**Meet the editor**

Dr S. C. Chakravarty did his Ph.D degree in space physics from the Physical Research Laboratory (PRL) , Ahmedabad, India and joined the Indian Space Research Organisation (ISRO) at Bangalore during 1973. He assumed various responsibilities to plan, organise and implement the coordinated atmospheric and space science research programmes sponsored by ISRO. As the Programme Di-

rector of Space Science research activities, Dr Chakravarty was involved in the planning and coordination of work related to India's first moon mission, Chandrayaan-1. He also coordinated large number of scientific research programmes involving scientific balloon and sounding rocket experiments, space science satellite missions including Astrosat, SROSS-C, and Megha-Tropiques. Dr Chakravarty also helped to enable Indian participation in internationally coordinated programmes like MAP, INDOEX, CAWSES etc. Presently his research activities include development of physics based and real time models on monsoon dynamics, the total electron content (TEC) of the ionosphere and aspects of mesospheric phenomena.

Contents

**Preface IX**

Chapter 1 **Analysis of Power Electronic**

Chapter 2 **Study of Inductive-Capacitive**

Chapter 3 **Fixed Transmission Media 41**  Rastislav Róka

Chapter 4 **Co-Simulation Procedure for**

Chapter 5 **S-Function Library for**

B. Umesh Rai

Chapter 6 **From Control Design to**

Marcus Müller,

Titu Niculescu

Juan Segundo-Ramirez and A. Medina

**PID and Fuzzy Logic Active Controls**

**Bond Graph Modeling 97**

**FPGA Implementation 129**

Chapter 7 **Describing Function Recording with Simulink and MATLAB 149**

Chapter 8 **Performance Evaluation of a Temperature**

**Controllers in Electric Power Systems Using Simulink 1**

**Series Circuits Using the Simulink Software Package 19**

**Strategies Applied to a Sprayers Boom Suspension 69**  Cristiano Okada Pontelli and Mario Francisco Mucheroni

Hans-Christian Schwannecke and Wolfgang Fengler

Krunoslav Horvat, Ognjen Kuljaca and Tomislav Sijak

**Sensor 3D Electro-Thermal Model Through Simulink® 167**

**Control Stage Used on a Semiconductor Gas**

E.N. Vázquez-Acosta, S. Mendoza-Acevedo, M.A. Reyes-Barranca, L.M. Flores-Nava, J.A. Moreno-Cadenas and J.L. González-Vidal

## Contents

## **Preface XI**

M.A. Reyes-Barranca, L.M. Flores-Nava,

J.A. Moreno-Cadenas and J.L. González-Vidal

X Contents

## Preface

Building on MATLAB (the language of technical computing), Simulink provides a platform for engineers to plan, model, design, simulate, test and implement complex electromechanical, dynamic control, signal processing and communication systems. Simulink-Matlab combination is very useful for developing algorithms, GUI assisted creation of block diagrams and realisation of interactive simulation based designs. The eleven chapters of the book demonstrate the power and capabilities of Simulink to solve engineering problems with varied degree of complexity in the virtual environment. The first four chapters provide details of design, analysis, simulation and implementation of flexible AC transmission system devices to control the power variables, multi-component electric circuits, signal transmission through power distribution lines of different kinds and PID/fuzzy logic control of a boom suspension. Details of enhanced capabilities in simulation effects constitute the subject matter of the next three chapters on S-functions for custom made subroutines and bond graph models, development of model and HDL code on FPGA system and harmonic linearization method of non-linear characteristics of modulation techniques. The last four chapters deal with examples of electromagnetic and dynamic systems of thermoelectrical micro hot plate with temperature control circuit, braked railed vehicle wheel slide protection, replacement in a car battery by a hybrid supply of battery and super capacitors and a novel speed control logic for electric vehicle. I hope that both newcomers and professionals in the field would benefit by going through the details given in the above chapters of the book.

**S.C. Chakravarty**

Scientist (retired), Indian Space Research Organisation (ISRO), Bangalore or Hon Senior Professor, Indian Centre for Space Physics (ICSP), Kolkata, India

**1**

*México*

**Analysis of Power Electronic Controllers in**

Flexible ac Transmission Systems (FACTS) devices have emerged in power system because of the development of power electronics components for high voltage and power. The FACTS devices provide higher controllability in power systems by means of power electronic devices. Several FACTS equipments have been already introduced for various applications worldwide, and new types of FACTS are in the stage of being introduced in practice. FACTS technology provides a better ability to varying operational conditions and

It can be seen that with growing line length and with higher power demand the opportunity for FACTS devices gets more important. The devices work electrically as fast current, voltage or impedance controllers. The power electronic allows very short reaction times down to far below one second (~ms). Detailed introductions in FACTS devices can also be found in the literature (Hingorani & Gyudyi, 2000) (Acha, et al. 2004) (Mathur & Varma, 2002)(Padiyar, 2007) (Zhang, Rehtanz, & Pal, 2006) with the main focus on new topologies,

Basically, there are two groups of FACTS, one is based on thyristor valve operation (Hingorani & Gyudyi, 2000), and the other is based on Voltage Source Converters (VSCs) (Segundo-Ramírez & Medina, 2008) (Segundo-Ramírez & Medina, 2009). A list of some FACTS devices are shown in Table 1. The FACTS devices based on VSCs provide a controllable voltage magnitude and phase angle due to a Pulse Width Modulation (PWM) technique (Mohan, Underland, & Robins, 1995). The Static Compensator (STATCOM) (Hingorani & Gyudyi, 2000) is a shunt connected device that is able to provide reactive power support at a network location far away from the generators. Through this reactive power injection, the STATCOM can regulate the voltage at the connection node. The Static Synchronous Series compensator (SSSC) (Hingorani & Gyudyi, 2000) is a series device which injects a voltage in series with the transmission line. The Unified Power Flow Controller (UPFC) (Hingorani & Gyudyi, 2000) is the most versatile device of the family of FACTS devices, since it is able to control the active and the reactive power, respectively, as

**1. Introduction**

modeling and control.

improves the usage of existing installations.

**2. Power electronic application in transmission systems**

**Electric Power Systems Using Simulink**

Juan Segundo-Ramirez1 and A. Medina2 *1Universidad Autónoma de San Luis Potosí*

*2Universidad Michoacana de San Nicolás de Hidalgo*

## **Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink**

Juan Segundo-Ramirez1 and A. Medina2

*1Universidad Autónoma de San Luis Potosí 2Universidad Michoacana de San Nicolás de Hidalgo México*

## **1. Introduction**

Flexible ac Transmission Systems (FACTS) devices have emerged in power system because of the development of power electronics components for high voltage and power. The FACTS devices provide higher controllability in power systems by means of power electronic devices. Several FACTS equipments have been already introduced for various applications worldwide, and new types of FACTS are in the stage of being introduced in practice. FACTS technology provides a better ability to varying operational conditions and improves the usage of existing installations.

## **2. Power electronic application in transmission systems**

It can be seen that with growing line length and with higher power demand the opportunity for FACTS devices gets more important. The devices work electrically as fast current, voltage or impedance controllers. The power electronic allows very short reaction times down to far below one second (~ms). Detailed introductions in FACTS devices can also be found in the literature (Hingorani & Gyudyi, 2000) (Acha, et al. 2004) (Mathur & Varma, 2002)(Padiyar, 2007) (Zhang, Rehtanz, & Pal, 2006) with the main focus on new topologies, modeling and control.

Basically, there are two groups of FACTS, one is based on thyristor valve operation (Hingorani & Gyudyi, 2000), and the other is based on Voltage Source Converters (VSCs) (Segundo-Ramírez & Medina, 2008) (Segundo-Ramírez & Medina, 2009). A list of some FACTS devices are shown in Table 1. The FACTS devices based on VSCs provide a controllable voltage magnitude and phase angle due to a Pulse Width Modulation (PWM) technique (Mohan, Underland, & Robins, 1995). The Static Compensator (STATCOM) (Hingorani & Gyudyi, 2000) is a shunt connected device that is able to provide reactive power support at a network location far away from the generators. Through this reactive power injection, the STATCOM can regulate the voltage at the connection node. The Static Synchronous Series compensator (SSSC) (Hingorani & Gyudyi, 2000) is a series device which injects a voltage in series with the transmission line. The Unified Power Flow Controller (UPFC) (Hingorani & Gyudyi, 2000) is the most versatile device of the family of FACTS devices, since it is able to control the active and the reactive power, respectively, as

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 3

The SSSC is a series device in which a synchronous voltage source injects a fundamental frequency voltage in series with the transmission line through a coupling transformer. The synchronous voltage source is supplied by a VSC. A schematic representation of the SSSC is

Ideally, the injected voltage is in quadrature with the line current. In this mode the VSC does not absorb or inject any real power. However, in practice, the VSC losses must be replenished by the ac system, in consequence a small phase lag is introduced for this purpose. The operating characteristics make this device very attractive for power transmission application. The main limitation of application is due to the losses and cost of the converter. The SSSC is a device which has so far not been built at transmission level because Series Compensator (fixed capacitor) and thyristor controlled series capacitor (TCSC) (Hingorani & Gyudyi, 2000) are fulfilling all the today's operational requirements at

This device contains two VSCs connected together through a dc link storage capacitor. One of the VSCs is connected in series with the transmission line, while the other VSC is connected in shunt with the transmission line. The UPFC can control the active and reactive power flow in the transmission line, and at the same time can regulate the voltage magnitude at the connection node. To control the real and reactive power flow in the series side, the UPFC allows interchange of real power between the shunt and the series converters. The main disadvantage of this device is the high cost level due to the complex systems setup (Zhang, Rehtanz, & Pal, 2006). A schematic representation of the UPFC is

Power System Blockset (PSB) for use with Matlab/Simulink employs state-variable analysis. In PSB complex control algorithms can be implemented into the models in an easy and fast way. Besides, PSB can use several Matlab toolboxes. In particular, it has the PWM generator and the VSC blocks. The main advantage of the PSB is that it is developed in Matlab/Simulink environment, this fact makes possible to use it together with several other control design tools. It is possible to use the Simulink Accelerator and the Real-Time Workshop to improve the PSB performance; a C code is generated. Additionally, the PSB can use several integration methods, which make it a powerful

This section is divided into three parts. Simulations relating to the STATCOM are presented first. This is followed by simulations carried out for the SSSC and then for the UPFC. The study case related to the STATCOM is described in detail using PSB/Simulink. For the cases

**2.2 Static synchronous series compensator (SSSC)**

**2.3 Unified power flow controller (UPFC)**

**3.1 Modeling and analysis of FACTS by PSB/SIMULINK**

of the SSSC and the UPFC, only the general implementation is described.

shown in Figure 1(b).

a low cost.

shown in Figure 1(c).

**3. PSB/SIMULINK**

simulation tool.

well as the voltage at the connection node. In Figure 1 a schematic representation of the STATCOM, the SSSC, and the UPFC are presented. The compensating FACTS devices under analysis in this chapter are briefly described below.

Table 1. Overview of the principal FACTS-Devices.

Fig. 1. FACTS devices based on VSCs. (a) STATCOM, (b) SSSC, and (c) UPFC.

## **2.1 Static compensator (STATCOM)**

It is a shunt device that does not require passive elements for reactive compensation. The STATCOM operation is based on a VSC, which is supplied by a dc storage capacitor. The VSC terminals are connected to the dc system through a coupling transformer. The VSC produces a quasi-sine wave voltage at the fundamental frequency (50 or 60 Hz). The STATCOM can generate or absorb reactive power. A schematic representation of the STATCOM is shown in Figure 1(a).

Assuming that the losses in the VSC and the coupling transformer are negligible, *vstatcom* is in phase with the voltage at the terminal bus *vk*. In this situation, the current *istatcom* is completely reactive. If the magnitude of the voltage *vk* is higher than the magnitude of *vstatcom*, the reactive current flows from the bus to the VSC, which means that the STATCOM absorbs reactive power. On the other hand, if the magnitude of *vstatcom* is higher than the magnitude of *vk*, the reactive current flows from the VSC to the ac system. Then, the STATCOM injects reactive power to the system. In practice, the power losses of the STATCOM are not negligible and must be drawn from the ac system to maintain constant the dc capacitor voltage.

## **2.2 Static synchronous series compensator (SSSC)**

The SSSC is a series device in which a synchronous voltage source injects a fundamental frequency voltage in series with the transmission line through a coupling transformer. The synchronous voltage source is supplied by a VSC. A schematic representation of the SSSC is shown in Figure 1(b).

Ideally, the injected voltage is in quadrature with the line current. In this mode the VSC does not absorb or inject any real power. However, in practice, the VSC losses must be replenished by the ac system, in consequence a small phase lag is introduced for this purpose. The operating characteristics make this device very attractive for power transmission application. The main limitation of application is due to the losses and cost of the converter. The SSSC is a device which has so far not been built at transmission level because Series Compensator (fixed capacitor) and thyristor controlled series capacitor (TCSC) (Hingorani & Gyudyi, 2000) are fulfilling all the today's operational requirements at a low cost.

## **2.3 Unified power flow controller (UPFC)**

This device contains two VSCs connected together through a dc link storage capacitor. One of the VSCs is connected in series with the transmission line, while the other VSC is connected in shunt with the transmission line. The UPFC can control the active and reactive power flow in the transmission line, and at the same time can regulate the voltage magnitude at the connection node. To control the real and reactive power flow in the series side, the UPFC allows interchange of real power between the shunt and the series converters. The main disadvantage of this device is the high cost level due to the complex systems setup (Zhang, Rehtanz, & Pal, 2006). A schematic representation of the UPFC is shown in Figure 1(c).

## **3. PSB/SIMULINK**

2 Technology and Engineering Applications of Simulink

well as the voltage at the connection node. In Figure 1 a schematic representation of the STATCOM, the SSSC, and the UPFC are presented. The compensating FACTS devices under

Shunt Connected Static Var Compensator (SVC) Static Compensator (STATCOM)

Compensator (TCSC)

FACTS Devices Thyristor-Based FACTS VSC-Based FACTS

> Static Series Synchronous Compensator (SSSC)

Unified Power Flow Controller (UPFC) HVDC VSC

analysis in this chapter are briefly described below.

