Resonante Mikrobauelemente mit elektrostatischem Antrieb finden seit einigen Jahren vermehrt Anwendung in vielen Bereichen der Technik. So beruhen beispielsweise Drehraten- oder Beschleunigungssensoren, die im Automobilbau eingesetzt werden auf diesem Prinzip. Neue Anwendungsfelder ergeben sich vor allem für Aktoren, beispielsweise für die am Fraunhofer IPMS entwickelten Mikroscannerspiegel mit Out-of-plane-comb-Antrieb. Sie dienen zur geometrischen Ablenkung von Licht und können zur Realisierung von hochintegrierten Systemen zur Ausgabe (Laser-Projektor) oder Aufnahme (Laser-Imager) von Daten genutzt werden. Zum Entwurf von Mikroaktoren gibt es eine Reihe von Arbeiten, die sich meist auf ein konkretes Antriebsprinzip beziehen oder den Entwurf im Allgemeinen behandeln. Die vorliegende Arbeit verfolgt daher das Ziel, speziell die Randbedingungen beim Entwurf resonanter Mikroaktoren mit Out-of-plane-comb-Antrieb zu identifizieren bzw. zu systematisieren sowie die gewonnenen Erkenntnisse in einem effizienten Entwurfsprozess umzusetzen. Dabei sollen möglichst auch relevante nichtlineare Effekte berücksichtigt werden, sodass sich neue Möglichkeiten zur Optimierung der Bauelemente und damit zur Erweiterung des Entwurfsraums ergeben. / Electrostatically driven microsystems are utilized in technical systems for several years. For instance, they are used in automotive applications as acceleration sensors or angular rate sensors. New fields of applications appear especially for actuators. The scanning micromirror of the Fraunhofer Institute for Photonic Microsystems is such an actuator. It is a micro-­optical-­electrical microsystem (MOEMS) which is driven resonantly by an electrostatic comb drive and can be used in scanning laser imaging systems or laser projectors. Several technical and scientific publications occupy with the design and the simulation of microactuators, which refer usually to a concrete drive principle or to the issues of design in general. The intention of this thesis is to identify and systematize particularly the boundary conditions of design regarding to resonant micro actuators with out-­of-­plane­comb drive. The findings are implemented in efficient design tools and design processes. One emphasis thereby is the investigation of nonlinear properties and effects. This includes geometrically non-­linearities of suspensions as well as non-linearities caused by fluid damping and the electrostatic comb drive. The findings are utilized in an analytical, nonlinear stability analysis of the device's equation of motion as well as in an object oriented software library for the MATLAB environment, which can be used to create nonlinear reduced order models of scanning micromirrors. With the developed techniques for design and optimization the available parameter range of scanning micromirrors can be extended. By that means, it is possible to improve the properties of existing devices as well as create new devices with outreaching performance.

In Deutschland und Europa ist im Zuge der Energiewende erforderlich, mehr Elektroenergie mit bestehenden Freileitungen zu transportieren. Eine technische Lösung, mit der dieses Ziel erreicht werden kann, ist das Umbeseilen der Freileitung mit Hochtemperatur-Leiterseilen mit geringem Durchhang (High Temperature Low Sag – HTLS conductors). Diese Leiterseile haben gegenüber konventionellen Leiterseilen (z. B. Aluminium/Stahl-Leiterseilen) höhere Bemessungsströme und temperaturen. Die stromführenden Verbindungen mit HTLS-Leiterseilen werden damit ebenfalls höher thermisch belastet. Diese sind für den zuverlässigen und sicheren Betrieb der Freileitung sehr wichtige Betriebsmittel. Neben anderen Verbindungstechnologien hat sich bei den stromführenden Verbindungen mit konventionellen Leiterseilen das Sechskantpressen seit Jahrzehnten bewährt. Aus der Literatur sind fast ausschließlich empirische Untersuchungen mit dieser Verbindungstechnologie bekannt. Das elektrische Kontaktverhalten von Pressverbindungen wurde bisher nur unzureichend untersucht. In dieser Arbeit wird dazu ein elektrisches Modell vorgestellt und weiterentwickelt, mit dem das elektrische Kontaktverhalten von Pressverbindungen genauer beschrieben werden kann. Weiterhin werden prinzipielle Zusammenhänge zwischen der Stromverteilung in den Kontaktpartnern und deren Einfluss auf den Verbindungswiderstand dargestellt. Als Ergebnis von theoretischen und experimentellen Untersuchungen konnten allgemeine Empfehlungen für das Dimensionieren von Pressverbindungen mit konventionellen und HTLS-Leiterseilen erarbeitet werden. Aus der prinzipiellen Funktionsweise einer Pressverbindung ist bekannt, dass der Form-, der Kraft- und der Stoffschluss in der Verbindung das elektrische Kontaktverhalten beeinflussen. Insbesondere der Kraftschluss wurde in der Literatur bislang nur näherungsweise berechnet. In den bekannten analytischen Modellen werden die Geometrie