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Electrical element
Electrical element
Electrical element
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In electrical engineering, electrical elements are conceptual abstractions representing idealized electrical components,[1] 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.

These ideal electrical elements represent actual, physical electrical or electronic components. Still, they do not exist physically and are assumed to have ideal properties. In contrast, actual electrical components have less than ideal properties, a degree of uncertainty in their values, and some degree of nonlinearity. To model the nonideal behavior of a real circuit component may require a combination of multiple ideal electrical elements to approximate its function. For example, an inductor circuit element is assumed to have inductance but no resistance or capacitance, while a real inductor, a coil of wire, has some resistance in addition to its inductance. This may be modeled by an ideal inductance element in series with a resistance.

Circuit analysis using electric elements is useful for understanding practical networks of electrical components. Analyzing how a network is affected by its individual elements makes it possible to estimate how a real network will behave.

Types

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Circuit elements can be classified into different categories. One is how many terminals they have to connect them to other components:

  • One-port elements – represent the simplest components, with only two terminals to connect to. Examples are
  • Two-port elements – are the most common multiport elements with four terminals consisting of two ports.
  • Multiport elements – these have more than two terminals. They connect to the external circuit through multiple pairs of terminals called ports. For example,
    • a transformer with three separate windings has six terminals and could be idealized as a three-port element; the ends of each winding are connected to a pair of terminals representing a port.

Elements can also be divided into active and passive:

  • Passive elements – These elements do not have a source of energy; examples are
    • diodes,
    • resistances,
    • capacitances,
    • and inductances.

Another distinction is between linear and nonlinear:

This graph shows the nonlinearity of the current versus voltage curve of diodes.
  • Nonlinear elements – these are elements in which the relation between voltage and current is a nonlinear function. An example is a diode, where the current is an exponential function of the voltage. Circuits with nonlinear elements are harder to analyse and design, often requiring circuit simulation computer programs such as SPICE.

One-port elements

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Only nine types of element (memristor not included), five passive and four active, are required to model any electrical component or circuit.[2] Each element is defined by a relation between the state variables of the network: current, ; voltage, ; charge, ; and magnetic flux, .

  • Two sources:
    • Current source, measured in amperes – produces a current in a conductor. Affects charge according to the relation .
    • Voltage source, measured in volts – 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.
Both these elements are necessarily non-linear elements. See #Non-linear elements below.
  • Three passive elements:
    • Resistance , measured in ohms – produces a voltage proportional to the current flowing through the element. Relates voltage and current according to the relation .
    • Capacitance , measured in farads – produces a current proportional to the rate of change of voltage across the element. Relates charge and voltage according to the relation .
    • Inductance , 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 .
  • Four abstract active elements:
    • 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 elsewhere in the circuit, with respect to a specified gain, used to model field-effect transistors and vacuum tubes (has infinite input impedance and infinite output impedance). The gain is characterised by a transfer conductance which will have units of siemens.
    • Current-controlled voltage source (CCVS) Generates a voltage based on an input current elsewhere in the circuit with respect to a specified gain. (has zero input impedance and zero output impedance). Used to model trancitors. The gain is characterised by a transfer impedance which will have units of ohms.
    • Current-controlled current source (CCCS) Generates a current based on an input current and a specified gain. Used to model bipolar junction transistors. (Has zero input impedance and infinite output impedance).
These four elements are examples of two-port elements.

Non-linear elements

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Conceptual symmetries of resistor, capacitor, inductor, and memristor.

In reality, all circuit components are non-linear and can only be approximated as linear over a certain range. To describe the passive elements more precisely, their constitutive relation is used instead of simple proportionality. Six constitutive relations can be formed from any two of the circuit variables. From this, there is supposed to be a theoretical fourth passive element since there are only five elements in total (not including the various dependent sources) 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. Hence, it is not included in linear time-invariant (LTI) circuit models. The constitutive relations of the passive elements are given by;[3]

  • 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 . Both independent voltage and independent current sources can be considered non-linear resistors under this definition.[3]

The fourth passive element, the memristor, was 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.[4][5][6][7] With the advent of the memristor, each pairing of the four variables can now be related.

Two special non-linear elements are sometimes used in analysis but are not the ideal counterpart of any real component:

  • Nullator: defined as
  • Norator: defined as an element that places no restrictions on voltage and current whatsoever.

