UUD data averaging circuit and waits for the next digital initiation signal from the UMC.

The UUD data averaging circuit, after receiving the initial digital signal from the USC, reads the number of samples n from memory unit n, and n data pairs from the BDP sorted data memory block. Then the UUD system averages the data pairs with the same time by replacing several data pairs with the same time with one data pair with the same time and a voltage value equal to the arithmetic mean of the averaged voltages of the data pairs, resulting in the element sequence m:

(A ’) '(4 Ά)' (G’ A)

PL 230 329 Β1 with properties m <n and = 0. After determining the averaged data sequence, the UUD system saves the obtained sequence in the BDU averaged data memory block and writes the m number of sequence elements in the memory unit m. Then the UUD system sends an initial digital signal to the determining system UWWC string values and waits for the next initial digital signal from the USC system.

The UWWC sequence evaluator determines the values of the elements of five sequences, three of which are used directly to determine the final value of the objective function, which is determined in the UWWKFC objective function terminator. UWWC uses digital buses to read the argument values of the objective function T _m from the memory unit Tm, the number of averaged data pairs m from the memory unit m and the averaged data pairs from the BDU averaged data memory block. Additionally, the UWWC system, using digital buses, reads and writes the values of three strings to memory blocks B1, B2, B3 and writes to memory blocks B4 and B5 the values of elements of the remaining two strings. At the same time, the UWWC system receives the digital data processing initiation signal from the UUD data averaging system and sends the digital initiation signal to the UWWKFC target function end calculator.

The UWWC string evaluator, after receiving the digital initiation signal from the UUD, reads the objective function argument T _m from the memory unit T m, the number of samples m from the memory unit m, and m data pairs from the BDU averaged data memory block. Then, based on the averaged times of data pairs and the argument T _m, the UWWC system determines m elements of the sequence At according to the formula:

Δ /, = t _M -t _t for i = 1,2,3 ... m-1

Δ / = T -t fil m nt and writes them to block B1. Then, using the same data of the averaged times of data pairs and the argument T _m, the UWWC system calculates m-) elements of the sequence A2t, according to the formula:

Δ / ₍ . ⁼ Æ + 2 “ύ for i = 1,2,3 ... m-2 which is written in block B2.

Then, based on the data held in the BDU block, the At and Ait sequences stored in B1 and B2 blocks, respectively, and the mi T _m data from mi Tm units, the UWWC system determines m +1 elements of the third sequence a according to the formula:

U. - u.

_ £ + and ---- μ. for / = 2,3,4 ... w -1 △ 2 ^ -1 _{a =} a = “'~ ^α and writes them to block B3.

Then, based on the elements of the sequence a \ At read respectively from blocks B3 and B1 and the number m from the unit m, the UWWC system determines m elements of the sequence b using the formula:

b _t for i = 1,3,4 ... m which he writes in block B4.

Then, based on the data in the BDU block, the sequences At and Ait stored in blocks B1 and B2, respectively, and the data mi T _m from mi Tm units, the UWWC system determines m elements of the fifth sequence c according to the formula:

PL 230 329 Β1

Cj = 0 ~ ΰ. ΛΤ + μ..ΔΛ_, ~, c _i = u, ——— ^and ---——— for? = 2,3,4 ... m -1 △ 2 Ci ~। + w. ΔΖ.

z> —7 / __ / τι — ln? I rtr — lw “ ^U m _A

Δ, /. from m-1 and writes them to block B5.

Then, on the basis of elements of the sequences b, c and At read respectively from blocks B4, B5 and B1 and the number m from the unit m, the UWWC system determines the average // according to the formula:

Α = 7-Σ (->) 'ί | + φ, the value of which is stored in the μ memory unit;

Then, on the basis of elements of the sequence c read from block B5 and B1 and the mean μζ of the memory unit μ and the number of m from the unit m, the UWWC system determines m elements of the sequence d using the formula:

d, = c, - (- 1 / μ for i = 1,3,4 ... m, the values of which are saved in the memory block B6;

After all five strings have been determined and stored, UWWC sends a digital initiation signal to the UWWKFC target function terminator and waits for another digital initiation signal from the UUD data averaging circuit to appear.

The last element of the UWWFC objective function evaluation system is the UWWKFC objective function terminator. The UWWKFC system, after receiving the initial digital signal from the UWWC string value determining system, reads the value of the variable m from the memory unit m. On this basis, it reads successively m values of the sequences At, and b from memory blocks Β1, B4 and B6, respectively, and then determines the final value of the objective function according to the formula :

((h λ A ¹ m <= l ^J / J which it writes to the memory unit f _c . Then the UWWKFC circuit sends the digital signal of completion of the determination of the objective function value to UWMINFC and waits for another initial digital signal from the UWWC chip.

The inventive measurement arrangement outlined above is to be considered as an exemplary arrangement. The individual elements of the circuit may be in the form of digital or analog circuits. It is obvious to a skilled person how to implement the individual blocks in order to fulfill their functionality. In one of the possible implementations, the measurement system may be implemented in the form of a processor controlled by a suitable software. In another embodiment, the measurement circuitry may be implemented as an FPGA circuit of programmable logic gates.

Claims (1)

1. A system for measuring the period T of voltage u (t), comprising an amplifier (W) whose input terminals are connected to points of the electrical circuit between which the measured voltage u (f) is present, and whose output is connected to the input of an analog-to-digital converter ( ADC), to which the sampling trigger input is connected a clock signal generated by a clock signal generator (CLK) with period T _s <0.57, triggering sampling at times h, t?, T3, ... t _n , where t _t = (7-1) T _s and t _n = (n-1) 7 _s > 27, while the output of the analog-to-digital converter (ADC), on which the result signals are sent for individual samples with the values ui = u (h), U2 = uff /), U3 = ufa), ... u _n = u (t _n ), connected to the measuring line controller (KTP) controlled by the interface controller PL 230 329 Β1 (KI) for communicating with the user via an interface (INT), a system, characterized in that it comprises: - measurement data memory block (BD) connected with the measuring track controller (KTP) for storing data read by the measuring track controller (KTP) from the analog-to-digital converter (ADC); - an objective function value evaluation circuit (UWWFC) attached to a measurement data memory (BD) block for determining and storing: values of the objective function f _c in a memory unit value of the objective function f _c for the argument value of the objective function T _m in the memory unit of the argument value of the objective function T _m ; - objective function minimum determination circuit (UWMINFC) attached to the objective function value memory unit (f _c ) and to the objective function argument value memory unit Tm for determining and storing the objective function argument value T _m , for which the objective function f _c value takes the minimum value , in memory unit T _u ; - where the interface controller (KI) shows as the measured value of the period T of the voltage u (f) read from the memory unit T _u after performing the calculations for the frequency search range from T _m in to T _ma x with the resolution T _step \ - where the objective function calculation system (UWWFC) is activated to determine the value of the objective function f _c for successive values of the arguments of the objective function T _m from the range Tmin to Tmax with the resolution T _ste pi includes: o time modification system (UMC), which reads a sequence of measurement data pairs from the data block (BD) in the form of measurement time and instantaneous voltage value at a given measurement time according to the formula: and modifies the measurement data string by modifying the measurement time values according to the formula