der Kontaktpartner sowie das Werkstoffverhalten vereinfacht und die mechanischen Belastungen beim Fügen der Verbindung nicht genau genug berücksichtigt. Aus den genannten Gründen wurde das Fügen von Pressverbindungen mit mehrdrähtigen Leiterseilen mit der Finite-Elemente-Methode (FEM) berechnet. Die Press- und Kontaktkräfte konnten damit für alle Kontaktflächen in einer Verbindung ermittelt werden. Es konnte gezeigt werden, dass insbesondere hohe Press- und Kontaktkräfte die Ursache für ein gutes elektrisches Kontaktverhalten einer Pressverbindung sind. Die physikalischen Ursachen dieses Zusammenhangs werden diskutiert. Das Langzeitverhalten von Pressverbindungen mit Verbund-Leiterseilen wurde experimentell untersucht. Die Ergebnisse der durchgeführten Langzeitversuche zeigen den Einfluss der Kontaktkraft im elektrischen Langzeitverhalten qualitativ auf. Bei den Pressverbindungen, für die nur sehr geringe Kontaktkräfte berechnet wurden, war das elektrische Langzeitverhalten weniger stabil. / In Germany and in Europe it is due to the “Energiewende” necessary to transmit more electrical ener-gy with existing overhead transmission lines. One possible technical solution to reach this aim is the use of high temperature low sag conductors (HTLS-conductors). Compared to the common Aluminium Conductor Steel Reinforced (ACSR), HTLS-conductors have higher rated currents and rated tempera-tures. Thus the electrical connections for HTLS-conductors are stressed to higher temperatures too. These components are most important for the safe and reliable operation of an overhead transmission line. Besides other connection technologies, hexagonal compression connections with ordinary transmis-sion line conductors have proven themselves since decades. From the literature, mostly empirical stud-ies with electrical tests for compression connections are known. The electrical contact behaviour, i.e. the quality of the electrical contact after assembly, of these connections has been investigated insuffi-ciently. This work presents and enhances an electrical model of compression connections, so that the electrical contact behaviour can be determined more accurate. Based on this, principal considerations on the current distribution in the compression connection and its influence on the connection re-sistance are presented. As a result from the theoretical and the experimental work, recommendations for the design of hexagonal compression connections for transmission line conductors were devel-oped. Furthermore it is known from the functional principle of compression type connections, that the elec-trical contact behaviour can be influenced from their form fit, force fit and cold welding. In particular the forces in compression connections have been calculated up to now by approximation. The known ana-lytical calculations simplify the geometry and material behaviour and do not consider the correct me-chanical load during assembly. For these reasons the joining process of hexagonal compression con-nections with stranded overhead transmission line conductors was calculated with the finite element method. The compression forces and the residual contact forces were determined for all contacts be-tween barrel and conductor as well as between the wires of the conductor. It was shown, that high compression and residual contact forces are the main cause of good electrical contact behaviour. The physical relations are discussed. The electrical long-term behaviour was furthermore investigated in experiments. The results confirm the influence of the residual contact force on the long-term behaviour qualitatively. Those compression connections, which had only small residual contact forces, were less stable in the long-term tests.

This work presents a 6.5-kV 1-kA SiC bipolar diode module for megawatt-range medium voltage converters. The study comprises a review of SiC devices and bipolar diodes, a description of the die and module technology, device characterization and modelling and benchmark of the device at converter level. The effects of current change rate, temperature variation, and different insulated-gate bipolar transistor (IGBT) modules for the switching cell, as well as parasitic oscillations are discussed. A comparison of the results with a commercial Si diode (6.5 kV and 1.2 kA) is included. The benchmark consists of an estimation of maximum converter output power, maximum switching frequency, losses and efficiency in a three level (3L) neutral point clamped (NPC) voltage-source converter (VSC) operating with SiC and Si diodes. The use of a model predictive control (MPC) algorithm to achieve higher efficiency levels is also discussed. The analysed diode module exhibits a very good performance regarding switching loss reduction, which allows an increase of at least 10 % in the output power of a 6-MVA converter. Alternatively, the switching frequency can be increased by 41 %.