These are sometimes used in models of components with more than two terminals: transistors, for instance.[3]

Two-port elements

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All the above are two-terminal, or one-port, elements except the dependent sources. Two lossless, passive, linear two-port elements are typically 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 ports is mapped by 1/n. On the other hand, the gyrator 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 just the basic linear elements are necessarily reciprocal, so they 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. The 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. However, transformer, capacitance, and inductance are normally retained in analysis because they are the ideal properties of the basic physical components transformer, inductor, and capacitor, whereas a practical gyrator must be constructed as an active circuit.[8][9][10]

Examples

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The following are examples of representations of components by way of electrical elements.

  • On a first degree of approximation, a battery 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).
  • On a first degree of approximation, a resistor 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 often used when representing semiconductors. For example, on a first degree of approximation, a bipolar transistor may be represented by a variable current source controlled by the input current.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Electrical element

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An electrical element is a fundamental component in circuit theory that serves as the basic building block for modeling electrical systems and devices. These elements represent idealized behaviors, such as energy storage, dissipation, or generation, and are interconnected to form complete electric circuits. The ideal basic circuit elements consist of the resistor, capacitor, inductor, voltage source, and current source. Passive elements, including resistors, capacitors, and inductors, do not generate electrical energy but instead store or dissipate it: resistors convert electrical energy to heat following Ohm's law (v=iRv = iR); capacitors store energy in an electric field proportional to voltage (i=Cdvdti = C \frac{dv}{dt}); and inductors store energy in a magnetic field proportional to current (v=Ldidtv = L \frac{di}{dt}). Active elements, such as voltage and current sources, supply energy to the circuit and are divided into independent sources, which provide constant voltage or current regardless of circuit conditions, and dependent sources, whose output depends on voltages or currents elsewhere in the circuit. These elements form the foundation for analyzing circuit performance using laws like Kirchhoff's voltage and current laws, enabling the design and simulation of complex electrical networks in applications ranging from electronics to power systems.

Basic Concepts

Definition

An electrical element is an idealized of a component in an electrical circuit that relates the voltage and current at its terminals through a characteristic equation. This simplifies the analysis of complex systems by focusing solely on the relationship between these two variables without regard to the underlying physical structure. Electrical elements represent specific behaviors in circuits, such as the dissipation of power as heat, the storage of energy in electric or magnetic fields, or the generation of electrical signals. These characteristics are defined independently of any particular material or construction, allowing for universal application in theoretical modeling. In circuit theory, electrical elements form the essential building blocks upon which fundamental principles like Kirchhoff's current and voltage laws are applied to perform network analysis. Ideal elements assume perfect adherence to their defining equations, whereas real-world implementations include parasitic effects and deviations that approximate but do not fully match these ideals. The concept of electrical elements emerged in the late within the framework of the , pioneered by in his work on transmission lines.

Terminal and Port Representation

Electrical elements are typically represented using two terminals, which serve as the points of connection in a circuit where voltage is measured across and current flows through. These terminals are nodes that allow the element to interact with the rest of the circuit, with the voltage defined as the potential difference between them and the current as the flow entering one terminal and exiting the other, in accordance with . A is formed by a pair of such terminals, providing a standardized way to describe the interface between an electrical element and the external circuit. For one-port elements, which characterize most basic components like resistors, capacitors, and inductors, the relationship between the voltage VV across the and the current II through it is expressed generally as V=f(I)V = f(I) or I=g(V)I = g(V), where ff and gg are functions specific to the element's behavior. This port-based modeling simplifies circuit analysis by focusing on the input-output characteristics at the interface. Graphical symbols for electrical elements are standardized to facilitate clear schematic representation, as defined in IEEE Std 315-1975. These symbols depict the element's terminals and basic form without detailing internal physics; for instance, a zigzag line represents a general , parallel lines denote a , and a coiled line indicates an . Such conventions ensure consistency in circuit diagrams across engineering practices. For more complex interactions, electrical elements can be extended to multi- networks, particularly two- networks, which model devices with separate input and output s. In two-port representations, the voltages and currents at the ports are related through parameter matrices, such as the impedance (Z) parameters, given by: [V1V2]=[Z11Z12Z21Z22][I1I2],\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}, where V1,I1V_1, I_1 are at port 1 and V2,I2V_2, I_2 at port 2, and the Z terms describe , transfer impedances, and . Alternative representations include (Y) or transmission (ABCD) parameters, enabling analysis of networks like amplifiers or filters. The representation of electrical elements relies on the lumped approximation, which treats components as point-like entities with no significant spatial variation in voltage or current across their dimensions. This assumption holds for low-frequency applications where the physical size of the element is much smaller than the of the signals (typically valid below radio frequencies, e.g., for elements on the order of centimeters at frequencies under 100 MHz), allowing Kirchhoff's laws to apply directly without accounting for wave propagation effects.

