Results of study of quantization and discretization error of digital tachometers with encoder


ACTA IMEKO 
ISSN: 2221-870X 
June 2023, Volume 12, Number 2, 1 - 6 

 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 1 

Results of study of quantization and discretization error of 
digital tachometers with encoder 

Vasyl Kukharchuk1, Oleksandr Vasilevskyi2, Volodymyr Holodiuk1 

1 Vinnytsia National Technical University, 95 Khmelnitsky Shose str., 21021, Vinnytsia, Ukraine  
2 University of Texas at Austin, Austin, Texas, USA  

 

 

Section: RESEARCH PAPER  

Keywords: angular velocity; encoder; quantization; sampling; microprocessor tachometer; quantization and sampling error equation; "adjoining interval" 

Citation: Vasyl Kukharchuk, Oleksandr Vasilevskyi, Volodymyr Holodiuk, Results of study of quantization and discretization error of digital tachometers with 
encoder, Acta IMEKO, vol. 12, no. 2, article 19, June 2023, identifier: IMEKO-ACTA-12 (2023)-02-19 

Section Editor: Francesco Lamonaca, University of Calabria, Italy  

Received January 20, 2023; In final form February 19, 2023; Published June 2023 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Vasyl Kukharchuk, e-mail: BKuch@ukr.net  

 

1. INTRODUCTION 

Currently, to intensify the testing of electric machines, the vast 
majority of research is focused on the acceleration of tests carried 
out in the no-load experiment [1]-[4]. The main one here is the 

transient characteristic (variable angular velocity over time 𝑛(𝑡)), 
which is obtained in the dynamic mode of operation of the 
measuring object (electric machine) with practically zero moment 

of resistance on its shaft (МС ≅ 0). To ensure the maximum 
number of measured values of angular velocity 𝑛 during the 
transition process (3 ÷ 5) 𝜏, most researchers [5]-[8] focused 
their choice on digital measuring channels with encoders, the 
main components of which are the following: measurements 
object OB, coupling shaft MC, encoder E, shaper F, device for 
the period selection T, f0 frequency generator G, "AND" logic 
gate for the TX period quantization, binary counter CT2, 
programmable PI interface (parallel or serial), MPS 
microprocessor system. In the generalized structural diagram 

shown in Figure 1, the hardware sequence of measurement data 
transformations is carried out in the vast majority [9]-[12]. 
1. Encoder E converts the non-electric value angular velocity n 

into a sequence of electrical signals, the frequency of which 
is determined as 

𝑓𝑥 =
𝑛 𝑍

60
 , (1) 

 

Figure 1. Generalized structural diagram of the angular velocity measuring 
channel.  

ABSTRACT 
The analysis of measuring channels of angular velocity with an encoder given by the authors made it possible for the first time to obtain 
an equation for estimating the quantization and sampling error for an exponential mathematical model describing the transient process 
of operation of electrical machines. The components of the mathematical model of this dynamic error are the sampling step and the 
derivative, which characterizes the rate of change of the measured value over time. It was found that the errors of quantization and 
sampling significantly depend on the value of the resolution z of the encoder. Moreover, an increase in z leads to a decrease in the 
sampling error, but the relative quantization error increases. To reconcile these components of errors, the laws of change in the 
distinguishing ability z of the encoder are adaptive to the dynamic properties of the change in angular velocity over time. 
Proved that to ensure the maximum speed of measuring the angular velocity during the transient process, it is advisable to implement 
the method of changing the distinguishing ability of the encoder on the internal timers of the microcontroller proposed adapt ive to its 
dynamic properties, and the quantization of informative periods proportional to the measured angular velocity should be carried out in 
"adjoining intervals". 

mailto:BKuch@ukr.net


 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 2 

where 𝑍 is the resolution of the encoder. 
2. The shaper F from the output quasi-sinusoidal encoder 

output signals forms pulses of a rectangular shape, the logic 
levels of which correspond to the levels of TTL logic. 

3. The counting trigger T from the output frequency signals 𝑓𝑥 
of the encoder, selects informative periods 𝑇𝑥 , proportional 
to the angular velocity 𝑛. 