In Geräten und Anlagen des Stromnetzes werden Steckverbinder mit hoher Stromtragfähigkeit eingesetzt, wenn bewegliche Teile kontaktiert werden oder Betriebsmittel mit geringem Aufwand montier- und demontierbar sein müssen. Die elektrische Verbindung der Leiter wird dabei oft mit federnden Kontaktelementen realisiert. Die Kontaktelemente müssen als Teil der Strombahn während der Lebensdauer des Geräts den Betriebsstrom im Kiloampere-Bereich und im Fehlerfall bis zu einige Sekunden lang den eine Größenordnung höheren Kurzschlussstrom tragen. In der vorliegenden Arbeit wurden Rechenmodelle für die innere Erwärmung von Hochstrom-Kontaktsystemen im stationären Dauerbetrieb und im transienten Kurzschlussfall entwickelt. Das elektrische und mechanische Langzeitverhalten im Temperaturbereich (105…180) °C wurde experimentell mit stromdurchflossenen, fettgeschmierten Modellsteckverbindern, die regelmäßig getrennt und neu gesteckt wurden, untersucht. Modellerstellung, Rechnungen und Versuche wurden beispielhaft mit Kontaktelementen vom Typ Multilam durchgeführt. Kontaktelemente und Leiter bestanden aus versilbertem Kupfer. Für das stationäre Betriebsverhalten wurden die mit der analytischen Spannungs-Temperatur-Beziehung nach Kohlrausch berechneten Ergebnisse mit einem Erwärmungsversuch verifiziert. Die Temperaturdifferenz zwischen Kontaktelement und Leiter ist bei Standardanwendungen, wie in Schaltanlagen, mit ≤ 3 K sehr klein. Deshalb ist die Leitertemperatur als Zielgröße beim Dimensionieren der Dauerstrombelastbarkeit ausreichend. Bei Kurzschlussstrombelastung wurde im Kontaktsystem eine schnelle, räumlich unterschiedlich ausgeprägte Erwärmung numerisch berechnet. Leiter und Kontaktelement erwärmen sich kontinuierlich, wobei die mittlere Endübertemperatur im Kontaktelement aufgrund des kleineren stromtragenden Querschnitts eine Größenordnung höher ist. Die Kontakte führen bei 50 Hz-Wechselstrom aufgrund ihrer vernachlässigbaren Wärmekapazität 100 Hz-Temperaturzyklen aus. Dabei können die Maximaltemperaturen noch deutlich größer als die mittlere Temperatur der Kontaktelemente sein. Im Langzeitversuch waren nach 16 000 h Betriebszeit bei 180 °C und regelmäßigen simulierten Steckvorgängen die Verbindungskräfte noch genügend groß, um die elektrischen Anforderungen eines neuen Kontaktsystems zu erfüllen. Allerdings führte bei einer Betriebstemperatur von 105 °C ein thermisch instabiles Schmierfett zum vorzeitigen elektrischen Ausfall eines Teils der Steckverbinder. / Switchgear and devices for the power grid use high-power connectors if moving parts have to be contacted or equipment shall be easily mountable and dismountable. The electrical connection of the conductors is often realized by spring-loaded contact elements. As part of the main circuit, contact elements must carry the full operating current in the kiloampere-range for the entire service life of the device. In case of a fault, the short-circuit current, which is one order of magnitude larger, has to be carried for up to several seconds. In this thesis, calculation models for the inner temperature rise of high-power contact systems in steady-state continuous operation, as well as for the transient short-circuit load case were developed. Electrical and mechanical long-term performance in the temperature range from 105 to 180 °C was experimentally investigated with current carrying, grease-lubricated model connectors which were regularly unplugged and replugged. Modelling, calculations and experiments were exemplarily carried out with Multilam contact elements. Conductors and contact element consisted of silver-plated copper. The analytical voltage-temperature relation was used to calculate the steady-state performance; calculations were verified with a temperature-rise test. The temperature difference from contact element to conductors is very small (≤ 3 K) for standard applications like switchgear. Thus, it is sufficient to use the conductor temperature as a criterion for the design of the continuous ampacity of high-power contact systems. At short-time load, a fast spatially inhomogeneous temperature rise was numerically calculated. Conductor and contact element continuously heat up; due to the smaller current carrying cross section, median final temperature rise in the contact element is one order of magnitude larger than in the conductors. Because of their negligible thermal capacity, contacts perform 100 Hz temperature cycles at 50 Hz AC load; the maximum contact temperatures may be significantly higher than the median temperature of the contact elements. In the long-term test, after 16 000 h operating time at 180 °C and regular plugging operations, contact elements maintained enough joint force to meet the requirements of a new contact system. At 105 °C however, a thermally instable grease led to electrical failure of part of the connectors.

