Passivity (engineering)

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 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.

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 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. 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:

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

where the notation sup_x→T≥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 E_A 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 E_A 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.

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 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" may be a more appropriate term than "passive device." In other cases, "solid state device" 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 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.

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.

Further reading

• 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. McGraw–Hill 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. McGraw–Hill 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.