Series Connected Thyristor Controlled Series

Connection Type

Table 1. Overview of the principal FACTS-Devices.

Back-to-Back Connected HVDC

**2.1 Static compensator (STATCOM)**

STATCOM is shown in Figure 1(a).

Fig. 1. FACTS devices based on VSCs. (a) STATCOM, (b) SSSC, and (c) UPFC.

must be drawn from the ac system to maintain constant the dc capacitor voltage.

It is a shunt device that does not require passive elements for reactive compensation. The STATCOM operation is based on a VSC, which is supplied by a dc storage capacitor. The VSC terminals are connected to the dc system through a coupling transformer. The VSC produces a quasi-sine wave voltage at the fundamental frequency (50 or 60 Hz). The STATCOM can generate or absorb reactive power. A schematic representation of the

Assuming that the losses in the VSC and the coupling transformer are negligible, *vstatcom* is in phase with the voltage at the terminal bus *vk*. In this situation, the current *istatcom* is completely reactive. If the magnitude of the voltage *vk* is higher than the magnitude of *vstatcom*, the reactive current flows from the bus to the VSC, which means that the STATCOM absorbs reactive power. On the other hand, if the magnitude of *vstatcom* is higher than the magnitude of *vk*, the reactive current flows from the VSC to the ac system. Then, the STATCOM injects reactive power to the system. In practice, the power losses of the STATCOM are not negligible and Power System Blockset (PSB) for use with Matlab/Simulink employs state-variable analysis. In PSB complex control algorithms can be implemented into the models in an easy and fast way. Besides, PSB can use several Matlab toolboxes. In particular, it has the PWM generator and the VSC blocks. The main advantage of the PSB is that it is developed in Matlab/Simulink environment, this fact makes possible to use it together with several other control design tools. It is possible to use the Simulink Accelerator and the Real-Time Workshop to improve the PSB performance; a C code is generated. Additionally, the PSB can use several integration methods, which make it a powerful simulation tool.

## **3.1 Modeling and analysis of FACTS by PSB/SIMULINK**

This section is divided into three parts. Simulations relating to the STATCOM are presented first. This is followed by simulations carried out for the SSSC and then for the UPFC. The study case related to the STATCOM is described in detail using PSB/Simulink. For the cases of the SSSC and the UPFC, only the general implementation is described.

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 5

peak line-to-neutral voltage at the bus 1 is shown in Figure 5(d). Figure 6 shows the steady state waveform of the voltage at the bus 1 at the top of the figure, and its harmonic spectrum at the bottom of the figure. The Figure 6 was obtained using the FFT Analysis button in the

In the Figure 7, some selected transient waveforms of the STATCOM with the passive filter are shown: Figure 7(a) presents the output of the phase controller; Figure 7(b) presents the output of the magnitude control; Figure 7(c) shows the voltage across the dc capacitor, and finally, the peak line-to-neutral voltage at the bus 1 is shown in Figure 7(d). Figure 8 shows the steady state waveform of the voltage at the bus 1 at the top of the figure, and its

(a) Control System (b) Bandpass Filter

(c) Vabc to Peak Value

Fig. 3. STATCOM PI control system. (a) Main control system. (b) Bandpass filter. (c)

Extraction of the peak value of a three-phase voltage in abc-reference.

powergui block.

harmonic spectrum at the bottom of the figure.

### **3.2 Static compensator (STATCOM)**

The test case is shown in Figure 2, where the STATCOM includes the control system described in (Mahyavanshi & Radman, 2006). The initial conditions are zero, the reference line to line voltage is 179.6292 volts at node 1, the reference voltage for the dc capacitor is 500 volts, and the modulation index is *mf =*15 (900 Hz). The shunt transformer, the line 1 and the line 2 are represented by *RL* branches, whose impedances areͲǤͲͷ ݆߱ ൈ ͲǤͲͲͷȳ, ͲǤʹ ݆߱ ൈ ͲǤͲͳͷȳ, and ͲǤͳͷ ݆߱ ൈ ͲǤͲͳȳ, respectively. The Figure 3 shows the PI control used in this analysis. On the other hand, Figure 4 shows some important blocks parameters of the system. Figure 4 shows some important block of parameters for some components shown in Figure 2 and Figure 3. Please notice that a passive filter has been connected at bus 1, in order to drain the harmonic currents.

Fig. 2. Test system including the STATCOM.

The test case is initially in periodic steady state, at ݐ ൌ ͲǤͲͷݏ, the voltage magnitude of the three-phase programmable voltage source is increased 15%, and finally at ݐ ൌ ͲǤ͵ͷݏ, the initial operating point is reestablished. The study for the STATCOM is divided in two parts, firstly the three-phase breaker is open, and consequently the passive filter is disconnected. In the second part, the three-phase breaker is closed allowing the filtering function. The passive filter is represented as a RC branch with ܥ ൌ ͺͲܨߤ and ܴ ൌ ͲǤͷȳ.

In the Figure 5, some selected transient waveforms of the STATCOM without the passive filter are shown: Figure 5(a) presents the output of the phase controller, alfaE, which is actually the phase angle of the voltage at the ac terminals of the VSC; Figure 5(b) presents the output of the magnitude control, which is the half part of the voltage magnitude at the ac terminal of the VSC; Figure 5(c) shows the voltage across the dc capacitor, and finally, the

The test case is shown in Figure 2, where the STATCOM includes the control system described in (Mahyavanshi & Radman, 2006). The initial conditions are zero, the reference line to line voltage is 179.6292 volts at node 1, the reference voltage for the dc capacitor is 500 volts, and the modulation index is *mf =*15 (900 Hz). The shunt transformer, the line 1 and the line 2 are represented by *RL* branches, whose impedances areͲǤͲͷ ݆߱ ൈ ͲǤͲͲͷȳ, ͲǤʹ ݆߱ ൈ ͲǤͲͳͷȳ, and ͲǤͳͷ ݆߱ ൈ ͲǤͲͳȳ, respectively. The Figure 3 shows the PI control used in this analysis. On the other hand, Figure 4 shows some important blocks parameters of the system. Figure 4 shows some important block of parameters for some components shown in Figure 2 and Figure 3. Please notice that a passive filter has been connected at bus

The test case is initially in periodic steady state, at ݐ ൌ ͲǤͲͷݏ, the voltage magnitude of the three-phase programmable voltage source is increased 15%, and finally at ݐ ൌ ͲǤ͵ͷݏ, the initial operating point is reestablished. The study for the STATCOM is divided in two parts, firstly the three-phase breaker is open, and consequently the passive filter is disconnected. In the second part, the three-phase breaker is closed allowing the filtering function. The

In the Figure 5, some selected transient waveforms of the STATCOM without the passive filter are shown: Figure 5(a) presents the output of the phase controller, alfaE, which is actually the phase angle of the voltage at the ac terminals of the VSC; Figure 5(b) presents the output of the magnitude control, which is the half part of the voltage magnitude at the ac terminal of the VSC; Figure 5(c) shows the voltage across the dc capacitor, and finally, the

passive filter is represented as a RC branch with ܥ ൌ ͺͲܨߤ and ܴ ൌ ͲǤͷȳ.

**3.2 Static compensator (STATCOM)**

1, in order to drain the harmonic currents.

Fig. 2. Test system including the STATCOM.

peak line-to-neutral voltage at the bus 1 is shown in Figure 5(d). Figure 6 shows the steady state waveform of the voltage at the bus 1 at the top of the figure, and its harmonic spectrum at the bottom of the figure. The Figure 6 was obtained using the FFT Analysis button in the powergui block.

In the Figure 7, some selected transient waveforms of the STATCOM with the passive filter are shown: Figure 7(a) presents the output of the phase controller; Figure 7(b) presents the output of the magnitude control; Figure 7(c) shows the voltage across the dc capacitor, and finally, the peak line-to-neutral voltage at the bus 1 is shown in Figure 7(d). Figure 8 shows the steady state waveform of the voltage at the bus 1 at the top of the figure, and its harmonic spectrum at the bottom of the figure.

Fig. 3. STATCOM PI control system. (a) Main control system. (b) Bandpass filter. (c) Extraction of the peak value of a three-phase voltage in abc-reference.

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time (s)

Fig. 5. Selected transient waveforms of the STATCOM without passive filter. (a) Phase angle of the voltage at the ac terminals of the VSC. (b) Amplitude modulation ratio. (c) Voltage

From Figure 5 and Figure 7 it is easy to notice that the passive filter has a positive impact on the reduction of the harmonic distortion in the power network without affecting the performance of the controller. In Figure 6, the waveform of the voltage at bus 1 is severely distorted with a total harmonic distortion (THD) of 39.88%. In this case, some harmonic components have a magnitude of almost 20% of the fundamental component, which is an undesirable operating condition in practical applications. On the other hand, the obtained voltage at bus 1 when the passive filter is on has a THD of 4.23% and the highest magnitude of the harmonic components is around 3.5% of the fundamental component, as shown in

In this simulation analysis, a discrete representation of the system given in Figure 2 was

used in order to carry-out the simulations, an integration step of ͳݏߤ was selected.

across the dc capacitor. (d) Peak line-to-neutral voltage at the bus 1.

Figure 8.

> 0.6 0.7 0.8

480 500 520

170

180

190

(a) alfaE

(b) mE

(c) Vdc (Volts)

(d) V1 (Volts)

peak L-n

Fig. 4. Blocks parameters of important components.

Fig. 4. Blocks parameters of important components.

Fig. 5. Selected transient waveforms of the STATCOM without passive filter. (a) Phase angle of the voltage at the ac terminals of the VSC. (b) Amplitude modulation ratio. (c) Voltage across the dc capacitor. (d) Peak line-to-neutral voltage at the bus 1.

From Figure 5 and Figure 7 it is easy to notice that the passive filter has a positive impact on the reduction of the harmonic distortion in the power network without affecting the performance of the controller. In Figure 6, the waveform of the voltage at bus 1 is severely distorted with a total harmonic distortion (THD) of 39.88%. In this case, some harmonic components have a magnitude of almost 20% of the fundamental component, which is an undesirable operating condition in practical applications. On the other hand, the obtained voltage at bus 1 when the passive filter is on has a THD of 4.23% and the highest magnitude of the harmonic components is around 3.5% of the fundamental component, as shown in Figure 8.

In this simulation analysis, a discrete representation of the system given in Figure 2 was used in order to carry-out the simulations, an integration step of ͳݏߤ was selected.

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 9

Fig. 8. Steady state waveform of the voltage at the bus 1 with passive filter (top), and its

The test system for the analysis of the SSSC is shown in Figure 9. The network parameters are the same of those of Figure 2. The series transformers are represented b an ideal model. The series active power reference Pref in the line 2 is equal to 5500 W; the dc voltage capacitor reference is equal to 500 V; the gains of the power control are Kp=0.000015 and Ki=0.004, and its limits are 0 and 1; the gains of the dc voltage control are Kp=0.03 and Ki=1,

The SSSC includes the control system shown in Figure 10. This control has two control objectives: to control the active power (P) that flows in the transmission line connected in series with the SSSC, and to maintain the dc voltage capacitor at its reference value. Remember that the SSSC does not interchange active power in steady-state, only the power loss; otherwise the dc capacitor is discharged. In transient state the SSSC has transient interchanges of active power with the network, which is reflected in variation of the dc voltage. Under this situation, the control regulates the interchange of active power in order to maintain the dc voltage capacitor at its steady-state. The initial

harmonic spectrum (bottom).

0 0.5 1 1.5 2 2.5 3 3.5

Mag (% of Fundamental)

0

100

200

and its limits are Ȃ ߨ and ߨ

**3.2.1 Static synchronous series compensator (SSSC)**

0 10 20 30 40 50 60 70 80

Harmonic order

1.184 1.186 1.188 1.19 1.192 1.194 1.196 1.198

Time (s)

Fundamental (60Hz) = 179.6 , THD= 4.23%

FFT window: 1 of 72 cycles of selected signal

Fig. 6. Steady state waveform of the voltage at the bus 1 without passive filter (top), and its harmonic spectrum (bottom).

Fig. 7. Selected transient waveforms of the STATCOM with passive filter. (a) Phase angle of the voltage at the ac terminals of the VSC. (b) Amplitude modulation ratio. (c) Voltage across the dc capacitor. (d) Peak line-to-neutral voltage at the bus 1.

1.184 1.186 1.188 1.19 1.192 1.194 1.196 1.198

Time (s)

Fundamental (60Hz) = 179.6 , THD= 39.88%

0 10 20 30 40 50 60 70 80

Harmonic order

FFT window: 1 of 72 cycles of selected signal

Fig. 6. Steady state waveform of the voltage at the bus 1 without passive filter (top), and its

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time (s)

Fig. 7. Selected transient waveforms of the STATCOM with passive filter. (a) Phase angle of the voltage at the ac terminals of the VSC. (b) Amplitude modulation ratio. (c) Voltage

across the dc capacitor. (d) Peak line-to-neutral voltage at the bus 1.

harmonic spectrum (bottom).

(a) alfaE

0.6

480

500

520

(b) mE

(c) Vdc (Volts)

(d) V1 (Volts)

peak L-n

0.8

0

5

10

Mag (% of Fundamental)

15

20

Fig. 8. Steady state waveform of the voltage at the bus 1 with passive filter (top), and its harmonic spectrum (bottom).

#### **3.2.1 Static synchronous series compensator (SSSC)**

The test system for the analysis of the SSSC is shown in Figure 9. The network parameters are the same of those of Figure 2. The series transformers are represented b an ideal model. The series active power reference Pref in the line 2 is equal to 5500 W; the dc voltage capacitor reference is equal to 500 V; the gains of the power control are Kp=0.000015 and Ki=0.004, and its limits are 0 and 1; the gains of the dc voltage control are Kp=0.03 and Ki=1, and its limits are Ȃ ߨ and ߨ

The SSSC includes the control system shown in Figure 10. This control has two control objectives: to control the active power (P) that flows in the transmission line connected in series with the SSSC, and to maintain the dc voltage capacitor at its reference value. Remember that the SSSC does not interchange active power in steady-state, only the power loss; otherwise the dc capacitor is discharged. In transient state the SSSC has transient interchanges of active power with the network, which is reflected in variation of the dc voltage. Under this situation, the control regulates the interchange of active power in order to maintain the dc voltage capacitor at its steady-state. The initial

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 11

Fig. 11. Selected transient waveforms of the SSSC without passive filter. (Top left) Series active power. (Top right) Amplitude modulation ratio. (Bottom left) Voltage across the dc

capacitor. (Bottom right) Phase angle of the voltage at the ac terminals.

