Electrical element
Encyclopedia
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistor
s, capacitor
s, and inductor
s, used in the analysis of electrical networks. Any electrical network can be analysed as multiple, interconnected electrical elements in a schematic diagram or circuit diagram
, each of which affects the voltage
in the network or current through the network. These ideal electrical elements represent real, physical electrical or electronic components
but they do not exist physically and they are assumed to have ideal properties according to a lumped element model
, while components are objects with less than ideal properties, a degree of uncertainty in their values and some degree of nonlinearity, each of which may require a combination of multiple electrical elements in order to approximate its function.
Circuit analysis using electric elements is useful for understanding many practical electrical networks using components. By analyzing the way a network is affected by its individual elements it is possible to estimate how a real network will behave.
s of the network: current,
; voltage
,
, charge
,
; and magnetic flux
,
.
The passive electrical elements and their variations are shown below in a form of a table where the rows represent the electrical properties and the columns represent the variations of these properties.
The fourth passive element
In reality, all circuit elements are non-linear and can only be approximated to linear over a certain range. To more exactly describe the passive elements, their constitutive relation is used instead of simple proportionality. From any two of the circuit variables there are six constitutive relations that can be formed. From this it is supposed that there is a theoretical fourth passive element since there are only five elements in total found in linear network analysis. This additional element is called memristor. It only has any meaning as a time-dependent non-linear element; as a time-independent linear element it reduces to a regular resistor. The constitutive relations of the passive elements are given by;
In some special cases the constitutive relation simplifies to a function of one variable. This is the case for all linear elements, but also' for example, an ideal diode, which in circuit theory terms is a non-linear resistor, has a constitutive relation of the form
.
The fourth passive element, the memristor, is proposed by Leon Chua in a 1971 paper, but a physical component demonstrating memristance was not created until thirty-seven years later. It was reported on April 30, 2008, that a working memristor had been developed by a team at HP Labs
led by scientist R. Stanley Williams
. With the advent of the memristor, each pairing of the four variables can now be related. Although memristors are analog memory elements, they are able to store one bit
of non-volatile memory
. They may see application in programmable logic
, signal processing
, neural network
s, and control systems
, among other fields. Because memristors are time-variant by definition, they are not included in linear time-invariant (LTI)
circuit models.
There are also two special non-linear elements which are sometimes used in analysis but which are not the ideal counterpart of any real component:
These are sometimes used in models of components with more than two terminals: transistors for instance.
elements that are normally introduced into network analysis. Their constitutive relations in matrix notation are;
Transformer:
Gyrator:
The transformer maps a voltage at one port to a voltage at the other in a ratio of n. The current between the same two port is mapped by 1/n. The gyrator
, on the other hand, maps a voltage at one port to a current at the other. Likewise, currents are mapped to voltages. The quantity r in the matrix is in units of resistance. The gyrator is a necessary element in analysis because it is not reciprocal. Networks built from the basic linear elements only are obliged to be reciprocal and so cannot be used by themselves to represent a non-reciprocal system. It is not essential, however, to have both the transformer and gyrator. Two gyrators in cascade are equivalent to a transformer but the transformer is usually retained for convenience. Introduction of the gyrator also makes either capacitance or inductance non-essential since a gyrator terminated with one of these at port 2 will be equivalent to the other at port 1.
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, 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, and 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, used in the analysis of electrical networks. Any electrical network can be analysed as multiple, interconnected electrical elements in a schematic diagram or circuit diagram
Circuit diagram
A circuit diagram is a simplified conventional graphical representation of an electrical circuit...
, each of which affects the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
in the network or current through the network. These ideal electrical elements represent real, physical electrical or electronic components
Electronic component
An electronic component is a basic electronic element and may be available in a discrete form having two or more electrical terminals . These are intended to be connected together, usually by soldering to a printed circuit board, in order to create an electronic circuit with a particular function...
but they do not exist physically and they are assumed to have ideal properties according to a lumped element model
Lumped element model
The lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions...
, while components are objects with less than ideal properties, a degree of uncertainty in their values and some degree of nonlinearity, each of which may require a combination of multiple electrical elements in order to approximate its function.
Circuit analysis using electric elements is useful for understanding many practical electrical networks using components. By analyzing the way a network is affected by its individual elements it is possible to estimate how a real network will behave.
One-port elements
Only nine types of two-terminal elements, five passive and four active, are required to model any electrical component or circuit. Each element is defined by a relation between the state variableState variable
A state variable is one of the set of variables that describe the "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour...
s of the network: current,
; voltageVoltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
,
, chargeElectric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
,
; and magnetic fluxMagnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
,
.Passive elements
- Resistors with resistanceElectrical resistanceThe electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
, measured in ohms – produces a voltage proportional to the current flowing through the element. Relates voltage and current according to the relation 
.