Passive Elements

Resistors

A is a passive electrical component that limits the flow of in a circuit by providing opposition to it, characterized by its resistance value measured in ohms (Ω). This opposition results in the dissipation of as , making resistors dissipative elements essential for controlling current levels and protecting other components from excessive flow. In circuit diagrams, resistors are typically represented by a two-terminal symbol, with voltage across the terminals and current through them. For linear resistors, the relationship between voltage VV and current II follows , expressed as V=IRV = I R, where RR is the constant resistance. The power dissipated as in the resistor can be calculated using P=I2RP = I^2 R or equivalently P=V2RP = \frac{V^2}{R}, highlighting the energy conversion inherent to its function. Resistance also varies with according to R(T)=R0(1+αΔT)R(T) = R_0 (1 + \alpha \Delta T), where R0R_0 is the resistance at a reference , α\alpha is the of resistance (typically positive for metals, indicating increased resistance with rising ), and ΔT\Delta T is the temperature change. Resistors are classified into several types based on their construction and adjustability. Fixed resistors, such as carbon composition or wire-wound varieties, maintain a constant resistance value throughout operation and are widely used for precise . Variable resistors, including potentiometers, allow manual or mechanical adjustment of resistance and are employed in applications like volume controls or tuning circuits. Special types, such as thermistors, are nonlinear resistors whose resistance varies significantly and nonlinearly with , used primarily for temperature sensing. An resistor exhibits purely resistive behavior independent of frequency, adhering strictly to across all conditions. In contrast, real resistors incorporate parasitic effects, including between leads and from wire elements, which cause deviations from ideal resistance at high frequencies, potentially altering circuit performance in RF applications.

Capacitors

A capacitor is a passive electrical component that stores electrical energy in an , consisting of two conductive plates separated by an insulating material known as a . The CC, measured in farads (F), is defined as the ratio of the charge QQ stored on the plates to the voltage VV across them: C=Q/VC = Q / V. This relationship allows capacitors to function in circuits for tasks such as , filtering signals, and coupling or decoupling AC and DC components. The dynamic behavior of a capacitor is governed by the equation i=Cdvdti = C \frac{dv}{dt}, where ii is the current through the capacitor and vv is the voltage across it. This indicates that current flows only when the voltage changes over time, as the capacitor charges or discharges. The stored in the of a is given by E=12CV2E = \frac{1}{2} C V^2, representing the work done to separate charges against the electric field. In time-domain analysis, a behaves as an open circuit in steady-state DC conditions, blocking since dv/dt=0dv/dt = 0 implies i=0i = 0. In AC circuits, it presents an impedance of Z=1jωCZ = \frac{1}{j \omega C}, where ω\omega is the and jj is the , allowing it to pass alternating signals while opposing changes in voltage. The capacitance value depends on the geometry and materials of the capacitor, particularly for parallel-plate configurations where C=ϵAdC = \epsilon \frac{A}{d}, with ϵ\epsilon as the permittivity of the dielectric, AA as the plate area, and dd as the separation distance. Dielectric materials, such as air (ϵr≈1\epsilon_r \approx 1), paper, or ceramics, increase capacitance by enhancing permittivity (ϵ=ϵrϵ0\epsilon = \epsilon_r \epsilon_0, where ϵ0\epsilon_0 is the vacuum permittivity), enabling higher charge storage for given dimensions. Larger plate areas and smaller separations yield higher capacitance, though practical limits arise from dielectric breakdown. Ideal capacitors are assumed lossless with infinite insulation, but real capacitors exhibit non-idealities including (ESR) from leads and plates, which introduces power losses as (P=I2RESRP = I^2 R_{ESR}), and leakage resistance in parallel, allowing a small DC current to flow through the dielectric over time. These parasitics affect performance, particularly at high frequencies or in precision applications, where ESR can limit ripple current handling and leakage impacts long-term charge retention. Manufacturers specify these parameters to guide selection for specific uses, such as low-ESR types for power supplies.

