Transient Analysis of LCLT Network With Two Switching Instants

8/2/2019 Transient Analysis of LCLT Network With Two Switching Instants

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Transient Analysis of Square-Wave Driven LCL-T Network

Advanced Power Systems II

Submitted By: Bryan Esteban (Group C) ID: 102275388

Submitted To: Dr. Govinda Raju

Submission Date: April 5, 2012


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Abstract

This paper presents the results obtained from the transient analysis of a hybrid series-

parallel inductor-capacitor-inductor resonant network, commonly known as an LCL-T

network. The network under consideration was subjected to the following two separate

transient conditions: 1) Turn-on transient, and 2) Short-circuit transient. The analysis of each

transient condition was completed by means of the following three separate approaches: 1)

Classical (analytical), 2) State-Space (numerical), and 3) LTspice circuit simulation

(numerical). All approaches showed to have good agreement with each other upon

comparison.


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Contents

1 Introduction1.1 Motivation and Background ............................................................................................. 1

1.2 Scope of Work & Project Deliverables ........................................................................... 5

2 Transient Analysis Via the Classical Approach

2.1 Classical Method Fundamentals Explained ..................................................................... 6

2.2 Application of Classical Method to LCL-Tranisent Analsys ......................................... 8

2.3 MATLAB Plots of Analytical Results ........................................................................... 12

3 Transient Analysis Via State-Space

3.1 State-Space Fundamentals Explained ............................................................................. 15

3.2 Application of State-Space Method to LCL-T Transient Analysis ............................ 17

3.3 MATLAB Plots of Numerical Results ........................................................................... 19

4 Transient Analysis Via LTspice Circuit Simulation

4.1 LTspice Fundamentals Explained ................................................................................... 21

4.2 Simmulation of Trasients for LCL-T Networks ........................................................... 22

4.3 MATLAB Plots of Simmulation Results ........................................................................ 23

5 Effect of Including Square Wave Harmonics

6 Conclusion ................................................................................................................................. 28

Appendix I Manual Derivation of Response Using Classical Method

Appendix II MATLAB Code for Plotting of Classical Method Results


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List of Figures1-1 Block Diagram of Typical Resonant IPT System............................................................................... 2

1-2 Physics Underlying Operation of Resonant IPT System .................................................................... 2

1-3 LCL-T Output Network for to be Analyzed for Transients at Two Switching Instants ....................... 5

2-1 Input Impedance Seen at the LCL-T Input Terminals by Incoming Signal ......................................... 82-2 Fourier Decomposition of Square-Wave and its Normalized Power Spectrum ................................... 9

2-3 Network at configuration at 0


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5-5 Total Response of Current through Inductor L2 ............................................................................ 26

5-6 Turn-On (Left) & Short-Circuit (Right) Transients for Current through L2 ..................................... 26


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Chapter 1

Introduction

1.1 Motivation and Background

Recent studies have shown the feasibility of using the near magnetic field to transfer

large amounts of power (i.e. >3.3 kW) across distances that span up to 10 inches with greater

than 92% efficiency across the full cascade of system components from the wall utilitysupply to the target device under charge [1-2]. This technology has come to be variously

known as Inductive Power Transfer (IPT), Resonant Magnetic Induction (RMI), or more

generally as Wireless Power Transfer (WPT). For this paper all of the foregoing terms will be

used interchangeably.

The fundamental difference between this type of IPT and the conventional magnetic

induction that has now been in use for nearly a century in such devices as transformers and

proximity chargers is the use of resonance at both the source and receiver inductive loops to facilitate

power transfer across larger distances (i.e. much smaller coupling coefficients) while still

maintaining a high transfer efficiency. As seen in figure 1-1, a typical resonant IPT system

generally has a source of high-frequency power anywhere from 20kHz to up to 145kHz

driving an inductive primary conductive loop, also commonly known as the track, through

a resonant network (i.e. in this context the term high frequency is intended to indicate that

the frequency of the signal driving the WPT system is very much higher than that of the

utility supply, which is 60Hz, and is not to be confused with the RF band). The resonant

network is made up of discrete capacitors and inductors that are chosen with a specific

operating frequency in mind, and interconnected in any of several available topologies.

