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Building of mathematical models of data transferring
through the interface RS-485 in industrial networks
Jelena Chaiko (Dr.sc.ing., Riga Technical University), Nadezhda Kunicina (Dr.sc.ing., Riga Technical University), Antons Patlins (M.sc.ing.,Riga Technical University) Alina Galkina (M.sc.ing., Riga Technical University).
Abstract – The mathematical model of data transfer with the use of the pulse response on interface RS-485 is developed in the paper. It allows defining frequency transfer function of the cable used at construction of network segment, and the pulse response for each segment.
Keywords – industrial network, mathematical model, transferring signal, receiving signal.
One of the most developing areas of a modern computer engineering is a microcontroller technique. Modern microcontrollers and their devices are the basis for so called “systems of small automation”, which are commonly used in technique- and instrument making industry, for an automation of a complicated industrial and common equipment. These systems are used as a managing apparatus a during the development and construction of new models of technique and automats. Due to the development of a microprocessor and microcontroller technique it became possible to develop some distributed managing systems. The distributed managing systems for data exchange mainly use local computer networks. The specialized local networks used in small automation systems also include some specific characteristics which are connected with using more simplified functional algorithms, high safety and productivity, low value, easy installation, regulation and maintenance [1,2]. For making effective a local computer network it is necessary to make some analysis of the future network segments. For making such analysis it is necessary to have some corresponding mathematical model. The complicity of the analysis is that such mathematical model isn’t described in literature. There is available data about the cable parameters used for network construction [3,4], as well as data exchange interface parameters in the network [2,5]. Thus summarizing and analyzing the available data about the transferring channels, an exchange interface, data transferring protocols, and as a result building of mathematical model of network segments are very important researching tasks.
When transferring data in the channel with noise it is possible to emphasize two cases.
The first case, when the transferring signal always makes the same receiving signal, i.e. received signal – it is defined (some) function from the receiving signal, this effect is called as a corruption. The second case, when the function has got an inverse view, none of transferred signals doesn’t coincide with the received signal, then the corruption might be corrected, making the inversion of received signal.
More interesting is the case, when the transferring signal is not always changed. In this case we can assume that received signal E is a function of transferred signal S and noise N:
Noise can be a stochastic process . There is shown that transferred signal can be described as in formula (1):
brx = S (b tx) + N(b tx) (2)
Where operation “+” is the operation of signal combining,
brx – received data bit;
b tx – transferred data bit.
Function S – data transferring data in communication channel;
Function N – signal corruption function by noise in communication channel.
So, if to define functions S, N and operation “+”, it is possible to get a mathematical model of data transferring.
Data transferring in communication channel (S function) is a signal passing through the linear system. Dynamical characteristics of the linear system with constant parameters can be described by weight function h(t), sometimes it is called impulse transition function, that is an impulse reaction in some time interval which has come to system input in m time interval up to this moment. The weight function is very useful for describing such systems, taking into account the following option. For random input signal x(t) the system output y(t) is defined by the convolution integral :
i.e. output signal value y(t) is suspended linear (infinite) sum for all input signal realization x(t).
If delta-function  is a single impulse function,
then a response of function (t) is (t).
The impulse transition function h(t) is defined by (t) in the following way :
where t – function impulse duration (t).
In (5) we can be seen that at relatively small t ,h(t). And equation (3) can see in :
Thus, if to define the function (t), we get a mathematical model of data channel, which is easily represented for writing a modeling program of data transferring in network segment (communication channel). Let’s suppose that the channel is a screened wire of 4 twisted pairs, class 5. It is possible to represent the channel through the analog scheme shown in figure 1. The electrical characteristics of the twisted pair as an ordinary system of electromagnetic oscillation are characterized with resistance R, conductor inductivity L, capacity C and insulation conductivity G. Values R and G define heat losses in copper and dielectrics. L and C define the system reactivity, or otherwise its frequency characteristics. It is necessary to emphasize that the screen usage helps to increase the volume on approx. 30% that decrease the operational characteristics of this cable .
