Passivity (engineering)
Encyclopedia
Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...

s. 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
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

(incremental passivity).

A component that is not passive is called an active component. An electronic circuit
Electronic circuit
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...

consisting entirely of passive components is called a passive circuit (and has the same properties as a passive component). Used 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.

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) .

## Thermodynamic passivity

In control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...

s 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 resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

s, transistor
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...

s, tunnel diode
Tunnel diode
A tunnel diode or Esaki diode is a type of semiconductor diode which is capable of very fast operation, well into the microwave frequency region, by using quantum mechanical effects....

s, glow tubes, capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s, 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. (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
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

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.

## Incremental passivity

In circuit design
Circuit design
The process of circuit design can cover systems ranging from complex electronic systems all the way down to the individual transistors within an integrated circuit...

, informally, passive components refer to ones that are not capable of power gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

. Under this definition, passive components include capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s, inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s, resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

s, diode
Diode
In electronics, a diode is a type of two-terminal electronic component with a nonlinear current–voltage characteristic. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material connected to two electrical terminals...

s, transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...

s, voltage sources, and current sources. They exclude devices like transistor
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...

s, vacuum tube
Vacuum tube
In electronics, a vacuum tube, electron tube , or thermionic valve , reduced to simply "tube" or "valve" in everyday parlance, is a device that relies on the flow of electric current through a vacuum...

s, relay
Relay
A relay is an electrically operated switch. Many relays use an electromagnet to operate a switching mechanism mechanically, but other operating principles are also used. Relays are used where it is necessary to control a circuit by a low-power signal , or where several circuits must be controlled...

s, tunnel diodes, and glow tubes. Formally, for a memoryless two-terminal element, this means that the current–voltage characteristic is monotonically increasing
Monotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

. 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 thermodynamically passive, under some conditions, the overall system will be incrementally passive, and therefore, 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 now almost universally considered incorrect. Here, devices like diodes would be considered active, and only very simple devices like capacitors, inductors, and resistors are considered passive. In some cases, the term "linear element
Linear element
In an electric circuit, a linear element is an electrical element with a linear relationship between current and [output[voltage]]. Resistors are the most common example of a linear element; other examples include capacitors, inductors, and transformers....

" may be a more appropriate term than "passive device." In other cases, "solid state device
Semiconductor device
Semiconductor devices are electronic components that exploit the electronic properties of semiconductor materials, principally silicon, germanium, and gallium arsenide, as well as organic semiconductors. Semiconductor devices have replaced thermionic devices in most applications...

" may be a more appropriate term than "active device."

## 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
LC circuit
An LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...

will have unbounded voltage output for a bounded voltage input, but will be stable in the sense of Lyapunov
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory 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
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...

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
Active filter
An active filter is a type of analog electronic filter that uses an amplifier stage. Amplifiers included in a filter design can be used to improve the performance, stability and predictability of a filter. An amplifier prevents the impedance of source or load stages from affecting the...

:
• Guaranteed stability
• Scale better to large signals (tens of amperes, hundreds of volts), where active devices are often impractical
• No power supply needed
• 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
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...

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
Power supply
A power supply is a device that supplies electrical energy to one or more electric loads. The term is most commonly applied to devices that convert one form of electrical energy to another, though it may also refer to devices that convert another form of energy to electrical energy...

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.