Electronic energy conversion with a switched series-resonant network operating at high frequency

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WITH A SWITCHED SERIES-RESONANT NETWORK

OPERATING AT HIGH FREQUENCY

SERIES-RESONANT NETWORK

OPERATING AT HIGH FREQUENCY

PROEFSCHRIFT

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, Prof. dr. J.M. Dirken, in het openbaar te verdedigen ten overstaan van het College van Dekanen op 14 oktober

1986 te 16.00 uur

door

Jan Berend Klaassens

elektrotechnisch ingenieur geboren te Assen

W

1986

DUTCH EFFICIENCY BUREAU - PIJNACKER

TR diss

1502

CIP-GESEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Klaassens, Jan Berend

Electronic energy conversion -with a switched

series-resonant network operating at high frequency / Jan Berend Klaassens. Pijnacker : Dutch Efficiency Bureau.

-111.

Proefschrift Delft. - Met lit. opg. - Met samenvatting in het Nederlands.

ISBN 90-6231-149-0

SISO 662 UDC 621.314(043.3)

Trefw.: vermogenselektronica / elektronische vermogensomzetters.

conversion

1.1. Introduction 1 1.2. Electric power conversion with series-resonant

converters 3 1.3. Critical components 12

1.4. Applications 17

2. Generalized series-resonant converter

2.1. Network configuration 18 2.2. Pulse forming network 23 2.3. Survey of conversion circuits 25

2.3.1. Dc-dc conversion 25 2.3.2. Dc-ac single phase conversion 29

2.3.3. Dc-ac three-phase conversion 32

2.3.4. Ac-ac conversion 33

3. Principles of series-resonant energy transfer

3.1. Series-resonant waveforms 34 3.2. Time domain analysis of switching networks 45

4. High frequency series-resonant energy conversion

4.1. Series-resonant energy conversion 50 4.2. Series-resonant converter for bipolar output

voltage and reversible power flow 58 4.3. Step-down series-resonant converter 64 4.4. Step-up series-resonant converter 80 4.5. Characteristics of the dc-dc series-resonant

converter 86 4.6. Energy storage in the resonant network 96

4.7. Conversion characteristics for a cyclic stable

operation 102 4.8. Characteristics of the output 111

5.3. Start-up conditions 148

6. Evaluation of experimental results

6.1. Experimental models 152 6.2. Dc series-resonant power converter for one quadrant

operation 152 6.3. Dc-dc series-resonant power converter for

four-quadrant operation 158 6.4. Dc-ac series-resonant power converter with a

single phase output 162 6.5. Dc-ac series-resonant power converter with a

three-phase output 165

6.6. Electronic control and protection system 169

7. Conclusions 174

List of notations and symbols 182

List of abbreviations 188

References 189

Samenvatting 203

1. INTRODUCTION TO THE STUDY OF SERIES-RESONANT POWER CONVERSION

1.1. Introduction

In the case of a load to be powered by electrical energy with­ drawn from a source, in almost every situation a method of energy conversion has to be applied. In the case of moderate specifications the energy conversion can be implemented by means of simple systems.

When the demands are high power converters have to be applied as a power interface between the source of energy and the load. The' power converter takes care of the conditioning of the flow of energy from the source*s) to the load(s) in which case the waveform of the electrical energy provided is prescribed by a reference signal(s).

The electrical system under consideration is in most cases com­ posed of smaller units symbolically represented in figure 1.1. A number of generators supply raw power to the centrally located power-processing system, which may consist of several individual and dissimilar units. The system provides con­ trolled electric power to the individual load subsystems as shown on the right side of the figure. This electrical power

Generators

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Storage

may be characterized by unipolar or bipolar potentials, contin­ uous or discontinuous waveforms or any combination thereof. Excess power may be transmitted into storage elements of electrical energy such as batteries and flywheels.

The stability of all the associated subsystems as well as the overall system has to be ascertained under all regular and irregular conditions of operation.

The power processor operates as a power interface between the sources of energy and the electrical loads and has to be able to reconcile conflicting characteristics. Neglecting these requirements may lead to improper system operation, which may in turn result in a catastrophic failure of subsystems due to the imposition of transient but nevertheless destructive stresses on component parts.

In terms of electrical losses (i.e. efficiency) it is evident that the control of the flow of energy have to take place with electronically controlled semiconductor switches, which means that pulse modulation processes are applied.

The properties of the source (dc-source, ac-source, etc.) and of the load (resistive or nonresistive load, dc- or ac-machines, electronic loads with or without rectifier input, radar tube, etc.) determine the configuration of the switching network.

Several classes of converters have been developed and industri­ ally applied. Chopper Gircuits used in traction applications and controlled rectifier bridges, in general applied as a power converter between three phase ac-systems and dc-loads, spring to mind.

A number of limitations and drawbacks have been recognized with respect to the above-mentioned converters which serves to stimulate the ongoing development of electronic power convert­ ers. Particularly in cases where the demands must meet high standards such as space and noncivilian applications, research

behorende bij het proefschrift van J.B. Klaasaens

De kwaliteit van processen, waarbij elektrische energie wordt omgezet met behulp van elektronische halfgeleiderschakelaars, die met een hoge interne pulsfrekwentie werken, kan worden verbeterd door het schakelen van stromen met vermogenshalfgeleiders te doen plaats vinden met een sterk begrensde waarde voor de snelheid van de stroomverandering.

Dit proefschrift.

F.C Schwarz, J.B. Klaassens,

A radar power supply without a voltage droop,

IEEE Power Electronics Specialists Conference, June 1983, Albuquerque, New Mexico, pp.377-384.

II

Het is onjuist te beweren, dat een verdere verhoging van de interne pulsfrekwentie van een elektronische vermogensomzetter tevens leidt tot een verdere vermindering van het specifieke gewicht van een dergelijke omzetter.

Dit proefschrift.

III

De introduktie van het begrip rendement zonder daarbij de bedrijfscondities te formuleren, kan gebruikt worden om het optreden van hoge verliezen bij de vermogensomzetting te flat­ teren. Het is aan te bevelen deze processen te kwalificeren aan de hand van de uitwisseling van energie onder gespecificeerde condities.

in een resonante kring gecombineerd met een transformator even­ eens laagfrekwente oscillaties op bij een conversie-verhouding kleiner dan een half.

R.J. King, T.A. Stuart,

Transformer Induced instability of the series-resonant con­ verter ,

IEEE Transactions on Aerospace and Electronic Systems, Vol.AES-19, No.3, May 1983, pp.474-482.

V

De door Stuart en King gegeven verklaring over het optreden van laagfrekwente oscillaties in een resonante kring gecombineerd met een transformator, is niet adequaat, aangezien deze verklaring instabiliteiten voorspelt, die niet optreden.

R.J. King, T.A. Stuart,

Transformer Induced instability of the series-resonant con­ verter ,

IEEE Transactions on Aerospace and Electronic Systems, Vol.AES-19, No.3, May 1983, pp.474-482.