Window decoding schedules are very attractive for message passing decoding of spatially coupled LDPC codes. They take advantage of the inherent convolutional code structure and allow continuous transmission with low decoding latency and complexity. In this paper we show that the decoding complexity can be further reduced if suitable message passing schedules are applied within the decoding window. An improvement based schedule is presented that easily adapts to different ensemble structures, window sizes, and channel parameters. Its combination with a serial (on-demand) schedule is also considered. Results from a computer search based schedule are shown for comparison.

In interference-limited wireless networks, interference management techniques are important in order to improve the performance of the systems. Given that spectrum and energy are scarce resources in these networks, techniques that exploit the resources efficiently are desired. We consider a set of base stations operating concurrently in the same spectral band. Each base station is equipped with multiple antennas and transmits data to a single-antenna mobile user. This setting corresponds to the multiple-input single-output (MISO) interference channel (IFC). The receivers are assumed to treat interference signals as noise. Moreover, each transmitter is assumed to know the channels between itself and all receivers perfectly. We study the conflict between the transmitter-receiver pairs (links) using models from game theory and microeconomic theory. These models provide solutions to resource allocation problems which in our case correspond to the joint beamforming design at the transmitters. Our interest lies in solutions that are Pareto optimal. Pareto optimality ensures that it is not further possible to improve the performance of any link without reducing the performance of another link. Strategic games in game theory determine the noncooperative choice of strategies of the players. The outcome of a strategic game is a Nash equilibrium. While the Nash equilibrium in the MISO IFC is generally not efficient, we characterize the necessary null-shaping constraints on the strategy space of each transmitter such that the Nash equilibrium outcome is Pareto optimal. An arbitrator is involved in this setting which dictates the constraints at each transmitter. In contrast to strategic games, coalitional games provide cooperative solutions between the players. We study cooperation between the links via coalitional games without transferable utility. Cooperative beamforming schemes considered are either zero forcing transmission or Wiener filter precoding. We characterize the necessary and sufficient conditions under which the core of the coalitional game with zero forcing transmission is not empty. The core solution concept specifies the strategies with which all players have the incentive to cooperate jointly in a grand coalition. While the core only considers the formation of the grand coalition, coalition formation games study coalition dynamics. We utilize a coalition formation algorithm, called merge-and-split, to determine stable link grouping. Numerical results show that while in the low signal-to-noise ratio (SNR) regime noncooperation between the links is efficient, at high SNR all links benefit in forming a grand coalition. Coalition formation shows its significance in the mid SNR regime where subset link cooperation provides joint performance gains. We use the models of exchange and competitive market from microeconomic theory to determine Pareto optimal equilibria in the two-user MISO IFC. In the exchange model, the links are represented as consumers that can trade goods within themselves. The goods in our setting correspond to the parameters of the beamforming vectors necessary to achieve all Pareto optimal points in the utility region. We utilize the conflict representation of the consumers in the Edgeworth box, a graphical tool that depicts the allocation of the goods for the two consumers, to provide closed-form solution to all Pareto optimal outcomes. The exchange equilibria are a subset of the points on the Pareto boundary at which both consumers achieve larger utility then at the Nash equilibrium. We propose a decentralized bargaining process between the consumers which starts at the Nash equilibrium and ends at an outcome arbitrarily close to an exchange equilibrium. The design of the bargaining process relies on a systematic study of the allocations in the Edgeworth box. In comparison to the exchange model, a competitive market additionally defines prices for the goods. The equilibrium in this economy is called Walrasian and corresponds to the prices that equate the demand to the supply of goods. We calculate the unique Walrasian equilibrium and propose a coordination process that is realized by the arbitrator which distributes the Walrasian prices to the consumers. The consumers then calculate in a decentralized manner their optimal demand corresponding to beamforming vectors that achieve the Walrasian equilibrium. This outcome is Pareto optimal and lies in the set of exchange equilibria. In this thesis, based on the game theoretic and microeconomic models, efficient beamforming strategies are proposed that jointly improve the performance of the systems. The gained results are applicable in interference-limited wireless networks requiring either coordination from the arbitrator or direct cooperation between the transmitters.

