# Passivity (engineering)

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Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems. A passive component, depending on field, may be either a component that consumes but does not produce energy (thermodynamic passivity) or a component that is incapable of power gain (incremental passivity).

A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large Industrial control systems which are used for controlling processes or machines.

In electronics, gain is a measure of the ability of a two-port circuit to increase the power or amplitude of a signal from the input to the output port by adding energy converted from some power supply to the signal. It is usually defined as the mean ratio of the signal amplitude or power at the output port to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units. A gain greater than one, that is amplification, is the defining property of an active component or circuit, while a passive circuit will have a gain of less than one.

## Contents

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. Used out-of-context and without a qualifier, the term passive is ambiguous. Typically, analog designers use this term to refer to incrementally passive components and systems, while control systems engineers will use this to refer to thermodynamically passive ones.

An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. To be referred to as electronic, rather than electrical, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.

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). [1]

## 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.

A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unlimited current. A voltage source is the dual of a current source. Real-world sources of electrical energy, such as batteries, generators, can be modeled for analysis purposes as a combination of an ideal voltage source and additional combinations of impedance elements.

A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.

A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active elements, and terminate transmission lines, among other uses. High-power resistors that can dissipate many watts of electrical power as heat, may be used as part of motor controls, in power distribution systems, or as test loads for generators. Fixed resistors have resistances that only change slightly with temperature, time or operating voltage. Variable resistors can be used to adjust circuit elements, or as sensing devices for heat, light, humidity, force, or chemical activity.

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. [2] 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:

In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside circuit, that meets the port condition; the currents flowing into the two nodes must be equal and opposite.

${\displaystyle E_{A}(x)=\sup _{x\to T\geq 0}\int _{0}^{T}-\langle v(t),i(t)\rangle \,{\mathord {\operatorname {d} }}t}$

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 ${\displaystyle \langle v(t),i(t)\rangle }$ 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; this means they cannot amplify signals. Under this definition, passive components include capacitors, inductors, resistors, diodes, transformers, voltage sources, and current sources. 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]

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

• A passive USB to PS/2 adapter consists of wires, and potentially resistors and similar passive (in both the incremental and thermodynamic sense) components. An active USB to PS/2 adapter consists of logic to translate signals (active in the incremental sense)
• A passive mixer consists of just resistors (incrementally passive), whereas an active mixer includes components capable of gain (active).
• In audio work one can also find both (incrementally) passive and active converters between balanced and unbalanced lines. A passive bal/unbal converter is generally just a transformer along with, of course, the requisite connectors, while an active one typically consists of a differential drive or an instrumentation amplifier.

## Other definitions of passivity

In electronic engineering, devices that exhibit gain or a rectifying function (such as diodes) are considered active. Only capacitors, inductors, and resistors are considered passive. [3] [4] 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. [5] United States Patent and Trademark Office is amongst the organisations classing diodes as active devices. [6]

## 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 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.

A passive filter has several advantages over an active filter:

• Guaranteed stability
• Scale better to large signals (tens of amperes, hundreds of volts), where active devices are often impractical
• No power supply needed
• Often less expensive in discrete designs (unless large coils are required)
• For linear filters, potentially greater linearity depending on components required

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.

## 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.

## Related Research Articles

An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.

Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter.

An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the power of a signal. It is a two-port electronic circuit that uses electric power from a power supply to increase the amplitude of a signal applied to its input terminals, producing a proportionally greater amplitude signal at its output. The amount of amplification provided by an amplifier is measured by its gain: the ratio of output voltage, current, or power to input. An amplifier is a circuit that has a power gain greater than one.

In electrical engineering, the power factor of an AC electrical power system is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit, and is a dimensionless number in the closed interval of −1 to 1. A power factor of less than one indicates the voltage and current are not in phase, reducing the instantaneous product of the two. Real power is the instantaneous product of voltage and current and represents the capacity of the electricity for performing work. Apparent power is the average product of current and voltage. Due to energy stored in the load and returned to the source, or due to a non-linear load that distorts the wave shape of the current drawn from the source, the apparent power may be greater than the real power. A negative power factor occurs when the device generates power, which then flows back towards the source.