VI

De opleiding tot elektrotechnisch ingenieur is ten zeerste gebaat bij het geven van een meer naar voren geschoven positie aan de grondslagen der energietechniek in het curriculum van de Facul­ teit der Elektrotechniek van de Technische Universiteit Delft.

VIII

Privatisering van overheidswerkzaamheden kan worden bevorderd door het negatieve verschil tussen salarissen van ambtenaren en die van medewerkers in het bedrijfsleven te vergroten.

IX

Het is een maatschappelijke inconsequentie, enerzijds te demon­ streren tegen het gebruik van kernenergie en te pleiten voor het sluiten van kerncentrales, terwijl anderzijds het roken van tabakswaren slechts beperkte weerstanden oproept en er geen aan­ leiding lijkt te zijn om te streven naar het sluiten van de tabaksindustrie.

X

In het kader van bezuinigingsoperaties van het ministerie van Onderwijs en Wetenschappen, verdient het aanbeveling voorstellen van de zijde van de onderwijsinstellingen serieus in het parle­ ment te behandelen, teneinde een evenwichtiger verhouding te bewerkstelligen tussen de omvang van het personeel van het departement en die van de onderwijsinstellingen.

activities are strongly being stimulated. Special emphasis is being placed on the improvement of reliability and the minia­ turization of power converters (particularly by increasing the internal pulse frequency) and the consequent decrease of inter­ nal losses.

The dynamic and static behavior in combination with the methods of analysis will have to be studied thoroughly. New methods of electrical energy conversion are being developed, founded on the ever-growing number of technological improvements on the applied components, which have led to new, industrially appli­ cable systems.

A power converter has to be able to draw its power from a power grid (indicated as a source) and to release it to the load (a passive or an active load as well). The character of the emit­ ted power will in general be related to one or more reference signals. Depending on the direction of the power transmission the functions indicated here as the source and load can be interchanged.

In general the character of emitted power indicates the exist­ ence of passive or active networks between (at least two) ports of the power converter.

Power processing includes distinct and interrelated criteria such as

a. low losses under all conditions of operation, b. low specific weight,

c. high reliability,

d. high reaction capability.

e. stability under all regular and irregular conditions of operation.

1.2. Electric power conversion with series-resonant converters The basic configuration of a power converter can be thought of as consisting of one or more matrices of lossless semiconductor

switches and equally lossless energy storage elements such as capacitors and inductors. Transformers have the ability to scale potentials and/or to provide galvanic insulation between the input and output ports of the power converter.

Clearly, the electrical components which are affected by sub­ stantial losses (such as a resistance) with respect to the criteria introduced, cannot be applied in the power conversion network to control the flow of energy.

A forced interrupt of current in the semiconductor switches will lead to excessive momentary values of the power compared to the conduction losses for the current conduction interval, which must be dissipated in those switching elements during the turn-on and turn-off intervals.

The switching losses are the fundamental reason for the in­ crease of the total losses in the semiconductor switches for an increased repetition frequency of the switches 111.

Depending on the electrical properties of the semiconductor switches the stated criteria can no longer be fulfilled for higher pulse repetition frequencies. It is shown that there is a maximum value for the pulse repetition frequency for which the dissipated energy in the semiconductor switch can no longer be transmitted to its environment with a limited temperature-rise.

For semiconductor switches suited to high voltages and/or high currents, the above-mentioned pulse repetition frequency is in the order of hundreds of herz.

The development of the theory and technology of electronic power converters with high internal frequencies satisfying the posed criteria is based on the special properties of resonant circuits in switching networks [2,4,5 1.

By programming a set of switches in a switching network config­ uration including a resonant circuit (with low electrical damping), one can manipulate the flow of energy through the resonant circuit.

This resonant circuit is inserted in the direct path of the energy transfer of the power converter. The energy can be withdrawn from a source (a single or polyphase voltage or a current source generating unipolar or bipolar waveforms) and transmitted to a network (load) in a specific form (a single or polyphase voltage or a current source generating unipolar or bipolar waveforms). The high frequency carrier that is gen­ erated can be manipulated by a pulse modulation process. The modulation process (the switching program of the semiconductor switches) controls the transmission of the electrical power from the source to the load and vice versa.

Low-pass filters designed for the (relatively high) switching frequencies can be applied to remove that specific part of the frequency spectrum that results from the switching process. The output waveform of the converter system embodies a frequen­ cy spectrum as desired by the load.

The waveform-transformation process is now realized by means of a switching process (pulse-modulation process) and not by means of passive filters, which are especially bulky and heavy for

low frequencies. a.

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converter generating a dc-output current with constant polarity for

a. a positive output voltage u ; o b. a negative output voltage u .

The principles of operation of a dc resonant converter system are discussed with reference to figure 1.2. A source of power is connected to the output terminals by a switched series-resonant network. A pulse-modulation process generates a pul­ sating, bipolar resonant current i , by exciting one single Beries-resonant circuit consisting of the passive elements C-^

and L.. The power system as a whole assumes the characteristics of a true secondary current source i^ as symbolically indicated in figure 1.2, since its primary power transfer and control mechanism is based on control of the transferred charge. This carrier is then demodulated by another set of semiconductor switches.

If the modulation process is accomplished by a rectification process of the high frequency carrier ij^, a rectified waveform is created at the output of the indicated diode rectifier in figure 1.2.a. The high frequency content is eliminated by a high frequency output capacitor C . The filter capacitor forms a short-circuit for the ac-component in the rectified resonant current i,. The result is a dc output current ïr^ü _1'av indicated in figure 1.3.

The average value of the output current through the load is controlled by varying the inter-pulse time T , =t,+-,—t, , where

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Figure 1.3 Simplified characteristic waveforms for a series-resonant dc-converter.

t^ is the firing instant of the kth current pulse.

By reversing the polarity of the output voltage UQ at the out­ put terminals of the converter, as indicated in figure 1.2.b, electrical power will be transmitted from the load through the resonant circuit to the source.

As a result of the positive biasing of the diode rectifiers indicated, electronically controlled semiconductor switches which can be turned both on and turned off have to be applied.

The application of two anti-parallel, controlled rectifier bridges as shown in figure 1.4, makes it possible to alter the polarity of the rectified current ir e c

-All combinations between the polarity of the output current i and voltage u , can be obtained by altering the polarity of the voltage at the output which is the tool necessary for the so-called four-quadrant operation.

If the demodulation process is accomplished by a selective rec­ tification of the resonant current i,, in which case the polarity of the rectified current ir e c is altered with a low frequency with respect to the carrier frequency, an alternating output current i is generated at the output terminals óf the power converter.

The high-frequency content in the output current is eliminated by a high-frequency output capacitor CQ.

A modulation process which modulates the inter-pulse time T . will control the wave shape of the alternating,

low-frequency output current i , as shown in figure 1.5.

The power converter employs a modulated train of power pulses to generate a synthesized, single-phase bipolar current wave­ form with bidirectional power flow for different characteristics of the load, varying from a passive capacitive load to a passive inductive load and active loads such as ac-machines.