Sie entwickeln Entwurfssysteme für elektronische Baugruppen? Dann gehören für Sie die mit der Pinzuordnung verbundenen Optimierungskriterien - die Verdrahtbarkeit im Elektronikentwurf - zum Berufsalltag. Um die Verdrahtbarkeit unter verschiedenen Gesichtspunkten zu verbessern, werden in diesem Buch neu entwickelte Algorithmen vorgestellt. Sie ermöglichen erstmals die automatisierte Pinzuordnung für eine große Anzahl von Bauelementen in hochkomplexen Schaltungen. Alle Aspekte müssen in kürzester Zeit exakt erfasst, eingeschätzt und im Entwurfsprozess zu einem optimalen Ergebnis geführt werden. Die beschriebenen Methoden reduzieren den Entwicklungsaufwand für elektronische Systeme auf ein Minimum und ermöglichen intelligente Lösungen auf der Höhe der Zeit. Die vorliegende Arbeit behandelt die Optimierung der Pinzuordnung und die dafür notwendige Verdrahtbarkeitsvorhersage im hierarchischen Layoutentwurf. Dabei werden bekannte Methoden der Verdrahtbarkeitsvorhersage aus allen Schritten des Layoutentwurfs zusammengetragen, gegenübergestellt und auf ihre Eignung für die Pinzuordnung untersucht. Dies führt schließlich zur Entwicklung einer Vorhersagemethode, die speziell an die Anforderungen der Pinzuordnung angepasst ist. Die Pinzuordnung komplexer elektronischer Geräte ist bisher ein vorwiegend manueller Prozess. Es existieren also bereits Erfahrungen, welche jedoch weder formalisiert noch allgemein verfügbar sind. In den vorliegenden Untersuchungen werden Methoden der Pinzuordnung algorithmisch formuliert und damit einer Automatisierung zugeführt. Besondere Merkmale der Algorithmen sind ihre Einsetzbarkeit bereits während der Planung des Layouts, ihre Eignung für den hierarchisch gegliederten Layoutentwurf sowie ihre Fähigkeit, die Randbedingungen differenzieller Paare zu berücksichtigen. Die beiden untersuchten Aspekte der Pinzuordnung, Verdrahtbarkeitsvorhersage und Zuordnungsalgorithmen, werden schließlich zusammengeführt, indem die neue entwickelte Verdrahtbarkeitsbewertung zum Vergleichen und Auswählen der formulierten Zuordnungsalgorithmen zum Einsatz kommt. / This work deals with the optimization of pin assignments for which an accurate routability prediction is a prerequisite. Therefore, this contribution introduces methods for routability prediction. The optimization of pin assignments, for which these methods are needed, is done after initial placement and before routing. Known methods of routability prediction are compiled, compared, and analyzed for their usability as part of the pin assignment step. These investigations lead to the development of a routability prediction method, which is adapted to the specific requirements of pin assignment. So far pin assignment of complex electronic devices has been a predominantly manual process. Hence, practical experience exists, yet, it had not been transferred to an algorithmic formulation. This contribution develops pin assignment methods in order to automate and improve pin assignment. Distinctive characteristics of the thereby developed algorithms are their usability during layout planning, their capability to integrate into a hierarchical design flow, and the consideration of differential pairs. Both aspects, routability prediction and assignment algorithms, are finally brought together by using the newly developed routability prediction to evaluate and select the assignment algorithms.

The number of secure data streams a relay-assisted interference channel can support has been an intriguing problem. The problem is not solved even for a fundamental scenario with a single antenna at each transmitter, receiver and relay. In this paper, we study the achievable secure degrees of freedom of instantaneous relay-assisted interference channels with real and complex coefficients. The study of secure degrees of freedom with complex coefficients is not a trivial multiuser extension of the scenarios with real channel coefficients as in the case for the degrees of freedom, due to secrecy constraints. We tackle this challenge by jointly designing the improper transmit signals and widely-linear relay processing strategies.

In this paper, we analyze the four-node relay wiretap channel, where the relay performs amplify-and-forward. There is no direct link between transmitter and receiver available. The transmitter has multiple antennas, which assist in securing the transmission over both phases. In case of full channel state information (CSI), the transmitter can apply information leakage neutralization in order to prevent the eavesdropper from obtaining any information about the signal sent. This gets more challenging, if the transmitter has only an outdated estimate of the channel from the relay to the eavesdropper. For this case, we optimize the worst case secrecy rate by choosing intelligently the beamforming vectors and the power allocation at the transmitter and the relay.