Fig. 10. SSSC control.

conditions are zero; the active power reference Pref is 5500 W, and the index modulation is 41 (2.46 kHz). Pref is the active power reference in the transmission line in series with the SSSC.

The test case has zero initial conditions for all the state variables, except for the voltage across the dc capacitor; its initial condition is 500 V. At ݐ ൌ ͲǤͶݏ the voltage magnitude of the three-phase programmable voltage source is increased 15%, and finally at ݐ ൌ ͲǤݏ, the initial operating point is reestablished. In this simulation analysis, a discrete representation of the system given in Figure 2 was used in order to carry-out the simulations, and an integration step of ʹݏߤ was selected. This analysis is also carried-out with and without the passive filter.

Figure 11 shows some selected transient waveforms of the SSSC without passive filter. In this figure it is possible to notice that the series active power decreases about 25% when the voltage magnitude of the three-phase programmable voltage source increases 15%, and the series active power increases about 25% when the same voltage goes back to the original value. The control objectives of the SSSC controller are achieved in six full cycles, as shown in this figure.

Fig. 9. Test system including the SSSC.

Fig. 10. SSSC control.

conditions are zero; the active power reference Pref is 5500 W, and the index modulation is 41 (2.46 kHz). Pref is the active power reference in the transmission line in series with

The test case has zero initial conditions for all the state variables, except for the voltage across the dc capacitor; its initial condition is 500 V. At ݐ ൌ ͲǤͶݏ the voltage magnitude of the three-phase programmable voltage source is increased 15%, and finally at ݐ ൌ ͲǤݏ, the initial operating point is reestablished. In this simulation analysis, a discrete representation of the system given in Figure 2 was used in order to carry-out the simulations, and an integration step of ʹݏߤ was selected. This analysis is also carried-out with and without the

Figure 11 shows some selected transient waveforms of the SSSC without passive filter. In this figure it is possible to notice that the series active power decreases about 25% when the voltage magnitude of the three-phase programmable voltage source increases 15%, and the series active power increases about 25% when the same voltage goes back to the original value. The control objectives of the SSSC controller are achieved in six full cycles, as shown

the SSSC.

passive filter.

in this figure.

Fig. 9. Test system including the SSSC.

Fig. 11. Selected transient waveforms of the SSSC without passive filter. (Top left) Series active power. (Top right) Amplitude modulation ratio. (Bottom left) Voltage across the dc capacitor. (Bottom right) Phase angle of the voltage at the ac terminals.

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 13

Fig. 13. Selected transient waveforms of the SSSC with passive filter. (Top left) Series active

Fig. 14. Steady state waveform of the voltage at the bus 1 with passive filter (top), and its

harmonic spectrum (bottom).

0

0.1

0.2

Mag (% of Fundamental)

0.3

0.4

0.5

0

100

200

power. (Top right) Amplitude modulation ratio. (Bottom left) Voltage across the dc

capacitor. (Bottom right) Phase angle of the voltage at the ac terminals.

FFT window: 1 of 60 cycles of selected signal

0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998

Time (s)

Fundamental (60Hz) = 191.7 , THD= 0.60%

0 10 20 30 40 50 60 70 80

Harmonic order

Figure 12 shows the steady state waveform of the voltage at bus 1 without passive filter. In this operating scenario, the voltage is highly distorted; the THD is around 22.85% and the harmonics 39 and 43 have a magnitude of around 12% of the fundamental component. Notice that THD is high even when the switching frequency is 2.46 kHz. On the other hand, of some selected variables, the transient and steady state solutions of the SSSC test system including the passive filter are shown in Figure 13 and Figure 14. Comparing Figure 11 and Figure 13, it is easy to notice that difference between these figures is negligible. The most remarkable difference is the reduction of the harmonic distortion in all the system, especially in the voltage at bus 1.

Fig. 12. Steady state waveform of the voltage at the bus 1 without passive filter (top), and its harmonic spectrum (bottom).

Figure 12 shows the steady state waveform of the voltage at bus 1 without passive filter. In this operating scenario, the voltage is highly distorted; the THD is around 22.85% and the harmonics 39 and 43 have a magnitude of around 12% of the fundamental component. Notice that THD is high even when the switching frequency is 2.46 kHz. On the other hand, of some selected variables, the transient and steady state solutions of the SSSC test system including the passive filter are shown in Figure 13 and Figure 14. Comparing Figure 11 and Figure 13, it is easy to notice that difference between these figures is negligible. The most remarkable difference is the reduction of the harmonic distortion in all the system,

Fig. 12. Steady state waveform of the voltage at the bus 1 without passive filter (top), and its

especially in the voltage at bus 1.

Mag (% of Fundamental)

0

200

400

0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998

Time (s)

Fundamental (60Hz) = 186.6 , THD= 22.85%

0 20 40 60 80 100 120 140 160

Harmonic order

FFT window: 1 of 60 cycles of selected signal

harmonic spectrum (bottom).

Fig. 13. Selected transient waveforms of the SSSC with passive filter. (Top left) Series active power. (Top right) Amplitude modulation ratio. (Bottom left) Voltage across the dc capacitor. (Bottom right) Phase angle of the voltage at the ac terminals.

Fig. 14. Steady state waveform of the voltage at the bus 1 with passive filter (top), and its harmonic spectrum (bottom).

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 15

Fig. 16. UPFC and its controllers.

Fig. 17. Series controller.

## **3.2.2 Unified power flow controller (UPFC)**

The UPFC test system of Figure 15 and Figure 16 shows the Simulink implementation of the UPFC. The network parameters are the same of those of Figure 2. The UPFC including the shunt control described in (Mahyavanshi & Radman, 2006) is schematically shown in Figure 18, and the series control proposed in (Fujita, Watanabe, & Akagi, 2001) is shown in Figure 17. The initial condition is zero, except for the dc capacitor voltage, which is 500 volts.

The series controller regulates the real (*Pref=*8500 watts) and reactive (*Qref=*0 vars) power flows by adjusting the injected series voltage. The shunt converter regulates the voltage across the dc capacitor and the sending end voltage V1 at bus 1. The modulation index is *mf=*27, which represents a commutation frequency of 1.61 kHz. The RC passive filter has a capacitance of �0 �� and a resistance of 0.2 Ω. The gains for the series controller are the following: Kp=1 and Ki=500 for the Discrete PI Controller 1, with no limits; Kp=2 and Ki=1000 for the Discrete PI Controller 2, with no limits. On the other hand, the gains for the shunt controller are: Kp=0.002 and Ki=1 for the Discrete PI Controller 1, with no limits; finally, Kp=0.0012 and Ki=0.09 for the Discrete PI Controller 2, with no limits. The UPFC analysis presented in this section only includes the case when the passive filter is connected, as shown in Figure 15.

Fig. 15. Test system including the UPFC.

Fig. 16. UPFC and its controllers.

The UPFC test system of Figure 15 and Figure 16 shows the Simulink implementation of the UPFC. The network parameters are the same of those of Figure 2. The UPFC including the shunt control described in (Mahyavanshi & Radman, 2006) is schematically shown in Figure 18, and the series control proposed in (Fujita, Watanabe, & Akagi, 2001) is shown in Figure 17. The initial condition is zero, except for the dc capacitor voltage, which is 500

The series controller regulates the real (*Pref=*8500 watts) and reactive (*Qref=*0 vars) power flows by adjusting the injected series voltage. The shunt converter regulates the voltage across the dc capacitor and the sending end voltage V1 at bus 1. The modulation index is *mf=*27, which represents a commutation frequency of 1.61 kHz. The RC passive filter has a capacitance of �0 �� and a resistance of 0.2 Ω. The gains for the series controller are the following: Kp=1 and Ki=500 for the Discrete PI Controller 1, with no limits; Kp=2 and Ki=1000 for the Discrete PI Controller 2, with no limits. On the other hand, the gains for the shunt controller are: Kp=0.002 and Ki=1 for the Discrete PI Controller 1, with no limits; finally, Kp=0.0012 and Ki=0.09 for the Discrete PI Controller 2, with no limits. The UPFC analysis presented in this section only includes the case when the passive filter is connected,

**3.2.2 Unified power flow controller (UPFC)**

volts.

as shown in Figure 15.

Fig. 15. Test system including the UPFC.

Fig. 17. Series controller.

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink 17

Fig. 20. Transient waveforms of selected variables: (Left) Voltage across the dc capacitor.

Fig. 21. Steady state waveform of the voltage at the bus 1 (top) and its harmonic spectrum

(Right) Reactive (purple) and active (yellow) power.

0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998

Time (s)

Fundamental (60Hz) = 170 , THD= 1.74%

0 20 40 60 80 100

Harmonic order

FFT window: 1 of 60 cycles of selected signal

(bottom).

0

0.2

0.4

0.6

Mag (% of Fundamental)

0.8

1

1.2

0

100

200

The dynamic response of UPFC controllers are presented in Figure 19. From this figure, it is possible to observe that the variation of the three-phase programmable voltage source significantly affects the response of the series controller. This is because the magnitude of the voltages at the ends of the line controlled by the UPFC affects the power flow across the line. On the other hand, Figure 20 shows the transient waveform of the voltage across the dc capacitor and the active and reactive power flow in the controlled line. Notice that the reactive and active power remains close to the reference during the step changes in the three-phase programmable voltage source. Finally, Figure 21 presents the periodic steady-state solution of the sending end voltage V1 and its harmonic spectrum, which has a low THD of 1.74%.

Fig. 18. Shunt controller.

Fig. 19. Transient response of the series and shunt controllers of UPFC.

The dynamic response of UPFC controllers are presented in Figure 19. From this figure, it is possible to observe that the variation of the three-phase programmable voltage source significantly affects the response of the series controller. This is because the magnitude of the voltages at the ends of the line controlled by the UPFC affects the power flow across the line. On the other hand, Figure 20 shows the transient waveform of the voltage across the dc capacitor and the active and reactive power flow in the controlled line. Notice that the reactive and active power remains close to the reference during the step changes in the three-phase programmable voltage source. Finally, Figure 21 presents the periodic steady-state solution of

the sending end voltage V1 and its harmonic spectrum, which has a low THD of 1.74%.

Fig. 19. Transient response of the series and shunt controllers of UPFC.

Fig. 18. Shunt controller.

Fig. 20. Transient waveforms of selected variables: (Left) Voltage across the dc capacitor. (Right) Reactive (purple) and active (yellow) power.

Fig. 21. Steady state waveform of the voltage at the bus 1 (top) and its harmonic spectrum (bottom).

**2**

Titu Niculescu *University of Petrosani*

*Romania*

**Study of Inductive-Capacitive Series Circuits**

This chapter presents a modern method for approaching the electrical circuits using the MATLAB-SIMULINK package programs. The simple series circuits which are switched on a DC voltage at the initial moment are presented below. For the young researchers these are very useful. We can determine the current variation forms and the reactive elements voltage, by using this virtual medium. Each presented case contains an analytical presentation of the problem, but it also contains electrical diagrams of electrical parameters.

We will consider the RL series circuit with concentrated parameters from Fig.1. At the initial moment, the k circuit switcher is closed and we intend to study the behavior and variation

The differential equation which corresponds to the transitory regime immediately after

*di Ri L E dt*

(1)

The diagrams were obtained by different methods which use this programs package.

**1. Introduction**

**2. Study of the inductive series circuits**

of circuit electrical parameters after connection.

Fig. 1. RL series circuits at closing

closing is the following:

**2.1 Theoretical study of the circuit**

**Using the Simulink Software Package**

## **4. Conclusions**

In this chapter, implementation in Simulink of three FACTS devices based on VSC has been presented. These implementations include the STATCOM, SSSC, and the UPFC. All the implementations have been carried-out taking into account the control systems. The presented models can be used in order to analyze the dynamic and the periodic steady-state behavior of these FACTS components.

## **5. Acknowledgements**

The authors thank the Univesidad Autónoma de San Luis Potosí (UASLP) and the Universidad Michoacana de San Nicolás de Hidalgo (UMSNH) for the facilities that were granted to carry out this chapter.

## **6. References**

## **Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package**

Titu Niculescu *University of Petrosani Romania*

## **1. Introduction**

18 Technology and Engineering Applications of Simulink

In this chapter, implementation in Simulink of three FACTS devices based on VSC has been presented. These implementations include the STATCOM, SSSC, and the UPFC. All the implementations have been carried-out taking into account the control systems. The presented models can be used in order to analyze the dynamic and the periodic steady-state

The authors thank the Univesidad Autónoma de San Luis Potosí (UASLP) and the Universidad Michoacana de San Nicolás de Hidalgo (UMSNH) for the facilities that were

Acha, E., Fuerte-Esquivel, C., Ambriz-Pérez, H., & Angeles-Camacho, C. (2004). *FACTS*

Fujita, H., Watanabe, Y., & Akagi, H. (2001). Transient Analisis of a Unified Power Flow

*on Power Electronics*, Vol. 16, No. 5, (september 2001), pp. 735-740, 0885-8993 Hingorani, N. G., & Gyudyi, L. (2000). *Undestanding FACTS,* IEEE Press, 078033455, New

Mahyavanshi, B., & Radman, G. (2006). A Study of Interaction Between Dynamic Load and

Mathur, R., & Varma, R. (2002). *Thyristor Based FACTS Controllers for Electrical Transmission Systems*, IEEE Computer Society Press, ISSN, Piscataway, New Jersey Mohan, N., Underland, T. M., & Robins, W. P. (1995). *Power Electronics: Converters,*

Padiyar, K. R. (2007). *FACTS Controller in Power Transmission and Distribution,* New Age, 978-

Segundo-Ramírez, J., & Medina, A. (2009). Modeling of FACTS Devices Based on SPWM

Segundo-Ramírez, J., & Medina, A. (2008). Periodic Steady-State Solution of Electric Systems

Zhang, X.-P., Rehtanz, C., & Pal, B. (2006). *Flexible AC Transmission Systems: Modelling and*

VSCs. *IEEE Transaction on Power Delivery*, Vol. 24, No. 4, (November 2009), pp.