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Figure 1.5 Simplified characteristic waveforms of a series-resonant converter generating a bipolar dc- or ac-output current.

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Figure 1.7 Simplified characteristic waveforms of a series-resonant converter generating a polyphase, bipolar output current.

A polyphase output configuration as illustrated in figure 1.6 can be thought of a multiple set of controlled rectifier bridges driven by one single ac-current source. The above-introduced methods of controlling the direction of the power flow through the series-resonant converter are applicable in combination with a selection procedure of one single recti­ fier out of the polyphase rectifier network at the output of the converter.

In the case of a variable selection of outputs resulting in a pulsed output current i for each individual pair of output terminals, in combination with a pulse-modulation process, it

is possible to generate polyphase, sinusoidal current waveforms at the output of the converter, as sketched in figure 1.7. The high-frequency content in the output current is eliminated by a polyphase high-frequency output capacitor C .

The process of waveform transformation described follows the principles of pulse area control. An aperiodic and non-uniform sampling process is applied which generates a required waveform after smoothing by a low-pass filter. The train of current pulses i-^ is modulated by varying the time spacing between the current pulses in order to obtain a current waveform of the programmed form. The power conversion process makes it possi­ ble to shape a current independently of the loading impedance and voltage polarity. The process described avoids the use of passive filters in order to implement a process of current wave shaping.

The (quasi) sinusoidal currents in the semiconductor switches allow the current through the switch to be interrupted at the zero crossing and, when combined with the limited value for the rate of rise of the current after turn-on and before turn-off, the switching losses can be reduced to a minimum. It is then possible to increase the above-mentioned barrier for the maxi­ mal pulse frequency up to hundreds of kHz. A high value for the efficiency can be maintained and no problems will occur

with respect to the dissipated power in the switches.

To be able to generate accurate waveforms, it appears to be greatly advantageous to create a sampling process with a high sampling frequency with respect to the fundamental output wave­ form. The existing resolution makes it possible to shape waveforms (for example a sinusoidal waveform with a frequency of 50 Hz) with a high degree of "high fidelity" (i.e. a low harmonic distortion). It is evident that as a consequence of the limited (minimum) value for the pulse repetition frequency, the bandwidth of the converter system is limited. As a result, more complex waveforms with spectral frequencies in the order of magnitude of the (minimum) pulse repetition frequency will not be generated without distortion.

Scaling of waveforms is possible by means of both a pulse modu­ lation process and a (current) transformer. The implementation of a transformer includes the possibility of galvanic insula­ tion between parts of the switching network. The distribution of energy through the transformer takes place with high fre­ quencies as well, so that its volume and weight can be decreased.

It is necessary to use modern magnetic materials suitable for high frequencies (ferrite materials and nickel-iron materials such as supermalloy). Problems concerning the thermal design of a transformer can be solved in this way while the efficiency of the high-frequency power converter will not be influenced negatively by the application of a transformer.

Clearly, this remark is also valid in the case of an inductor being applied in a switching network. However, for high power applications (>100 kW) it is advisable to use air-core inductors to improve the conversion efficiency.

The attractive properties of the quasi-sinusoidal high-frequency waveforms as they appear in this class of conversion networks are particularly based on

a. the absence of switching losses [1],

b. high reliability as characterized by the inherent short-circuit proof of the converter output,

c. the limited frequency spectrum of the resonant current, which means an advantage with respect to EMC, losses in mag­ netic materials, capacitors and galvanic connections,

d. the reduction in the weight (and volume) of the components, made feasible by the input and output filters resulting from the high pulse repetition frequency,

e. a high efficiency in the critical area of design, based on the almost constant current form factor of the resonant cur­ rent,

f. a great resolution for the pulse-modulated control system resulting from the high internal pulse repetition frequency.

The disadvantages can be summarized as follows:

a. optimization is more complex resulting from the fact that the switching frequency has to be related to the resonant frequency of the series-resonant network,

b. maximum values of voltages and currents reach values higher than the source and output voltages,

c. losses in the magnetic components are difficult to relate to the published data,

d. pulse-frequency modulation interferes with the analysis of the control aspects of the conversion system.

1.3. Critical components

A series-resonant converter embodies a switching network with active and passive components. In general, the resonant net­ work will be constructed of capacitors and inductors. This passive circuit will be excited by a set of semiconductor

r reliable operation of the power converter, namely power thyristors and transistors,

power diodes, resonant capacitors, resonant inductors, filter capacitors, transformers. . Thyristors.

In dc-converters it is permitted to apply asymmetrical semiconductor switches having the characteristic property that the maximum value of the positive anode-cathode voltage which can be blocked by the asymmetrical switch is much higher than the maximum value of the negative anode-cathode voltage.

In ac-converters only symmetrical semiconductor switches can be applied.

Critical switch parameters are degraded with respect to the parameters of asymmetrical semiconductor switches.

Advanced gate-drive techniques which generate a negative gate current just after the anode current in the thyristor switches have turned negative will influence the critical parameters in a positive way.

The critical parameters can be listed as follows 1. circuit turn-off time (t ).

The circuit turn-off time is one of the important factors which limits the maximum value of the pulse repetition frequency. A general rule of thumb for the maximum pulse repetition frequency f,,™-,, in the case of a practical design can be stated as f_.m_ =l/8t_.

A series-resonant conversion process with a pulse-repetition frequency of 10 kHz makes it necessary to apply thyristors with a maximum turn-off time t =12.5 ps under all regular and irregular conditions of operation. For practical design cases thyristors with a circuit turn-off time of t <15 ys under all conditions

have to be selected.

To guarantee the conditions for turning off the thyris­ tors under all modes of operation, the parameters indicated such as circuit turn-off time and the polarity of the voltage across the semiconductor switch which has to be turned off are continuously, electronically meas­ ured and evaluated.

2. The allowable rate of rise of the anode-cathode voltage must be as high as possible: dv/dt>500 V/ys, to limit the losses in the low-pass filter thereby protecting the semiconductor switch against excessive values of the dv/dt.

These losses are of course proportional to the time con­ stants of the low-pass RC-network, the square of the positive switch voltage and the pulse repetition fre­ quency. For high-frequency power converters the conventional RC-networks make it impossible to obtain an efficient operation. Efficient low-pass filters contain­ ing non-linear inductances are utilized, particularly for applications at high voltage levels.

3. The turn-on time of the thyristor is not critical because of the moderate values for the rate of rise of the anode current. Nevertheless the turn-on charac­ teristics, and especially the so-called spreading time, has to fulfil specific demands.

The turn-on behavior can be improved by injecting an optimal positive gate current.

The properties of the switches with respect to the rate of rise of the anode current di/dt are not critical in a series-resonant converter. Especially when symmetrical thyristors are applied the properties with respect to the behaviour of the reverse current are important in obtaining a reliable operation of the switch in a series-resonant network.

a2. Power transistors.

Faster semiconductor switches (such as bipolar transistors, power-FET's) make it possible to design and construct efficient series-resonant converters with pulse repetition frequencies from 10 kHz to more than 100 kHz [10,11,16,19, 23,24,25,261.