An AC system’s voltage unbalance by a fundamental frequency negative sequence system is usually the main cause for the emission of non-characteristic harmonics by current source converters as used in conventional HVDC systems. This emission takes place on both sides of each 12-pulse converter. On the DC side mainly a 2nd harmonic voltage appears driving a 2nd harmonic current. The magnitude of this harmonic current can exceed the magnitudes of the characteristic harmonics even if no low order resonance exists. Further non-characteristic harmonics generated by the converter under such unbalanced supply voltage conditions have frequencies with a frequency distance to the characteristic harmonics of 2 times the fundamental frequency. The main technical drawbacks are the unintended coupling between both AC systems and the risk of thyristor over-stresses by DC current discontinuities at low power transfer levels. On both AC sides the largest 2 non-characteristic current harmonics generated by a 12-pulse HVDC converter under unbalanced supply voltage conditions are a negative sequence system of the fundamental harmonic and a positive sequence system of the 3rd harmonic. Also on the AC sides further harmonics are emitted by the converter with a order number distance of 2 to the orders of the characteristic harmonics. However, in practical AC system operation special attention has to be paid to the 3rd harmonic distortion level, in particular when low order resonance appears between the system impedance and the impedance of the converter station AC filters. In order to avoid the above mentioned problems, large smoothing reactors and sometimes large blocking filters are installed on the DC side and the voltage distortion on the AC sides is reduced by AC filters. However, these filters require an expensive high component rating if they are tuned to the 2nd or 3rd harmonic respectively. The work shows that a modification of the valve firing can reduce the levels of the 2nd and 3rd harmonic without investment into additional primary equipment. Furthermore, this offers the chance to reduce the minimum power transfer level since also the risk of an intermittent DC current can be reduced. A corresponding algorithm and a control strategy are proposed. However, the calculation of an appropriate firing pattern requires a detailed modelling of the processes within the converters, especially the formation of the harmonics and the harmonic transfer between AC and DC sides. The work proposes a component vector model for the calculation of the harmonics. This model assumes that each harmonic consists of a first component representing the ideal conversion process, a 2nd component representing the impact of different commutation angles and in the case of the modified firing a 3rd component considering the impact of the intended non-equidistant firing. The work shows, that the harmonic component vectors resulting from voltage unbalance and from firing modulation can be treated separately and superimposed linearly. The calculation of the harmonic component vectors is performed applying the method of switching functions. For the consideration of the commutation and firing angle differences the modelling of switching functions based on differential impulses is proposed. However, especially an accurate representation of the above mentioned 2nd component vector requires a correct calculation of the commutation angles and their valve-specific differences. The investigations of this work have revealed that the conventional method of calculating the commutation angles – assuming an ideal smoothed DC current - may not produce results of sufficient accuracy. This is especially true in the case of a high ripple of the DC current, e.g. smoothed with a small smoothing reactor. A small smoothing reactor is typical for HVDC back-to-back applications. Therefore a new analytical method for the calculation of the commutation angles has been developed which in particular considers the typical pulse form of the DC current and additionally the impacts of the voltage unbalance and of the proposed modification of the firing on the ripple shape of the DC current. Moreover, as this analytical method requires the instantaneous values of the DC current at the instants of valve firing, a further analytical method for the calculation of these discrete current values has been developed. The equations are valid under the same conditions as the new ones for calculation of the commutation angles, i.e. resistive-inductive AC system fundamental frequency impedances, any degree of DC current smoothing between ideal smoothing and a ripple at the limit for current discontinuities. Symmetrical conditions, supply voltage unbalances and non-equidistant firing as proposed are applied. It is shown that, using this method, also the discrete values of the DC current at the end of the commutation intervals can be determined. In practice one of these discrete current values indicates the minimum value during one period of the fundamental frequency. This offers the chance for a more exact analytical determination of the limit for the appearance of DC current discontinuities. For typical parameters of a back-to-back installation the new methods and the new analytical equations have been compared with simulation results showing excellent correlation for typical voltage unbalances of not more than 1...2% and firing angle differences of not more than 2.5°. This verification is performed for the harmonics, the commutation angles and the discrete values of the DC current at the firing instants as well.