Across the large air gap on the secondary side, there is a capture loop that is also part of asimilar, but not necessarily identical, resonant network of discrete capacitors and inductors

which are in turn chosen to match the primarys resonant frequency.


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Figure 1-1: Block Diagram of Typical Resonant IPT System

As shown in figure 1-2, when the primary inductive loop is driven with a time-varying

current alternating at the output networks resonant frequency (fo), a time-varying magnetic

field with the same frequency will be induced about its surrounding by virtue of the Ampere-Maxwell circuital law. In turn, when a secondary loop is brought within the vicinity of the

primarys inductive near field it will capture some of the flux produced by the primary and by

Faradays law of induction this captured flux will cause an AC potential difference in the

receiving loop which will in turn drive a time-varying current at the same frequency of the

inducing flux. By ensuring that the secondarys input network is also tuned to the sources

resonant frequency, power transfer efficiency is thus made to be maximum. It has recently

been shown that upwards of 98% transfer efficiency can be achieved from source loop to

receiver loop when transferring power levels in excess of 4 kW [3]. The received power can

then be rectified, regulated, and used to power a device or charge a battery.

Figure 1-2: Physics Underlying Operation of Resonant IPT System


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To date, the type of power source used most often in high power applications, such as

electric vehicle (EV) charging, is a resonant full-bridge converter topology with square wave

output [4-7]. The converter is termed as resonant because the fundamental frequency of the

square wave it produces at its output terminals drives the output network to which it is

connected at its inherent resonant frequency. This type of topology and operation is essential

to the efficiency of the entire system because it minimizes the instantaneous switching power

losses that are normally associated with hard switched power electronic converters by

ensuring that the switching instants of the converter switches take place when the current

drawn by the output network is very nearly zero, hence this type of switching is often called

Zero-Current and/or Zero-Voltage switching (i.e. ZCS/ZVS) [8].

In late 2010 the Society of Automotive Engineers (SAE) assembled an internationaltaskforce, formally known as SAEJ2954, to develop a working industry standard that

establishes the minimum performance and safety criterea for wireless charging of electric andplug-in vehicles [9]. There are a number of topics under the overall scope of SAEJ2954,including:

1. Classification of different charging types and minimum efficiency per charging type2. Interoperability including center operating (resonant) frequency of charging3. Communications & software (harmonize with SAE conductive charging)4.Validation testing (vehicle, charger, system)5. Parking alignment between the vehicle secondary coil and the primary coil of the

wireless charging unit (EV Supply Equipment, EVSE)

6. Location on vehicle and orientation of charger7. Safety items, including obstacle detection, both organic and inorganic; magnetic field

levels; charging battery state of charge levels and rate; temperature development tests;and electric shock

8. Design validation test and wireless charging verification test.The aforementioned developments in the state of the art clearly demonstrate that this

type of technology holds much research potential. In view of the foregoing, the author ofthis paper has chosen this growing area of research as the general topic of interest for hisgraduate studies. More specifically, the research topic under study is the Investigation ofPrimary Converter Topologies for IPT Systems. Research of this topic calls for theinvestigation and comparison of converter topologies used in IPT systems for low and highpower applications. The ultimate aim of the study is to propose and develop a new type ofconverter suitable for handling a range of applications. The research would start byinvestigating all possible types of primary converter topologies for generating a highfrequency magnetic field. A thorough comparison between all of them would be made, interms of complexity (cost and size), efficiency, frequency, etc. In addition, operationinvolving variations in both magnetic coupling and loading would be considered. This


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project will involve the construction and design of multiple primary converters to use as acomparative study and determine the optimal. To date, no such work has been thoroughlycarried out in terms of an application study.

In view of the fact that there are many different choices for output networks (i.e.

Series-Resonant, Parallel-Resonant, Series-Parallel-Resonant, etc.), a key part of the authorsgraduate work will be the determination of the optimal output network topology andconverter type combination. The first topology chosen by the investigator to begin hiscomparative work is the LCL-T topology. The next section of this report will present thenature of the proposed analysis to be carried out on the LCL-T topology and its expectedoutcome.