Power diodes.

Fast switching diodes with adequate qualifications with respect to the reverse current have to be applied in series-resonant converters.

Resonant capacitor.

The resonant capacitor has to be designed for a high rms-current: 10 to 50 A/yF. This indicates that the capa­ citor losses are important for good functioning of the resonant circuit. A critical aspect of the capacitor con­ struction .is the electrical connection inside the capac­ itor. The connections to the metal foil must be lossless and suitable for high current values.

Capacitors with plastic foil (such as polypropylene, polyester, polycarbonate) can be applied in the power con­ verter. The thermal design of the capacitor has to allow the removal of the losses with a limited temperature rise with respect to environmental temperature of the capacitor. The maximum value for the amplitude of the ac-voltage on the capacitor is, in the case of a conventional converter design approximately equal to 2.5 times the source voltage. In certain configurations such as the half-bridge configu­ ration with a center-tapped resonant capacitor [5,61, there will be a dc-component on which the above-mentioned ac-component is superimposed.

The capacitor need not be of a low-inductance type because of its position in the resonant circuit.

d. Resonant inductor.

The resonant inductor is constructed with high-frequency litze wire with an appropriate cross-section to avoid extra losses resulting from the high frequencies (skin effect, proximity effect) 117].

e. Filter capacitor.

In general the value of the rms-current through the filter capacitor is smaller than the rms-current through the reso­ nant capacitor. Therefore, the losses are less significant. What^s more, the value of the filter capacitor CQ is larger compared to that of the resonant capacitor

C-^ , which results in a smaller ratio of the rms-current

over the value of the capacitance.

In the case of a transformerless converter the value of the filter capacitance can be estimated by C =40*C,. For an adequate filter function the series resistance and the parasitic series inductance have to be as low as possible. Particularly for low-voltage converters electrolytic capa­ citors with a suitable frequency bandwidth can be applied.

f. Transformer.

Those aspects brought up by the resonant inductor can be applied to the transformer as well, except for the remarks with respect to the air core.

Recall that a large magnetizing current compared with the resonant current can disturb the stability of the oscilla­ tion. Consequently, measures with respect to the control have to be taken [28,481.

Because of the high internal pulse frequencies attention has to be paid to the mechanical structure, in particular to intercon­ nections and the way in which the components form electromagnetic subsystems.

1.4. Applications

The series-resonant techniques in electronic power converters can be applied in

a. dc-dc converters:

single quadrant dc-motor drives [6,7,151; four-quadrant dc-motor drives [18,221; power supply for electronic systems [111.

b. dc-ac converters:

single phase ac-motor drives with variable output voltage and frequency [271;

single phase power supply for electrical and electronic equipment with a conventional frequency (50, 60 or 400 H z ) ; single phase ac-traction;

three-phase ac-motor drives with variable output voltages and frequency [391.

c. ac-ac converters: ac-motor drives;

coupling of polyphase, asynchronous systems (wind turbines, hydro-generators);

power factor compensation.

d. high-voltage power supply with a conventional (low-voltage) current transformer in combination with a passive, series-resonant multiplier for radar tubes [16,19,23,24,25,261; laser tubes;

röntgen-diffraction apparatus; HVDC.

Prototypes up to 150 kW have been designed and constructed as a dc-converter with a series-resonant circuit, while those up to 12 kVA have been designed and constructed as an ac-converter with a series-resonant circuit.

2. GENERALIZED SERIES-RESONANT CONVERTER

2.1. Network configuration

The principles of series-resonant energy conversion are based on the properties of a non-linear, active network consisting of an undamped series-resonant LC-circuit and a number of semicon­ ductor switches, which transform the voltage waveform of a unipolar or bipolar, single or polyphase source of energy into

another waveform as required by the load. Scaling and stabili­ zation and wave shaping require the processess of inversion, ac-scaling with the aid of a transformer, selective and con­ trolled rectification and filtering. Inversion can include forms of pulse modulation and voltage stabilization and scaling can be implemented concurrently. The series-resonant converter employs an underdamped series-resonant circuit consisting of an inductor L. and a capacitor C,, in which the load current must cease before the switching elements can be opened. The power dissipation in the switching element is therefore near zero during its opening process. The series inductor L, limits the rate of rise of current after current conduction is initiated in any of the switching elements such that this current cannot attain a value of significance during the closing process of the switch.

The resonant network is excited from a source with a voltage e1 by a set of semiconductor switches as symbolically indicat­ ed in figure 2.1.

Energy will be extracted from the series-resonant circuit and delivered to a source with a voltage &j by a second set of

semiconductor switches.

The collection of semiconductor switches is indicated as SWITCHING MATRIX SMI and SM2. The voltage sources e± and

e. are connected to the PORTS 1-2 and 3-4, respectively of these switching matrices.

Each individual port is composed of two POLES and is indicated by the indices of the associated poles.

The pulsating flow of energy at both sides of the switching network must be converted in an almost continuous flow of energy by electric filters on the two sides of the converter. Both high frequency filters and the associated switching matrices are equally ranked in the system. Both voltage sources e, and e? can be interchanged.

Power losses in these voltage sources are represented by the resistors R 1 and Rs?» as indicated in figure 2.1.

Each individual switching matrix is a collection of so-called SWITCHING LEGS. Each switching leg is basically constructed from two series connected semiconductor switches.

Figure 2.1 Schematic representation of a series-resonant power converter. SM 1 SM 2 L ' S M l j

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The outside connections of each switching leg are linked to the series-resonant network, while the center tap of the switching leg is connected to a voltage source.

The basic configuration of a series-resonant converter is dis­ played in figure 2.2 and contains two switching matrices SMI and SM2, while the individual switching matrices are construct­ ed from two switching legs SL1, SL2 and SL3, SL4 respectively. Switching matrix SMI is constructed from electronically controlled semiconductor switches and its gate 1-2 is connected to a (dc) voltage source e..

Excitation of the series-resonant circuit with the elements L and C by the switching matrix SMI generates a modulated high frequency current i..

Diode rectifiers will rectify the resonant current i., thus creating a pulsating rectified current waveform i_ „=|i.|.

Finally the electric energy will be distributed to the (dc) voltage source e2 which is connected to the gate 3-4 of the switching matrix SM2.

Four basic configurations of a switching leg can be distin­ guished and are indicated in figure 2.3.

The centre tap of a switching leg with index k is connected to

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¥

Thkl 'p,k *

© - o

V D k 3 Thk1 Thk2 !p,k

¥

?

Dk4 Thk2 A VThk3

©

k : V AThkA a) b) c) d) Figure 2.3 Typical configurations of a switching leg.

a voltage source e symbolizing a source of energy (power grid) or a load (an electric machine). The parameter i , is the cur­ rent from this voltage source to the switching leg as shown in figure 2.3.

Both sides of each switching leg are connected to the series-resonant circuit which generates an alternating current i,. The tap where the positive resonant current will leave the switching leg is denoted by r+.

