# Electrical network

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An electrical network is an interconnection of electrical components (e.g. batteries, resistors, inductors, capacitors, switches) or a model of such an interconnection, consisting of electrical elements (e.g. voltage sources, current sources, resistances, inductances, capacitances). 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 (voltage or current), linear lumped elements (resistors, capacitors, inductors), and linear distributed elements (transmission lines), 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.

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 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.

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 coil around a core.

## Contents

A resistive circuit is a circuit containing only resistors and ideal current and voltage sources. Analysis of resistive circuits is less complicated than analysis of circuits containing capacitors and inductors. If the sources are constant (DC) sources, the result is a DC circuit. The effective resistance and current distribution properties of arbitrary resistor networks can be modeled in terms of their graph measures and geometrical properties. [1]

A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are many different techniques for calculating these values. However, for the most part, the applied technique assumes that the components of the network are all linear. The methods described in this article are only applicable to linear network analysis, except where explicitly stated.

Direct current (DC) is the unidirectional flow of electric charge. A battery is a good example of a DC power supply. Direct current may flow in a conductor such as a wire, but can also flow through semiconductors, insulators, or even through a vacuum as in electron or ion beams. The electric current flows in a constant direction, distinguishing it from alternating current (AC). A term formerly used for this type of current was galvanic current.

A network that contains active electronic components is known as an electronic circuit . Such networks are generally nonlinear and require more complex design and analysis tools.

Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. The identification of the electron in 1897, along with the invention of the vacuum tube, which could amplify and rectify small electrical signals, inaugurated the field of electronics and the electron age.

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.

## Classification

### By passivity

An active network is a network that contains an active source – either a voltage source or current source.

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 passive network is a network that does not contain an active source.

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 or a component that is incapable of power gain.

An active network contains one or more sources of electromotive force. It consists of active elements like a battery or a transistor. Active elements can inject power to the circuit, provide power gain, and control the current flow within the circuit.

Electromotive force, abbreviated emf, is the electrical intensity or "pressure" developed by a source of electrical energy such as a battery or generator. A device that converts other forms of energy into electrical energy provides an emf as its output.

Passive networks do not contain any sources of electromotive force. They consist of passive elements like resistors and capacitors. These elements are not capable of the same functions as active elements.

### By linearity

A network is linear if its signals obey the principle of superposition; otherwise it is non-linear.

### By lumpiness

Discrete passive components (resistors, capacitors and inductors) are called lumped elements because all of their, respectively, resistance, capacitance and inductance is assumed to be located ("lumped") at one place. This design philosophy is called the lumped element model and networks so designed are called lumped element circuits. This is the conventional approach to circuit design. At high enough frequencies the lumped assumption no longer holds because there is a significant fraction of a wavelength across the component dimensions. A new design model is needed for such cases called the distributed element model. Networks designed to this model are called distributed element circuits.

A distributed element circuit that includes some lumped components is called a semi-lumped design. An example of a semi-lumped circuit is the combline filter.

## Classification of sources

Sources can be classified as independent sources and dependent sources.

### Independent

An ideal independent source maintains the same voltage or current regardless of the other elements present in the circuit. Its value is either constant (DC) or sinusoidal (AC). The strength of voltage or current is not changed by any variation in the connected network.

### Dependent

Dependent sources depend upon a particular element of the circuit for delivering the power or voltage or current depending upon the type of source it is.

## Electrical laws

A number of electrical laws apply to all electrical networks. These include:

• Kirchhoff's current law: The sum of all currents entering a node is equal to the sum of all currents leaving the node.
• Kirchhoff's voltage law: The directed sum of the electrical potential differences around a loop must be zero.
• Ohm's law: The voltage across a resistor is equal to the product of the resistance and the current flowing through it.
• Norton's theorem: Any network of voltage or current sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor.
• Thévenin's theorem: Any network of voltage or current sources and resistors is electrically equivalent to a single voltage source in series with a single resistor.
• Superposition theorem: In a linear network with several independent sources, the response in a particular branch when all the sources are acting simultaneously is equal to the linear sum of individual responses calculated by taking one independent source at a time.

Other more complex laws may be needed if the network contains nonlinear or reactive components. Non-linear self-regenerative heterodyning systems can be approximated. Applying these laws results in a set of simultaneous equations that can be solved either algebraically or numerically.