- linear (resistorResistorA 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...
) - nonlinear (varistorVaristorA varistor is an electronic component with a "diode-like" nonlinear current–voltage characteristic. The name is a portmanteau of variable resistor...
) - time-dependent nonlinear (memristorMemristorMemristor is a passive two-terminal electrical component envisioned by Leon Chua as a fundamental non-linear circuit element relating charge and magnetic flux linkage...
)
- linear (resistor
- Capacitors with capacitanceCapacitanceIn electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
, measured in faradFaradThe farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...
s – produces a current proportional to the rate of change of voltage across the element. Relates charge and voltage according to the relation
.
- linear (capacitorCapacitorA 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...
) - nonlinear (varactor)
- time-dependent nonlinear (memcapacitor)
- linear (capacitor
- Inductors with inductanceInductanceIn electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
, measured in henries – produces the magnetic flux proportional to the rate of change of current through the element. Relates flux and current according to the relation 
.
- linear (inductorInductorAn 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...
) - nonlinear inductor
- time-dependent nonlinear (meminductor)
- linear (inductor
The passive electrical elements and their variations are shown below in a form of a table where the rows represent the electrical properties and the columns represent the variations of these properties.
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... |
Varistor A varistor is an electronic component with a "diode-like" nonlinear current–voltage characteristic. The name is a portmanteau of variable resistor... |
Memristor Memristor is a passive two-terminal electrical component envisioned by Leon Chua as a fundamental non-linear circuit element relating charge and magnetic flux linkage... |
|
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... |
|||
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... |
The fourth passive element
In reality, all circuit elements are non-linear and can only be approximated to linear over a certain range. To more exactly describe the passive elements, their constitutive relation is used instead of simple proportionality. From any two of the circuit variables there are six constitutive relations that can be formed. From this it is supposed that there is a theoretical fourth passive element since there are only five elements in total found in linear network analysis. This additional element is called memristor. It only has any meaning as a time-dependent non-linear element; as a time-independent linear element it reduces to a regular resistor. The constitutive relations of the passive elements are given by;
- Resistance: constitutive relation defined as
. - Capacitance: constitutive relation defined as
. - Inductance: constitutive relation defined as
. - Memristance: constitutive relation defined as
.
- where
is an arbitrary function of two variables.
In some special cases the constitutive relation simplifies to a function of one variable. This is the case for all linear elements, but also' for example, an ideal diode, which in circuit theory terms is a non-linear resistor, has a constitutive relation of the form
.The fourth passive element, the memristor, is proposed by Leon Chua in a 1971 paper, but a physical component demonstrating memristance was not created until thirty-seven years later. It was reported on April 30, 2008, that a working memristor had been developed by a team at HP Labs
HP Labs
HP Labs is the exploratory and advanced research group for Hewlett-Packard. The lab has some 600 researchersin seven locations throughout the world....
led by scientist R. Stanley Williams
R. Stanley Williams
Richard Stanley Williams is research scientist in the field of nanotechnology and a Senior Fellow and the founding director of the Quantum Science Research laboratory at HP. He has over 57 patents, with 40 more patents pending...
. With the advent of the memristor, each pairing of the four variables can now be related. Although memristors are analog memory elements, they are able to store one bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
of non-volatile memory
Non-volatile memory
Non-volatile memory, nonvolatile memory, NVM or non-volatile storage, in the most basic sense, is computer memory that can retain the stored information even when not powered. Examples of non-volatile memory include read-only memory, flash memory, ferroelectric RAM, most types of magnetic computer...
. They may see application in programmable logic
Programmable logic device
A programmable logic device or PLD is an electronic component used to build reconfigurable digital circuits. Unlike a logic gate, which has a fixed function, a PLD has an undefined function at the time of manufacture...
, signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, neural network
Neural network
The term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...
s, and control systems
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, among other fields. Because memristors are time-variant by definition, they are not included in linear time-invariant (LTI)
LTI system theory
Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...
circuit models.
Sources
- Current sourceCurrent sourceA current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
, measured in ampereAmpereThe ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
s – produces a current in a conductor. Affects charge according to the relation
. - Voltage sourceVoltage sourceIn electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...
, measured in voltVoltThe volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
s – produces a potential difference between two points. Affects magnetic flux according to the relation
.