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1.2 Scope of Work and Project Deliverables

As stated in the previous section of this report, the LCL-T network of figure 1-3 is just

one of several possible output network topologies that are presently being used for IPT

applications. The input source Vp represents square wave output which is fed from thesquare-wave inverter terminals. The inductive element in series with the resistance and

impedance is normally referred to as the magnetic pad when used for EV charging

applications, and it is what couples magnetically to a secondary coil when wireless power

transfer is being carried out. The resistor Rt models the very small equivalent series resistance

(ESR) of the magnetic pad and the impedance Z models the reflected impedance of the

secondary side. It is relevant to note that when the system is driven at its inherent resonant

frequency the reflected impedance is purely real [10-14].

Figure 1-3: LCL-T Output Network to be Analyzed for Transients at Two Switching Instants

When an IPT system is initially energized (i.e. turned-on) transients will inevitably take

place for a brief period of time leading to the systems steady-state operation. Moreover, if

during operation at steady state the load to which power is being wirelessly supplied is

removed (i.e. the vehicle is abruptly moved away from the magnetic pad), then the reflected

real resistance drops to zero. In turn, this abrupt change in the networks configuration leads

to transients also. This project has as its aim to analyze and model the transients that arise as

a result of the foregoing two conditions when a square-wave such as the one shown in figure

1-2 of the previous section is used to drive the system. The two switches in the circuit of

figure 1-3 model the foregoing two conditions. Specifically, SW1 models the effect of

energizing the system, while SW2 models the effect of abruptly removing the load on thesecondary side.

As the deliverables of the project, the current through each inductor and voltage across

the capacitor will be obtained by means of the following three methods: 1) Classical

approach, 2) State-Space Approach, and 3) SPICE circuit simulation.


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Chapter 2

Transient Analysis Via The Classical Approach

2.1 Classical Method Fundamentals Explained

The classical method of transient analysis is based on the determination of the

ordinary differential equations (ODEs) that describe the dynamic behavior of the system

under study, in this particular case an electrical circuit with three energy storage elements.The foregoing differential equations are obtained from the first principles or laws that

characterize lumped element circuit operation. These laws have come to be known as

Kirchoffs Voltage and Current Laws.

Once the ODEs that describe the circuit have been ascertained, the complete

solution is obtained by applying the superposition principle and summing the solution to

two separate equations, or system of equations. The first system solved for is the unforced

system which is characterized by a homogeneous set of differential equations that results

from setting all inputs to zero. The solution obtained from this initial step is commonly

referred to as the natural or transient response and is independent of the nature of the

forcing function acting upon the system, being completely determined by the circuits

topology and any previously stored energy.

The second system solved for is the system that includes all forcing functions. The

solution for this system is known as the forced or steady-state response and its general formis

solely dependent on the form on the input, while its actual values are affected by the wave-

shaping effects of the network elements.

Application of the general classical approach calls for the completion of five general

steps which can be summarized as follows [15]:

Determination of

1. characteristic equation and evaluation of its roots2. forced response


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3. independent initial conditions4. dependent initial conditions5. integration constants

The next section of this report will show how this five-step process was applied to the

LCLT network under study.


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2.2 Application of Classical Method to LCL-T Transient

Analysis

Following the first step of the five-step process previously outlined the characteristic

equation of the LCL-T network under study can be obtained by shorting the source and

calculating the s-domain equivalent impedance function seen at the terminals that connect

to the source as illustrated in figure 2-1.

Figure 2-1: Input Impedance Seen at the LCL-T Input Terminals by Incoming Signal

This step yields the following input impedance and characteristic equation

respectively:

|| 1 1.1

01.2

Upon substitution of the actual component values into (1.2) the characteristic

equation produces the following three modes:

759,600 , 10,200 119,670With the obtained modes the general form of the natural response is readily shown to

be:

1.3

1.316 98.03Next the forced response is obtained by carrying out simple phasor analysis with all

impedances being computed at the fundamental frequency of the input signal, namely 20

kHz. The force response is thus computed to be:


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4 arg0||arg 9.87 sin 37.521.4It is important to note that the reason why only the fundamental frequency

component of the square-wave input is considered in the computation of the forced

response is that it simplifies the analytical solution process greatly while still providing a highlevel of accuracy. This later statement concerning accuracy is better understood by noting

that most of the power of this type of signal is carried in the fundamental component as

seen from the normalized power spectrum shown figure 2-2.

