# Passivity (engineering)

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Passivity (engineering)

Passivity is a property of engineering systems, most commonly used in electronic engineering and control systems. A passive component, depending on field, may either refer to a component that consumes (but does not produce) energy, or to a component that is incapable of power gain. A component that is not passive is called an active component. An electronic circuit consisting entirely of passive components is called a passive circuit (and has the same properties as a passive 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 transistors, resistors, tunnel diodes, glow tubes, capacitors, and other dissipative and energy-neutral components are considered passive. For memoryless two-terminal elements, this means that the current-voltage characteristics lie in the first and third quadrant. 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). We will state a correct, formal definition, taken from Wyatt [IEEE Transactions on Circuits and Systems, CAS-28, No 1. January 1981, page 48-61. "Energy Concepts in the State-Space Theory of Nonlinear n-Ports: Part I -- Passivity", by Wyatt, et al.] (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 $E_A$ as:

:

Where the notation $sup_\left\{x ightarrow T>0\right\}$ indicates that the supremum is taken over all $T>0$ and all admissible pairs $left\left\{v\left(cdot\right), i\left(cdot\right) ight\right\}$. A system is considered passive if $E_A$ is finite for all initial states $x$. Otherwise, the system is considered active.

Incremental passivity

In circuit design, informally, passive components refer to ones that are not capable of power gain. Under this definition, passive components include capacitors, inductors, resistors, transformers, voltage sources, and current sources. They exclude devices like transistors, relays, glow tubes, tunnel diodes, and similar devices. 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. [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 passive, under some conditions, the overall system will be stable. This needs to be verified. ]

Other definitions of passivity

In some very informal settings, passivity may refer to the simplicity of the device, although this definition is almost universally considered incorrect (indeed, maybe should not even be included here). Here, devices like diodes would be considered active, [Young EC, "passive", "The Penguin Dictionary of Electronics", 2nd ed, ISBN 0140511873] and only very simple devices like capacitors, inductors, and resistors are considered passive. In some cases, the term "linear element" may be a more appropriate term than "passive device." In other cases, "solid state device" may be a more appropriate term than "active device."

tability

Passivity, in most cases, can be used to demonstrate that passive circuits will be stable under specific criteria. Note that this only works if only one of the above definitions of passivity is used -- if components from the two are mixed, the systems will, in general, not be stable 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. 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 elements -- 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.

A passive filter has several advantages over an active filter:
* Guaranteed stability
* Passive filters scale better to large signals (tens of amps, hundreds of volts), where active devices are often impractical
* No power consumption (aside from possibly taking some power out of the signal)
* Inexpensive
* For linear filters, generally, more linear than filters including active (and therefore non-linear) elements

They are commonly used in speaker crossover design (due to the moderately large voltages and currents, and the lack of easy access to power), 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 less common in integrated circuit design, where active devices are comparatively inexpensive compared to resistors and capacitors, and inductors are prohibitively expensive.

Notes

References

*cite book|last=Khalil|first=Hassan|title=Nonlinear Systems (3rd Edition)|year=2001|publisher=Prentice Hall|isbn=0130673897 The most readable introductory discussion on passivity in control systems.
*cite book|last=Chua|first=Leon|authorlink=Leon O. Chua|coauthors=Desoer, Charles; Kuh, Ernest|title=Linear and Nonlinear Circuits|year=1987|publisher=McGraw-Hill Companies|isbn=0070108986 Good collection of passive stability theorems, but restricted to memoryless one-ports. Readable and formal.
*cite book|last=Desoer|first=Charles|coauthors=Kuh, Ernest|title=Basic Circuit Theory|year=1969|publisher=McGraw-Hill Education|isbn=0070851832 Somewhat less readable than Chua, and more limited in scope and formality of theorems.
*cite book|last=Cruz|first=Jose|coauthors=Van Valkenberg, M.E.|title=Signals in Linear Circuits|year=1974|publisher=Houghton Mifflin|isbn=0395169712 Gives a definition of passivity for multiports (in contrast to the above), but the overall discussion of passivity is quite limited.
*cite book|last=Wyatt|first=J.L.|coauthors=Chua, L.O.; Gannett, J.; Göknar, I.C.; Green, D.|title=Foundations of Nonlinear Network Theory, Part I: Passivity|year=1978|publisher=Memorandum UCB/ERL M78/76, Electronics Research Laboratory, University of California, Berkeley
cite book|last=Wyatt|first=J.L.|coauthors=Chua, L.O.; Gannett, J.; Göknar, I.C.; Green, D.|title=Foundations of Nonlinear Network Theory, Part II: Losslessness|year=1980|publisher=Memorandum UCB/ERL M80/3, Electronics Research Laboratory, University of California, Berkeley A pair of memos that have good discussions of passivity.

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