Including UPFCs by Extrapolation to the Limit Cycle. *IEEE Transaction on Power*

*Modelling and Simulation in Power Networks,* John Wiley & Sons,<PHONE_NUMBER>712,

Controller and its Application to Design of the DC-Link Capacitor. *IEEE Transaction*

STATCOM. *Proceedings of the 38th southeastern symposium on system theory*, 0-7803-

**4. Conclusions**

behavior of these FACTS components.

**5. Acknowledgements**

England

York

9457-7, Cookeville, TN, march 2006

*Delivery*, Vol. 23, No. 3, pp. 1506-1512

81-224-2142-2, New Delhi

1815-1823, 0885-8977

*Applications, and Design,* Wiley,<PHONE_NUMBER>, New York

*Control,* Springer-Verlag. 978-3-540-30606-1, Germany

**6. References**

granted to carry out this chapter.

This chapter presents a modern method for approaching the electrical circuits using the MATLAB-SIMULINK package programs. The simple series circuits which are switched on a DC voltage at the initial moment are presented below. For the young researchers these are very useful. We can determine the current variation forms and the reactive elements voltage, by using this virtual medium. Each presented case contains an analytical presentation of the problem, but it also contains electrical diagrams of electrical parameters. The diagrams were obtained by different methods which use this programs package.

## **2. Study of the inductive series circuits**

We will consider the RL series circuit with concentrated parameters from Fig.1. At the initial moment, the k circuit switcher is closed and we intend to study the behavior and variation of circuit electrical parameters after connection.

Fig. 1. RL series circuits at closing

## **2.1 Theoretical study of the circuit**

The differential equation which corresponds to the transitory regime immediately after closing is the following:

$$Ri + L\frac{di}{dt} = E\tag{1}$$

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 21

we will obtain the diagrams from Fig.3 and Fig.4.

Fig. 3. Variation of the coil voltage after closing

Fig. 4. Variation of the circuit current

electrical parameters of the circuit have the values:

**2.3 Analysis of RL series circuits using the SimPowerSistems software package**

The SimPowerSystems software package is a component of the MATLAB program which allows analyzing an electrical circuit by drawing it in an editing window. The SimPowerSystems model is presented in Fig.5 and allows the visualization of the circuit current and the coil voltage if we apply a step voltage signal and a unitary impulse. The

The current expression through the circuit after closing is the solution of the differential equation of the circuit (1).

> ( ) (1 ) - *t <sup>T</sup> <sup>E</sup> it e R* (2)

where:

$$T = \frac{L}{R} \tag{3}$$

and it represents the time constant of the circuit.

The voltage expression on the coil after connection is:

$$
\mu(t) = L\frac{di}{dt} = E \cdot e^{-\frac{t}{T}}\tag{4}
$$

#### **2.2 SIMULINK model of the circuit**

The SIMULINK model of the circuit after closing was done on grounds of equation (1) where the derivative of the current was separated.

$$\frac{di}{dt} = \frac{1}{L}(E \text{ - } Ri) \tag{5}$$

The SIMULINK model obtained in this way is shown in Fig.2 and was created in order to allow the drawing of the current diagrams through the circuit and the voltage on the coil, for different values of the R resistance and the E D.C. voltage.

Fig. 2. SIMULINK model of the RL circuit at closing

The k switcher is realized using a voltage step signal which is applied at the terminal of the circuit, to simulate the closing of the k switcher.

If we consider the following values of the electric parameters

20 Technology and Engineering Applications of Simulink

The current expression through the circuit after closing is the solution of the differential

( ) (1 ) -

*<sup>T</sup> <sup>E</sup> it e*

*<sup>L</sup> <sup>T</sup>*

( )

The SIMULINK model of the circuit after closing was done on grounds of equation (1)

The SIMULINK model obtained in this way is shown in Fig.2 and was created in order to allow the drawing of the current diagrams through the circuit and the voltage on the coil,

Product <sup>1</sup>

The k switcher is realized using a voltage step signal which is applied at the terminal of the

1 s Integrator

*<sup>T</sup> di ut L E e*

*t*

*t*

*R* (2)

*<sup>R</sup>* (3)

*dt* (4)

*dt L* (5)

L

du/dt Derivative

Scope U

Scope I

equation of the circuit (1).

and it represents the time constant of the circuit.

**2.2 SIMULINK model of the circuit**

Step

50 R

The voltage expression on the coil after connection is:

where the derivative of the current was separated.

Add

Fig. 2. SIMULINK model of the RL circuit at closing

If we consider the following values of the electric parameters

circuit, to simulate the closing of the k switcher.

for different values of the R resistance and the E D.C. voltage.

1 1/L

where:

we will obtain the diagrams from Fig.3 and Fig.4.

Fig. 3. Variation of the coil voltage after closing

Fig. 4. Variation of the circuit current

## **2.3 Analysis of RL series circuits using the SimPowerSistems software package**

The SimPowerSystems software package is a component of the MATLAB program which allows analyzing an electrical circuit by drawing it in an editing window. The SimPowerSystems model is presented in Fig.5 and allows the visualization of the circuit current and the coil voltage if we apply a step voltage signal and a unitary impulse. The electrical parameters of the circuit have the values:

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 23

The selection *of U\_Voltage Measurement or I\_Current Measurement* is followed by the obtaining of the coil voltage diagram and of the circuit current upon applying the step

Fig. 8. The coil voltage variation upon applying a step signal and a unitary impulse.

signal and a unitary impulse. These are presented in Fig.8 and Fig.9.

Fig. 7. The *LTI Viewer* window

R = 50[Ω]; L = 1[H].

Fig. 5. SimPowerSystems model of an inductive circuit

The *Continous powergui* analyze block is required to be present in the drawing window for launching the simulation model. We will open an analysis window by double-clicking on this block, Fig. 6.

Fig. 6. The analysis window of the circuit

The selection of the *Use LTI Viewer* button will lead to the opening of the window in Fig.7, from which we can select the current diagram or the coil voltage diagram.

Fig. 7. The *LTI Viewer* window

Current Measurement <sup>s</sup>

The *Continous powergui* analyze block is required to be present in the drawing window for launching the simulation model. We will open an analysis window by double-clicking on

The selection of the *Use LTI Viewer* button will lead to the opening of the window in Fig.7,

from which we can select the current diagram or the coil voltage diagram.

v + - Voltage Measurement

R L Scope I

<sup>i</sup> <sup>+</sup> -

Scope U

R = 50[Ω]; L = 1[H].

Step

this block, Fig. 6.

Continuous powergui

Fig. 6. The analysis window of the circuit

Fig. 5. SimPowerSystems model of an inductive circuit

> The selection *of U\_Voltage Measurement or I\_Current Measurement* is followed by the obtaining of the coil voltage diagram and of the circuit current upon applying the step signal and a unitary impulse. These are presented in Fig.8 and Fig.9.

Fig. 8. The coil voltage variation upon applying a step signal and a unitary impulse.

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 25

We will consider the RC series circuit with concentrated parameters from Fig.12. At the initial moment, the k circuit switcher is closed and we want to study the current variation

The integral equation which corresponds to the transitory regime immediately after closing

<sup>1</sup> *Ri idt E C*

(6)

through the circuit and the capacitor voltage variation, after connection.

Fig. 11. The frequency current diagrams.

Fig. 12. RC series circuit at closing

is given by:

**3.1 Theoretical study of the circuit**

Equation (6) is equivalent with differential equation:

**3. Study of the capacitive series circuits**

Fig. 9. The current variation upon applying a step signal and a unitary impulse.

We can use the *Edit/Plot Configurations* and *Edit/ Viewer Preferences* options, in order to create these diagrams. We can also select the *Bode* option from these menus, which allows the obtaining of the current frequency diagrams of the RL series circuit.

Fig. 10. The amplitude frequency diagram of the voltage coil.

Fig. 11. The frequency current diagrams.

Fig. 9. The current variation upon applying a step signal and a unitary impulse.

obtaining of the current frequency diagrams of the RL series circuit.

Fig. 10. The amplitude frequency diagram of the voltage coil.

We can use the *Edit/Plot Configurations* and *Edit/ Viewer Preferences* options, in order to create these diagrams. We can also select the *Bode* option from these menus, which allows the

## **3. Study of the capacitive series circuits**

We will consider the RC series circuit with concentrated parameters from Fig.12. At the initial moment, the k circuit switcher is closed and we want to study the current variation through the circuit and the capacitor voltage variation, after connection.

Fig. 12. RC series circuit at closing

#### **3.1 Theoretical study of the circuit**

The integral equation which corresponds to the transitory regime immediately after closing is given by:

$$i\text{ Ri} + \frac{1}{C} \int \quad idt = E \tag{6}$$

Equation (6) is equivalent with differential equation:

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 27

Fig. 14. The capacitor voltage variation

Fig. 15. The current variation after closing

impulse.

**3.3 Analysis of RC series circuits using the SimPowerSistems software package**

The SimPowerSystems model is presented in Fig.16 and allows the visualization of the circuit current and the capacitor voltage if we apply a step voltage signal and a unitary

we will obtain the diagrams from Fig.14 and Fig.15.

$$RC\frac{du}{dt} + u = E \tag{7}$$

The expression of the capacitor voltage after closing is the solution of differential equation (7):

$$
\mu(t) = E(1 \cdot e^{-\frac{t}{T}}) \tag{8}
$$

where *T RC* , represents the time constant of the circuit.

The circuit current after closing is given by:

$$
\dot{u} = \mathbf{C} \frac{du}{dt} = \frac{\mathbf{E}}{R} e^{-\frac{\mathbf{f}}{T}} \tag{9}
$$

#### **3.2 SIMULINK model of the circuit**

The SIMULINK model of the circuit after closing was done on grounds of equation (7) where the voltage derivative was separated.

$$\frac{du}{dt} = \frac{1}{RC}(E \text{ -} u) \tag{10}$$

The SIMULINK model obtained is shown in Fig.13 and was created in order to allow the drawing of the current diagrams through the circuit and the voltage capacitor, for different values of the R resistance, of the E D.C. voltage and of the C capacitor.

Fig. 13. SIMULINK model of the RC circuit at closing

The k switcher is realized using the same voltage step signal which is applied at the terminal of the circuit, to simulate the closing of the k switcher.

If we consider the following values of the electric parameters:

26 Technology and Engineering Applications of Simulink

*du RC u E dt*

The expression of the capacitor voltage after closing is the solution of differential equation

( ) (1 ) -

*<sup>T</sup> du E iC e*

The SIMULINK model of the circuit after closing was done on grounds of equation (7)

The SIMULINK model obtained is shown in Fig.13 and was created in order to allow the drawing of the current diagrams through the circuit and the voltage capacitor, for different

Step Scope U

1 s Integrator

Scope I <sup>1500</sup>

The k switcher is realized using the same voltage step signal which is applied at the terminal

values of the R resistance, of the E D.C. voltage and of the C capacitor.

Divide

1 1 Product1

where *T RC* , represents the time constant of the circuit.

The circuit current after closing is given by:

where the voltage derivative was separated.

Add

Product

Fig. 13. SIMULINK model of the RC circuit at closing

of the circuit, to simulate the closing of the k switcher.

If we consider the following values of the electric parameters:

**3.2 SIMULINK model of the circuit**

R

*t*

*t*

(7):

(7)

*ut E e <sup>T</sup>* (8)

*dt R* (9)

*dt RC* (10)

Product2

du/dt Derivative

we will obtain the diagrams from Fig.14 and Fig.15.

Fig. 14. The capacitor voltage variation

## **3.3 Analysis of RC series circuits using the SimPowerSistems software package**

The SimPowerSystems model is presented in Fig.16 and allows the visualization of the circuit current and the capacitor voltage if we apply a step voltage signal and a unitary impulse.

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 29

Fig. 18. The current variation upon applying a step signal and a unitary impulse.

presented bellow:

Fig. 19. The capacitor voltage frequency diagrams

The Bode diagrams which allow the visualization of the frequency behavior of the circuit are

Fig. 16. SimPowerSystems model of the capacitive circuit

The electrical parameters of the circuit have the values:

To study the voltage variation on the capacitor, we follow the steps:

This method allows us to obtain the voltage capacitor diagram upon applying the step signal and the unitary impulse (Fig.17).

The current variation through the circuit upon applying a step signal and a unitary impulse was obtained by selecting *I\_Current Measurement* option (Fig.18).

Fig. 17. The capacitor voltage variation upon applying a step signal and a unitary impulse.

Current Measurement <sup>s</sup>

<sup>v</sup> <sup>+</sup> - Voltage Measurement

R Scope I

i + -

Scope U

Continuous powergui

C ontrolled Voltage Source <sup>C</sup>

Fig. 16. SimPowerSystems model of the capacitive circuit

The electrical parameters of the circuit have the values:

 Select *Use LTI Wiewer* in the new window; We will select U Voltage *Measurement*.

signal and the unitary impulse (Fig.17).

To study the voltage variation on the capacitor, we follow the steps:

was obtained by selecting *I\_Current Measurement* option (Fig.18).

Select by double clicking the *Continous powerguy* block from the simulation model.

This method allows us to obtain the voltage capacitor diagram upon applying the step

The current variation through the circuit upon applying a step signal and a unitary impulse

Fig. 17. The capacitor voltage variation upon applying a step signal and a unitary impulse.

Step

Fig. 18. The current variation upon applying a step signal and a unitary impulse.