Figure 2-2: Fourier Decomposition of Square-Wave and its Normalized Power Spectrum

With both the natural response and the forced response in hand, the total response is:

9.87

sin

37.52

1.5

This last expression can be simplified into a more compact one by means of phasors:

9.87 sin 37.52 0 251.6

The final step that remains is the computation of the constants A1 through A3.

Because this is a third order system, we need an initial condition as well its first two

derivatives at the switching instant. The initial condition is simply taken to be zero because

the system has zero energy (i.e. zero current through L1) when it is first turned on. The

remaining derivatives needed must be obtained through auxiliary KCL & KVL equations

that relate the first and second derivatives of the signal of interest to known quantities.


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Figure 2-3: Network at configuration at 0


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The complete total response for the current through inductor L1 is thus:

0.002 351.93 142.6 9.87 sin 37.521.12Next the capacitor voltage and inductor current are easily obtained with the aid of

equations (1.7) and (1.9) as:

0.0087 238.8 23.438 227.3 118.311.13 0.086 61.31 48.07 57.5684 127.451.14

All currents and voltages obtained are valid for the time interval 0 25 .Evaluation of equations (1.12) to (1.14) at the 25ms boundary of this latter interval yields the

initial conditions to be used in obtaining the transient response of the system after the

second switching instant takes place. The foregoing initial conditions are as follows:

25 216.4035A 25 45.7023 25 200.1348VBy following the same procedure used to obtain the transient response for the first

switching instant the transient response for the second switching instant is readily found to

be:

4.3213 300.03 44.34 34.45 sin 3.7991.15 18.54 203.59 50.51 50.513 28.891.16 181.82 52.26 138.32 72.43 93.371.17

1.347 51.59These expressions are valid for t>25ms.


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2.3 MATLAB Plots of Analytical Results

The transient response components for both switching instants along with the steady-

state component are shown plotted individually on the same time axis in figures 2-4 and 2-5

respectively. From the first plot (first switching instant) we see that the first transient term isinfinitesimally small in amplitude and dies out extremely quickly (i.e. 5 = 6.58us). The

second term, which is a damped sinusoid has a significantly larger amplitude and remains for

490us. It is worth noting that there is a net 180.12o phase shift between the second transient

term and the steady-state term, which results in a reduction of the greater steady-state terms

amplitude for the duration of the transient term. This in turn results in a smoother transition

to steady-state.

Figure 2-4: Individual IL1 Components for 0


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Figure 2-5: Individual IL1 Components for t>25ms

The following are the plots of the total response for the two inductor currents and the

capacitor voltage shown for 1) the entire 50ms window, 2) for the first 800us f the first

transient, and 3) for the time interval spanning the duration of the second transient (i.e.

.0246 ms


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Figure 2-8: Total Response of Voltage across Capacitor C

Figure 2-9: Capacitor Voltage Turn-On (Left) & Short-Circuit (Right) Transients

Figure 2-10: Total Response of Current through Inductor L2

Figure 2-11: Turn-On (Left) & Short-Circuit (Right) Transients for Current through L2


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Chapter 3

Transient Analysis Via State-Space

3.1 State-Space Fundamentals Explained

The state-space method is a powerful mathematical tool that readily lends itself for

rapid numerical solution of complex systems of differential equations by means of fast

computers or digital processors. This approach is based on the notion that the future state ofa system can be fully described by a non-unique set of time-dependent system variables

formally known as state variables [16]. To understand the fundamental concepts underlying

this powerful tool, the following basic definitions must be formally established:

System State. The state of a dynamic system is the smallest set of variables (called statevariables) such that knowledge of these variables at some initial instant of time, along withknowledgeof the future system inputs, completely determines the behavior of the system foranyfuture time instant.

State Space.The n-dimensional space whose coordinate axes consist of the xl axis, x2 axis, .. . , x, axis, where xl , x2,. . . , x, are state variables; is called a state space. Any state can berepresented by a point in the state space.