Inductors

An inductor is a passive electrical component that stores energy in a magnetic field when electric current flows through it. The inductance LL, measured in henries (H), is defined as the ratio of the magnetic flux linkage Φ\Phi to the current II producing it, given by L=Φ/IL = \Phi / I. This property arises from the coil's geometry and the material's magnetic permeability, enabling the inductor to oppose changes in current through induced electromotive force. The voltage VV across an is related to the rate of change of current by V=LdIdtV = L \frac{dI}{dt}, which describes its dynamic behavior in circuits. The energy stored in the is E=12LI2E = \frac{1}{2} L I^2, analogous to the energy in a but based on current rather than voltage. In (AC) circuits, an exhibits impedance Z=jωLZ = j \omega L, where ω\omega is the and jj is the , causing the voltage to lead the current by 90 degrees. At (DC), after initial transients decay, the behaves as a with zero impedance, as steady current produces no changing . The inductance of a solenoid inductor is approximated by L≈μN2A/lL \approx \mu N^2 A / l, where μ\mu is the magnetic permeability of the core material, NN is the number of turns, AA is the cross-sectional area, and ll is the length. Core materials like ferrite or iron increase μ\mu compared to air (μ0\mu_0), enhancing inductance but introducing nonlinear effects. An ideal inductor has no resistance or capacitance, storing energy reversibly without losses. In contrast, real inductors include series winding resistance, which dissipates power as heat, and core saturation, where high currents exceed the material's magnetic limit, reducing effective inductance.

Active Elements

Independent Sources

Independent sources are active electrical elements that generate a specified voltage or current, which can be constant (DC) or time-varying (AC), without depending on other voltages or currents in the circuit. These sources provide power to the circuit autonomously, serving as fundamental building blocks in circuit and . An ideal independent voltage source maintains a constant or specified time-varying voltage Vs(t)V_s(t) across its terminals, regardless of the current flowing through it, and is characterized by zero internal impedance. This model implies that the source can supply or absorb any amount of current while keeping the terminal voltage fixed at V=Vs(t)V = V_s(t). In circuit theory, any linear network containing voltage sources and impedances can be equivalently represented by a Thevenin consisting of an independent in series with an equivalent impedance. An ideal independent current source maintains a constant or specified time-varying current Is(t)I_s(t) through its terminals, regardless of the voltage across it, and is characterized by infinite internal impedance. This ensures that the source delivers exactly I=Is(t)I = I_s(t) to the circuit, with the voltage adjusting as needed to sustain the current flow. Similarly, any linear network with current sources and impedances can be represented by a Norton , comprising an independent in parallel with an equivalent impedance. In practice, independent voltage sources are implemented using devices such as batteries for DC applications, which approximate ideal behavior by providing a nearly constant voltage output. For AC signals, function generators serve as independent voltage sources, producing programmable waveforms like sine waves at specified amplitudes and frequencies. However, real-world implementations deviate from ideality due to , which causes a under load and limits the source's ability to maintain exact Vs(t)V_s(t). Standard circuit symbols for independent sources follow IEEE conventions: the is depicted as a circle with a plus (+) and minus (−) indicating polarity, while the is shown as a circle with an arrow pointing in the direction of conventional current flow. These symbols facilitate clear representation in diagrams, with the arrow for current sources denoting the reference direction for positive current.