## Design methods

Linear network analysis
Elements

Components

Series and parallel circuits

Impedance transforms

Generator theorems Network theorems

Network analysis methods

Two-port parameters

To design any electrical circuit, either analog or digital, electrical engineers need to be able to predict the voltages and currents at all places within the circuit. Simple linear circuits can be analyzed by hand using complex number theory. In more complex cases the circuit may be analyzed with specialized computer programs or estimation techniques such as the piecewise-linear model.

Circuit simulation software, such as HSPICE (an analog circuit simulator), [2] and languages such as VHDL-AMS and verilog-AMS allow engineers to design circuits without the time, cost and risk of error involved in building circuit prototypes.

## Network simulation software

More complex circuits can be analyzed numerically with software such as SPICE or GNUCAP, or symbolically using software such as SapWin.

### Linearization around operating point

When faced with a new circuit, the software first tries to find a steady state solution, that is, one where all nodes conform to Kirchhoff's current law and the voltages across and through each element of the circuit conform to the voltage/current equations governing that element.

Once the steady state solution is found, the operating points of each element in the circuit are known. For a small signal analysis, every non-linear element can be linearized around its operation point to obtain the small-signal estimate of the voltages and currents. This is an application of Ohm's Law. The resulting linear circuit matrix can be solved with Gaussian elimination.

### Piecewise-linear approximation

Software such as the PLECS interface to Simulink uses piecewise-linear approximation of the equations governing the elements of a circuit. The circuit is treated as a completely linear network of ideal diodes. Every time a diode switches from on to off or vice versa, the configuration of the linear network changes. Adding more detail to the approximation of equations increases the accuracy of the simulation, but also increases its running time.

## Related Research Articles

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.

As originally stated in terms of DC resistive circuits only, Thévenin's theorem holds that:

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.

Practical capacitors and inductors as used in electric circuits are not ideal components with only capacitance or inductance. However, they can be treated, to a very good degree of approximation, as being ideal capacitors and inductors in series with a resistance; this resistance is defined as the equivalent series resistance (ESR). If not otherwise specified, the ESR is always an AC resistance measured at specified frequencies, 100 kHz for switched-mode power supply components, 120 Hz for linear power-supply components, and at the self-resonant frequency for general-application components. Audio components may report "Q factor", incorporating ESR among other things, at 1000 Hz.

Kirchhoff's laws are two equalities that deal with the current and potential difference in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws. These laws can be applied in time and frequency domains and form the basis for network analysis.

In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly, that is a simplest form of a more complex circuit in order to aid analysis. In its most common form, an equivalent circuit is made up of linear, passive elements. However, more complex equivalent circuits are used that approximate the nonlinear behavior of the original circuit as well. These more complex circuits often are called macromodels of the original circuit. An example of a macromodel is the Boyle circuit for the 741 operational amplifier.

The electronic–hydraulic analogy is the most widely used analogy for "electron fluid" in a metal conductor. Since electric current is invisible and the processes at play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. Electricity was originally understood to be a kind of fluid, and the names of certain electric quantities are derived from hydraulic equivalents. As with all analogies, it demands an intuitive and competent understanding of the baseline paradigms.

Electronic filters are circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both. Electronic filters can be:

Mathematical methods are integral to the study of electronics.

The superposition theorem for electrical circuits states that for a linear system the response in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any steady-state output of the circuit is also sinusoidal with frequency f. Note that the output need not be in phase with the input.

An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.

The impedance analogy is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.

Mechanical–electrical analogies are the representation of mechanical systems as electrical networks. At first, such analogies were used in reverse to help explain electrical phenomena in familiar mechanical terms. James Clerk Maxwell introduced analogies of this sort in the 19th century. However, as electrical network analysis matured it was found that certain mechanical problems could more easily be solved through an electrical analogy. Theoretical developments in the electrical domain that were particularly useful where the representation of an electrical network as an abstract topological diagram using the lumped element model and the ability of network analysis to synthesise a network to meet a prescribed frequency function.

The mobility analogy, also called admittance analogy or Firestone analogy, is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.

## References

1. Kumar, Ankush; Vidhyadhiraja, N. S.; Kulkarni, G. U . (2017). "Current distribution in conducting nanowire networks". Journal of Applied Physics. 122: 045101. Bibcode:2017JAP...122d5101K. doi:10.1063/1.4985792.
2. "HSPICE" (PDF). HSpice. Stanford University, Electrical Engineering Department. 1999.