State-Space Equations. State-space analysis is concerned with three types of variables thatplay a prominent role in modeling dynamic systems, these variables are: input variables,output variables, and state variables. The state-space representation for a given system is notunique, except that the number of state variables is the same for any of the different state-space representations of the same system. The system being considered is required to have

elements that maintain a history of the values of the input for t > t1. Since energy storageelements such as capacitors and inductors in an electrical system serve as memory devices,the outputs of such elements can be considered as the variables that define the internal stateof the system. The number of state variables to completely define the dynamics of thesystem is equal to the number of integrators involved in the system. It is common to expressthe system of differential equations that describe the system under study in the followingcompact matrix notation:


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B (3.1) D (3.2)where x is the state vector,y is the output vector, u is the input vector, A is the state matrix,B is the input matrix, C is the output matrix, and D is the direct transmission matrix. The

block diagram of figure 3.1 illustrates the relationship between all of the foregoingmatrices/vectors pictorially [17].

Figure 3-1: Block Diagram of a General State-Space Model


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3.2 Application of State-Space Method to LCL-T Transient

Analysis

Using the state-space paradigm summarized in the previous section, the state variables

for the LCL-T network under study are chosen as the two inductor currents and the

capacitor voltage. With these variables in mind, a set of three equations can be obtained

from the circuit at each switching instant by writing two KVL equation, one about each of

the two loops, and one KCL equation about the central node. This process is illustrated for

the 0


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With the above system of equations expressed in state-space form the solution can

now be obtained numerically by using Simulink or any other desired computational software

package. For this particular project Simulink was used. The following are the functional

Simulink blocks that represent the LCL-Ts state-space model:

Figure 3-3: Simulink Implementation of State-Space Equation (3.6)

Figure 3-4: Simulink Implementation of State-Space Equation (3.11)


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3.3 MATLAB Plots of Numerical Results


capacitor voltage shown for 1) the entire 50ms window, 2) for the first 800us, and 3) for the

time interval spanning the second switching instant (i.e. .0246 ms


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Figure 3-10: Turn-On (Left) & Short-Circuit (Right) Transients for Current through L 2


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Chapter 4

Transient Analysis Via LTspice Simulation

4.1 LTspice Fundamentals Explained

Since its introduction in 1971, SPICE (Simulation Program with Integrated CircuitEmphasis) has become the most popular analog circuit simulation tool in use today. In the

last two decades, we have witnessed an exponential growth in the use of SPICE, with theaddition of Berkley SPICE 3 enhancements, and support for C code model and mix-modesimulation. Each vendor of SPICE simulation software has added features such as MonteCarlo analysis, schematic entry, and post simulation waveform processing, as well asextensive model libraries. In most cases, the manufacturers have modified the algorithms forcontrolling numerical convergence and have added new parameters or syntax for componentmodels. As a result, each electronic design automation (EDA) tool vendor has the basicBerkley SPICE 2 features and a unique set of capabilities and performance enhancements[18].

Circuit simulation programs, of which SPICE and derivatives are the most prominent,take a text netlist describing the circuit elements (transistors, resistors, capacitors, etc.) andtheir connections, and translate this description into equations to be solved. The generalequations produced are nonlinear differential algebraic equations which are solvedusing implicit integration methods, Newton's method and sparse matrix techniques.

Linear Technologys (LT) LTspice IV is a high performance SPICE simulator,

schematic capture and waveform viewer with enhancements and models for easing the

simulation of switching regulators. LTs enhancements to SPICE have made simulation of

switching regulators extremely fast compared to conventional SPICE simulators, allowingthe user to view waveforms for most switching regulators in just a few minutes. The

standard LTspice IV library includes Macro Models for 80% of Linear Technology's

switching regulators, over 200 op amp models, as well as resistors, transistors and MOSFET

models. The next section will explain how this powerful tool was utilized to analyze the

transient behavior of the LCL-T network under study.


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4.2 LTspice Simulation of LCL-T Network

Figure 4-1 shows a snapshot of the LTspice Schematic Editor implementation of the

LCL-T network under study along with the driving square wave inverter and line supply

rectifier. The two boxes in blue are voltage controlled switches that have been defined so as

to model the switching action at start-up and the short-circuit condition that takes place

25ms after. All elements in the circuit with the exception of the voltage sources are actual

components available in the market. Their high order SPICE models have been obtained

from the vendor and loaded into the LTspice library. The foregoing fact makes the

simulation of this circuit as close as it can possibly be to a real life implementation. The next

section of this report will present the simulation results obtained from this model.

