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Electrical network
Electrical network
Electrical network
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A simple electric circuit made up of a voltage source and a resistor. Here, , according to Ohm's law.

An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) 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. Thus all circuits are networks, but not all networks are circuits (although networks without a closed loop are often referred to as open circuits).

A resistive network is a network containing only resistors and ideal current and voltage sources. Analysis of resistive networks is less complicated than analysis of networks containing capacitors and inductors. If the sources are constant (DC) sources, the result is a DC network. 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 that contains active electronic components is known as an electronic circuit. Such networks are generally nonlinear and require more complex design and analysis tools.

Classification

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By passivity

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An active network contains at least one voltage source or current source that can supply energy to the network indefinitely. A passive network does not contain an active source.

An active network contains one or more sources of electromotive force. Practical examples of such sources include a battery or a generator. Active elements can inject power to the circuit, provide power gain, and control the current flow within the circuit.

Passive networks do not contain any sources of electromotive force. They consist of passive elements like resistors and capacitors.

By linearity

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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.

Passive networks are generally taken to be linear, but there are exceptions. For instance, an inductor with an iron core can be driven into saturation if driven with a large enough current. In this region, the behaviour of the inductor is very non-linear.

By lumpiness

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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, or for long enough circuits (such as power transmission lines), 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

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Sources can be classified as independent sources and dependent sources.

Independent

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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

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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.

Applying electrical laws

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A number of electrical laws apply to all linear resistive 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.
  • 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.

Applying these laws results in a set of simultaneous equations that can be solved either algebraically or numerically. The laws can generally be extended to networks containing reactances. They cannot be used in networks that contain nonlinear or time-varying components.

Design methods

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Linear network analysis
Elements

ResistanceReactanceImpedance
Conductance

Components

Series and parallel circuits

Impedance transforms

Y-Δ transform Δ-Y transform star-polygon transforms

Generator theorems Network theorems

Network analysis methods

Two-port parameters

z-parametersy-parametersh-parametersg-parametersS-parameters

See also: Network analysis (electrical circuits)

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

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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

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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

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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.

See also

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Wikimedia Commons has media related to Electrical circuits.
Look up electrical circuit in Wiktionary, the free dictionary.

Representation

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Design and analysis methodologies

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Measurement

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Analogies

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Specific topologies

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Electrical network

View on Grokipedia
from Grokipedia
An electrical network is an interconnection of electrical elements, including passive components such as resistors, capacitors, and inductors, as well as active sources like voltage and current generators, connected via ideal wires to enable the distribution of and current flow. These networks form the foundational structure for analyzing and designing systems where is harnessed, stored, or transformed, adhering to physical laws that ensure and charge. The core quantities in electrical networks are voltage (or potential difference), which drives current flow between nodes, and current, the rate of charge movement through branches. Passive elements store or dissipate —resistors via according to (v=Riv = Ri), capacitors by accumulating charge, and inductors by magnetic fields—while active elements supply , either independently or dependently on other network variables. Networks can be classified by , such as series (elements sharing current), parallel (sharing voltage), or complex combinations like ladders and meshes, each influencing the overall behavior under DC or AC conditions. Analysis of electrical networks relies on fundamental laws and theorems to solve for voltages and currents. Kirchhoff's current law (KCL) states that the algebraic sum of currents at any node is zero, reflecting , while Kirchhoff's voltage law (KVL) mandates that the sum of voltages around any closed loop is zero, ensuring . Key theorems simplify complex circuits: the superposition theorem allows linear networks to be analyzed by considering each source independently; reduces any linear network to an equivalent in series with impedance; and uses a current source in parallel with equivalent resistance. Methods like nodal (KCL-based) and mesh (KVL-based) analysis provide systematic equation-solving approaches for practical computation. Electrical networks underpin diverse engineering applications, from power distribution grids that deliver electricity over long distances to integrated circuits in microelectronics for computing and communication. In signal processing, they filter and amplify waveforms; in control systems, they stabilize feedback loops; and in biomedical devices like MRI scanners, they enable precise imaging through high-current configurations. Advances in network theory continue to support innovations in renewable energy integration and semiconductor design, emphasizing efficiency and reliability.