The thyristor switches are identified by the symbol ThkS. and are associated with the relevant output port by a compound

index kl which consists of two integers.

The first index k indicates the switching leg SLk. The second index l is related to the polarity of the resonant current i.:

index i is equal to 1 or 4 for a positive resonant current i,,

index £ is equal to 2 or 3 for a negative resonant current i... Those thyristor switches which conduct the current from the node k are denoted by the second index with a numerical value of 1 or 2 and those conducting the current to the node k by the second index with a numerical value of 3 or 4.

A switching matrix is in principle composed of at least one switching leg.

In combination with a voltage source (or load) a switching leg creates a so-called half-bridge configuration.

Similarly, in combination with a voltage source (or load) two switching legs create a so-called full-bridge configuration.

The switching leg as presented in figure 2.3.a, only utilizes the operation of the diodes Dk3 and Dk4 as a rectifier.

Figure 2.3.b depicts a switching leg which can excite a series-resonant circuit from a voltage source e by an alter­ nating turning-on and -off of the thyristor switches Thkl and Thk3 [2J.

Figure 2.3.c shows a switching leg which, in cooperation with a series-resonant circuit, can exchange electric energy as stored in the resonant circuit with a voltage source in two direc­ tions. This is the tool necessary for avoiding extreme voltages on the resonant capacitor C and extreme currents through the resonant inductor L as well as the semiconductor switches [51, as explained in chapter 4.

The switching legs as depicted in figures 2.3.a and c are only to be applied in combination with a unipolar voltage source. Figure 2.3.d depicts the generalized configuration of a switch­ ing leg applicable in combination with a bipolar voltage source

Emanating from the general configuration of a switching leg as shown in figure 2.3.d, the network of a generalized series-resonant converter connected to a pair of distinctive poly­ phase ac-voltage sources is depicted in figure 2.4.

The resonant circuit composed of the inductor L and capacitor C, is connected to an m-phase ac voltage source and an n-phase ac voltage source by means of two switching matrices SMI and SM2. Clearly, the equal rank of both matrices in the

conver-Thll Th13 Th21

©

i inn in u \ inn ir

®

Th23

J

* ^TThlZ JThIA ■ ■ Th22 Tl

J

? *

Th24

J

I

t-, Óte

2

Ót

J T T W .

u

° i

Thlm.112 Th(m

(nED

1

a

I'p/THl

s i si

(m+n) l'p,m*' p,rm2 I'p/n+n em«vnl V _ J

sion network allows the direction of the power flow to be mutu­ ally exchanged. It is possible to exchange electric energy between both sources in two directions.

Galvanic insulation can be realized between switching matrices by placing a transformer in the resonant circuit. Moreover, the transformer makes it possible to scale the currents and voltages without essentially affecting the qualities of the converter. This transformer operates with a high pulse fre­ quency which reduces the volume and weight substantially with respect to a 50 Hz transformer.

2.2. Pulse forming network

The series-resonant circuit composed of the elements L. and C, is connected to one (or more) voltage source(s) by turning on one (or more) semiconductor switches at time t=t.. The se­ ries-resonant circuit is excited by the voltage source(s) as presented by uT„ in figure 2.5. As a consequence of the low damping of the resonant circuit a sinusoidal current i will be

1 generated as shown in figure 2.5.

W i t , CDjtjtrt _{CO, t,}

00, t

The damped resonant frequency w. is determined by the value of the resonant inductor L., the resonant capacitor C, and the resistor R. representing the total ohmic losses in the series-resonant circuit.

During the current-conducting period [0,ir] an exchange of elec­ tric energy will take place between the resonant capacitor, the resonant inductor and the voltage source from which the series-resonant circuit is powered.

For an undamped series-resonant circuit the magnetic energy stored in the resonant inductor L, will reach its maximum value for w1 (t-t, ) =TT/2 . The amplitude of the resonant current is de­

pendent on the magnitude of the excitation voltage at the moment t=t, and the amount of stored energy in the resonant ca­ pacitor C, and inductor L, at the indicated time.

The resonant current i, will change polarity at time w,(t-t,)=TT and because of the unipolar character of the semiconductor switch, the resonant network will cease to oscillate.

The current-carrying semiconductor switches will change to the non-conducting state in a natural way (natural commutation). By virtue of the low value for the speed of change of the current in the semiconductor switches during the zero crossing of the resonant current, the switching losses in the semi­ conductor switches will be negligibly small. The stored energy in the resonant inductor will be equal to zero at time w1<t-t1)=ir.

After the oscillation has settled down, it can be restarted by turning on another set of semiconductor switches at the time t=t2 which will connect the series-resonant circuit to a voltage source with an opposite polarity.

A continuous excitation of the series-resonant circuit will generate a train of pulses which can be characterized as a high-frequency carrier. This carrier is constructed from dis­ crete current pulses of a quasi-sinusoidal waveform.

A control process can be implemented by changing, the time interval between the discrete current pulses (the interpulse time) which indicates that the control process is an aperiodic pulse modulation process in spite of a theoretically constant resonant frequency.

The shape of the current pulse might differ from pulse to pulse which indicates a non-uniform pulse modulation process.

As a result of the control of the interpulse time the control process for the flow of charge through the series-resonant cir­ cuit is characterized as a non-uniform^ aperiodic pulse modulation process.

2.3. Survey of conversion circuits 2.3.1. Dc-dc conversion

Characteristic configurations of the series-resonant converter can be presented, depending on the specific application.

The network topology of the series-resonant converter used to implement the power conversion process with a dc-source and a dc-load is shown in figure 2.6 [2].

The converter network comprises the following elements

a. a switching matrix SMI constructed from two switching legs containing thyristor switches (see figure 2.3.b);

b. a switching matrix SM2 constructed from two switching legs

containing diodes (see figure 2.3.a); a series-ret

capacitor C.

c. a series-resonant circuit consisting of an inductor L and a

r

The switching matrix SMI with an input current i , is connect-SM1

ed to a dc-voltage source e . s

A continuously operating switching process of the thyristors generates an ac-voltage e which excites the series-resonant ^ sac

circuit.

The sinusoidal resonant current i 1 is now rectified by the action of the switching matrix SM2 to obtain a unipolar current through the load Z.. The load is short-circuited with respect to the high-frequency components in the rectified resonant cur­ rent icM2='ii' by bypassing the load with a (filter) capacitor. The dc-voltage U obtained on the load is seen by the series-resonant circuit as an alternating voltage u . The

^ ^ oac voltage uL C controlling the double excited LC-circuit is now

equal to uT/Jt> = e <t> - u (t) (2.1)

LC sac oac

It can be easily understood that the polarity of the dc-current through the load I =|i,| and thus the polarity of the

o 1 av

dc-output voltage U cannot be changed. The converter is lim­ ited with respect to the flow of electric power from the source to the load (single quadrant operation).

