Passive Filter
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Passivity is a property of engineering systems, most commonly encountered in
analog electronics Analogue electronics ( en-US, analog electronics) are electronic systems with a continuously variable signal, in contrast to digital electronics where signals usually take only two levels. The term "analogue" describes the proportional relati ...
and
control system 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 c ...
s. Typically, analog designers use ''passivity'' to refer to incrementally passive components and systems, which are incapable of
power gain The power gain of an electrical network is the ratio of an output power to an input power. Unlike other signal gains, such as voltage and current gain, "power gain" may be ambiguous as the meaning of terms "input power" and "output power" is not alw ...
. In contrast, control systems engineers will use ''passivity'' to refer to thermodynamically passive ones, which consume, but do not produce, energy. As such, without context or a qualifier, the term ''passive'' is ambiguous. An
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. It is a type of electrical ...
consisting entirely of passive components is called a passive circuit, and has the same properties as a passive component. If a component is ''not'' passive, then it is an active component.


Thermodynamic passivity

In
control system 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 c ...
s and circuit network theory, a passive component or circuit is one that consumes energy, but does not produce energy. Under this methodology,
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
and
current sources A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it. A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed ...
are considered active, while
resistor 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 el ...
s,
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s,
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s,
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ...
s,
tunnel diode A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively "negative resistance" due to the quantum mechanical effect called tunneling. It was invented in August 1957 by Leo Esaki, Yuriko Kurose, and Takashi Suzuki ...
s,
metamaterial A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
s 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 A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as Ham ...
''R'' with a state representation ''S'', and initial state ''x'', define available energy ''E''''A'' as: :E_A(x)=\sup_ \int_0^T -\langle v(t),i(t)\rangle \, \mathordt where the notation sup''x''→''T''≥0 indicates that the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
is taken over all ''T'' ≥ 0 and all admissible pairs 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 In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
\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 The process of circuit design can cover systems ranging from complex electronic systems down to the individual transistors within an integrated circuit. One person can often do the design process without needing a planned or structured design ...
, informally, passive components refer to ones that are not capable of
power gain The power gain of an electrical network is the ratio of an output power to an input power. Unlike other signal gains, such as voltage and current gain, "power gain" may be ambiguous as the meaning of terms "input power" and "output power" is not alw ...
; this means they cannot amplify signals. Under this definition, passive components include
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s,
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s,
resistor 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 el ...
s,
diode A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other. A diode ...
s,
transformer A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
s, voltage sources, and current sources. They exclude devices like
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ...
s,
vacuum tube A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied. The type kn ...
s,
relay A relay Electromechanical relay schematic showing a control coil, four pairs of normally open and one pair of normally closed contacts An automotive-style miniature relay with the dust cover taken off A relay is an electrically operated switch ...
s, tunnel diodes, and
glow tube A voltage-regulator tube (VR tube) is an electronic component used as a shunt regulator to hold a voltage constant at a pre-determined level. Physically, these devices resemble vacuum tubes, but there are two main differences: * Their glass env ...
s. Formally, for a memoryless two-terminal element, this means that the
current–voltage characteristic A current–voltage characteristic or I–V curve (current–voltage 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 p ...
is
monotonically increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
. 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. 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
balun A balun (from "balanced to unbalanced", originally, but now dated from "balancing unit") is an electrical device that allows balanced and unbalanced lines to be interfaced without disturbing the impedance arrangement of either line. A balun c ...
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
diode A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other. A diode ...
s) are considered active. Only resistors, capacitors, inductors, transformers, and gyrators are considered passive. 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.
United States Patent and Trademark Office The United States Patent and Trademark Office (USPTO) is an agency in the U.S. Department of Commerce that serves as the national patent office and trademark registration authority for the United States. The USPTO's headquarters are in Alexa ...
is amongst the organisations classing diodes as active devices. Systems for which the
small signal model Small-signal modeling is a common analysis technique in electronics engineering used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC si ...
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).


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 An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
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 Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components ...
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 An active filter is a type of analog circuit implementing an electronic filter using active components, typically an amplifier. Amplifiers included in a filter design can be used to improve the cost, performance and predictability of a 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 Speaker may refer to: Society and politics * Speaker (politics), the presiding officer in a legislative assembly * Public speaker, one who gives a speech or lecture * A person producing speech: the producer of a given utterance, especially: ** I ...
crossover design (due to the moderately large voltages and currents, and the lack of easy access to a power supply), filters in
power distribution Electric power distribution is the final stage in the delivery of electric power; it carries electricity from the transmission system to individual consumers. Distribution substations connect to the transmission system and lower the transmissio ...
networks (due to the large voltages and currents),
power supply A power supply is an electrical device that supplies electric power to an electrical load. The main purpose of a power supply is to convert electric current from a source to the correct voltage, current, and frequency to power the load. As a r ...
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 An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny M ...
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 circuit A hybrid integrated circuit (HIC), hybrid microcircuit, hybrid circuit or simply hybrid is a miniaturized electronic circuit constructed of individual devices, such as semiconductor devices (e.g. transistors, diodes or monolithic ICs) and pa ...
s. Indeed, it may be the desire to incorporate a passive filter that leads the designer to use the hybrid format.


Energic and non-energic passive circuit elements

Passive circuit elements may be divided into energic and non-energic kinds. When current passes through it, an energic passive circuit element converts some of the energy supplied to it into
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
. It is
dissipative In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to a ...
. When current passes through it, a non-energic passive circuit element converts none of the energy supplied to it into heat. It is non-dissipative. Resistors are energic. Ideal capacitors, inductors, transformers, and gyrators are non-energic.Nordholt, E.H. (1983). Design of High-Performance Negative Feedback Amplifiers, Elsevier Scientific Publishing Company, Amsterdam, , p. 15.


Notes


References


Further reading

* —Very readable introductory discussion on passivity in control systems. * —Good collection of passive stability theorems, but restricted to memoryless one-ports. Readable and formal. * —Somewhat less readable than Chua, and more limited in scope and formality of theorems. * —Gives a definition of passivity for multiports (in contrast to the above), but the overall discussion of passivity is quite limited. *

— A pair of memos that have good discussions of passivity. * —A complete exposition of dissipative systems, with emphasis on the celebrated KYP Lemma, and on Willems' dissipativity and its use in Control. {{refend Engineering concepts