The Bode diagrams which allow the visualization of the frequency behavior of the circuit are presented bellow:

Fig. 19. The capacitor voltage frequency diagrams

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 31

*d u du LC RC u E dt dt*

> 2 *R L*

> > 1 *LC*

> 2 2 0

*<sup>E</sup> rt rt i t e e*

We will consider the situation when δ <ω0 or *R LC* 2 . The following notation was made:

*<sup>E</sup> <sup>t</sup> it e t L*

1 2 *r r*

0

Consequently, the solving of the differential equation gives the following solutions:

*<sup>t</sup> ut E e t*

*rt rt ut E re re*

1 2 2 2 0 <sup>1</sup> ( ) [1 ( )]

2 1

(17)

(18)

' (19)

(21)

1 2

 <sup>0</sup>

0

Consequently, the solving of the differential equation gives the following solutions:

2

2

*L*

where r1 and r2 are the roots of the characteristic equation:

( )

(12)

(13)

(14)

(15)

(16)

(20)

or *R LC* 2 , which will be checked by the

2 2

or:

The following notations are made:

The circuit amortization:

the circuit personal pulsation:

resistor in the circuit.

We will consider the situation when 0

Fig. 20. The current frequency diagrams.

## **4. Study of the inductive-capacitive series circuits**

We will consider the RLC series circuit with concentrated parameters from Fig.21. At the initial moment, the k circuit switcher is closed and we intend to study the behavior and variation of circuit electrical parameters after connection.

Fig. 21. RLC series circuit at connecting

## **4.1 Theoretical study of the circuit**

The integral-differential equation which corresponds to the transitory regime of the considered circuit is the following:

$$\left[Ri + L\frac{di}{dt} + \frac{1}{C}\right]idt = E\tag{11}$$

or:

30 Technology and Engineering Applications of Simulink

We will consider the RLC series circuit with concentrated parameters from Fig.21. At the initial moment, the k circuit switcher is closed and we intend to study the behavior and

The integral-differential equation which corresponds to the transitory regime of the

1 *di Ri L idt E*

*dt C* (11)

Fig. 20. The current frequency diagrams.

Fig. 21. RLC series circuit at connecting

**4.1 Theoretical study of the circuit**

considered circuit is the following:

**4. Study of the inductive-capacitive series circuits**

variation of circuit electrical parameters after connection.

$$\text{LC}\frac{d^2u}{dt^2} + \text{RC}\frac{du}{dt} + u \,\, = \text{E} \tag{12}$$

The following notations are made:

$$
\delta = \frac{R}{2L} \tag{13}
$$

The circuit amortization:

$$
\alpha\_0 = \frac{1}{\sqrt{LC}}\tag{14}
$$

the circuit personal pulsation:

$$
\alpha = \sqrt{\delta^2 \cdot \alpha\_0^2} \tag{15}
$$

We will consider the situation when 0 or *R LC* 2 , which will be checked by the resistor in the circuit.

Consequently, the solving of the differential equation gives the following solutions:

$$u(t) = E\left[1 \cdot \frac{1}{2\sqrt{\delta^2 \cdot \alpha\_0^2}} (r\_1 e^{r\_2 t} \cdot r\_2 e^{r\_1 t})\right] \tag{16}$$

$$i(t) = \frac{E}{2L\sqrt{\delta^2 \cdot \alpha\_0^2}} \left( e^{\gamma\_1 t} \cdot e^{\gamma\_2 t} \right) \tag{17}$$

where r1 and r2 are the roots of the characteristic equation:

$$\begin{aligned} r\_1 &= \text{-}\mathcal{S} + \mathcal{O} \\ r\_2 &= \text{-}\mathcal{S} \cdot \mathcal{O} \end{aligned} \tag{18}$$

We will consider the situation when δ <ω0 or *R LC* 2 . The following notation was made:

$$
\alpha \alpha\_0^2 \cdot \delta^2 = \alpha^{\cdot 2} \tag{19}
$$

Consequently, the solving of the differential equation gives the following solutions:

$$\mu(t) = E\left[ \left[ 1 - \frac{\alpha\_0}{\alpha'} \bullet e^{\cdot \delta t} \bullet \sin(\phi' t + \beta') \right] \right] \tag{20}$$

$$\dot{a}(t) = \frac{E}{a^\prime L} e^{\therefore \delta t} \sin \alpha^\prime t \tag{21}$$

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 33

we get the following MATLAB diagrams (Fig.23 and Fig.24) according to the equations (16) and (17) for a-periodic mode, and according to the equations (20) and (21) for

oscillating mode (Fig.25 and Fig.26).

Fig. 23. The voltage capacitor variation in the a-periodic mode

Fig. 24. The current variation in the a-periodic mode

Fig. 25. The capacitor voltage variation in the oscillating regime

where

$$
\beta' = \arccos \frac{\delta}{a\_0} \tag{22}
$$

### **4.2 SIMULINK model of the circuit**

The SIMULINK model of the circuit after closing was done on grounds of the second order differential equation (12), which is put in the form (23) where the higher order derivative is separated:

$$\frac{d^2u}{dt} = \frac{1}{LC} \left[ \left[ E \cdot RC \frac{du}{dt} \cdot u \right] \right] \tag{23}$$

The SIMULINK model from Fig.22 generates the voltage capacitor and the current through the circuit during the transitory regime. To this purpose, two values of the R resistor are considered, which correspond to two important regimes:

For each regime the capacitor voltage and the circuit current variation diagrams are plotted.

Fig. 22. SIMULINK model of the circuit

If we consider:

' arccos

The SIMULINK model of the circuit after closing was done on grounds of the second order differential equation (12), which is put in the form (23) where the higher order derivative is

> <sup>2</sup> <sup>1</sup> [ ] *d u du E RC u dt LC dt*

The SIMULINK model from Fig.22 generates the voltage capacitor and the current through the circuit during the transitory regime. To this purpose, two values of the R resistor are

For each regime the capacitor voltage and the circuit current variation diagrams are

1 s Integrator

0

(22)

(23)

ScopeU

ScopeI

du/dt

C

Derivative

1 s Integrator1

C1

where

separated:

plotted.

Step

1500

R

If we consider:

 Aperiodic regime; Oscillatory regime.

**4.2 SIMULINK model of the circuit**

considered, which correspond to two important regimes:

1/LC

RCdu/dt

Fig. 22. SIMULINK model of the circuit

Add

we get the following MATLAB diagrams (Fig.23 and Fig.24) according to the equations (16) and (17) for a-periodic mode, and according to the equations (20) and (21) for oscillating mode (Fig.25 and Fig.26).

Fig. 23. The voltage capacitor variation in the a-periodic mode

Fig. 24. The current variation in the a-periodic mode

Fig. 25. The capacitor voltage variation in the oscillating regime

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 35

R L

pow ergui <sup>v</sup> <sup>+</sup>

The a-periodic loading of the capacitor leads to the obtaining of the capacitor voltage variation diagrams when the unit step and the unitary impulse are applied to the input.

The POWERGUY analysis block, with the above values, allows the obtaining of the

ScopeU

ScopeI

C

Voltage Measurement

i + - Current Measurement

Continuous

Step

s -

1. *Voltage capacitor*

These charts are presented below.

following Matlab diagrams (Fig.28 and Fig.29):

For a-periodic regime:

For oscillating regime:

+

Fig. 27. SimPowerSystems model of the circuit

The following values of electrical parameters were considered:

To study the capacitor voltage variation, we follow the steps:

Controlled Voltage Source

Fig. 26. The current variation in the oscillating regime

The k switcher is realized by using a voltage step signal which is applied at the terminal of the circuit, to simulate the closing of the k switcher. Changing the value of resistance, automatically leads to the updating of the voltage and current charts.

## **4.3. Analysis of inductive-capacitive circuits using the SimPowerSystems software package**

The SimPowerSystems software package allows the study of the circuit behavior, when it is connected to a voltage step or a unitary impulse, using *Continous powerguy* block. The current variation in the circuit, the voltage variation of the capacitive element, or the frequency behavior of the circuit for different values of circuit elements can be studied.

## **4.3.1 SimPowerSystems model of the circuit**

The MATLAB software package contains SimPowerSystems of Simulink, which can simulate the electrical circuits and analyze different operating regimes. Because the signal step response or pulse generates a transitory regime, the circuit behavior in these conditions can be studied. The simulation is shown in Fig.27:

The advantage of this simulation method is that the circuit can be tested at both signals, unit step and unitary impulse, and leads to several types of typical circuit diagrams. Two sets of values were chosen in this case for circuit parameters, which correspond to two special regimes.

In the simulation model from Fig.27 we used a *Controlled Voltage Source* controlled by a step voltage signal. In this way, the switcher closing can be simulated.

The SimPowerSystems software package allows the obtaining of a lot of diagrams for a circuit. So, we will study the capacitor voltage diagrams and current diagrams for the two occurring regimes:

Fig. 27. SimPowerSystems model of the circuit

1. *Voltage capacitor*

34 Technology and Engineering Applications of Simulink

The k switcher is realized by using a voltage step signal which is applied at the terminal of the circuit, to simulate the closing of the k switcher. Changing the value of resistance,

**4.3. Analysis of inductive-capacitive circuits using the SimPowerSystems software**

The SimPowerSystems software package allows the study of the circuit behavior, when it is connected to a voltage step or a unitary impulse, using *Continous powerguy* block. The current variation in the circuit, the voltage variation of the capacitive element, or the frequency behavior of the circuit for different values of circuit elements can be studied.

The MATLAB software package contains SimPowerSystems of Simulink, which can simulate the electrical circuits and analyze different operating regimes. Because the signal step response or pulse generates a transitory regime, the circuit behavior in these conditions

The advantage of this simulation method is that the circuit can be tested at both signals, unit step and unitary impulse, and leads to several types of typical circuit diagrams. Two sets of values were chosen in this case for circuit parameters, which correspond to two special

In the simulation model from Fig.27 we used a *Controlled Voltage Source* controlled by a step

The SimPowerSystems software package allows the obtaining of a lot of diagrams for a circuit. So, we will study the capacitor voltage diagrams and current diagrams for the two

Fig. 26. The current variation in the oscillating regime

**4.3.1 SimPowerSystems model of the circuit**

can be studied. The simulation is shown in Fig.27:

voltage signal. In this way, the switcher closing can be simulated.

**package**

regimes.

occurring regimes:

 A-periodic regime: Oscillating regime.

automatically leads to the updating of the voltage and current charts.

The a-periodic loading of the capacitor leads to the obtaining of the capacitor voltage variation diagrams when the unit step and the unitary impulse are applied to the input. These charts are presented below.

The following values of electrical parameters were considered:

To study the capacitor voltage variation, we follow the steps:

The POWERGUY analysis block, with the above values, allows the obtaining of the following Matlab diagrams (Fig.28 and Fig.29):

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 37

Fig. 30. Frequency diagrams of the voltage swing on the capacitive element

selecting the *U voltage Measurement, I Current Measuremen*t must be selected.

So, we will study the current variation diagrams for the two occurring regimes:

the unit step and the unitary impulse are applied to the input.

To study the circuit current variation the same procedure is used, mentioning that instead of

A-periodic loading of the capacitor leads to obtaining the current variation diagrams when

The SimPowerSystems software package allows the obtaining of two diagrams for a circuit.

The a-periodical and oscillating regime according to these values are obtained in the

2. Study of circuit current

For this, we will follow these steps:

following charts (Fig.31 and Fig.32):

 A-periodical regime: Oscillating regime.

Fig. 28. The capacitor voltage variation in the a-periodic mode

Fig. 29. The capacitor voltage variation in the oscillating regime

The POWERGUY block permits the analysis and plotting of the frequency characteristics of the capacitor voltage, where the frequency in logarithmic coordinates was considered on the horizontal axis (BODE diagram).

For this, we follow these steps:

Fig. 28. The capacitor voltage variation in the a-periodic mode

Fig. 29. The capacitor voltage variation in the oscillating regime

horizontal axis (BODE diagram). For this, we follow these steps:

The POWERGUY block permits the analysis and plotting of the frequency characteristics of the capacitor voltage, where the frequency in logarithmic coordinates was considered on the

Fig. 30. Frequency diagrams of the voltage swing on the capacitive element

2. Study of circuit current

To study the circuit current variation the same procedure is used, mentioning that instead of selecting the *U voltage Measurement, I Current Measuremen*t must be selected.

A-periodic loading of the capacitor leads to obtaining the current variation diagrams when the unit step and the unitary impulse are applied to the input.

For this, we will follow these steps:

The SimPowerSystems software package allows the obtaining of two diagrams for a circuit. So, we will study the current variation diagrams for the two occurring regimes:

The a-periodical and oscillating regime according to these values are obtained in the following charts (Fig.31 and Fig.32):

Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package 39

Fig. 33. Frequency diagrams of the current swing through the circuit

POWERGUY block, other electrical circuit parameters can be determined.

influence of each parameter variation in the final diagrams can be studied.

In the diagram from Fig.33 we can observe a maximum of current which correspond to a resonance regime. We can determine the resonance frequencies for an inductive-capacitive circuit using the simulation circuit model and the SimPowerSystems software package. With

An important conclusion of this chapter is that the electrical circuits, regardless of configuration, can be studied with this modern method which involves the use of virtual

Three elementary circuits are presented here, but the study modality can be extended to

If we consider an electric circuit, a first problem is that of writing the characteristic integral differential equations, which can be integrated with a simulation model and which uses the Simulink software package. These models can be conceived so that the electric circuit parameters must be electric input values in the simulation model. Thus, the obtained diagrams are automatically updated to any changes of circuit electrical parameters, and the

**5. Conclusions**

other circuits configuration as well.

medium.

Fig. 31. The current variation for signal step and impulse in a-periodic mode

Fig. 32. The current variation for signal step and impulse in the oscillating regime

The POWERGUY block allows the analysis and plotting of frequency current characteristics, where the frequency in logarithmic coordinates was considered on the horizontal axis (BODE diagram). For this, we will follow these steps:

Fig. 31. The current variation for signal step and impulse in a-periodic mode

Fig. 32. The current variation for signal step and impulse in the oscillating regime

(BODE diagram). For this, we will follow these steps:

The POWERGUY block allows the analysis and plotting of frequency current characteristics, where the frequency in logarithmic coordinates was considered on the horizontal axis

Fig. 33. Frequency diagrams of the current swing through the circuit

In the diagram from Fig.33 we can observe a maximum of current which correspond to a resonance regime. We can determine the resonance frequencies for an inductive-capacitive circuit using the simulation circuit model and the SimPowerSystems software package. With POWERGUY block, other electrical circuit parameters can be determined.