Fundamental Concepts

Definition and Scope

An electrical network is an interconnection of electrical components, such as resistors, capacitors, inductors, and sources, connected by wires to direct or control the flow of electric current. These networks are mathematically modeled as graphs, consisting of nodes (points of connection) and branches (elements or wires linking nodes), which facilitate the analysis of voltages and currents throughout the system. Understanding electrical networks requires familiarity with foundational quantities: voltage, current, and power. Voltage represents the potential difference between two points, serving as the driving force for charge movement, measured in volts (V). Current is the rate of flow of through a conductor, quantified in amperes (A). Power, the rate of energy transfer, is calculated as the product of voltage and current, expressed in watts (). The conceptual origins of electrical network theory trace back to the mid-19th century, with Gustav Kirchhoff's formulation of his circuit laws in 1845, which provided the basis for analyzing currents and voltages in interconnected circuits. Subsequent advancements in the late , building on electromagnetic theory by James Clerk Maxwell, enabled analysis of AC networks and transmission lines by figures like . The field evolved significantly in the 20th century, incorporating applications, synthesis techniques by Ronald Foster and Wilhelm Cauer, and computational tools like for simulation, transforming it from rudimentary circuit rules to a cornerstone of modern . In scope, electrical networks encompass both theoretical models for analyzing circuit behavior and practical implementations in devices ranging from simple electronics to complex systems. Electrical networks include applications in large-scale power systems, where aids in designing efficient grids for generation, distribution, and stability at utility levels. Similarly, networks form the basis for specialized circuits, applying general interconnection principles that can include frequency-specific filtering or amplification.

Basic Components

Electrical networks are constructed from fundamental passive and active components that govern the flow, storage, and manipulation of . Passive components, namely resistors, capacitors, and inductors, dissipate or store energy without requiring external power input, serving as the building blocks for circuit behavior. Active components, including diodes, transistors, and operational amplifiers, enable signal amplification and control by drawing power from external sources. Resistors are passive elements that impede current flow and dissipate electrical energy as heat, with their defining relation given by : V=IRV = IR, where VV is the , II is the current, and RR is the resistance measured in ohms (Ω\Omega). This linear relationship allows resistors to limit current, divide voltages, and stabilize circuits, such as in voltage dividers or current-sensing applications. Power dissipation in a resistor follows P=I2RP = I^2 R or P=V2RP = \frac{V^2}{R}, highlighting their role in thermal management within networks. Capacitors function as passive charge-storage devices, consisting of two conductive plates separated by an insulator, with the stored charge QQ related to the applied voltage VV by Q=CVQ = CV, where CC is the in farads (F). They oppose rapid changes in voltage, enabling applications like , filtering noise in signals, and timing in RC circuits, where the time constant is τ=RC\tau = RC. Typical values range from picofarads (pF) for high-frequency uses to microfarads (μ\muF) for power smoothing. Inductors are passive components that store energy in through coiled wire, often around a core, and resist changes in current according to V=LdidtV = L \frac{di}{dt}, where LL is the in henrys (H) and didt\frac{di}{dt} is the rate of current change. This property makes inductors essential for smoothing current fluctuations, such as in power supplies or filters, and for creating in transformers. Common values are in millihenrys (mH) for low-frequency circuits. Active components introduce control and amplification capabilities to networks. Diodes are two-terminal devices that conduct current preferentially in one direction (forward bias) while blocking it in the reverse, exhibiting a nonlinear current-voltage characteristic with a typical forward of 0.7 V for diodes. They serve roles in rectification, protection against reverse currents, and signal clipping. Transistors, typically bipolar junction (BJT) or field-effect (FET) types, are three-terminal active devices that amplify signals or act as switches by controlling a large collector or drain current with a small base or gate input. For instance, NPN BJTs offer high-frequency performance and are used in amplification stages where output power exceeds input, drawing from an external supply. Their key is current gain (β\beta), efficient in networks. Operational amplifiers (op-amps) are high-gain, differential active devices with two inputs and one output, producing an output voltage proportional to the difference between inverting and non-inverting inputs, often with open-loop gains exceeding 10510^5. Requiring dual power supplies (e.g., ±12 V), they perform amplification, integration, and summation without dissipating input signal energy, forming the core of analog circuits like filters and comparators. In theoretical analysis, components are modeled as ideal—resistors with purely constant resistance, capacitors and inductors without losses, and wires with zero resistance—to simplify computations. Real components, however, include parasitic effects that alter performance, particularly at high frequencies; for example, resistors exhibit series inductance (e.g., ~15 nH from leads) and shunt (~1 pF), capacitors have (ESR, e.g., ~1 Ω) and inductance, while inductors include series resistance (e.g., ~1 Ω from wire) and inter-winding (~1 pF). These parasitics lead to self-resonance, limiting usable frequency ranges. Networks are assembled by interconnecting these components in series or parallel configurations, which determine equivalent properties. In series, components share the same current, yielding additive effects like total resistance Req=R1+R2R_{eq} = R_1 + R_2 or inductance Leq=L1+L2L_{eq} = L_1 + L_2; in parallel, they share voltage, resulting in reciprocal sums such as 1Ceq=1C1+1C2\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} for capacitors. These combinations form the basis for complex topologies like filters and oscillators.