4. In the logical "AND" gate, quantization takes place [10] by 

comparing the measured physical value of the period 𝑇𝑥  and 
the sample period 𝑇0. As a result of a such comparison, 
conversion equations for the frequency measurement 
channel of the instantaneous values given by such a 
transformation function are obtained 

𝑁 =
𝑇𝑥
𝑇0

= 𝑇𝑥 ∙ 𝑓0 =
𝑓0
𝑓𝑥

 . (2) 

5. Binary counter CT2 carries out the procedure of counting the 

number of 𝑁 sample periods 𝑇0, which were quantized by 
periods 𝑇𝑥 . 

6. The programmable interface provides the transfer of binary 

codes of the number of pulses 𝑁[00. . .15] from the parallel 
outputs of the binary counter CT2 to the accumulator of the 
MPS microprocessor system, the main components of which 
are the MCU microcontroller, RAM, and permanent ROM 
memory. The exchange of measurement information 
between the programmable interface and the microprocessor 
system accumulator is implemented in program mode, 
interrupt mode or direct memory access. 

7. An array of measurement information about the transient 
characteristic of the measurement object is accumulated and 
memorized in the operational RAM of the MPS in the form 
of binary codes N, proportional to the instantaneous values 

of the periods 𝑇𝑥  of the frequency 𝑓𝑥 from the output of the 
encoder E. 

8. To present the measured information in angular velocity 
values, the conversion equation for this type of non-electric 

quantity measuring channel is obtained by substituting the 𝑓𝑥 
value from equation (1) into (2), which unambiguously links 

the input angular velocity 𝑛𝑋 with the output value, the 
number of pulses N on digital outputs of binary counter CT2 

𝑁 =
60 ⋅ 𝑓0

𝑛𝑥  𝑧
 . (3) 

9. From the conversion equation (3), the array of instantaneous 
angular velocity values is calculated by software 

𝑛𝑥 =
60 ⋅ 𝑓0

𝑁 𝑧
 . (4) 

The imperfection of the given approach is explained by the 
shortcomings [13]-[17] inherent in digital measuring devices, the 
circuitry of which is implemented according to a "hard" control 
scheme. 

2. MEASURING CHANNELS WITH MICROPROCESSOR 
CONTROL 

The duality of the hardware and software implementation of 
microprocessor devices (Figure 2) potentially provides greater 
flexibility and adaptability to the dynamic properties of 
measurement objects [10], [16], [18], which significantly expands 
their functional capabilities and improves static and dynamic 
metrological characteristics. 

The disadvantage of this development direction of 

microprocessor tachometers is the quantization of only even 𝑇𝑥  
or only odd periods. In the above "quasi-instantaneous" means, 
measurement information is obtained not in each period, but 
after a period, which does not allow for the transient 

characteristic 𝑛(𝑡) with the required accuracy: 
- determine the numerical values of the amplitude and duration 

of synchronous dips in angular velocity; 
- to differentiate the experimentally obtained numerical values 

of the periods to construct the functional dependence of the 
change in acceleration over time; 

- to indirectly determine the value of the dynamic moment and 
dynamic mechanical characteristic - the phase "portrait" of 
the object of measurement.  

The solution to the problems highlighted above is the 
implementation of quantization [10], [12], [19]-[21] of each 

even 𝑇𝑥  and each odd periods (Figure 3), which are 
proportional to the instantaneous value of the angular velocity, 
which is implemented by the hardware and software of the 
microcontroller. 

It is advisable to implement the quantization method in 
"adjacent" intervals on two internal timers of the 
microcontroller, and the analysis of metrological characteristics 
will allow to determine the parameter, knowledge of which will 
ensure its adaptation to the dynamic properties of the 
measurement object. 

3. MAIN METROLOGICAL CHARACTERISTICS 

For further research, as an initial mathematical model of the 
angular velocity of electric machines, we will use a trivial 
(Figure 4) exponential model  

𝑛(𝑡) =  Ω ∙ (1 − e
 − 

𝑡
𝜏), (5) 

where Ω – synchronous speed of the electric motor rotation, 𝜏 – 
electromechanical constant. 

As a result of the interaction of the hardware and software of 

the microcontroller, the analogue value 𝑛 is replaced, which has 

 

Figure 2. Generalized structural diagram of the microprocessor tachometer.  

 

Figure 3. To the issue of quantization in "adjacent" intervals.  