-
in this relationship does not necessarily represent anything physically meaningful. In the case of the current generator, 
, the time integral of current, represents the quantity of electric charge physically delivered by the generator. Here 
is the time integral of voltage but whether or not that represents a physical quantity depends on the nature of the voltage source. For a voltage generated by magnetic induction it is meaningful, but for an electrochemical source, or a voltage that is the output of another circuit, no physical meaning is attached to it.
Controlled sources
- Voltage-controlled voltage source (VCVS) Generates a voltage based on another voltage with respect to a specified gain. (has infinite input impedance and zero output impedance).
- Voltage-controlled current source (VCCS) Generates a current based on a voltage with respect to a specified gain, used to model field-effect transistorField-effect transistorThe field-effect transistor is a transistor that relies on an electric field to control the shape and hence the conductivity of a channel of one type of charge carrier in a semiconductor material. FETs are sometimes called unipolar transistors to contrast their single-carrier-type operation with...
s and vacuum tubeVacuum tubeIn 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 (has infinite input impedance and infinite output impedance). - Current-controlled voltage source (CCVS) Generates a voltage based on an input current with respect to a specified gain. (has zero input impedance and zero output impedance).
- Current-controlled current source (CCCS) Generates a current based on an input current and a specified gain. Used to model bipolar junction transistorBipolar junction transistor|- align = "center"| || PNP|- align = "center"| || NPNA bipolar transistor is a three-terminal electronic device constructed of doped semiconductor material and may be used in amplifying or switching applications. Bipolar transistors are so named because their operation involves both electrons...
s. (Has zero input impedance and infinite output impedance).
There are also two special non-linear elements which are sometimes used in analysis but which are not the ideal counterpart of any real component:
- NullatorNullatorIn electronics, a nullator is a theoretical linear, time-invariant one-port defined as having zero current and voltage across its terminals. Nullators are strange in the sense that they simultaneously have properties of both a short and an open circuit...
: defined as
- NoratorNoratorIn electronics, a norator is a theoretical linear, time-invariant one-port which can have an arbitrary current and voltage between its terminals...
: defined as an element which places no restrictions on voltage and current whatsoever.
These are sometimes used in models of components with more than two terminals: transistors for instance.
Two-port elements
All the above are two-terminal, or one-port elements. There are two lossless, passive, linear two-portTwo-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
elements that are normally introduced into network analysis. Their constitutive relations in matrix notation are;
Transformer:
Gyrator:
The transformer maps a voltage at one port to a voltage at the other in a ratio of n. The current between the same two port is mapped by 1/n. The gyrator
Gyrator
A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by 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...
, on the other hand, maps a voltage at one port to a current at the other. Likewise, currents are mapped to voltages. The quantity r in the matrix is in units of resistance. The gyrator is a necessary element in analysis because it is not reciprocal. Networks built from the basic linear elements only are obliged to be reciprocal and so cannot be used by themselves to represent a non-reciprocal system. It is not essential, however, to have both the transformer and gyrator. Two gyrators in cascade are equivalent to a transformer but the transformer is usually retained for convenience. Introduction of the gyrator also makes either capacitance or inductance non-essential since a gyrator terminated with one of these at port 2 will be equivalent to the other at port 1.
Examples
The following are examples of representation of components by way of electrical elements.- On a first degree of approximation, a batteryBattery (electricity)An electrical battery is one or more electrochemical cells that convert stored chemical energy into electrical energy. Since the invention of the first battery in 1800 by Alessandro Volta and especially since the technically improved Daniell cell in 1836, batteries have become a common power...
is represented by a voltage source. A more refined model also includes a resistance in series with the voltage source, to represent the battery's internal resistance (which results in the battery heating and the voltage dropping when in use). A current source in parallel may be added to represent its leakage (which discharges the battery over a long period of time). - On a first degree of approximation, a resistorResistorA 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...
is represented by a resistance. A more refined model also includes a series inductance, to represent the effects of its lead inductance (resistors constructed as a spiral have more significant inductance). A capacitance in parallel may be added to represent the capacitive effect of the proximity of the resistor leads to each other. A wire can be represented as a low-value resistor - Current sources are more often used when representing semiconductorSemiconductorA semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s. For example, on a first degree of approximation, a bipolar transistorTransistorA 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...
may be represented by a variable current source that is controlled by the input current.
See also
- Electronic componentElectronic componentAn electronic component is a basic electronic element and may be available in a discrete form having two or more electrical terminals . These are intended to be connected together, usually by soldering to a printed circuit board, in order to create an electronic circuit with a particular function...
- Lumped element modelLumped element modelThe lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions...
- Distributed element modelDistributed element modelIn electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...
- Transmission lineTransmission lineIn communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...