The indicated method of distribution of electric energy through a series-resonant circuit has a number of drawbacks which will be analyzed in chapter 4.1:

1. the dependence of the maximal value of the resonant current M m a x °^ t n e ele c tric damping of the resonant circuit,

2. the dependence of the maximal value of the resonant current i^m a x of the magnitude of the excitation voltage u ,

pil

lrms ' V a v

(2.2)

resulting from 2. in the case of varying output voltages and a constant current.

In general it can be stated that the control of the flow of electric energy through the series-resonant circuit, especially the accumulation of electric energy stored in the resonant capacitor, is the main problem.

Several solutions to this problem are known from literature [2,5]. A more practical solution not negatively affecting the qualities of the series-resonant circuit is to replace the thyristor switches in the switching matrix SMI by a combination of

a. a thyristor and an anti-parallel diode (see figure 2.3.c), b. a combination of two anti-parallel thyristors (see figure

2.3.d).

The principles of this solution are based on the possibility to transmit electric energy from the series-resonant circuit back into the source with a voltage E and/or into the output with a

SMI

©

11 £ VD13 C s D21A V

MI

I l h

^ ^ V ^ _ _ H C i _ JL X ' - , L 0 J V, , 14 7A 2Th12 ^ — Th24A £ D Th23 — JLC

©

Ï

SM2 ■sac uoac

©

22 DM r D52

liHh

1

4

©

AD43 u0 D5IV

voltage ü , which prevents an undesirable accumulation of elec­ tric energy in the resonant circuit.

We begin an analysis of the qualities and waveforms of this class of series-resonant power converters by discussing the topology of the conversion network for the transfer of electric energy from a dc-source to a dc-load as shown in figure 2.7. By replacing the diodes in the switching matrix SM2 (in figure 2.7 the diodes D43, D44, D51 and D52) by a switching matrix of thyristor switches (in figure 2.8 the thyristors Thij with i=4,5 and j=l,2,3,4), we achieve the possibility of transfer­ ring electric energy from the load with a dc-voltage U0 to the

source with a dc-voltage E .

The selection of the switching elements in switching matrix SM2 determines the polarity of the current igM2 a t t n e o u tPu t o f

the switching matrix SM2.

The current iS M2> 0 i f t h e switching matrix operates with the

switching elements Th43, Th44, Th51 and Th52. Similarly, the current iS M2< 0 i f t n e switching matrix operates with the

switching elements Th41, Th42, Th53 and Th54.

The power system assumes the characteristics of a true secon­ dary current source, since its primary power transfer and control mechanism are based on the control of the transferred

SM2 Th« ,.° Th52 S142.

Hh

Ï

'.ILQJ

Th41 ^ Th53 VTh54

i

VTh51

J

Figure 2.8 Dc-dc series-resonant converter for a reversible power flow.

charge.

Within the physical design limits the output current is inde­ pendent of the value and polarity of the voltage across the

load connected to the output terminals. In the case of a power flow from the output to the source, the voltage on the active switching elements will become positive immediately after the zero crossing of the current. Special measures with respect to the switching pattern of the resonant current will be intro­ duced (see chapter 4.4).

2.3.2. Dc-ac single phase conversion

The series-resonant converter as shown in figure 2.8 is capable of exchanging electric energy between a dc-source E and a dc-load with a voltage u . This means that on the converter side connected to the load, the product of voltage u over and current i through the load can obtain positive and negative values as well, independent of the polarity of the voltage u and of the character of the load. This method of power con­ trol is in general defined as the four-quadrant mode of operation [14,18].

Figure 2.9 schematically depicts the way to connect a source of energy with a voltage Es to an output port with a voltage u0 by using two matrices of switching elements in combination with a resonant circuit.

The source of energy is connected to the switching matrix SMI by a high-frequency filter Cs. Excitation of the resonant cir­ cuit with the switching matrix SMI generates a modulated high-frequency pulse carrier ij. The interpulse time will be varied in accordance with the information sent to the electron­

ic control system. This carrier is generated in the form of discrete pulses, where each pulse possesses the shape of a

lightly damped sinusoid.

Next, this carrier will be rectified by another switching matrix SM2. In this way a rectified current iqMO will be

generated at the output of the switching matrix SM2 with a polarity depending on the configuration of the active elements in the switching matrix SM2, but independent of the polarity of the output voltage u , as well as of the polarity of the resonant current i .

By changing the configuration of the switching matrix SM2 with a frequency low with respect to the frequency of the resonant current, a current with a low frequency will be synthesized at the output port of SM2.

The high-frequency content of iS M 2 i s eliminated by a high fre­ quency output filter C .

The reconstruction of the low-frequency signal from the high-frequency carrier i. after it has been generated and modu­

lated by means of the power conversion process can be described

es —*- —"-Cs SWITCHING MATRIX SM1 L, |esac

n

c

'

uoac[ SWITCHING MATRIX S M 2 'SM2 '0 =c0 'SMI

Figure 2.9 Symbolic representation of the dc-ac

as a process of demodulation.

This results in a low-frequency, single-phase, bipolar output current i which develops an ac-voltage u on the load. The

o o direction of the power transfer is reversible [27].

Both high-frequency filters and the associated controlled switching matrices are equally ranked in the demodulation according to the momentary direction of the power flow, as des­ cribed by a fixed time-varying reference program.

The power conversion process makes it possible to start, terminate and shape each current waveform, independent of the voltage polarity and magnitude and the loading impedance. The process avoids the use of passive filter elements in order to implement such a process of current wave shaping.

064

<J045 uo 5 6 uo64

Figure 2.10 Symbolic representation of the dc-ac polyphase series-resonant converter system with symbolic waveforms.

2.3.3. Dc-ac three-phase conversion

A simplified version of the process of conversion is represent­ ed in figure 2.10. A source of energy with a voltage e is connected to the output port with a voltage u O Da <P<3=45 , 56 , 64 )

by two switching matrices SMI and SM2 and a series-resonant circuit.

A source of dc power is connected to the switching matrix SMI. The switching matrix SMI generates a modulated high-frequency carrier i, by exciting the resonant circuit with the elements L| and C j . This carrier is then demodulated by another switch­ ing matrix SM2, a process which is accomplished by a selective rectification of the resonant current i and a distribution of this current over the three output ports.

A rectified current i (p=4,5,6) is created at the output poles p of the switching matrix SM2. The polarity of the current i_ is dependent on the configuration of the switches of the matrix SM2, independent of the polarity of the output voltage

u and the polarity of the resonant current i,. The high-_{opq c- i i} frequency content of i „M, is eliminated by a

high-frequency output capacitor. The result is the low high-frequency, three-phase, bipolar output current i _n n which creates a

three-PM

phase output voltage u over the load, as illustrated in figure 2.10.

The direction of the power flow is reversible and is suitable for four-quadrant operation. The train of current pulses i., as indicated in figure 2.10, can be altered by applying a modu­ lation process, in which the amplitude of and/or the time spacing between the pulses concerned are varied in order to obtain a current waveform of the programmed form. The power-conversion process makes it possible to synthesize a cur­ rent waveform independent of the loading impedance and polarity of the output voltage. This process avoids the use of passive filter elements in order to implement the current wave-shaping

process. _ Thus the electric power in an ac-load can be controlled by pro­

the available mechanism [391. By a repeated variation of the polarity of the output current i and by programming the momen­ tary value of the output current or voltage, the power converter as shown in figure 2.8 is capable of generating a single-phase or polyphase ac-current to a passive as well as an active load, independent of the character of this load.

