Angewandte Mathematik mit Mathcad, Band 2. Komplexe Zahlen, by Josef Trölß

By Josef Trölß

Computer-Algebra-Systeme (CAS) und computerorientierte numerische Verfahren (CNV) vereinfachen den praktischen Umgang mit der Mathematik ganz entscheidend und werden in immer weiteren Bereichen angewendet.

Mathcad stellt dazu eine Vielfalt an Werkzeugen zur Verfügung und verbindet mathematische Formeln, Berechnungen, Texte, Grafiken usw. in einem einzigen Arbeitsblatt. So lassen sich Berechnungen und ihre Resultate besonders einfach illustrieren, visualisieren und kommentieren.

Dieses Lehr- und Arbeitsbuch, aus dem vierbändigen Werk „Angewandte Mathematik mit Mathcad“, richtet sich vor allem an Schülerinnen und Schüler höherer Schulen, Studentinnen und Studenten, Naturwissenschaftlerinnen und Naturwissenschaftler sowie Anwenderinnen und Anwender – speziell im technischen Bereich –, die sich über eine computerorientierte Umsetzung mathematischer Probleme im Bereich komplexer Zahlen, komplexer Funktionen, Vektor- und Matrizenrechnung, Vektoranalysis informieren und dabei die Vorzüge von Mathcad möglichst effektiv nützen möchten.

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Extra info for Angewandte Mathematik mit Mathcad, Band 2. Komplexe Zahlen, Matrizenrechnung, Vektoranalysis

Sample text

12 Für eine Spule gilt das Induktionsgesetz u = us = L ˜ d i ( t) , (2-37) dt und in komplexer Darstellung (Permanenzprinzip - die Differentiation verläuft nach den gleichen Regeln wie im Reellen) u = L˜ d dt i. (2-38) Daraus erhält man: j ˜Z˜t 2˜ U˜ e = L˜ d dt =j˜ j ˜Z˜t 2˜ I˜ e j˜Z˜t 2˜ Z ˜ L˜ I˜ e . (2-39) Durch Kürzen folgt schließlich das Ohmsche Gesetz in komplexer Formulierung: U= j ˜ Z ˜ L˜ I. (2-40) Damit wird ein induktiver Widerstand durch den imaginären Widerstandsoperator dargestellt: ZL = j ˜ Z ˜ L = j ˜ X L .

622j V Û j˜Z˜t u = ۘ e komplexer Scheitelwert resultierende Wechselspannung in komplexer Form 3. 05 100 u 3( t) V 200 300 t s Abb. 3 Berechnungen im Wechselstromkreis Weist ein aus Widerständen R, Kondensatoren C und Spulen L bestehendes elektrisches Netzwerk nur Quellen einer festen Frequenz auf, müssen alle Ströme i(t) und Spannungen u(t) Schwingungen derselben Frequenz sein, da die Zusammenhänge sich entweder durch Proportionalität (R), Integration (C) oder Ableitung (L) ergeben, die wieder zu trigonometrischen Zeitfunktionen führen.

T1 , M = Mu - Mi Phasenverschiebungswinkel (Phasenwinkel oder Nullphasenwinkel) in rad oder Grad. Die Beschränkung auf Sinusschwingungen ist deshalb gerechtfertigt, weil sich Kosinusschwingungen immer auf Sinusschwingungen zurückführen lassen: § © y ( t ) =  ˜ cos ( Z ˜ t  M ) =  ˜ sin ¨ Z ˜ t  M  S· 2¹ Abb. B. bei t = 0): j˜Z˜t = Û ˜ e j˜M u ˜ e j˜Z˜t = Û ˜ e j˜Z˜t = Î ˜ e j˜M i ˜ e j˜Z˜t = Î ˜ e j˜Z˜t Â= ˜e Î = Θe j˜M u j˜M i = = 2 ˜ Ueff ˜ e 2 ˜ Ieff ˜ e Û = ۘ e j˜M u Î = Θe j˜M i bzw.

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