Passivity (engineering)

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Passivity is a property of engineering systems, most commonly encountered in analog electronics and control systems. Typically, analog designers use passivity to refer to incrementally passive components and systems, which are incapable of power gain. In contrast, control systems engineers will use passivity to refer to thermodynamically passive ones, which consume, but do not produce, energy. As such, without context or a qualifier, the term passive is ambiguous.

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An electronic circuit consisting entirely of passive components is called a passive circuit, and has the same properties as a passive component.

If a component is not passive, then it is an active component.

Thermodynamic passivity

In control systems and circuit network theory, a passive component or circuit is one that consumes energy, but does not produce energy. Under this methodology, voltage and current sources are considered active, while resistors, capacitors, inductors, transistors, tunnel diodes, metamaterials and other dissipative and energy-neutral components are considered passive. Circuit designers will sometimes refer to this class of components as dissipative, or thermodynamically passive.

While many books give definitions for passivity, many of these contain subtle errors in how initial conditions are treated and, occasionally, the definitions do not generalize to all types of nonlinear time-varying systems with memory. Below is a correct, formal definition, taken from Wyatt et al., [1] which also explains the problems with many other definitions. Given an n-port R with a state representation S, and initial state x, define available energy EA as:

where the notation supxT≥0 indicates that the supremum is taken over all T  0 and all admissible pairs {v(·), i(·)} with the fixed initial state x (e.g., all voltage–current trajectories for a given initial condition of the system). A system is considered passive if EA is finite for all initial states x. Otherwise, the system is considered active. Roughly speaking, the inner product is the instantaneous power (e.g., the product of voltage and current), and EA is the upper bound on the integral of the instantaneous power (i.e., energy). This upper bound (taken over all T  0) is the available energy in the system for the particular initial condition x. If, for all possible initial states of the system, the energy available is finite, then the system is called passive. If the available energy is finite, it is known to be non-negative, since any trajectory with voltage gives an integral equal to zero, and the available energy is the supremum over all possible trajectories. Moreover, by definition, for any trajectory {v(·), i(·)}, the following inequality holds:

.

The existence of a non-negative function EA that satisfies this inequality, known as a "storage function", is equivalent to passivity. [2] For a given system with a known model, it is often easier to construct a storage function satisfying the differential inequality than directly computing the available energy, as taking the supremum on a collection of trajectories might require the use of calculus of variations.

Incremental passivity

In circuit design, informally, passive components refer to ones that are not capable of power gain; this means they cannot amplify signals. Under this definition, passive components include capacitors, inductors, resistors, diodes, transformers, voltage sources, and current sources. [3] They exclude devices like transistors, vacuum tubes, relays, tunnel diodes, and glow tubes. Formally, for a memoryless two-terminal element, this means that the current–voltage characteristic is monotonically increasing. For this reason, control systems and circuit network theorists refer to these devices as locally passive, incrementally passive, increasing, monotone increasing, or monotonic. It is not clear how this definition would be formalized to multiport devices with memory as a practical matter, circuit designers use this term informally, so it may not be necessary to formalize it. [nb 1] [4]

This term is used colloquially in a number of other contexts:

Other definitions of passivity

In electronic engineering, devices that exhibit gain or a rectifying function (such as diodes) are considered active. Only resistors, capacitors, inductors, transformers, and gyrators are considered passive. [5] [6] In terms of abstract theory, diodes can be considered non-linear resistors, but non-linearity in a resistor would not normally be directional, which is the property that leads to diodes being classified as active. [7] United States Patent and Trademark Office is amongst the organisations classing diodes as active devices. [8]

Systems for which the small signal model is not passive are sometimes called locally active (e.g. transistors and tunnel diodes). Systems that can generate power about a time-variant unperturbed state are often called parametrically active (e.g. certain types of nonlinear capacitors). [9]

Stability

Passivity, in most cases, can be used to demonstrate that passive circuits will be stable under specific criteria. This only works if only one of the above definitions of passivity is used if components from the two are mixed, the systems may be unstable under any criteria. In addition, passive circuits will not necessarily be stable under all stability criteria. For instance, a resonant series LC circuit will have unbounded voltage output for a bounded voltage input, but will be stable in the sense of Lyapunov, and given bounded energy input will have bounded energy output.

Passivity is frequently used in control systems to design stable control systems or to show stability in control systems. This is especially important in the design of large, complex control systems (e.g. stability of airplanes). Passivity is also used in some areas of circuit design, especially filter design.

Passive filter

A passive filter is a kind of electronic filter that is made only from passive components in contrast to an active filter, it does not require an external power source (beyond the signal). Since most filters are linear, in most cases, passive filters are composed of just the four basic linear elements resistors, capacitors, inductors, and transformers. More complex passive filters may involve nonlinear elements, or more complex linear elements, such as transmission lines.