2.3.4. Ac-ac conversion

The connection of the switching matrix SMI fully equipped with anti-parallel thyristor switches (see figure 2.3.d), to a polyphase ac-source, as indicated in section 2.3.3, makes it possible to control electric energy from or to the ac-source via the series-resonant circuit.

Conventional ac-ac converters are equipped with a so-called dc-voltage or dc-current link. The dc-link is realized by heavy and voluminous low-pass filters resulting from the low frequency content of the ac-source.

W h a f s more, the reaction speed will be low as a consequence of the low time constant of such filters.

The ac series-resonant converter can be classified as an ac-converter containing a high-frequency ac-current link.

The freedom to initiate and terminate the current waveforms at will and to modulate their harmonic content at and near power frequencies opens up possibilities to avoid and mitigate sub­ stantially the need for low-frequency filters and transformers. This method opens the path to relatively high internal frequencies as required for high-quality waveform transforma­ tion, dynamic-stability enhancement and increased power density.

3. PRINCIPLES OF SERIES-RESONANT ENERGY TRANSFER

3.1. Series-resonant waveforms

The principles of the series-resonant energy conversion are based on the unique qualities of the series-resonant circuit in switching networks when transferring electric energy from a voltage source e to a load Z. with a voltage u as

schemati-3 s a ^ o

cally represented in figure 3.1.

In general such a converter consists of a series-resonant circuit with the elements L1 and C , excited by a number of electronically controlled (semiconductor) switches. The usual input and output filters complete the conversion network.

An analysis of the phenomena in a series-resonant circuit driven from a dc-source yields a deeper understanding of the phenomena in the more complex switching networks of series-resonant converters.

A series-resonant network is connected to a dc-voltage source by an ideal switch S which will be turned on at t=0, as symbol­ ically indicated in figure 3.2.

Electric energy is distributed from the source with a voltage e to an output port with a voltage u .

The voltage which excites the series-resonant LC-circuit is for t>0 equal to

*LC e - u s o

<3.1)

Figure 3.1 Schematic presentation of the distribution of electric energy between a source and a load.

Furthermore it ia assumed that the voltages of both sources are c o n s t a r

l y , so

c o n s t a n t , which i m p l i e s t h a t de / d t = 0 and du / d t = 0 , r e s p e c t i v e

-d uL C/ d t = 0 ( 3 . 2 )

The impedances of the voltage source with a voltage e as well as that with a voltage u are supposed to be zero for any value of the pulse repetition frequency. In practical applications the above-mentioned supposition will be realized by parallel­ ling the voltage sources by high-frequency capacitors.

The discrete components which are the building blocks of the series-resonant network and the associated resonant phenomena are denoted by a subscript 1.

The resistor R. represents the ohmic losses in the resonant network (proportional to the square of the rms-value of the resonant current i. „ ) .

1 rms

The electric losses proportional to the average value of the resonant current per half cycle li, I _,, are set by the power as

i a v delivered by the source e .

The dynamic switching losses in the semiconductor switch S can be neglected.

Figure 3.2 Excitation of a series-resonant circuit from a voltage source.

The state equation for a linear time-invariant system, describ­ ing the series-resonant circuit as depicted in figure 2.2, is written as

dy(t)

—SfT = A y(t) + B u (t) (3.3)

where the state vector y(t) contains the state variables y. < t ) , yo(t) , . .. . A and B are matrices with constant coefficients while u(t) is the input vector.

The general solution of the state equation (3.3) is formulated as

A(t-t«) t

y(t) = e ° y(tQ) + eA CB ïï(?)d? (3.4)

fc0

The vector y(tg) is defined as the state variable at the begin­ ning of the time interval t = tn.

For symplicity of presentation the initial time is chosen as zero.

The complexity of the solution of the state equation is optim­ ized by a careful selection of the state variables.

The state variables are selected as

y{(t) = ij (t) (3.5)

y,(t) = u <t) = L.di./dt (3.6)

The system matrix is defined as 0 l/L \

A = | (3.7)

1/C,

R . / L J

The input vector u and the matrix B are equal to zero.

We next introduce the characteristic parameters of a series-resonant network:

■fê

resonant impedance : Z. = \l ^— (3.8) L. time constant : T| = ^— (3.9) Rl 1 damping coefficient : A| = 2L = Tx— (3.10)

undamped resonant frequency : ojn = 1 / y L C (3.11)

damped resonant frequency : u. = ü)„\/! - - (3.12)

V <2< V , >

q u a l i t y : Qj = ^ - = WQT j ( 3 . 1 3 )

p h a s e s h i f t : tp. = a r c t g l ^ ) ( 3 . 1 4 )

r 1 ' 210.T.

At

A series expansion of the state transition matrix e results

At e l W , t / COS((Ü t-ip ) f s i n f u t) .

e = Zl ' I (3.15)

cosdPj) 1-ZjSindüjt) cos(wjt+(p])i

If the switch is implemented as a programmable power semicon­ ductor, the initial conditions have to fulfill the general term for the existence of a resonant current

i (0) > 0 (3.16)

uL 1<0> = O (3.17)

We subsequently introduce a normalized system of variables:

time : x = w]t (3.18) voltage : u(x) = ~^~ (3.19) UL C Zxi(x) current : Z.i(x) = — (3.20) LC

For the newly introduced parameters the energy stored in the resonant capacitor C. is expressed by

wc l = u ^ (3.21)

while the energy stored in the resonant inductor L] is expressed by

wT , = zf±f (3.22)

Li 1 I I

By so doing, we introduce new dimensionless variables, which are therefore independent of the physical values of these vari­ ables.

The introduction of normalized variables is comparable to the situation in which the converter is operating under the physi­ cal conditions:

uL C . 1 ,

w = 1 s

The set of equations is now of course not dimensionless, but the calculated variables are, with respect to the magnitudes, equal to the values of the normalized set of equations.

From now on all equations will be formulated under the above-mentioned conditions, so the variables will not

necessar-ily be denoted by a new symbol.

The transformation of the functions representing the solutions to the real physical parameters can easily take place by

a. multiplying a voltage u(x) and a current Z.i(x) by the value of the source voltage uT(-,;

b. multiplying electric energy w(x) by ^C u ; c. dividing time x by to .

The general solution of the state equation (3.3) is written as

y~(x) = c y<0) (3.23) where y(x) = ( Zjijfx) u Li( x ) ' (3.24) -T y(0) = ( Z ,i ,(0) uT ,(0) ) (3.25) 1 I L! _, . /cos(x + ip ) sinx C = e ' I 1 (3.26) costp, \ . 1 \ sinx -cos(x-(j> )

while the voltage on the resonant capacitor C. is equal to

uc |< 0 ) = 1 - uL )( 0 ) - 2Z,i jtOsiiKP, (3.27)

and the phase shift 'P. is defined by (3.14).