Fig. 1. Analog scheme of twisted pair class 5 and frequency dependence of electrical characteristics of the twisted pair.
According to the analog scheme shown in figure 1 it is easy to get a mathematical dependence of input and output, this dependence is shown :
Solution of equations for voltage and current at the optional point of x line has got :
Where Zв() – cable’s complex wave resistance, – line distribution coefficient (constant):
= + j= (R jL)(G jC) (10)
As it is seen in formulas (7)-(10) the modeling and model construction using the analog scheme is a very complicated process. First, it is difficult to build an analog scheme in concordance with the real system; second, solution of differential equations in concordance with the analog scheme is also a very complicated process. Thus it is necessary to build a model using the impulse response.
We got the dependence experimentally, now we’re comparing the received results. Making response digitization and its transformation using Fast Fourier Transform (FFT) , we get function h(t), and the model for no-load twisted pair (communication channel without transmitters and receivers).
Impulse response is shown in figure 2.
Then we quantize an impulse response, build function h(t), using FFT. Mathematical pack MatCad 9.0 in calculations has been used there.
Fig. 2. Cable impulse response.
The impulse response also can be found using Fourier inversion of the frequency transition function of cable:
Knowing the impulse response it is possible to build a channel model using formulas (3) or (6).
Then we get a mathematical model for no-load channel.
Then we are building a model with receivers and transmitters of EIA RS-422A/RS-485 standard, using the above mentioned principle.
The interface RS-485 is stable to in-phase disturbances, and therefore it is more effectively used in building industrial communication systems. In according to the standard EIA RS-422A/RS-485 an analog scheme of communication system at in-phase disturbances is shown in figure 3.
Fig. 3. Analog scheme of communication system at in-phase disturbances.
The stability of communication systems to electromagnetic disturbances appeared in the result of parasitic inductive or capacitive communications of disturbance resources with exchange environment, it is partly defined by asymmetry level (or misbalance) of distributed and concentrated parameters of communication line in relation to the land.
Intensity of disturbances between two cable conductors is defined by the level of a full impedance asymmetry in relation to the land, if to suggest that the disturbance source has got the same parasite connection to each of the conductors .
Therefore, if we know the state of direct and inverse channel at receiver’s input, then according to the standard EIA RS-422A/RS-485 it is possible to define the output state. Knowing the communication channel value which includes some receiving transmitters, it is possible to make modelling of the built network using the interface EIA RS-422A/RS-485.
We are building a simple model which includes from 1 to 3 receivers and from 1 to 3 transmitters, the distance between them is 180 m. We are measuring the impulse response on both wires of the twisted pair (figures 4-6).
Fig. 4. Cable’s impulse response with one receiver and one transmitter at distance 180 m.
Fig. 5. Cable’s impulse response with two receivers and transmitters at the distance 180 m.
Fig. 6. Cable’s impulse response with three receivers and transmitter at the distance 180 m.
It is possible to see in oscillograms shown in figures 4-6 that impulse responses of segments with 1, 2, 3 receivers and the channel without receivers are the same. Therefore in the first approximation there can be used the channel model as the segment’s model.
The recommendations on building the mathematical model of the segment using the impulse response in the above mentioned analysis are given. The spectres of the impulse responses without receivers and transmitters as well as with both of them heve been analysed and compared in the article.
On the basis of the made analysis of the spectres of the impulse responses it is possible to suppose the following steps at building and analyzing the segments of command-information network that used the interface of EIA RS-422A/RS-485 standard and contained no more than 3 components:
She is a member COST Antenna Systems & Sensors for Information Society Technologies ASSIST (IC0603).
She is a member COST Action IC0902 „Cognitive Radio and Networking for Cooperative Coexistence of Heterogeneous Wireless Networks"
She is a member of IEEE.
Dr.sc.ing., senior researcher, Yelena Chaiko
Business Address: Riga Technical University, 1, Kalku Latvia, LV 1658.
Private Phone:+371 22046003
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