Classifications

By Passivity

Electrical networks are classified by passivity into passive and active categories based on their ability to handle . Passive networks consist exclusively of passive elements, such as resistors, capacitors, and inductors, which cannot generate but instead dissipate, store, or release it. These networks adhere to the passivity theorem, which requires that the total input to the network over any finite time interval is non-negative, ensuring that output does not exceed input. For linear time-invariant multiports, passivity is equivalent to the impedance matrix being positive real, a condition that guarantees the network's -dissipative behavior. In contrast, active networks include active elements, such as transistors or operational , or independent power sources that enable energy generation or amplification beyond the input signal. These networks can produce outputs with greater power than the input, facilitating functions like signal amplification in electronic circuits. A representative example of a passive network is an RC low-pass filter, where resistors and capacitors attenuate high-frequency components without adding energy. Conversely, a transistor-based common-emitter exemplifies an active network, as it boosts the input signal's using the transistor's gain properties. The distinction by passivity carries significant implications for network behavior and design. Passive networks exhibit inherent stability, as their inability to generate prevents unbounded growth in responses, making them suitable for applications requiring reliability without external control. Active networks, however, offer versatility for power delivery and amplification but introduce the risk of , such as oscillations, necessitating careful feedback mechanisms to ensure stable operation.

By Linearity

Electrical networks are classified by based on whether their response to inputs is proportional and additive, a property that determines the applicability of certain techniques. Linear networks consist of elements where the output is directly proportional to the input, adhering to the principles of homogeneity and additivity. This classification is fundamental in circuit theory, as it influences the mathematical models used for and . In linear networks, the applies, allowing the total response to multiple inputs to be calculated as the sum of responses to each input individually. This principle holds because the network's behavior satisfies the conditions of a , where scaling the input scales the output proportionally and combining inputs combines outputs additively. Such networks are typically described by linear differential equations, which model the relationships between voltages and currents using constant coefficients for passive elements like resistors, capacitors, and inductors. For instance, the current i(t)i(t) through a linear network can be expressed as i(t)=f(v(t))i(t) = f(v(t)), where ff is a linear operator representing the system's transformation. Nonlinear networks, in contrast, incorporate components such as diodes or transistors whose characteristics do not follow proportional relationships, leading to outputs that are not simply scalable or additive with respect to inputs. In these networks, the fails because the response to combined inputs cannot be decomposed into individual contributions without accounting for interactions. Analysis of nonlinear networks often requires iterative numerical methods, such as Newton-Raphson for solving steady-state conditions, due to the absence of closed-form solutions from linear algebra. Linear network models are essential in small-signal analysis, where circuits are approximated as linear around an to evaluate stability and in amplifiers and filters. Nonlinear networks are prevalent in applications, such as converters and inverters, where devices operate over wide ranges to handle high voltages and currents efficiently.

By Lumpiness

Electrical networks are classified by lumpiness based on whether components are modeled as idealized point-like elements or as distributed parameters along their physical extent, which affects the accuracy of analysis particularly at varying frequencies. The treats circuit components, such as resistors (R), inductors (L), and capacitors (C), as discrete elements with negligible physical size relative to the of the signals involved. This approximation assumes that the time taken for electromagnetic waves to propagate across a component is insignificant, allowing the voltage and current to be considered uniform at any instant. It is valid primarily at low frequencies, where the circuit dimensions are much smaller than the signal , enabling the use of standard circuit laws like Kirchhoff's without accounting for delays. In contrast, the is employed when component sizes become comparable to the , necessitating consideration of wave propagation effects along the structure. This approach models networks using parameters distributed per unit length, such as resistance and per unit length for series elements, and conductance and per unit length for shunt elements. It is essential for high-frequency applications, where signals behave as traveling waves, leading to phenomena like reflections and standing waves. The foundational equations for this model are the in the phasor domain, which describe the relationship between voltage VV and current II along a : ∂V∂x=−(R+jωL)I\frac{\partial V}{\partial x} = -(R + j\omega L) I ∂I∂x=−(G+jωC)V\frac{\partial I}{\partial x} = -(G + j\omega C) V Here, RR, LL, GG, and CC are the per-unit-length parameters, ω\omega is the angular frequency, and xx is the position along the line. These coupled partial differential equations capture the distributed nature of the network and can be solved to yield wave equations for voltage and current propagation. The key criterion for applying the lumped-element approximation is that the physical size of each component must be significantly smaller than the λ\lambda of the operating , typically satisfying the condition where the component dimension l≪λ/10l \ll \lambda / 10. This ensures that phase shifts across the component are minimal (less than about 36 degrees), preserving the validity of lumped assumptions; beyond this threshold, distributed effects dominate, and the lumped model introduces errors in predicting network behavior. Frequencies corresponding to this limit depend on the circuit scale—for instance, for a 1 cm component, the transition occurs around 3 GHz, as λ=c/f\lambda = c / f with c≈3×108c \approx 3 \times 10^8 m/s in free space. A practical example of this transition is observed in (PCB) traces, which function as lumped elements at low frequencies but shift to distributed behavior at high frequencies, such as in RF applications above several hundred MHz. For a typical trace on a PCB with a constant around 4, the guided shortens due to the slower (v=c/ϵrv = c / \sqrt{\epsilon_r}
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