Name: continuous and discrete system models
Uploaded: 2024-09-26T13:16:54.000Z
Duration: 1 min 31 s
Channel: JoVE

Zeitkontinuierliche Systeme haben kontinuierliche Ein- und Ausgangssignale, wobei die Zeit kontinuierlich gemessen wird, und werden im Allgemeinen durch Differentialgleichungen oder algebraische Gleichungen definiert.
In einer RC-Schaltung wird beispielsweise die Beziehung zwischen Eingangs- und Ausgangsspannung durch eine Differentialgleichung ausgedrückt, die unter Verwendung des Ohmschen Gesetzes und des Spannungs-Strom-Verhältnisses des Kondensators abgeleitet wird.
Zeitdiskrete Systeme verarbeiten Ein- und Ausgangssignale in bestimmten Intervallen, und ihr Verhalten wird durch Differenzengleichungen zu unterschiedlichen Zeitinstanzen beschrieben. Ein Beispiel für ein solches System ist ein einfaches Modell für den Saldo auf einem Bankkonto von Monat zu Monat.
Eine andere Art von System ist zeitvariabel und zeitinvariant.
Ein zeitveränderliches System hat Parameter, die sich im Laufe der Zeit ändern. Zeitkontinuierliche, zeitveränderliche Systeme werden durch Differentialgleichungen mit zeitabhängigen Koeffizienten beschrieben, während zeitdiskrete, zeitveränderliche Systeme durch Differenzengleichungen mit zeitabhängigen Parametern beschrieben werden.
Zeitinvariante Systeme erzeugen einen Ausgang, der sich bei zeitversetztem Eingangssignal identisch verschiebt, ohne die Eigenschaften des Systems zu verändern.
Eine RC-Schaltung ist zeitinvariant, wenn die Widerstands- und Kapazitätswerte konstant bleiben. Wenn diese schwanken, wird das System zeitvariant, da die Versuchsergebnisse je nach Zeitpunkt der Durchführung des Experiments variieren.

klassifizierung von systemen ii

Kontinuierliche Systeme haben kontinuierliche Eingangs- und Ausgangssignale, wobei die Zeit kontinuierlich gemessen wird. Diese Systeme werden im Allgemeinen durch Differential- oder algebraische Gleichungen definiert. Beispielsweise wird in einem RC-Schaltkreis die Beziehung zwischen Eingangs- und Ausgangsspannung durch eine Differentialgleichung ausgedrückt, die aus dem Ohmschen Gesetz und der Kondensatorbeziehung abgeleitet wird,

<img alt="Equation1" src="/files/ftp_upload/16046/16046_PT%20EQUATION_1.svg" style="height: 30px;">

Diskrete Systeme haben Eingangs- und Ausgangssignale in bestimmten Intervallen, die zu bestimmten Zeitpunkten durch Differenzgleichungen definiert werden. Ein Beispiel ist ein einfaches Modell für den Kontostand eines Bankkontos von Monat zu Monat, dargestellt als

<img alt="Equation2" src="/files/ftp_upload/16046/16046_PT%20EQUATION_2.svg" style="height: 14px;">

wobei B[n] der Kontostand im Monat n, D[n] die Einzahlung und W[n] die Abhebung ist.

Systeme können auch als zeitvariabel oder zeitinvariant kategorisiert werden. Ein zeitabhängiges System hat Parameter, die sich im Laufe der Zeit ändern. Kontinuierliche zeitabhängige Systeme werden durch zeitabhängige Differentialgleichungen beschrieben, während diskrete zeitabhängige Systeme durch zeitabhängige Differenzgleichungen definiert werden.

Ein zeitinvariantes System weist identische Zeitverschiebungen in den Ausgangssignalen auf, wenn das Eingangssignal zeitverschoben ist. Mathematisch gesehen ist y(t) die Ausgabe für den Eingang x(t), dann ist für ein zeitinvariantes System y(t−t_0) die Ausgabe für den Eingang x(t−t_0). Beispielsweise ist ein RC-Schaltkreis zeitinvariant, wenn die Werte für Widerstand R und Kapazität C konstant bleiben. Wenn diese Werte schwanken, wird das System zeitvariant, was bedeutet, dass die Ergebnisse je nach Zeitpunkt der Durchführung des Experiments variieren.

Das Verständnis dieser Unterschiede zwischen kontinuierlichen und diskreten Systemen sowie zwischen zeitvariablen und zeitinvarianten Systemen ist für die Analyse und den Entwurf einer breiten Auswahl technischer Systeme von entscheidender Bedeutung.

Klassifizierung von Systemen II

Zeitkontinuierliche Systeme haben kontinuierliche Ein- und Ausgangssignale, wobei die Zeit kontinuierlich gemessen wird, und werden im Allgemeinen durch differentielle oder algebraische ...

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Introduction to Signals and Systems

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Classification of Systems-II

continuous and discrete system models

Continuous-time and discrete-time systems are two common ways to model signals and their outputs. In continuous-time systems, both input and output signals change smoothly as time moves continuously. These systems are often written with differential or algebraic equations.

An RC circuit is one example of a continuous-time system. Its input and output voltage can be related by a differential equation based on Ohm’s law and the capacitor relation. Discrete-time systems work differently. Their inputs and outputs are defined at specific time steps, and they are described by difference equations.

A bank account balance from month to month is a simple discrete-time model. In that example, B[n] is the balance at month n, D[n] is the deposit, and W[n] is the withdrawal. Systems are also grouped as time-varying or time-invariant. A time-varying system has parameters that change with time, while a time-invariant system keeps the same shift in output when the input is shifted.

For a time-invariant system, if y(t) is the output for input x(t), then y(t − t0) is the output for x(t − t0). An RC circuit is time-invariant when the resistance R and capacitance C stay constant. If those values change, the system becomes time-variant. Then the result depends on when the experiment is done. Knowing these categories helps in analyzing and designing many engineering systems.

Continuous and Discrete System Models

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classification of systems-ii

Continuous-time and discrete-time systems are two common ways to model signals and their outputs. In continuous-time systems, both input and output signals change smoothly as time moves continuously. These systems are often written with differential or algebraic equations. An RC circuit is one example of a continuous-time system. Its input and output voltage can be related by a differential equation based on Ohm’s law and the capacitor relation. Discrete-time systems work differently. Their...

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Learn how continuous-time and discrete-time systems differ, plus how time-varying and time-invariant behavior is identified using RC circuits and bank account examples.

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JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Continuous and Discrete System Models

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