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State Space Representation01:27

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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
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The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
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Semiclassical TEM image formation in phase space.

Axel Lubk1, Falk Röder1

  • 1Triebenberg Laboratory, Institute of Structure Physics, Technische Universität Dresden, 01062 Dresden, Germany.

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|January 13, 2015
PubMed
Summary
This summary is machine-generated.

This study explores semiclassical methods for describing electron beam aberrations in transmission electron microscopy (TEM). It uses phase space and Wigner functions to model wave optics for advanced TEM imaging.

Keywords:
Aberration correctionElectron opticsSemiclassicsWave optics

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Area of Science:

  • Physics
  • Materials Science
  • Microscopy

Background:

  • Advanced transmission electron microscopy (TEM) requires accurate wave optical descriptions.
  • Current quantum mechanical solutions for aberration theory are incomplete.

Purpose of the Study:

  • To explore semiclassical image formation in TEM using quantum mechanical phase space.
  • To provide accurate wave optical descriptions for advanced TEM imaging techniques.

Main Methods:

  • Utilizing Miller's semiclassical algebra and the frozen Gaussian method.
  • Applying the Wigner function representation of phase space.
  • Investigating both coherent and incoherent aberrations.

Main Results:

  • Successfully generalized arbitrary geometric aberrations, including nonisoplanatic and slope aberrations.
  • Demonstrated the suitability of Wigner functions for incoherent aberrations and partial coherence.
  • Identified relationships between classical phase space distortions and quantum mechanical purity.

Conclusions:

  • Semiclassical methods offer a viable approach for describing wave optics in advanced TEM.
  • The Wigner function is a powerful tool for analyzing aberrations and coherence in electron microscopy.
  • This work advances the theoretical understanding of image formation in TEM.