Figure 4-1: LTspice Implementation of Square-Wave Driven LCL-T Network


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4.3 MATLAB Plots of Simulation Results


capacitor voltage shown for 1) the entire 50ms window, 2) for the first 800us f the first

transient, and 3) for the time interval spanning the duration of the second transient (i.e..0246 ms


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Chapter 5

Effects of Including Square-Wave HarmonicsIn section 2.2 it was stated that transient analysis of the LCL-T network with only the

fundamental component of the input signal being considered was acceptable due its carrying

the vast majority of the power. All of the plots to follow further corroborate the soundness

of the foregoing assertion. Accordingly, each plot shows the signals thus far considered (i.e.

inductor currents and capacitor voltage) for the case where all harmonics are considered and

for when only the fundamental is considered. The plots for both cases are drawn on the

same axis so as to facilitate the visual comparison.




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Figure 5-3: Total Response of Voltage Across Capacitor C

Figure 5-4: Turn-On (Left) & Short-Circuit (Right) Transients for Capacitor Voltage




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From the foregoing plots we clearly see that for all cases the signal outputs with and

without harmonics are almost identical, as was initially asserted on the basis of the spectral

power distribution. The only case where there is an appreciable, though small, difference in

the response wave shape is in the capacitor voltage, figure 5-2, where high frequency

components are appreciable at the instants that coincide with the sharp transitions between

the positive and negative edges of the square-wave input.


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Chapter 6

ConclusionThe completion of transient analysis for two switching instants on the hybrid LCL-T

network by means of the three separate analysis techniques used (i.e. Classical, State-Space,

& LTspice Simulation) has been very useful in bringing to life the theory that was learned

throughout the Advanced Power Systems II course. Moreover, since the author of this paper

will eventually have to build and implement the network in question and others similar to

it as part of his graduate work, the analysis carried out advances his progress towards his

ultimate goal and sheds helpful practical insights into the operation of the circuit. The

aforesaid insight will in turn be of great value in the selection of component ratings that

ensure safe operation despite the sharp transient spikes that could potentially take place as

showcased by the results obtained from the analysis of the second switching instant.


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References

[1] J.M. Miller, M.B. Scudiere, J.W. McKeever, C. White. Wireless Plug-In ElectricVehicle (PEV) Charging, Oak Ridge National Labs Power Electronics Symposium,Oak Ridge, Tennessee, July 22, 2011.

[2] N. A. Keeling, G. A. Covic and J. T. Boys, "A Unity-Power-Factor IPT Pickup for

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[3] Brian Maffly. USU Invention Powers Electric Motors Without Wires. Internet:

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historic-milestone-for-electric-vehicles-125005864.html, July 26, 2011 [September 27,

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[4] H. Hao, G. Covic, M. Kissin and J. Boys, "A parallel topology for inductive powertransfer power supplies," in Twenty-Sixth Annual IEEE Applied Power ElectronicsConference and Exposition (APEC), 2011 2011, pp. 2027-2034.

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[7] J. Franke, "Contactless power transfer in electric vehicles in the Bavarian Technology

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Third Edition. Bill Sobrist, ED. New York: John Wiley & Sons Inc., pp. 249-291.


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[9] Brian Maffly. SAE taskforce J2954 on wireless charging and positioning standards

looking to have final draft of guideline this year; significant industry involvement.

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22, 2012 [March 26, 2012]

[10] M. B. Borage, K. V. Nagesh, M. S. Bhatia and S. Tiwari, "Characteristics and Designof an Asymmetrical Duty-Cycle-Controlled LCL-T Resonant Converter," IEEETransactions on Power Electronics, vol. 24, no. 10, pp. 2268-2275, 2009.

[11] M. Borage, S. Tiwari and S. Kotaiah, "LCL-T Resonant Converter With ClampDiodes: A Novel Constant-Current Power Supply With Inherent Constant-VoltageLimit," IEEE Transactions on Industrial Electronics, vol. 54, no. 2, pp. 741-746, 2007.