The characteristic waveforms for a series-resonant network are indicated in figure 3.3 in combination with the composite sig­ nals

uL ](0)sinx and Z j i g ( 0 )cos( x + <Pj ) ,

- uL j ( 0 )cos ( x-(p j ) and Z,i,(0)sinx, respectively.

As a result of applying semiconductor switches the resonant current will be fixed at the moment x at which the polarity of the current changes its polarity.

'l 11 m a x -i,(o)7 / / 1 \ - T t + Xs UC 1 1-^* -^ UL1 ~~yr~^ 1 \ X 0 x' 0 / -uc,(o) 0

Figure 3.3 Characteristic waveforms for the series-resonant network a. the resonant current i , b. the voltage u on the

Cl

resonant capacitor C , c. the voltage u on the

* LI resonant inductor L .

For a graphical interpretation of the resonant current one has to distinguish between

I. the initial condition of the resonant current i.(0)=0, in which case a resonant current can only be generated if after turn-on the voltage on the resonant inductor Uji<0)>0, which implies that u ( O K I .

In all other cases no resonant current will be generated. The current-conduction angle of the switch x =ir (see

s figure 3.4.a).

II. the initial conditions for the resonant current i,(0)>0, in which case a resonant current can be generated for each polarity of the voltage on the resonant inductor u-.tO) after turn-on. The polarity of the initial value of the

inductor voltage determines the characteristic boundaries for the current-conduct:

figure 3.4.b, c, and d) is

for the current-conduction angle x_ of the switch (see

3 s UL .> 0 uL 1 = 0 uL,>0 2< Xs< 1 T 0<xs <-<3.28) (3.29) (3.30) c. d. 2 -- f i 2 2 2 2 2 -Z.ii \ s UL 1 O S \ \ \ \ \' \ \ UCI S-IT, / I X ' X5 TC 2--*\ 2 2 2 2 0 -Z,i, \ UL1

V

"

c

v

• s X 2 -- < P , 2 2 2 2 2 -Z,i, UL1 *P1

y

s X Xs TC

Figure 3.4 Conditions of operation for the resonant network a. i (0)=0 and u (0)>0,

b. i (0)>0 and uL 1( 0 ) > 0 ,

c. i1(0)>0 and uL 1( 0 ) = 0 ,

III. the initial conditions for the resonant current i ( 0 X 0 after turn-on, which implies that at time t=0 the switch S will carry a negative source current.

The case of a semiconductor switch which introduces the condition will be based on the property that the polarity of the current has to be positive (like cases I and I I ) ; this mode of operation will never occur.

At the moment of time x=0 the series-resonant circuit is excit­ ed by an input voltage

uf(0) = 1-u (0) (3.31)

For each resonant current pulse an amount of electric energy is emanated from the source and distributed to the circuit ele­ ments .

The final conditions are reached at time x=x for the next zero s

crossing of the resonant current i . At that moment

a. the energy level of the resonant circuit is changed,

b. electric energy is dissipated in the resistive element R., c. electric energy is delivered by the voltage source u =1 V;

LC for the situation where u =e -e electric energy is

distri-LC 1 Z

buted from a voltage source e to a voltage source e .

The amplitude of the resonant current and the voltage on the resonant inductor can be written as

^lmax* M l ' V = M l O

6

*

™

<3

'

32)

UL 1(V = 1 ~ uC l( xs > ( 3-3 3 )

. - Xlxs

z

i V

Z ^ l o ) cos ip.

U f l o l - Z j i j l o l s i n ip

Figure 3.5 Goniometric relation between the phase angle <P, , the current conduction angle xs and the network

parameters.

where the parameters introduced are formulated as symbolically depicted in figure 3.5:

current-conduction angle coscp arctg

sinip, - uf (0)/Z i (0)

time for the amplitude of resonant current

( 3 . 3 5 ) m s <P, - 7 T / 2 ( 3 . 3 6 ) t h e a m p l i t u d e o f t h e r e s o n a n t c u r r e n t Z . i . , . f o r A =0 2 2 2 2 2 Z , i ,n = l u ^ O ) + Z . i , ( 0 ) - 2 u J 0 ) Z i ( O ) s i n ( p ] ( 3 . 3 7 ) 1 l u £ 1 1 f 1 1 1

The distributed charge Q 'through the resonant circuit is now

|Qf = [uC ](xs> - u (0)] costp (3.38)

writ-t e n i n a n o r m a l i z e d f o r m a s

ws = 2 l uc l( xs) - uc ]< 0 > ] ( 3 . 3 9 )

This amount of energy will be partially dissipated in the resistive element R.

"loss = [ l - uc l( 0 ) ]2- [ l - uC ]( xs) ]2 + Z ^ i ^ O ) (3.40)

The remaining part of the energy w will increase the energy level in the series-resonant circuit from

w (0) = Z2i2(0) + u 2(0) (3.41)

LC 1 1 C1 to an energy level equal to

V

x

s » =

u

a

l x

s '

<3

-

42>

The balance of energy is formulated as

WL C< Xs> = WL C( 0 ) + ws " wl o s s < 3-4 3 >

The series-resonant circuit is capable of withdrawing a charge Qf from a voltage source E (with a voltage E =1 V o l t ) . The charge Q^ will be transmitted to the series-resonant network and will change its level of energy.

An efficient transmission is only obtained if the loss of elec­ tric energy as dissipated in the resistive elements is small with respect to the transmitted power.

This aspect of power distribution is treated in section 4.1. It is demonstrated that for a cyclic stable operation for the power transfer the method introduced will have a drawback with respect to efficiency.

A careful look at the balance of energy (3.43) raises the suspicion of a low efficiency for the power transfer for the cyclic stable mode of operation.

Accumulation of energy in the series-resonant network has to be avoided during operation to be capable of limiting the ampli­ tudes of internal voltage and current waveforms.

The minimal amount of stored energy in the series-resonant network has to be limited as well to maintain an oscillation.

The excitation voltage ULQ which drives the resonant network

has an alternating polarity to allow a continuous oscillation of the series-resonant current.

The tools available to limit the maximum and minimum amount of energy stored in the series-resonant network under all regular and irregular conditions will be analyzed in subsequent chapters.

3.2. Time domain analysis of switching networks

Properties of networks composed of switches, both active and passive components, can be analyzed based on the solution of a set of differential equations describing the switching network. The solutions of the set of differential equations present the course of such functions as voltage, current and energy in relation to the time for unaltered positions of the switching elements.

A switching network can be described by the succession of a number of unvarying network configurations, which are them­ selves created by programming the switching elements.

The course of the voltages and currents in a switching network within a closed interval txk'xk+l' c a n ^e described at the

points of time x^, at which one or multiple switches will change their state of conduction.

During this interval txwxk + i l ^ e state of conduction for all

switching elements will not change.

The discrete values for the voltages and currents at the points of time xj.can be found as partial solutions of the differential