Extended Higher-Order Elements with Frequency-Doubled Parameters: The Hysteresis Loops Are Always of Type II
Abstract
:1. Introduction
2. MEMS Modeling via Extended HOEs: Type I and Type II Pinched Hysteresis Loops
3. Extended HOE: PHL Type Evaluation
4. Characteristics of Extended HOE: PHL, PIL, and SIL
5. Extended Higher-Order Element with Frequency-Doubled Parameter
5.1. PIL Features
5.2. PHL Features
- Axial symmetry changes the orientation of the curve. A composition of two axial symmetries leading from one lobe of the PHL to the other will preserve the orientation of the original lobe. The PHL will be of type II. □
5.3. Illustrative Example: The Graetz Bridge with Inductive Load
6. Extended Higher-Order Element with Frequency-Doubled State
6.1. PHL and PIL Features in the Case of a Generic Element
6.2. PHL Type in the Case of an Extended Element
- The native PHL of the extended element with frequency-doubled state is therefore of the NCT(1) type, but what the actual type is depends on the specific form of the function P(x,u). □
6.3. Illustrative Example: Extended Memristor
7. Implications of New Theorems
- If the steady PHL is of type I or is centrally asymmetric regardless of the loop type, then it is not an element with a frequency-doubled parameter (implied by Theorem 2).
- If the generic element produces a steady PHL of type I or the PHL is centrally asymmetric regardless of the loop type, then it is not an element with a frequency-doubled state (implied by Theorem 3).
- If an element generates a PHL of the classical CT(0) type, it means that at least one of its state variables does not oscillate at twice the frequency of the excitation signal (following from Theorem 4).
- If the Parameter vs. Input Loop (PIL) consists of two lobes, this does not mean that a type II PHL will be generated (see the illustrative example in Section 6.3). Thus, the current classification of loops into types I and II based on the number of PIL loops [5] does not always correspond to reality.
- If the state variable oscillates at twice the frequency of the excitation variable, it does not follow that the PHL must be of type II. However, this always follows for generic elements.
- If the state variable oscillates at the same frequency as the excitation variable, it does not mean that the PHL must be of type I.
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Biolek, Z.; Biolek, D.; Biolková, V.; Kolka, Z. Extended Higher-Order Elements with Frequency-Doubled Parameters: The Hysteresis Loops Are Always of Type II. Sensors 2023, 23, 7179. https://doi.org/10.3390/s23167179
Biolek Z, Biolek D, Biolková V, Kolka Z. Extended Higher-Order Elements with Frequency-Doubled Parameters: The Hysteresis Loops Are Always of Type II. Sensors. 2023; 23(16):7179. https://doi.org/10.3390/s23167179
Chicago/Turabian StyleBiolek, Zdeněk, Dalibor Biolek, Viera Biolková, and Zdeněk Kolka. 2023. "Extended Higher-Order Elements with Frequency-Doubled Parameters: The Hysteresis Loops Are Always of Type II" Sensors 23, no. 16: 7179. https://doi.org/10.3390/s23167179