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Power Transfer System," 2006.

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[14] A.K.S.Bhat, .Analysis and Design of LCL-Type Series Resonant Converter,. IEEEINTELEC, pp172-178, (1994).

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[18] S.M. Sandler, C. Hymowitz. SPICE Circuit Handbook. New York: McGraw-Hill, 2006, pp.1-2.


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Appendix I Manual Derivation ofResponse Using Classical Method


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Analysis for First Switching Instant (t = 0)

Fourier series of Input Signal:

4 1,,,.. sin

4 sin 1.273 sin 4 cos 1.273 cos1 4 cos 225.3510 cosKVL1

(1)

KVL2

(2)KCL1

(3)


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Placing equations 1 3 in standard form:

(4) 1 1 Choosing the systems state variables to be the inductor currents and capacitor voltage the state-space model of

the model under study is:

0 0 10 11 1 0

100

0 0 0 0

00

System ofOutput Equations for remaining voltages & currents:

In Matrix form:

1100001001

1000


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Page | 35

11000010

01 1000

Differentiating (4) & then using (3) to sub for the derivative of the capacitor voltage

NATURAL RESPONSE:

||

1 0

0 (Characteristic equation) 5.65 5 12.6 3.9 7.808 10 2.99 10 1.09 10 0 759600 , 10200 119670Setting 759600 10200 119670


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FORCED RESPONSE:

4 arg0||arg 9.87sin 37.52|| arg 0.1289arg37.51 TOTAL RESPONSE:

9.87sin 37.52 9.87 sin 37.52 0 25 9.87 cos 37.52

9.87 cos 37.52 2 9.87 sin 37.52When the system is first energized all currents and voltages are zero, therefore:

0 6.0110 7.830 2 225.410 6.011


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1 1 0 26.0117.83225.3510 6.011

7.83 6.011 1 1 cos

sin

7.83 cos90 6.011 sin90 Initial conditions for next switching instant:

216.4035 45.7023 -200.1348

Analysis for Second Switching Instant (t = 25ms)


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Page | 38

NATURAL RESPONSE:

|| 1 0 0 (Characteristic equation)

5.65

5

12.6

0.3

7.808 10 2.99 10 8.43 10 0 742040 , 19380 27580Setting

742040 19380 27580

FORCED RESPONSE:

4 arg0||arg 0.957sin 3.799|| arg 1.333.799TOTAL RESPONSE:

0.957sin 3.799 0.957sin 3.799


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Page | 39

1


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Page | 40

Appendix II MATLAB Code forPlotting of Classical Method Results


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Page | 41

clc

a = 759600; %decay constant for transient term 1, tau1 = 1.31us (5*tau1 = 6.57us)

tau1 = 1/a;

b = 10200; %decay constant for second transient term, tau2 = 97.7us (5*tau2 =

488.5us)

w1 = 119670; %radian frequency of oscillating component of second transient term (f1

= 18.98kHz, T1 =52.69us)

wo = 2*pi*20000; %radian frequency of input signal ((f1 = 20kHz, T1 =50us)

Vp = 36; %magnitude of imput signal

L1 = 5.65*10^(-6);

L2 = 5*10^(-6);

C = 12.6*10^(-6);

to = .025;

A = [1 1 0; -a -b w1; a^2 (b^2 - w1^2) -2*b*w1];%system of equations for solving

ODE constants

B = [6.011*Vp; -7.83*wo*Vp; ((4*wo*Vp)/(pi*L1))-(6.011*Vp*wo^(2))];

X = inv(A)*B; %solving for constants via matrx inversion

M1 = sqrt(X(2)^2 +X(3)^2);

theta1 = atan2(X(2),X(3));

theta1_deg = 180*theta1/pi;

M2 = sqrt(b^2 + w1^2);

theta2 = atan2(w1, -b);


k1 = -7.83*L1*wo;

k2 = (4/pi)-6.011*L1*wo;M3 = sqrt(k1^2 + k2^2);

theta3 = atan2(k1, k2);


