TECHNOLOGY AND ENGINEERING APPLICATIONS OF SIMULINK pot

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.. In the Figure 5, some s

TECHNOLOGY AND

ENGINEERING APPLICATIONS OF

SIMULINK Edited by S.C Chakravarty

Technology and Engineering Applications of Simulink

As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications

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First published May, 2012

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Technology and Engineering Applications of Simulink, Edited by S.C Chakravarty

p cm

ISBN 978-953-51-0635-7

Contents

Preface IX

Chapter 1 Analysis of Power Electronic

Controllers in Electric Power Systems Using Simulink 1

Juan Segundo-Ramirez and A Medina Chapter 2 Study of Inductive-Capacitive

Series Circuits Using the Simulink Software Package 19

Titu Niculescu Chapter 3 Fixed Transmission Media 41

Rastislav Róka Chapter 4 Co-Simulation Procedure for

PID and Fuzzy Logic Active Controls Strategies Applied to a Sprayers Boom Suspension 69

Cristiano Okada Pontelli and Mario Francisco Mucheroni Chapter 5 S-Function Library for

Bond Graph Modeling 97

B Umesh Rai Chapter 6 From Control Design to

FPGA Implementation 129

Marcus Müller, Hans-Christian Schwannecke and Wolfgang Fengler Chapter 7 Describing Function Recording with

Simulink and MATLAB 149

Krunoslav Horvat, Ognjen Kuljaca and Tomislav Sijak Chapter 8 Performance Evaluation of a Temperature

Control Stage Used on a Semiconductor Gas Sensor 3D Electro-Thermal Model Through Simulink ® 167

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

Chapter 9 Matlab Simulink Model of

a Braked Rail Vehicle and Its Applications 189

Grażyna Barna Chapter 10 Using of Hybrid Supply for Electric or Hybrid Vehicles 219

N Rizoug, G Feld, B Barbedette and R Sadoun Chapter 11 The Uses of Artificial Intelligence for

Electric Vehicle Control Applications 239

Brahim Gasbaoui and Abdelfatah Nasri

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 thermo-electrical 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

Analysis of Power Electronic Controllers in Electric Power Systems Using Simulink

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

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, v statcom is in

phase with the voltage at the terminal bus v k In this situation, the current i statcom is completely

reactive If the magnitude of the voltage v k is higher than the magnitude of v statcom, 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 v statcom is higher than the magnitude of v k, 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

Thyristor-Based FACTS VSC-Based FACTS Shunt Connected Static Var Compensator (SVC) Static Compensator (STATCOM) Series Connected Thyristor Controlled Series

Compensator (TCSC)

Static Series Synchronous Compensator (SSSC) Back-to-Back Connected HVDC

Unified Power Flow Controller

(UPFC) HVDC VSC

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

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

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 m f =15 (900 Hz) The shunt transformer, the line 1 and the line 2 are represented by RL branches, whose impedances are 0.05 + × 0.005 Ω, 0.2 + × 0.015 Ω, and 0.15 + × 0.01 Ω, 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 = 0.05 , the voltage magnitude of the three-phase programmable voltage source is increased 15%, and finally at = 0.35 , 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 = 80 and = 0.5 Ω

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

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

(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

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 1 was selected

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 -400

0 10 20 30 40 50 60 70 80 0

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 – and

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

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 = 0.4 the voltage magnitude of the three-phase programmable voltage source is increased 15%, and finally at = 0.7 , 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 2 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

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

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)

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)

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

m f =27, which represents a commutation frequency of 1.61 kHz The RC passive filter has a

capacitance of 80 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

Fig 17 Series controller

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

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)

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

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

Modelling and Simulation in Power Networks, John Wiley & Sons, 9780470852712,

England

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

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

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

York

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

STATCOM Proceedings of the 38th southeastern symposium on system theory,

0-7803-9457-7, Cookeville, TN, march 2006

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,

Applications, and Design, Wiley, 0471226939, New York

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

978-81-224-2142-2, New Delhi

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

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

1815-1823, 0885-8977

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

Including UPFCs by Extrapolation to the Limit Cycle IEEE Transaction on Power Delivery, Vol 23, No 3, pp 1506-1512

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

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

Study of Inductive-Capacitive Series Circuits

Using the Simulink Software Package

Titu Niculescu

University of Petrosani

Romania

1 Introduction

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:

di

dt

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

equation of the circuit (1)

( ) (1  )

-t T

and it represents the time constant of the circuit

The voltage expression on the coil after connection is:

( )   

t T

di

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

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

Step

Scope U Scope I

du/dt Derivative Add

1 1/L

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

- D.C voltage applied: E = 24[V];

- The resistance value: R = 50[Ω];

- The circuit inductivity: L = 1[H],

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:

R = 50[Ω]; L = 1[H]

C ontinuous

powe rgui

v + - Voltage Measurement Step

Scope U

Scope I

i + -

C urrent Measurement s

C ontrolled Voltage Source

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

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

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

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

-t T

where T RC , represents the time constant of the circuit

The circuit current after closing is given by:



t T

du E

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

1

du

E u

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

Scope I 1500

R

Product2 Product1

Product

1 s Integrator

Divide

du/dt Derivative

-C-C

Add

1 1

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:

- D.C voltage applied: E = 24[V];

- The circuit resistance: R = 50[Ω];

- The circuit capacity: C = 100[µF],

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

Fig 14 The capacitor voltage variation

Fig 15 The current variation after closing

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

powergui

v + -

C urrent Measurement

s

- +

Fig 16 SimPowerSystems model of the capacitive circuit

The electrical parameters of the circuit have the values:

- R = 50[Ω]; C = 10[µF]

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

 Select by double clicking the Continous powerguy block from the simulation model

 Select Use LTI Wiewer in the new window;

 We will select U Voltage Measurement

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

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

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:

1

 di  