## **5. Conclusions**

An important conclusion of this chapter is that the electrical circuits, regardless of configuration, can be studied with this modern method which involves the use of virtual medium.

Three elementary circuits are presented here, but the study modality can be extended to other circuits configuration as well.

If we consider an electric circuit, a first problem is that of writing the characteristic integral differential equations, which can be integrated with a simulation model and which uses the Simulink software package. These models can be conceived so that the electric circuit parameters must be electric input values in the simulation model. Thus, the obtained diagrams are automatically updated to any changes of circuit electrical parameters, and the influence of each parameter variation in the final diagrams can be studied.

**3**

Rastislav Róka

*Slovakia*

**Fixed Transmission Media**

For successful understanding of the signal transmission in access networks that utilized fixed transmission media, it is necessary exactly to recognize essential negative influences in the real environment of metallic homogeneous symmetric lines, power distribution cables and optical fibers. This chapter discusses features and frequency characteristics of negative influences on signals transmitted by means of the VDSL technology, the PLC technology and PON networks. For the expansion of communication systems on fixed transmission media, it is necessary to have a detailed knowledge of their transmission environments and

A main attention of the metallic transmission media's parts is focused on the description of the proposed VDSL and PLC simulation models and on the explanation of simulation methods for substantial negative influences. Presented simulation models represent a reach enough knowledgebase for the extended digital signal processing techniques of the VDSL and PLC signal transmissions that can be extremely helpful for various tests and

A main attention of the optical transmission media's part is focused on the description of the proposed optical fiber's simulation model and on the explanation of simulation methods for its substantial linear effects - transmission factors. The presented simulation model represents a reach enough knowledgebase that can be helpful for various tests and performance comparisons of various novel modulation techniques suggested and intended

Propagation loss and linear distortions (distortions of the module and the phase characteristics and the group delay characteristic) are linear negative influences dependent on physical and constructional parameters, such as a line length, a core diameter of the wire, a mismatch of impedances in cross-connecting points of sections, a frequency bandwidth

We first discuss the propagation loss *LdB* in a perfectly terminated line. If *R*, *L*, *G* and *C* are

*.f*, where *f* is the frequency, then

to be used at signal transmissions in the transmission environment of optical fibers.

negative influences in the real developing of customer installations.

**2. The environment of metallic homogeneous lines 2.1 Linear negative influences on transmitted signals**

> = *2.*

**1. Introduction**

performance comparisons.

and so forth (Róka, 2004).

primary constants of the line and

*Slovak University of Technology, Bratislava*

In order to realize the simulating model which integrates the differential equations, the basic idea is to separate in the left member of equation, the higher order derivative. The analytical expressions obtained in the right member of the equation underlie the achievement of the simulating model by means of specific blocks of the virtual medium. In the case of complex circuits, the simulating model is conceived on a differential equations system.

Another given facility of this package programs, is the possibility to study the circuit behavior upon applying the standard signals (step unit, unitary impulse). The specific analysis block *Continous Powerguy* allows the circuit response to these signals. It also allows the study of circuit frequency characteristics and the other specific diagrams: the coil hysteresis diagrams, line parameters, FTF analysis, Bode Diagrams, Nyquist diagrams, Nichols diagrams and others.

This study intends to be a starting point in the approach of more complex circuits.

## **6. References**

## **Fixed Transmission Media**

## Rastislav Róka

*Slovak University of Technology, Bratislava Slovakia*

## **1. Introduction**

40 Technology and Engineering Applications of Simulink

In order to realize the simulating model which integrates the differential equations, the basic idea is to separate in the left member of equation, the higher order derivative. The analytical expressions obtained in the right member of the equation underlie the achievement of the simulating model by means of specific blocks of the virtual medium. In the case of complex

Another given facility of this package programs, is the possibility to study the circuit behavior upon applying the standard signals (step unit, unitary impulse). The specific analysis block *Continous Powerguy* allows the circuit response to these signals. It also allows the study of circuit frequency characteristics and the other specific diagrams: the coil hysteresis diagrams, line parameters, FTF analysis, Bode Diagrams, Nyquist diagrams,

Niculescu, T.; Pasculescu, D.; Pana, L. (2010). *Study of the operating states of intrinsic safety*

Pasculescu, D.; Niculescu, T.; Pana, L. (2010). *Uses of Matlab software to size intrinsic safety*

EQUIPMENT (EEETE '10), ISBN: 978-960-474-181-6, Bucharest, Romania Niculescu, T.; Niculescu, M. (2010). *The study of fundamental electrical circuits transitory*

Ghinea, M.; Firţeanu, V. (1999*). MATLAB calcul numeric, grafică, aplcatii,* Teora, ISBN 973-601-

Halunga-Fratu, S.; Fratu, O. (2004) *Simularea sistemelor de transmisiune analogice şi digitale*

Tudorache, T. (2006). *Medii de calcul in ingineria electrica MATLAB,* MatrixRom, ISBN 973-

Niculescu, T.;Costinaş, S. (1998) *Electrotehnică,* Printech, ISBN 973-9402-14-3, Bucharest,

ComputerEngineering, ISSN 1842-4449 Petrosani, Romania

*Barriers of the electric equipment intended for use in atmospheres with explosion hazard*,Main Page of the Journal WSEAS TRANSACTIONS on CIRCUITS and

*barriers of the electric equipment intended for use in atmospheres with explosion,*  International Conference on ENERGY and ENVIRONME TECHNOLOGIES and

*Phenomena using MATLAB software, UNIVERSITARIA SIMPRO*  ElectricalEngineering and Energetics System Control, Applied Informatics and

*folosind mediul MATLAB/SIMULINK,* MatrixRom, ISBN 973-685-716-6, Bucharest,

circuits, the simulating model is conceived on a differential equations system.

This study intends to be a starting point in the approach of more complex circuits.

Nichols diagrams and others.

SYSTEMS, ISSN: 1109-2734,

275-1, Bucharest, Romania

775-005-6, Bucharest, Romania

Romania

Romania

**6. References**

For successful understanding of the signal transmission in access networks that utilized fixed transmission media, it is necessary exactly to recognize essential negative influences in the real environment of metallic homogeneous symmetric lines, power distribution cables and optical fibers. This chapter discusses features and frequency characteristics of negative influences on signals transmitted by means of the VDSL technology, the PLC technology and PON networks. For the expansion of communication systems on fixed transmission media, it is necessary to have a detailed knowledge of their transmission environments and negative influences in the real developing of customer installations.

A main attention of the metallic transmission media's parts is focused on the description of the proposed VDSL and PLC simulation models and on the explanation of simulation methods for substantial negative influences. Presented simulation models represent a reach enough knowledgebase for the extended digital signal processing techniques of the VDSL and PLC signal transmissions that can be extremely helpful for various tests and performance comparisons.

A main attention of the optical transmission media's part is focused on the description of the proposed optical fiber's simulation model and on the explanation of simulation methods for its substantial linear effects - transmission factors. The presented simulation model represents a reach enough knowledgebase that can be helpful for various tests and performance comparisons of various novel modulation techniques suggested and intended to be used at signal transmissions in the transmission environment of optical fibers.

## **2. The environment of metallic homogeneous lines**

## **2.1 Linear negative influences on transmitted signals**

Propagation loss and linear distortions (distortions of the module and the phase characteristics and the group delay characteristic) are linear negative influences dependent on physical and constructional parameters, such as a line length, a core diameter of the wire, a mismatch of impedances in cross-connecting points of sections, a frequency bandwidth and so forth (Róka, 2004).

We first discuss the propagation loss *LdB* in a perfectly terminated line. If *R*, *L*, *G* and *C* are primary constants of the line and = *2..f*, where *f* is the frequency, then

Fixed Transmission Media 43

The term "crosstalk" generally refers to the interference that enters a communication channel through some coupling paths. On Fig. 1, a kind of generating and propagating of

At the input of pair *j*, the information signal *us (t)* is generated. This signal, when propagating through the line, can generate two types of crosstalk signals arising in pair *i*. A crosstalk signal *xNEXT (t)* is called the near-end crosstalk NEXT. A crosstalk signal *xFEXT (t)* is called the far-end crosstalk FEXT. From a data communication point of view, the NEXT crosstalk is generally more damaging than the FEXT crosstalk, because the NEXT does not necessarily propagate through the line length and thus does not experience a propagation

If either single or multiple interferers generate a crosstalk signal, we can define a gain of the NEXT crosstalk path according to (Werner, 1991), (Róka, 2002, 2004) using a following

> 2 2 2 3 4. ( ). /2 . . (, ) <sup>1</sup> . ( ) *NEXT f l*

2 22 2. ( ). 2 2 ( , ) 4. . . . . . .3280. . ( , ) *f l FEXT FEXT FEXT l f f k le K l f l f*

where variables are given as *KFEXT* = 3,083.10-20, *l* is the line length in km, *f* is the frequency in Hz and **H** (*l*, *f*) expresses the transfer function of a metallic homogeneous symmetric line.

**H H** (9)

**H** (8)

 *(*

*f* . We can also derive a gain of the FEXT crosstalk path

*)* of the propagation

*NEXT NEXT f k l f e Kf <sup>f</sup>*

where variables are given as *KNEXT* = 0,882.10-14.*Nd*0,6, *Nd* is the number of disturbing pairs (disturbers), *f* is the frequency in Hz. An approximation on the right in (8) is valid when the

line length *l* is large and for frequency regions where the real part

in a similar manner using a following relation

*<sup>e</sup>* ≈

( ) ( ) ( ) ( ) *<sup>e</sup>*

*d*

*e* and the phase delay

= 5,4 s/km.

*d* (7)

are approximately

Fig. 1. Types of crosstalks in the environment of metallic homogeneous lines

At higher frequencies, the group envelope delay

**2.2 Near-end and far-end crosstalk signals**

loss of the signal (Werner, 1991).

constant is proportional to

relation

frequency-independent and equal to the value about

two crosstalk types in a multipair cable is presented.

$$\gamma'(o) = \alpha(o) + j. \; \beta(o) = \sqrt{(R+j.o.L). (G+j.o.C)} \tag{1}$$

and

$$\mathcal{Z}\left(\boldsymbol{\alpha}\right) = \sqrt{\frac{\left(\boldsymbol{R} + \boldsymbol{j}.\,\boldsymbol{\alpha}.\,\boldsymbol{L}\right)}{\left(\boldsymbol{G} + \boldsymbol{j}.\,\boldsymbol{\alpha}.\,\boldsymbol{C}\right)}}\tag{2}$$

where  *()* denotes the propagation constant of the line,  *()* is the specific constant of the attenuation,  *()* is the specific constant of the phase shift and **Z** *()* is the characteristic impedance of the line. For a perfectly terminated line with the length *l*, the transfer function **H** (*l*, *f*) of metallic homogeneous symmetric lines is given by

$$\mathcal{H}(l,f) = e^{-l.\gamma(f)} = e^{-l.\alpha(f)} \cdot e^{-j.l.\beta(f)} \tag{3}$$

and the propagation loss *LdB* is given as

$$\left| L\_{dB}(l,f) = -20.\log\_{10} \left| \mathcal{M}(l,f) \right| = \frac{20}{\ln 10}.l.\alpha(f) = a\_{\text{line}}(l,f) \left[ d\mathcal{B} \right] \tag{4}$$

We must place emphasis on the interchangeable use of terms - the line attenuation *aline (l, f)*  and the propagation loss *LdB (l, f)* to designate the quantity in (4) only for the case of a perfectly terminated line. We can see that a dependency of the propagation loss *LdB* on the line length *l* is linear and is also an increasing function of the frequency *f* as should be apparent from the expression for the propagation constant  *()* in (1). A power level of transmitted signals is also influenced by other important parameters - a diameter and constructional material of the core.

For lower frequency regions, for which *.L << R* is valid and *G* can be neglected, the propagation constant expressed in (1) can be simplified to

$$\gamma(o) \approx \sqrt{\frac{o o.R.C}{2}} \cdot \left[1 - \frac{o o.L}{2.R}\right] + j \cdot \sqrt{\frac{o o.R.C}{2}} \cdot \left[1 + \frac{o o.L}{2.R}\right] \tag{5}$$

For frequencies less than 20 kHz, both the real and imaginary parts  *()* and  *()* are approximately proportional to *f* . At higher frequencies, frequency dependencies of the primary constants *R* and *L* (except for *C*) become noticeable and the propagation constant in (1) can be approximated by

$$\gamma(o) \approx \frac{R(o)}{2} \cdot \sqrt{\frac{C}{L(o)}} + j \cdot o \cdot \sqrt{C \cdot L(o)}\tag{6}$$

In this case, the imaginary part  *()* is approximately a linear function of the frequency. Major variations for the real part  *()* are due to the frequency dependency of *R*, which becomes proportional to *f* . because of the skin effect. Therefore, it is necessary to take into account increased signal attenuation in the area of higher frequencies.

The phase and envelope *<sup>e</sup>* delays of the line can be expressed as following

( . .) ( ) ( ..) *Rj L Gj C*

impedance of the line. For a perfectly terminated line with the length *l*, the transfer function

. ( ) . ( ) .. ( ) (, ) . *<sup>l</sup> <sup>f</sup> <sup>l</sup> f j <sup>l</sup> <sup>f</sup> lf e e e*

<sup>20</sup> ( , ) 20.log ( , ) . . ( ) ( , )[ ] <sup>10</sup> ln 10 *L l <sup>f</sup> <sup>l</sup> <sup>f</sup> <sup>l</sup> <sup>f</sup> a l <sup>f</sup> dB dB line* **<sup>H</sup>**

We must place emphasis on the interchangeable use of terms - the line attenuation *aline (l, f)*  and the propagation loss *LdB (l, f)* to designate the quantity in (4) only for the case of a perfectly terminated line. We can see that a dependency of the propagation loss *LdB* on the line length *l* is linear and is also an increasing function of the frequency *f* as should be

transmitted signals is also influenced by other important parameters - a diameter and

.. . .. . ( ) .1 . .1 2 2. 2 2.

approximately proportional to *f* . At higher frequencies, frequency dependencies of the primary constants *R* and *L* (except for *C*) become noticeable and the propagation constant in

> ( ) () . . . .() 2 () *<sup>R</sup> <sup>C</sup> j CL <sup>L</sup>*

becomes proportional to *f* . because of the skin effect. Therefore, it is necessary to take

*<sup>e</sup>* delays of the line can be expressed as following

(6)

*)* is approximately a linear function of the frequency.