Television signal splitter consisting of a passive high-pass filter (left) and a passive low-pass filter (right). The antenna is connected to the screw terminals to the left of center. Hilofilter.agr.jpg
Television signal splitter consisting of a passive high-pass filter (left) and a passive low-pass filter (right). The antenna is connected to the screw terminals to the left of center.

A passive filter has several advantages over an active filter:

They are commonly used in speaker crossover design (due to the moderately large voltages and currents, and the lack of easy access to a power supply), filters in power distribution networks (due to the large voltages and currents), power supply bypassing (due to low cost, and in some cases, power requirements), as well as a variety of discrete and home brew circuits (for low-cost and simplicity). Passive filters are uncommon in monolithic integrated circuit design, where active devices are inexpensive compared to resistors and capacitors, and inductors are prohibitively expensive. Passive filters are still found, however, in hybrid integrated circuits. Indeed, it may be the desire to incorporate a passive filter that leads the designer to use the hybrid format.

Energic and non-energic passive circuit elements

Passive circuit elements may be divided into energic and non-energic kinds. When current passes through it, an energic passive circuit element converts some of the energy supplied to it into heat. It is dissipative. When current passes through it, a non-energic passive circuit element converts none of the energy supplied to it into heat. It is non-dissipative. Resistors are energic. Ideal capacitors, inductors, transformers, and gyrators are non-energic. [10]

Notes

  1. This is probably formalized in one of the extensions to Duffin's Theorem. One of the extensions may state that if the small signal model is thermodynamically passive, under some conditions, the overall system will be incrementally passive, and therefore, stable. This needs to be verified.

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<span class="mw-page-title-main">Electrical network</span> Assemblage of connected electrical elements

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<span class="mw-page-title-main">Amplifier</span> Electronic device/component that increases the strength of a signal

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<span class="mw-page-title-main">Rectifier</span> Electrical device that converts AC to DC

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<span class="mw-page-title-main">Negative resistance</span> Property that an increasing voltage results in a decreasing current

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<span class="mw-page-title-main">Gyrator</span> Two-port non-reciprocal network element

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Chua's circuit is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never "repeats". It was invented in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system, leading some to declare it "a paradigm for chaos".

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<span class="mw-page-title-main">Chua's diode</span> Electronic resistor

In electronics and chaos theory, Chua's diode is a type of two-terminal, nonlinear active resistor which can be described with piecewise-linear equations. It is an essential part of Chua's circuit, a simple electronic oscillator circuit which exhibits chaotic oscillations and is widely used as an example for a chaotic system. It is implemented as a voltage-controlled, nonlinear negative resistor.

The following outline is provided as an overview of and topical guide to electronics:

This glossary of electrical and electronics engineering is a list of definitions of terms and concepts related specifically to electrical engineering and electronics engineering. For terms related to engineering in general, see Glossary of engineering.

References

  1. Wyatt Jr., John L.; Chua, Leon O.; Gannett, Joel W.; Göknar, Izzet C.; Green, Douglas N. (January 1981). "Energy Concepts in the State-Space Theory of Nonlinear n-Ports: Part IPassivity" (PDF). IEEE Transactions on Circuits and Systems. CAS-28 (1): 48–61. doi:10.1109/TCS.1981.1084907.
  2. Khalil, Hassan (2001). Nonlinear Systems (3rd ed.). Prentice Hall. ISBN   0-13-067389-7.
  3. Rath, Subh (29 April 2022). "What are the Main Differences Between Active and Passive Components in Electronics?". Electronic Components CSE. Archived from the original on Aug 15, 2022. Retrieved 6 July 2022.
  4. Loría, Antonio; Nijmeijer, Henk. "Passivity based control" (PDF). Control Systems, Robotics, and Automation. Vol. XIII. Encyclopedia of Life Support Systems. Retrieved 6 July 2022.
  5. E C Young, "passive", The New Penguin Dictionary of Electronics, 2nd ed, p. 400, Penguin Books ISBN   0-14-051187-3.
  6. Louis E. Frenzel, Crash Course in Electronics Technology, p. 140, Newnes, 1997 ISBN   9780750697101.
  7. Ian Hickman, Analog Electronics, p. 46, Elsevier, 1999 ISBN   9780080493862.
  8. Class 257: Active Solid-state Devices", U.S. Patent and Trademark Office: Information Products Division, accessed and archived 19 August 2019.
  9. Tellegen's Theorem and Electrical Networks. Penfield, Spence, and Duinker. MIT Press, 1970. pg 24-25.
  10. Nordholt, E.H. (1983). Design of High-Performance Negative Feedback Amplifiers, Elsevier Scientific Publishing Company, Amsterdam, ISBN   0 444 42140 8, p. 15.

Further reading