Dependent Sources

Dependent sources, also known as controlled sources, are active electrical elements whose output voltage or current value is determined by a voltage or current measured at another location within the same circuit, enabling modeling of interactions between distant parts of a network. Unlike independent sources, which provide fixed outputs regardless of circuit conditions, dependent sources adjust dynamically based on sensing elements elsewhere. They are essential for representing amplification and control in circuit analysis, particularly in linear models of complex devices. There are four fundamental types of linear dependent sources, each defined by the nature of the controlling and controlled quantities. The voltage-controlled voltage source (VCVS) produces an output voltage proportional to a controlling , expressed as vout=μvinv_\text{out} = \mu v_\text{in}, where μ\mu is the dimensionless voltage gain. This configuration is widely used to model operational amplifiers (op-amps), where the output is vo=A(v+−v−)v_o = A (v_+ - v_-) with AA typically ranging from 10610^6 to 10810^8, assuming infinite and ideal differential input behavior under . The voltage-controlled current source (VCCS) generates an output current proportional to the controlling voltage, given by iout=gmvini_\text{out} = g_m v_\text{in}, where gmg_m is the measured in (S). In ideal models, voltage-controlled sources like VCVS and VCCS exhibit infinite , ensuring no loading effect on the controlling signal. The current-controlled voltage source (CCVS) delivers an output voltage proportional to the controlling current, modeled as vout=rmiinv_\text{out} = r_m i_\text{in}, with rmr_m denoting the transresistance in ohms (Ω\Omega). Similarly, the current-controlled current source (CCCS) produces an output current proportional to the controlling current, iout=βiini_\text{out} = \beta i_\text{in}, where β\beta is the dimensionless current gain. These dependent source models are crucial for abstractly representing the behaviors of semiconductor devices such as transistors in circuit simulations, allowing analysis of amplification without delving into physical device details; for instance, a bipolar junction transistor may be approximated as a CCCS with β≈100\beta \approx 100 to 300300, while a MOSFET often uses a VCCS representation. In practice, they facilitate techniques like nodal analysis by incorporating control dependencies into system matrices, ensuring accurate prediction of circuit performance in applications ranging from signal processing to power control.

Non-linear Elements

Diodes

A is a two-terminal consisting of a p-n junction, which permits to flow preferentially in one direction while exhibiting non-linear conduction characteristics dependent on the polarity of the applied voltage. The p-n junction forms when p-type material (with excess holes) is joined to n-type material (with excess electrons), creating a at the interface where mobile charges are depleted. In forward bias, with the p-side () at a higher potential than the n-side (), the narrows, and if the voltage exceeds approximately 0.7 V for diodes, majority carriers cross the junction, enabling substantial current flow. In reverse bias, the widens, establishing a potential barrier that blocks current, limiting it to a small reverse due to minority carrier . The ideal model treats the device as a switch: conducting with negligible voltage drop in forward bias and non-conducting with zero current in reverse bias. The quantitative relationship between current and voltage in a p-n junction is given by the : I=Is(eV/(nVT)−1)I = I_s \left( e^{V / (n V_T)} - 1 \right) where II is the current through the , IsI_s is the (typically on the order of 10−1210^{-12} to 10−1510^{-15} A for at ), VV is the voltage across the , nn is the ideality factor (ranging from 1 for diffusion-dominated transport to 2 for recombination-dominated), and VT=kT/qV_T = kT/q is the thermal voltage (kk is Boltzmann's constant, TT is absolute temperature, and qq is the , yielding VT≈26V_T \approx 26 mV at 300 K). This equation derives from the physics of carrier injection and recombination across the junction under low-level injection assumptions. The current-voltage (I-V) characteristics of a diode feature an exponential rise in forward current with increasing voltage, reflecting the dominant exponential term in the Shockley equation for V>0V > 0, while reverse current remains nearly constant and negligible until a is reached. In reverse bias, beyond a critical voltage, the diode enters breakdown; in heavily doped junctions (typical for with breakdown voltages below about 5-6 V), this occurs via the involving quantum mechanical tunneling of carriers through the thin , while in lightly doped junctions (higher breakdown voltages), dominates due to . These mechanisms allow controlled reverse conduction in specialized diodes without permanent damage. Diodes are categorized into several types based on their junction structure and intended function, including rectifier diodes, which are conventional p-n junction devices optimized for high current handling in AC-to-DC conversion circuits; light-emitting diodes (LEDs), p-n junctions in direct-bandgap semiconductors like that emit photons during forward-biased electron-hole recombination; and Schottky diodes, which employ a metal-semiconductor junction to achieve a lower forward (typically 0.2-0.4 V) and faster recovery times compared to p-n diodes. For small-signal analysis, where voltage variations are superimposed on a point, the diode's non-linear behavior is linearized, yielding an equivalent small-signal resistance rd=dVdI≈VTIr_d = \frac{dV}{dI} \approx \frac{V_T}{I}, with II as the DC operating current; this dynamic resistance decreases with increasing bias current, enabling approximation of the diode as a resistor in AC circuit models.