*)* are due to the frequency dependency of *R*, which

*RC L RC L <sup>j</sup> R R*

*)* is the specific constant of the phase shift and **Z** *(*

**Z** (2)

*)* is the specific constant of the

*)* is the characteristic

(4)

*)* in (1). A power level of

(5)

( ) ( ) . ( ) ( . . ).( . . ) *j Rj LGj C* (1)

 *(*

 **H** (3)

> *(*

> >

*.L << R* is valid and *G* can be neglected, the

 *()* and  *()* are

*)* denotes the propagation constant of the line,

**H** (*l*, *f*) of metallic homogeneous symmetric lines is given by

apparent from the expression for the propagation constant

propagation constant expressed in (1) can be simplified to

 *(*

> *(*

into account increased signal attenuation in the area of higher frequencies.

For frequencies less than 20 kHz, both the real and imaginary parts

and

where  *(*

attenuation,

 *(*

and the propagation loss *LdB* is given as

constructional material of the core.

(1) can be approximated by

In this case, the imaginary part

The phase

Major variations for the real part

and envelope

For lower frequency regions, for which

$$
\tau\_{\phi} \text{ (}\phi\text{)} = \sqrt{\frac{\beta \left(\alpha\right)}{\alpha}} \qquad \tau\_{e} \text{ (}\alpha\text{)} = \sqrt{\frac{d\beta \left(\alpha\right)}{d\alpha}} \tag{7}
$$

At higher frequencies, the group envelope delay *e* and the phase delay are approximately frequency-independent and equal to the value about *<sup>e</sup>* ≈ = 5,4 s/km.

#### **2.2 Near-end and far-end crosstalk signals**

The term "crosstalk" generally refers to the interference that enters a communication channel through some coupling paths. On Fig. 1, a kind of generating and propagating of two crosstalk types in a multipair cable is presented.

Fig. 1. Types of crosstalks in the environment of metallic homogeneous lines

At the input of pair *j*, the information signal *us (t)* is generated. This signal, when propagating through the line, can generate two types of crosstalk signals arising in pair *i*. A crosstalk signal *xNEXT (t)* is called the near-end crosstalk NEXT. A crosstalk signal *xFEXT (t)* is called the far-end crosstalk FEXT. From a data communication point of view, the NEXT crosstalk is generally more damaging than the FEXT crosstalk, because the NEXT does not necessarily propagate through the line length and thus does not experience a propagation loss of the signal (Werner, 1991).

If either single or multiple interferers generate a crosstalk signal, we can define a gain of the NEXT crosstalk path according to (Werner, 1991), (Róka, 2002, 2004) using a following relation

$$\left| \left| \mathfrak{M}\_{\rm NEX}(l\_{\prime}f) \right| \right|^{2} = \frac{\pi^{2} \cdot f^{2} \cdot k\_{\rm NEX}}{a(f)} \left[ 1 - e^{-4.a(f) \cdot l} \right] \approx K\_{\rm NEX} \cdot f^{3/2} \tag{8}$$

where variables are given as *KNEXT* = 0,882.10-14.*Nd*0,6, *Nd* is the number of disturbing pairs (disturbers), *f* is the frequency in Hz. An approximation on the right in (8) is valid when the line length *l* is large and for frequency regions where the real part  *()* of the propagation constant is proportional to *f* . We can also derive a gain of the FEXT crosstalk path in a similar manner using a following relation

$$\left| \left| \mathfrak{M}\_{\text{FEXT}}(l, f) \right| ^2 = 4 \cdot \pi^2 \cdot f^2 \cdot k\_{\text{FEXT}} \cdot l \cdot e^{-2a(f) \cdot l} \approx K\_{\text{FEXT}} \cdot l \cdot 3280 \cdot f^2 \cdot \left| \mathfrak{M} \left( l, f \right) \right|^2 \tag{9}$$

where variables are given as *KFEXT* = 3,083.10-20, *l* is the line length in km, *f* is the frequency in Hz and **H** (*l*, *f*) expresses the transfer function of a metallic homogeneous symmetric line.

Fixed Transmission Media 45

The lower frequency *fLOW* is given by the spectral compatibility with narrowband services POTS and BA ISDN. The VDSL frequency plan depends on installation variations, on crosstalks from different sources and on the existence of narrowband services. Therefore, PSD masks of the VDSL signal are defined on the base of specific criteria. By (ETSI, 1999, 2001), we can discriminate between the VDSL deployment with or without the existence of narrowband services in the same cable and with or without the possibility for creating frequency apertures. The FTTEx variation seems to be the most probably variation of the VDSL deployment scenario where the line termination transceiver is placed in the central office exchange. Therefore, in our analysis, we used the FTTEx power spectral density mask. A graphical representation of the selected variation is shown on Fig. 3. If necessary, it is

possible to assign any other standard spectral mask for the considered VDSL signal.

Fig. 3. The PSD mask of the VDSL signal for the FTTEx variation with narrowband services

Because crosstalk signals from the POTS service in disturbing pairs don't have significant influences on the VDSL signal in the disturbed pair, we supposed both disturbing and disturbed signals to have the same power spectral densities *PSD1 (f)* = *PSD2 (f)*. This situation is happened when in neighboring pairs are also VDSL signals – we can talk about

For local subscriber loops, we assigned the power spectral density of NEXT and FEXT

2 23/2 *PSD f PSD f l f NEXT* ( ) ( ). ( , ) , ( , ) . *NEXT NEXT NEXT* **H H** *lf K f* (10)

0.6 <sup>49</sup> . <sup>10</sup> *d*

*<sup>N</sup> K K* (12)

2 2 22 ( ) ( ). ( , ) , ( , ) . .3280. . ( , ) *PSD f PSD f l f FEXT FEXT FEXT FEXT* **H H** *lf K l f lf* **H** (11)

Equations (10) – (11) allow very well approximation of practically observed kind of multiple interferer crosstalks. Before starting of the simulation, we determined values of variables *KNEXT* and *KFEXT* in the simulation model as a function of the number of disturbing pairs *Nd*

*NEXT NEXT*

**2.5 Spectral characteristics of near-end and far-end crosstalk signals**

the self-NEXT crosstalk and the self-FEXT crosstalk.

where variables *KNEXT* and *KFEXT* are functions of disturbed pairs.

crosstalks using relations (8) and (9) as follows

and the ADSL presence

for typical 50-pairs cable as

## **2.3 Impulse noise signal**

Due to the important effect of this negative influence, we took into account also this type of noise. The most common and the most damaging type of impulse noise seems to occur when a disturbed pair shares a common cable sheath with switched disturbing pairs – that is usual in the local access network. Sharp voltage changes can occur on analog pairs because of the opening and closing of relays. These voltage changes when are coupled into neighboring pairs through the NEXT and FEXT coupling path, create spurious, impulsivelike voltages whose amplitudes can be quite significant (Werner, 1991).

In unshielded twisted pairs, various equipments and environmental disturbances such as signaling circuits, transmission and switching gear, electrostatic discharges, lightning surges and so forth can generate an impulse noise. The impulse noise has some reasonably welldefined characteristics. Features of the typical impulse noise can be summarized as follows:

Of course, mentioned features don't characterize all possible impulse noise signals. In the simulation model, therefore, characteristics of the impulse noise signal can be randomly varied.

## **2.4 Spectral characteristics of the VDSL signals**

The VDSL modem uses a frequency bandwidth up to 20MHz. From this reason, the VDSL transmitter must solve situations that are not emergent in other xDSL modems. To these problems belong a spectral compatibility and cooperation with installed xDSL systems and a high level of different crosstalk signals.

In VDSL modems, various types of duplex methods and proposed modulation techniques are considered in (Cherubini et al., 2000), (Mestagh et al., 2000) and (Oksman & Werner, 2000). The ETSI recommendation binds producers to keep an established frequency plan (European Telecommunications Standards Institute [ETSI], 1999). The VDSL system can work in a frequency band bounded by frequencies *fLOW* and *fHIGH* (Fig. 2). In given frequency areas, the power of the VDSL signal must be adjusted to a level that can ensure the spectral compatibility with older xDSL systems. Alike, the signal level must be decreased to obviate undesirable emissions RFI, concretely caused by amateur radio stations.

Fig. 2. The general frequency plan for the VDSL system

Due to the important effect of this negative influence, we took into account also this type of noise. The most common and the most damaging type of impulse noise seems to occur when a disturbed pair shares a common cable sheath with switched disturbing pairs – that is usual in the local access network. Sharp voltage changes can occur on analog pairs because of the opening and closing of relays. These voltage changes when are coupled into neighboring pairs through the NEXT and FEXT coupling path, create spurious, impulsive-

In unshielded twisted pairs, various equipments and environmental disturbances such as signaling circuits, transmission and switching gear, electrostatic discharges, lightning surges and so forth can generate an impulse noise. The impulse noise has some reasonably welldefined characteristics. Features of the typical impulse noise can be summarized as follows:

Of course, mentioned features don't characterize all possible impulse noise signals. In the simulation model, therefore, characteristics of the impulse noise signal can be randomly

The VDSL modem uses a frequency bandwidth up to 20MHz. From this reason, the VDSL transmitter must solve situations that are not emergent in other xDSL modems. To these problems belong a spectral compatibility and cooperation with installed xDSL systems and

In VDSL modems, various types of duplex methods and proposed modulation techniques are considered in (Cherubini et al., 2000), (Mestagh et al., 2000) and (Oksman & Werner, 2000). The ETSI recommendation binds producers to keep an established frequency plan (European Telecommunications Standards Institute [ETSI], 1999). The VDSL system can work in a frequency band bounded by frequencies *fLOW* and *fHIGH* (Fig. 2). In given frequency areas, the power of the VDSL signal must be adjusted to a level that can ensure the spectral compatibility with older xDSL systems. Alike, the signal level must be decreased to obviate

like voltages whose amplitudes can be quite significant (Werner, 1991).

occurs about 1-5 times per minute (on an average 4 times per minute),

undesirable emissions RFI, concretely caused by amateur radio stations.

Fig. 2. The general frequency plan for the VDSL system

has peak values in the range 2 - 33 mV,

has time duration in the range 30 - 150 s.

has most of its energy concentrated below 40 kHz,

**2.4 Spectral characteristics of the VDSL signals**

a high level of different crosstalk signals.

**2.3 Impulse noise signal**

varied.

The lower frequency *fLOW* is given by the spectral compatibility with narrowband services POTS and BA ISDN. The VDSL frequency plan depends on installation variations, on crosstalks from different sources and on the existence of narrowband services. Therefore, PSD masks of the VDSL signal are defined on the base of specific criteria. By (ETSI, 1999, 2001), we can discriminate between the VDSL deployment with or without the existence of narrowband services in the same cable and with or without the possibility for creating frequency apertures. The FTTEx variation seems to be the most probably variation of the VDSL deployment scenario where the line termination transceiver is placed in the central office exchange. Therefore, in our analysis, we used the FTTEx power spectral density mask. A graphical representation of the selected variation is shown on Fig. 3. If necessary, it is possible to assign any other standard spectral mask for the considered VDSL signal.

Fig. 3. The PSD mask of the VDSL signal for the FTTEx variation with narrowband services and the ADSL presence

#### **2.5 Spectral characteristics of near-end and far-end crosstalk signals**

Because crosstalk signals from the POTS service in disturbing pairs don't have significant influences on the VDSL signal in the disturbed pair, we supposed both disturbing and disturbed signals to have the same power spectral densities *PSD1 (f)* = *PSD2 (f)*. This situation is happened when in neighboring pairs are also VDSL signals – we can talk about the self-NEXT crosstalk and the self-FEXT crosstalk.

For local subscriber loops, we assigned the power spectral density of NEXT and FEXT crosstalks using relations (8) and (9) as follows

$$PSD\_{\rm NEXT}(f) = PSD(f) \cdot \left| \mathcal{H}\_{\rm NEX}(l, f) \right|^2, \ \left| \mathcal{H}\_{\rm NEX}(l, f) \right|^2 \approx K\_{\rm NEX} \cdot f^{3/2} \tag{10}$$

$$PSD\_{\rm FET}(f) = PSD(f) \cdot \left| \mathcal{H}\_{\rm FET}(l, f) \right|^2 / \left| \mathcal{H}\_{\rm FET}(l, f) \right|^2 \approx K\_{\rm FET} . 1.3280 . f^2 \cdot \left| \mathcal{H} \left(l, f\right) \right|^2 \tag{11}$$

where variables *KNEXT* and *KFEXT* are functions of disturbed pairs.

Equations (10) – (11) allow very well approximation of practically observed kind of multiple interferer crosstalks. Before starting of the simulation, we determined values of variables *KNEXT* and *KFEXT* in the simulation model as a function of the number of disturbing pairs *Nd* for typical 50-pairs cable as

$$K\_{NEXT} = K\_{NEXT-49} \cdot \frac{N\_d^{0.6}}{10} \tag{12}$$

Fixed Transmission Media 47

Inverse precoder Demodulator QAM demapp./

The transmitted message carried to the receiving part is generated as a random binary chain with the given length. This message is also saved (for the BER calculation), encoded by a particular type of the FEC codes (RS, BCH) and modulated. It can be chosen from two kinds of modulations - the first one is the classical QAM modulation and the second one is a combination of the QAM and the convolutional coding, i.e. the TCM. The QAM modulation is chosen because of its compatibility with considered precoding techniques, a low distortion resistance and an easy implementation in the Matlab program environment.