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 3 

an infinite number of values in the specified measurement range 

(from 𝑛min to 𝑛max), due to the limited number of its 
instantaneous values (because 𝑇Д ≠ 0), a discretization error 
occurs [22] 

𝛥Д(𝑡) =
1

2
𝑇Д ⋅

d𝑛

d𝑡
 (6) 

Thus, obtain a mathematical model for estimating the 
sampling error. The angular acceleration of the shaft movement 
of the object of measurement in the transient mode of its 
operation is determined as 

ε(𝑡) =
d𝑛

d𝑡
=

Ω ⋅ e
− 

𝑡
𝜏

𝜏
 

(7) 

The sampling step for a digital tachometer with 
microprocessor control is determined as follows [10] 

𝑇Д = 𝑡ADC + 𝑡FL + 𝑡DR . 

Here, 𝑡ADC is the duration of the analog-to-digital conversion, 
which is equal to the measured period 𝑡ADC = 𝑇𝑋 , 𝑡FL is the time 
required to execute the Flag subroutine waiting for the flag, 𝑡DR 
is the time to execute the Driver software driver. 

Considering the fact that during the measurement of the 
angular velocity in the "adjacent" intervals, the waiting 
subroutines for the flag and the Driver software driver are 

executed after the completion of 𝑇𝑥  quantization, then 

𝑇Д = 𝑡ADC = 𝑇𝑥  . (8) 

In this regard, the sampling frequency [23] of the angular 
velocity measuring channel is defined as 

𝑓Д =
1

𝑇Д
=

1

𝑇𝑋
 . (9) 

Now present the encoder conversion equation (1) in the 
following form: 

𝑓Д =
Ω ∙ 𝑧

60
 (10) 

and the discretization step, respectively 

𝑇Д(𝑡) =
 60 

𝛺 ∙ 𝑧
=  

60

𝛺 ∙ (1 − e
− 

𝑡
𝜏) ∙ 𝑧

 . 
(11) 

Substitute (7), (8), (11) into (6) and obtain a mathematical 
model for estimating (Figure 5) the sampling error of the digital 
angular velocity measurement channel in "adjacent" intervals 

𝛥Д(𝑡) =
1

2
𝑇Д ⋅

d𝜔

d𝑡
=  

30  e
− 

𝑡
𝜏

(1 − e
 − 

𝑡
𝜏) ∙ 𝑧  𝜏

 (12) 

To analyse the relative error of quantization, we will first 
obtain the transfer function of the microprocessor tachometer 

𝑁(𝑡) =  
𝑓0
𝑓𝑥

=  
60 ∙ 𝑓0

𝛺 ∙ (1 − e
− 

𝑡
𝜏)  𝑧

 
(13) 

Considering (13), the mathematical model for estimating the 
quantization error in the transient mode of operation of the 
measurement object will have the form 

𝛿К(𝑡) =  
1

𝑁
∙ 100% =  

5 ∙ 𝛺 ∙ (1 − e
−

𝑡
𝜏) ∙ 𝑧

3 ∙ 𝑓0
. 

(14) 

Analysis of the quantization error equation (Figure 6) shows 
that its reduction can be ensured by two methods: 

1. Reducing resolution z of the encoder; 

2. Increasing the quantization frequency 𝑓0 of the quartz 
resonator G. 

The second approach has the limitation 𝑓0 ≤ 𝑓𝑔𝑟 . The 

quantization frequency is limited by the limit frequency of the 
crystal resonator of the microcontroller MCU. 

In turn, decreasing in the resolution z of the encoder leads to 
an increasing in the sampling error (Figure 6), which is not 
acceptable for dynamic measurements. 

The following conclusion can be drawn from the above 
graphic dependences of quantization and discretization errors: 

Quantization and discretization errors significantly depend on 
the resolution value z of the encoder. Moreover, increasing z 
leads to decreasing in the sampling error, but the relative 
quantization error increases. To reconcile these component 
errors, it is necessary to obtain the laws of change of the 
encoder’s resolution z, adapted to the law of angular velocity 
change in time. 

Using (12), we obtain the law of change in the resolution of 
the encoder in the process of increasing the angular velocity from 

0 to Ω, compliance with which will ensure the value of the 
dynamic sampling error, which does not exceed the normalized 

value ∆Д≤ ∆ДН 

ZД(t) =  
30 e

−
𝑡
𝜏

∆ДН  𝜏 ∙ (1 −  e
−

𝑡
𝜏)

. (15) 

 

Figure 4. Dependence 𝑛 = (𝑡): 𝜏 = 0.5;  𝛺 = 1500 rpm.  

