Electrical network

Electrical network

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An electrical network is an interconnection of electrical element
Electrical element
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks...

s such as 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, 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, 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, transmission line
Transmission line
In 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...

s, voltage source
Voltage source
In 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...

s, current source
Current source
A 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...

s and switch
In electronics, a switch is an electrical component that can break an electrical circuit, interrupting the current or diverting it from one conductor to another....

es. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current. Electrical networks that consist only of sources (voltage or current), linear lumped elements (resistors, capacitors, inductors), and linear distributed elements (transmission lines) can be analyzed by algebraic and transform methods to determine DC response
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

, AC response
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....

, and transient response
Transient response
In electrical engineering and mechanical engineering, a transient response or natural response is the response of a system to a change from equilibrium. The transient response is not necessarily tied to "on/off" events but to any event that affects the equilibrium of the system...


A network that contains active electronic
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

 components is known as 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...

. Such networks are generally nonlinear and require more complex design and analysis tools.

Design methods

To design any electrical circuit, either analog
Analogue electronics
Analogue electronics are electronic systems with a continuously variable signal, in contrast to digital electronics where signals usually take only two different levels. The term "analogue" describes the proportional relationship between a signal and a voltage or current that represents the signal...

 or digital
Digital circuit
Digital electronics represent signals by discrete bands of analog levels, rather than by a continuous range. All levels within a band represent the same signal state...

, electrical engineers
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

 need to be able to predict the voltages and currents at all places within the circuit. Linear circuit
Linear circuit
A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any output of the circuit is also sinusoidal with frequency f...

s, that is, circuits with the same input and output frequency, can be analyzed by hand using complex number theory
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

. Other circuits can only be analyzed with specialized software programs or estimation techniques such as the piecewise-linear model.

Circuit simulation software, such as HSPICE, and languages such as VHDL-AMS
VHDL-AMS is a derivative of the hardware description language VHDL . It includes analog and mixed-signal extensions in order to define the behavior of analog and mixed-signal systems ....

 and verilog-AMS
Verilog-AMS is a derivative of the Verilog hardware description language. It includes analog and mixed-signal extensions in order to define the behavior of analog and mixed-signal systems. It extends the event-based simulator loops of Verilog/SystemVerilog/VHDL, by a continuous-time simulator,...

 allow engineers to design circuits without the time, cost and risk of error involved in building circuit prototypes.

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
    Ohm's law
    Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

    : The voltage across a resistor is equal to the product of the resistance and the current flowing through it.
  • Norton's theorem
    Norton's theorem
    Norton's theorem for linear electrical networks, known in Europe as the Mayer–Norton theorem, states that any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R...

    : Any network of voltage and/or current sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor.
  • Thévenin's theorem
    Thévenin's theorem
    In circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem...

    : Any network of voltage and/or current sources and resistors is electrically equivalent to a single voltage source in series with a single resistor.

  • See also Analysis of resistive circuits
    Analysis of resistive circuits
    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 a number of different techniques for achieving this...


Other more complex laws may be needed if the network contains nonlinear or reactive components. Non-linear self-regenerative heterodyning
Heterodyning is a radio signal processing technique invented in 1901 by Canadian inventor-engineer Reginald Fessenden where high frequency signals are converted to lower frequencies by combining two frequencies. Heterodyning is useful for frequency shifting information of interest into a useful...

 systems can be approximated. Applying these laws results in a set of simultaneous equations
Simultaneous equations
In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...

 that can be solved either algebraically or numerically.

Network simulation software

More complex circuits can be analyzed numerically with software such as SPICE
SPICE is a general-purpose, open source analog electronic circuit simulator.It is a powerful program that is used in integrated circuit and board-level design to check the integrity of circuit designs and to predict circuit behavior.- Introduction :Unlike board-level designs composed of discrete...

GNU Circuit Analysis Package
Gnucap is a general purpose circuit simulator started by Albert Davis in 1993. The latest stable version is 0.35 from 2006. The latest development version is from December 2009 and is quite usable....

, or symbolically using software such as SapWin
SAPWIN is proprietary a symbolic circuit simulator written in C++ for the Windows environment. Unlike more common numerical circuit simulators , SAPWIN can generate analytical Laplace domain expressions for arbitrary network functions of linear analog circuits...


Linearization around operating point

When faced with a new circuit, the software first tries to find a steady state solution
Steady state
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...

, 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
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...


Piecewise-linear approximation

Software such as the PLECS
PLECS is a Simulink toolbox for system-level simulations of electrical circuits developed by Plexim.It is especially designed for power electronics but can be used for any electrical network....

 interface to Simulink
Simulink, developed by MathWorks, is a commercial tool for modeling, simulating and analyzing multidomain dynamic systems. Its primary interface is a graphical block diagramming tool and a customizable set of block libraries. It offers tight integration with the rest of the MATLAB environment and...

 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.

See also