A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction.

The electrical resistance of an object is a measure of its opposition to the flow of electric current. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S).

Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumped element circuit model. In other cases infinitesimal elements are used to model the network in a distributed element model.

A switched-mode power supply is an electronic power supply that incorporates a switching regulator to convert electrical power efficiently. Like other power supplies, an SMPS transfers power from a DC or AC source to DC loads, such as a personal computer, while converting voltage and current characteristics. Unlike a linear power supply, the pass transistor of a switching-mode supply continually switches between low-dissipation, full-on and full-off states, and spends very little time in the high dissipation transitions, which minimizes wasted energy. A hypothetical ideal switched-mode power supply dissipates no power. Voltage regulation is achieved by varying the ratio of on-to-off time. In contrast, a linear power supply regulates the output voltage by continually dissipating power in the pass transistor. This higher power conversion efficiency is an important advantage of a switched-mode power supply. Switched-mode power supplies may also be substantially smaller and lighter than a linear supply due to the smaller transformer size and weight.

In electronics, a varicap diode, varactor diode, variable capacitance diode, variable reactance diode or tuning diode is a type of diode designed to exploit the voltage-dependent capacitance of a reverse-biased p–n junction.

In electronics, negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.

A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.

Small-signal modeling is a common analysis technique in electronics engineering which is used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC signals, the time-varying currents and voltages in the circuit, have a small magnitude compared to the DC bias currents and voltages. A small-signal model is an AC equivalent circuit in which the nonlinear circuit elements are replaced by linear elements whose values are given by the first-order (linear) approximation of their characteristic curve near the bias point.

An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not to be confused with electrical elements, which are conceptual abstractions representing idealized electronic components.

A current–voltage characteristic or I–V curve is a relationship, typically represented as a chart or graph, between the electric current through a circuit, device, or material, and the corresponding voltage, or potential difference across it.

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".

A capacitor is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.

A linear circuit is an electronic circuit which obeys the superposition principle. This means that the output of the circuit F(x) when a linear combination of signals ax1(t) + bx2(t) is applied to it is equal to the linear combination of the outputs due to the signals x1(t) and x2(t) applied separately:

Capacitors have many uses in electronic and electrical systems. They are so ubiquitous that it is rare that an electrical product does not include at least one for some purpose.

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:

## References

1. Tellegen's Theorem and Electrical Networks. Penfield, Spence, and Duinker. MIT Press, 1970. pg 24-25.
2. 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.
3. E C Young, "passive", The New Penguin Dictionary of Electronics, 2nd ed, p. 400, Penguin Books ISBN   0-14-051187-3.
4. Louis E. Frenzel, Crash Course in Electronics Technology, p. 140, Newnes, 1997 ISBN   9780750697101.
5. Ian Hickman, Analog Electronics, p. 46, Elsevier, 1999 ISBN   9780080493862.
6. Class 257: Active Solid-state Devices", U.S. Patent and Trademark Office: Information Products Division, accessed and archived 19 August 2019.
• Khalil, Hassan (2001). Nonlinear Systems (3rd Edition). Prentice Hall. ISBN   0-13-067389-7.Very readable introductory discussion on passivity in control systems.
• Chua, Leon; Desoer, Charles; Kuh, Ernest (1987). Linear and Nonlinear Circuits. McGrawHill Companies. ISBN   0-07-010898-6.Good collection of passive stability theorems, but restricted to memoryless one-ports. Readable and formal.
• Desoer, Charles; Kuh, Ernest (1969). Basic Circuit Theory. McGrawHill Education. ISBN   0-07-085183-2.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   0-395-16971-2.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.
• Brogliato, Bernard; Lozano, Rogelio; Maschke, Bernhard; Egeland, Olav (2007). Dissipative Systems: Analysis and Control, 2nd edition. Springer Verlag London. ISBN   1-84628-516-X.A complete exposition of dissipative systems, with emphasis on the celebrated KYP Lemma, and on Willems' dissipativity and its use in Control.