 

Figure 5. Laws of encoder resolution changing.  



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 4 

Similarly, from (14) we will get the law of the encoder 
resolution changing, which implementation will ensure the 

normalization 𝛿К ≤ 𝛿КН of the quantization error 

𝑍К(𝑡) =  
3 ∙  𝛿КН ∙ 𝑓0

5 ∙ 𝛺 (1 −  e
−

𝑡
𝜏)

 . 
(16) 

Graphic dependences of the change in encoder resolution 

𝑧 = 𝑓(𝑡) during dynamic measurements of angular velocity 𝑛 =
𝑓(𝑡) during the transient process of the measurement object are 
shown in Figure 5. 

To present the quality of measurement results based on the 
concept of measurement uncertainty, which is currently 
recommended by international standards [24]-[28], the following 
method of calculating the standard uncertainty of type B, which 

arises due to the existence of the discredit error 𝑢Bd(𝑡), is 
proposed in the assumption on the normal distribution law of 
the components of the digital angular velocity measurement 
channel 

𝑢Bd(𝑡) =
𝛥Д(𝑡)

𝑘p
=

1

2
𝑇Д

d𝜔

d𝑡
𝑘p

−1, (17) 

where 𝑘p is the coverage coefficient, which for a normal 
distribution law is taken as equal to 1.96 at a confidence level of 
p=0.95 [29], [30]; 

𝛥Д(𝑡) is the discrediting error of the digital angular velocity 
measurement channel. 

Substituting the maximum value of the sampling error ∆ДН ≤

0.2 rpm (Figure 6, b) into equation (17), we obtain the standard 
sampling uncertainty of type B, which is equal to 𝑢Bd(𝑡) =
0.1 rpm for a confidence level of 0.95. 

The relative standard uncertainty of discretization can be 
estimated by the expression [29], [31] 

𝑢�̃� =
𝑢Bd(𝑡)

𝑛(𝑡)
100 % =

0.1

1500
100 % ≈ 0.01 % . (18) 

As follows from the conducted studies of the quantization 
error (Figure 6, a), its value does not exceed 0.5 % at an angular 
speed of 1500 rpm. The standard uncertainty of type B, due to 
the presence of the quantization error [32], assuming its uniform 
distribution law, is calculated using the expression 

𝑢𝐵𝑘 (𝑡) =
𝛿𝑘 (𝑡)

100 % √12
 𝑛max(𝑡) =

0.5 %

346 %
1500 rpm

= 2.17 rpm . 

(19) 

For the given object of measurement (for example, a three-
phase asynchronous motor UAD-34 with a nominal rotation 

speed Ω = 1500 rpm and 𝜏 = 0.5, a coupling of the membrane 
type, an encoder LIR-120A (z=65536) and a debugging a board 
based on a dual-core 32-bit microcontroller TMS320F28379D, 
containing 1 MB of flash memory, 128 kB of RAM and having a 

sample frequency 𝑓0=8 MHz formed from the clock frequency 
of a quartz resonator, a method of normalizing the error [22] of 

the 𝑍Д(𝑡) discretization and a method of normalization is 

proposed [33], [34] quantization errors 𝑍K(𝑡) by determining the 
change in resolution 𝑧 of the encoder, which corresponds to the 
change in the time coordinate 𝑡 in the transient process of 
measurement object: 

𝑍Д(𝑡) =  
300 e

− 
2 𝑡
𝜏

(1 −  e
− 

2 𝑡
𝜏 )

 (20) 

𝑍К(𝑡) =  
1600

(1 −  e
− 

2 𝑡
𝜏 )

. 
(21) 

A microcontroller such as the TMS320F28379D has twelve 
32-bit general-purpose timers in its structure, which is quite 
enough to carry out this kind of dynamic measurements with the 
adaptation of the z resolution of the encoder to the dynamic 
properties of the measurement object: 

- timer 𝑇0 is programmed to the mode of the real-time 
counter, which every 0.01 s generates a control signal at its 
output, according to which the value of the binary code of the 

coefficient k of its list is recorded in the counter of timer 𝑇1; 
- every 0.01 s, the value of the list coefficient k is recorded in 

timer 𝑇1 to form at its output a frequency signal 
𝑓𝑋

𝑘
⁄ , 

proportional to the resolution 𝑍Д(𝑡) or 𝑍К(𝑡) according to (20) 
or (21); 

- in timer 𝑇2, even 𝑇𝑥  periods of the frequency signal 
𝑓𝑋

𝑘
⁄  

from the direct output of timer 𝑇1 are quantized. 