Utilizing of local subscriber loops for the broadband access of subscribers by means of the VDSL technology assumes a replacing of the significant part of metallic lines by optical fibers. This will bring a subscriber distribution point (SDP) unit closer to subscribers, when metallic lines will distribute signals to subscriber premises. Although metallic lines will comprise only a small part of the transmission path, their influences on transmitted signals

Negative influences of the VDSL environment at the signal transmission depend on parameters of metallic homogeneous lines (a core material, a cable insulation, a core diameter, a number of neighboring lines in the cable binder, a cable length). If we want to achieve exact results from simulations, all these factors must be accepted. Of course, this acceptance leads to a complicated and complex simulation model. For modeling of all these influences, a theoretical description together with simulation methods is introduced (Róka & Cisár, 2002).

 The block scheme of the VDSL simulation model

$$K\_{EXT} = K\_{EXT-49} \frac{N\_d^{0.6}}{10} \tag{13}$$

The value of the *KNEXT-49* variable is given as 8,8.10-14, the value of the *KFEXT-49* variable is empirically estimated as 8.10-20/3280 (Aslanis & Cioffi, 1992).

## **3. The simulation model for the VDSL technology**

For considering of the signal transmission on metallic homogeneous lines by means of the VDSL technology, it is necessary comprehensively to know characteristics of negative environmental influences and features of applied modulation techniques. It is difficult to realize of the exact analytical description of complex systems such as the VDSL system in the real environment of local access networks. In addition, due to dynamical natures of some processes, it is not suitable. For analyzing of various signal processing techniques used by the VDSL technologies, a suitable and flexible enough tool are computer simulations and modeling schemes of real environmental conditions at the signal transmission.

For modeling of the VDSL transmission path, we used the software program *Matlab* together with additional libraries like *Signal Processing Toolbox* and *Communication Toolbox*. A proposed and realized modeling scheme represents a transmission of high-speed data signals in downstream and upstream directions by means of the VDSL technology utilizing of metallic homogeneous lines. The realized model (Fig. 4) represents the signal transmission in the VDSL environment utilizing metallic homogenous lines for high-speed data signals in the downstream and upstream direction. This VDSL environment model is the enhanced version of the ADSL environment model introduced (Róka & Cisár, 2002). New features of this simulation model are VDSL transmission characteristics and applications of precoding techniques and trellis coded modulations.

Basic functional blocks realized in our simulation model are shown on Fig. 4. The VDSL simulation model (Fig. 4) can be divided into the three main parts:

The analysis can be based on computer simulations that cover the most important features and characteristics of the real transmission environment for the VDSL technology and result in searching for the best combination of coding and precoding techniques.

The value of the *KNEXT-49* variable is given as 8,8.10-14, the value of the *KFEXT-49* variable is

For considering of the signal transmission on metallic homogeneous lines by means of the VDSL technology, it is necessary comprehensively to know characteristics of negative environmental influences and features of applied modulation techniques. It is difficult to realize of the exact analytical description of complex systems such as the VDSL system in the real environment of local access networks. In addition, due to dynamical natures of some processes, it is not suitable. For analyzing of various signal processing techniques used by the VDSL technologies, a suitable and flexible enough tool are computer simulations and modeling schemes of real environmental conditions at the signal

For modeling of the VDSL transmission path, we used the software program *Matlab* together with additional libraries like *Signal Processing Toolbox* and *Communication Toolbox*. A proposed and realized modeling scheme represents a transmission of high-speed data signals in downstream and upstream directions by means of the VDSL technology utilizing of metallic homogeneous lines. The realized model (Fig. 4) represents the signal transmission in the VDSL environment utilizing metallic homogenous lines for high-speed data signals in the downstream and upstream direction. This VDSL environment model is the enhanced version of the ADSL environment model introduced (Róka & Cisár, 2002). New features of this simulation model are VDSL transmission characteristics and

Basic functional blocks realized in our simulation model are shown on Fig. 4. The VDSL

1. A transmitting part - it is responsible for the encoding (using the FEC technique) and for the modulation of signals into a form suitable for the transmission channel. 2. A transmission channel (the metallic homogenous line) - this part of the model realized negative influences on the transmitted signal. Above all, it goes about a propagation loss, a signal distortion, crosstalk noises, white and impulse noises, the radio interference. Because these negative influences expressively interfere into the communication and represent its main limiting factors, they present a critical part of the model and, therefore, it is necessary exactly to recognize and express their

3. A receiving part - it is conceptually inverted in a comparison with the transmitter. Its main functions are the signal amplification, the removing of the ISI, the demodulation

The analysis can be based on computer simulations that cover the most important features and characteristics of the real transmission environment for the VDSL technology and result

in searching for the best combination of coding and precoding techniques.

*FEXT FEXT*

empirically estimated as 8.10-20/3280 (Aslanis & Cioffi, 1992).

**3. The simulation model for the VDSL technology**

applications of precoding techniques and trellis coded modulations.

simulation model (Fig. 4) can be divided into the three main parts:

characteristics by correct parameters.

and the correction of errored information bits.

transmission.

0.6

*<sup>N</sup> K K* (13)

<sup>49</sup> 10 *d*

Fig. 4. The block scheme of the VDSL simulation model

## **3.1 The transmitting part of the model**

The transmitted message carried to the receiving part is generated as a random binary chain with the given length. This message is also saved (for the BER calculation), encoded by a particular type of the FEC codes (RS, BCH) and modulated. It can be chosen from two kinds of modulations - the first one is the classical QAM modulation and the second one is a combination of the QAM and the convolutional coding, i.e. the TCM. The QAM modulation is chosen because of its compatibility with considered precoding techniques, a low distortion resistance and an easy implementation in the Matlab program environment.

## **3.2 The transmission line**

Utilizing of local subscriber loops for the broadband access of subscribers by means of the VDSL technology assumes a replacing of the significant part of metallic lines by optical fibers. This will bring a subscriber distribution point (SDP) unit closer to subscribers, when metallic lines will distribute signals to subscriber premises. Although metallic lines will comprise only a small part of the transmission path, their influences on transmitted signals will not negligible.

Negative influences of the VDSL environment at the signal transmission depend on parameters of metallic homogeneous lines (a core material, a cable insulation, a core diameter, a number of neighboring lines in the cable binder, a cable length). If we want to achieve exact results from simulations, all these factors must be accepted. Of course, this acceptance leads to a complicated and complex simulation model. For modeling of all these influences, a theoretical description together with simulation methods is introduced (Róka & Cisár, 2002).

Fixed Transmission Media 49

The influence of the transmission channel that we can derive from its transmission function is expressed above all at the attenuation of the transmitted signal. The signal attenuation is more accentuated for areas of higher frequency components of power spectral density characteristics. This influence is more expensive for longer line lengths. However, we can find out that the influence is decreased with increasing of the core diameter of wires. This results from a change of values for the primary constant of the line, concretely *R* and *L*.

We can create the NEXT crosstalk noise signal by forming of the white noise spectrum (with constant PSD = 0 dB/Hz) that is generated by a random number generator. First, we must calculate a frequency characteristic of the **H***NEXT* (*l*, *f*) crosstalk transmission function using (8). Its parameters are the number of disturbing pairs and the appropriate value of the variable *KNEXT*. On Fig. 6, the NEXT crosstalk transmission function for various numbers of disturbers is presented. For modeling of the NEXT negative influence, the NEXT crosstalk noise signal acquired by filtering is added to the transmitted signal entering the

The FEXT crosstalk signal is created in a similar manner as the NEXT crosstalk signal (its spectrum is presented on Fig. 6). Because this type of a crosstalk must be propagated through a disturbing line, we included into calculating the **H***FEXT* (*l*, *f*) crosstalk transmission function also the transmission function of the line **H** (*l*, *f*) using (9) with given parameters. The FEXT crosstalk signal is added to the transmitted information signal attenuated at

> Nd = 49 Nd = 25 Nd = 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 **f (MHz)**

Fig. 6. The frequency characteristic of NEXT and FEXT crosstalk transmission functions for

0 -220

**HFEXT (dB)**

The influence of the FEXT crosstalk transmission function is characterized by the power spectral density of the FEXT crosstalk signal and by the FEXT crosstalk path. This FEXT crosstalk path is depend on the line length, on the frequency of signal and on the transmission function of the transmission line because of propagating of crosstalk signals through the disturbing pair. For longer line lengths, the influence of the FEXT crosstalk can be neglected. On other side, this influence is accentuated at higher frequency components of the transmitted signal. Therefore, it is necessary to take into account of the FEXT crosstalk

a transmission through the metallic homogeneous line.

Nd = 49 Nd = 25 Nd = 1

transmission line.

various numbers of disturbers

**HNEXT (dB)**

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

**f (MHz)**

#### **3.3 The receiving part of the model**

At the receiver side, the distorted and attenuated signal is first amplified, next demodulated and then shifted into the inverse precoder that removes constellation changes introduced by the precoder at the transmitter side. If the TCM is used, the TCM decoder follows and the Viterbi algorithm is searching the most probably binary sequence. Otherwise, the QAM demapping block converts the constellation points sequence into the binary data sequence that is corrupted by transmission errors. They are consequently removed in the FEC decoder. Finally, the corrected sequence is compared with the original transmitted message and the bit error rate is calculated.

We should notice that individual precoding techniques suppose conceptually different manner of the signal regeneration, therefore this process is not here exactly described. For both methods, it is essential to know a transmission function of metallic lines. This information can be extracted from the signal transmission of predefined symbol sequences at the initialization process.

Before starting of the simulation, we can calculate the transmission function of the line for given parameters *(l, )*. Values of this function are sampled in equal proportioned frequency intervals in the range from 0 Hz up to the half of the sampling frequency (*fsamp /2*). A number of samples is optional. For signal processing, it is desirable to choice the number of samples equal to *2.N*, where *N* is an integer number. Using sampled values, the impulse characteristic of the transmission line *h (t)* is calculated using the inverse Fourier transformation with the same number of samples (from practical viewpoint, the number of 512 samples is adequate). This sampled impulse characteristic is used as coefficients for the digital filter. A simulation of the signal transmission through the line itself is executed by digital filtering of the sampled modulated signal using the proposed filter. On Fig. 5, there are shown frequency characteristics of the transmission line ( = 0,4 mm, Cu) for various line lengths.

Fig. 5. Frequency characteristics of the transmission line for various line lengths

At the receiver side, the distorted and attenuated signal is first amplified, next demodulated and then shifted into the inverse precoder that removes constellation changes introduced by the precoder at the transmitter side. If the TCM is used, the TCM decoder follows and the Viterbi algorithm is searching the most probably binary sequence. Otherwise, the QAM demapping block converts the constellation points sequence into the binary data sequence that is corrupted by transmission errors. They are consequently removed in the FEC decoder. Finally, the corrected sequence is compared with the original transmitted message

We should notice that individual precoding techniques suppose conceptually different manner of the signal regeneration, therefore this process is not here exactly described. For both methods, it is essential to know a transmission function of metallic lines. This information can be extracted from the signal transmission of predefined symbol sequences

Before starting of the simulation, we can calculate the transmission function of the line for

intervals in the range from 0 Hz up to the half of the sampling frequency (*fsamp /2*). A number of samples is optional. For signal processing, it is desirable to choice the number of samples equal to *2.N*, where *N* is an integer number. Using sampled values, the impulse characteristic of the transmission line *h (t)* is calculated using the inverse Fourier transformation with the same number of samples (from practical viewpoint, the number of 512 samples is adequate). This sampled impulse characteristic is used as coefficients for the digital filter. A simulation of the signal transmission through the line itself is executed by digital filtering of the sampled modulated signal using the proposed filter. On Fig. 5, there

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fig. 5. Frequency characteristics of the transmission line for various line lengths

**f (MHz)**

*)*. Values of this function are sampled in equal proportioned frequency

l = 3.0 km l = 3.5 km l = 4.0 km l = 4.5 km l = 5.0 km

= 0,4 mm, Cu) for various

**3.3 The receiving part of the model**

and the bit error rate is calculated.

are shown frequency characteristics of the transmission line (

**H (dB)**

at the initialization process.

given parameters *(l,*

line lengths.

The influence of the transmission channel that we can derive from its transmission function is expressed above all at the attenuation of the transmitted signal. The signal attenuation is more accentuated for areas of higher frequency components of power spectral density characteristics. This influence is more expensive for longer line lengths. However, we can find out that the influence is decreased with increasing of the core diameter of wires. This results from a change of values for the primary constant of the line, concretely *R* and *L*.

We can create the NEXT crosstalk noise signal by forming of the white noise spectrum (with constant PSD = 0 dB/Hz) that is generated by a random number generator. First, we must calculate a frequency characteristic of the **H***NEXT* (*l*, *f*) crosstalk transmission function using (8). Its parameters are the number of disturbing pairs and the appropriate value of the variable *KNEXT*. On Fig. 6, the NEXT crosstalk transmission function for various numbers of disturbers is presented. For modeling of the NEXT negative influence, the NEXT crosstalk noise signal acquired by filtering is added to the transmitted signal entering the transmission line.

The FEXT crosstalk signal is created in a similar manner as the NEXT crosstalk signal (its spectrum is presented on Fig. 6). Because this type of a crosstalk must be propagated through a disturbing line, we included into calculating the **H***FEXT* (*l*, *f*) crosstalk transmission function also the transmission function of the line **H** (*l*, *f*) using (9) with given parameters. The FEXT crosstalk signal is added to the transmitted information signal attenuated at a transmission through the metallic homogeneous line.

Fig. 6. The frequency characteristic of NEXT and FEXT crosstalk transmission functions for various numbers of disturbers

The influence of the FEXT crosstalk transmission function is characterized by the power spectral density of the FEXT crosstalk signal and by the FEXT crosstalk path. This FEXT crosstalk path is depend on the line length, on the frequency of signal and on the transmission function of the transmission line because of propagating of crosstalk signals through the disturbing pair. For longer line lengths, the influence of the FEXT crosstalk can be neglected. On other side, this influence is accentuated at higher frequency components of the transmitted signal. Therefore, it is necessary to take into account of the FEXT crosstalk

Fixed Transmission Media 51

a chosen spectral mask. Alike, it is easy to keep frequency apertures in given subbands at occurrences of disturbing frequencies of the RFI type. Therefore, the MCM allows a high flexibility of the PSD transmitted signal and tries as effective as possible utilizations of the available bandwidth. We can therefore suppose that the transmitted VDSL signal will achieve maximum allowable levels in particular subchannels according to the FTTEx