# Passivity (engineering): Wikis

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# Encyclopedia

### From Wikipedia, the free encyclopedia

Passivity is a property of engineering systems, most commonly used in electronic engineering and control systems. A passive component, depending on field, may be either a component that consumes (but does not produce) energy, or 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 and cannot control the flow of electrons, but does not produce energy. Under this methodology, voltage and current sources are considered active, while resistors, tunnel diodes, glow tubes, capacitors, metamaterials 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 (i.e., voltage and current have the same sign). 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:

$E_A(x)=\sup_{x \to T \geq 0} \int_0^T -\langle v(t),i(t)\rangle \, \mathord{\operatorname{d}}t$

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 $\langle v(t),i(t) \rangle$ 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.

## 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, diodes, and transformers. They exclude devices like transistors, vacuum tubes, relays, glow tubes, energy sources like current- or voltage sources, 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.[nb 1]

## 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. Here, devices like diodes would be considered active,[2] 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."

To be fair to this view of passivity, a non-linear device will inevitably include a generator in its small-signal equivalent circuit and so will be modelled as a source of energy. This arises because a linear approximation to a small section of the transfer function is unlikely to pass through the origin and such an offset requires a generator in the equivalent circuit to produce the value of the offset current or voltage. For instance, a forward biased diode can be modelled as a resistor in series with a DC offset as far as small signals are concerned.

Keeping component function value variation as the parameter for definition, a relay consists of a solenoid or induction coil combined with a switching mechanism. Going by previous examples, both coil and switch are passive components, and so a relay is a passive component. It merely changes or routes the path of current without any gain as a switch, while the coil resistance or impedance does not change throughout its useful life.

## Stability

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.

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:

• Guaranteed stability
• Passive filters scale better to large signals (tens of amperes, hundreds of volts), where active devices are often impractical
• No power consumption, but the desired signal is invariably attenuated. If no resistors are used, the amount of signal loss is directly related to the quality (and the price) of the components used.
• Inexpensive (unless large coils are required)
• 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 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.

## 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 passive, under some conditions, the overall system will be stable. This needs to be verified.

## 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 I—Passivity", IEEE Transactions on Circuits and Systems CAS-28 (1): 48–61
2. ^ Young EC, passive, The Penguin Dictionary of Electronics, 2nd ed, ISBN 0140511873

## Further reading

• Khalil, Hassan (2001). Nonlinear Systems (3rd Edition). Prentice Hall. ISBN 0130673897.  — Very readable introductory discussion on passivity in control systems.
• Chua, Leon; Desoer, Charles; Kuh, Ernest (1987). Linear and Nonlinear Circuits. McGraw–Hill Companies. ISBN 0070108986.  — Good collection of passive stability theorems, but restricted to memoryless one-ports. Readable and formal.
• Desoer, Charles; Kuh, Ernest (1969). Basic Circuit Theory. McGraw–Hill Education. ISBN 0070851832.  — Somewhat less readable than Chua, and more limited in scope and formality of theorems.
• Cruz, Jose; Van Valkenberg, M.E. (1974). Signals in Linear Circuits. 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.
• Wyatt, J.L.; Chua, L.O.; Gannett, J.; Göknar, I.C.; Green, D. (1978). Foundations of Nonlinear Network Theory, Part I: Passivity. Memorandum UCB/ERL M78/76, Electronics Research Laboratory, University of California, Berkeley.
Wyatt, J.L.; Chua, L.O.; Gannett, J.; Göknar, I.C.; Green, D. (1980). Foundations of Nonlinear Network Theory, Part II: Losslessness. 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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