- in timer 𝑇3, odd 𝑇𝑥  periods of the frequency signal 
𝑓𝑋

𝑘
⁄  

from the inverse output of timer 𝑇1 are quantized. 
Therefore, the method of quantization in "adjacent" intervals 

takes place in two timers 𝑇2 and 𝑇3 of the microcontroller. 

Quantization of 𝑇𝑥  and 𝑇𝑥  periods occurs in both timers. First, 
as a result of counting the periods 𝑇0 of the sample frequency of 
the quartz resonator 𝑓0 from the leading edge to the trailing edge 
of the even period 𝑇𝑥  in the second 𝑇2 timer, and then from the 

leading edge to the trailing edge of the odd period 𝑇𝑥  in the third 
𝑇3 timer. Thus, each of the programmable timers of the 
microcontroller works in two modes: 

- Quantization (from rising edge to falling edge); 
- Transfer and memorization of measurement information 

(from falling edge to rising edge); 
- Additional speed of the proposed method is provided by the 

transfer and recording of measured information to the RAM in 
DMA mode (direct access to memory) without the participation 
of the computing core, freeing its resources for other, priority 
tasks during the measurement process. 

This hardware and software implementation of quantization 
in " adjacent " intervals ensures maximum speed of measurement 
and guarantees sufficient time for transferring and memorizing a 

 

Figure 6. To the issue of quantization and discretization error.  



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 5 

large volume of quantization results in the MPS RAM. And the 
implementation of the method of error normalization [22] of 

discretization 𝑍Д(𝑡) or the quantization error [23], [34] 𝑍К(𝑡) by 
changing the resolution z of the encoder in the real-time 
measuring mode of operation allows to increase the accuracy of 
dynamic angular velocity measurements. 

4. CONCLUSIONS 

The analysis of well-known digital angular velocity 
measurement channels with an encoder made it possible to solve 
two extremely important tasks for metrology - increasing the 
accuracy and speed of dynamic measurements of the angular 
velocity of the measurement object operating in real time. 

For the first time, an equation for estimating the quantization 
and discretization error was obtained for an exponential 
mathematical model that describes the transient process of the 
electric machines operation. The components of the 
mathematical model of these dynamic errors are the quantization 
and discretization steps and the derivative, which characterizes 
the rate of change of the measured quantity over time. 

It is proved that quantization and discretization errors 
significantly depend on the value of encoder resolution z. 
Moreover, an increasing in z leads to a decreasing in the sampling 
error, but the relative quantization error increases. In order to 
reconcile these component errors, the laws of changing the 
resolution z of the encoder were obtained, which make it 
possible to adapt to the dynamic properties of the change in 
angular velocity over time 

For the selected object of measurement (for example, a three-
phase asynchronous machine UAD-34 with a nominal speed of 

rotation Ω = 1500 𝑟𝑝𝑚 and τ = 0.5), a coupling coupling of 
the membrane type, an encoder LIR-120 (z=65536) and dual-
core 32-bit microcontroller TMS320F28379D, methods of 

normalizing the value of the sampling error ∆ДН and the 

quantization error 𝛿КН are proposed. The normalization of these 
error components is carried out in the process of dynamic 
measurements of the angular velocity n during the transient 
process of the measurement object, changing the resolution z of 

the encoder, thus ensuring the fulfilment of the condition: ∆Д≤

∆ДН or 𝛿К ≤ 𝛿КН 
It was established that in order to ensure the maximum speed 

of the angular velocity measuring during the transient process of 
the measurement object, it is advisable to implement the 
proposed adaptive to its dynamic properties method of changing 
the resolution of encoder on the internal timers of the 
microcontroller, and to carry out the quantization of informative 
periods proportional to the measured angular velocity in 
"adjacent intervals". 

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