M4 = M2;

theta4 = atan2(-w1, b);


theta5 = theta2+theta4;

k3 = 1-C*L1*M2^2*cos(theta5);

k4 = -C*L1*M2^2*sin(theta5);

M5 = sqrt(k3^2 + k4^2);


k5 = 7.83-C*wo*M3*cos((pi/2)+theta3);

k6 = -(6.011+C*wo*M3*sin((pi/2)+theta3));

M6 = sqrt(k5^2 + k6^2);



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Ts = .1*tau1;

t1 = 0:Ts:.025;

i_L1_1 = X(1)*exp(-a*t1)+M1*exp(-b*t1).*sin(w1*t1+theta1)+9.87*Vp*sin(wo*t1-

(37.52*pi/180));

Vc_1 = a*L1*X(1)*exp(-a*t1)-L1*M2*M1*exp(-

b*t1).*sin(w1*t1+theta1+theta2)+Vp*M3*sin(wo*t1+theta3);

i_L2_1 = X(1)*(1+a^2*C*L1)*exp(-a*t1)+M5*M1*exp(-

b*t1).*sin(w1*t1+theta1+theta6)+Vp*M6*sin(wo*t1+theta7);

%subplot(3, 1, 1);

%plot(t1, i_L1_1);

%title('IL1 Current');

%axis([0 .025 -450 450]);

%grid on

%subplot(3, 1, 2);

%plot(t1, Vc_1);%title('Capacitor Voltage');

%axis([0 .025 -300 300]);

%grid on

%subplot(3, 1, 3);

%plot(t1, i_L2_1);

%title('IL2 Current');

%axis([0 .025 -80 80]);

%grid on

t2 = .025:Ts:.050;

u1 = 0.066*Vp-216.4035;u2 = 35.42*10^6-0.955*wo*Vp;

u3 = (4*wo*Vp/pi*L1)-(0.063*Vp*wo^(2))+(170.7012/L1*C);

E = [1 1 0; -a -b w1; a^2 (b^2 - w1^2) -2*b*w1];%system of equations for solving

ODE constants

F = [u1; u2; u3];

Y = inv(E)*F; %solving for constants via matrx inversion

N1 = sqrt(Y(2)^2 +Y(3)^2);

alpha1 = atan2(Y(2),Y(3));

N2 = sqrt(b^2 + w1^2);

alpha2 = atan2(w1, -b);

c1 = -.955*L1*wo;

c2 = (4/pi)-0.063*L1*wo;

N3 = sqrt(c1^2 + c2^2);

alpha3 = atan2(c1, c2);

N4 = N2;


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alpha4 = atan2(-w1, b);

alpha5 = alpha2+alpha4;

c3 = 1-C*L1*N2^2*cos(alpha5);

c4 = -C*L1*N2^2*sin(alpha5);

N5 = sqrt(c3^2 + c4^2);


c5 = 0.955-C*wo*N3*cos((pi/2)+alpha3);

c6 = -(0.0634+C*wo*N3*sin((pi/2)+alpha3));

N6 = sqrt(c5^2 + c6^2);


i_L1_2 = Y(1)*exp(-a*(t2-to))+N1*exp(-b*(t2-to)).*sin(w1*(t2-

to)+alpha1)+0.957*Vp*sin(wo*(t2-to)-(3.799*pi/180));

Vc_2 = a*L1*Y(1)*exp(-a*(t2-to))-L1*N2*N1*exp(-b*(t2-to)).*sin(w1*(t2-

to)+alpha1+alpha2)+Vp*N3*sin(wo*(t2-to)+alpha3);i_L2_2 = Y(1)*(1+a^2*C*L1)*exp(-a*(t2-to))+N5*N1*exp(-b*(t2-to)).*sin(w1*(t2-

to)+alpha1+alpha6)+Vp*N6*sin(wo*(t2-to)+alpha7);

t3 = [t1 t2];

i_L1 = [i_L1_1 i_L1_2];

Vc = [Vc_1 Vc_2];

i_L2 = [i_L2_1 i_L2_2];

plot(t3, i_L2, '-r', 'LineWidth', 2);

title('Current Through L2');

xlabel('Time (Seconds)');ylabel('Current (Amps)');

grid on;

Documents

Transient Analysis of LCLT Network With Two Switching Instants