Transistors

A is a three-terminal that controls a larger current or voltage with a smaller input signal, serving as a fundamental building block for amplification and switching in electronic circuits. In bipolar junction transistors (BJTs), the terminals are base, emitter, and collector, while in metal-oxide-semiconductor field-effect transistors (MOSFETs), they are , drain, and source. These devices operate on principles of injection and field-induced modulation, enabling high gain and efficient power handling compared to earlier vacuum tubes. Bipolar junction transistors (BJTs) are constructed from three alternately doped layers forming two p-n junctions, available in NPN (n-type base between p-type emitter and collector) or PNP configurations. In forward-active mode, the base-emitter junction is forward-biased to inject carriers, while the base-collector junction is reverse-biased to collect them, resulting in a collector current ICI_C much larger than the base current IBI_B; the current gain is defined as β=IC/IB\beta = I_C / I_B, typically 50 to 300 for devices. The Ebers-Moll model provides a large-signal representation by treating the BJT as interconnected diodes and controlled current sources, accounting for both forward and reverse injection currents across the junctions. Metal-oxide-semiconductor field-effect transistors (MOSFETs) function through voltage-controlled channel formation between the drain and source, insulated by an layer under the . Enhancement-mode MOSFETs require a gate-source voltage VGSV_{GS} exceeding the VthV_{th} (usually 0.5–1 for ) to induce an inversion channel of opposite carriers (electrons for n-channel, holes for p-channel). In saturation region, where VDS>VGS−VthV_{DS} > V_{GS} - V_{th}, the drain current follows ID=k(VGS−Vth)2I_D = k (V_{GS} - V_{th})^2, with k=12μCoxWLk = \frac{1}{2} \mu C_{ox} \frac{W}{L} as the parameter incorporating carrier mobility μ\mu, capacitance CoxC_{ox}, and channel dimensions W/LW/L. Depletion-mode MOSFETs conduct at zero gate voltage and require negative VGSV_{GS} to deplete the channel. Transistor characteristics are depicted via transfer curves (output current versus input voltage or current) and output curves (output current versus output voltage at fixed ), delineating operating regions. For BJTs, occurs with both junctions reverse-biased (negligible current), forward-active provides linear amplification, and saturation has both junctions forward-biased (high output current). MOSFETs similarly feature (VGS<VthV_{GS} < V_{th}), triode (linear, VDS<VGS−VthV_{DS} < V_{GS} - V_{th}), and saturation (constant ) regions, with transfer curves showing quadratic dependence in enhancement mode. The approximates BJT small-signal behavior around a DC operating point using a pi-network , with gm=dICdVBE≈ICVTg_m = \frac{dI_C}{dV_{BE}} \approx \frac{I_C}{V_T} (where VT≈26V_T \approx 26 mV at is the voltage) representing current control by base-emitter voltage, and input resistance rπ=βgmr_\pi = \frac{\beta}{g_m} modeling base-emitter dynamics; an output resistance ror_o accounts for modulation. This model facilitates analysis of AC performance in amplifiers, treating the akin to a voltage-controlled as in dependent source abstractions.

Advanced and Distributed Elements

Memristors

A , short for memory , is the fourth fundamental passive circuit element, alongside the , , and , that relates the integral of the current through it () to the integral of the voltage across it ( linkage) in a history-dependent manner. Mathematically, it is defined by the constitutive relation M=dΦdQM = \frac{d\Phi}{dQ}, where MM is the memristance, Φ\Phi is the , and QQ is the charge, leading to the voltage-current relationship v(t)=Mi(t)v(t) = M i(t) with MM varying based on the device's internal state influenced by prior current flow. The behavior of a is characterized by its resistance, or memristance, which changes dynamically with the amount and direction of charge passed through it, resulting in a pinched loop in the current-voltage (I-V) characteristic curve that passes through the origin and scales with the of the applied signal. This arises from the device's ability to "remember" its resistance state even after the power is removed, distinguishing it from purely linear elements. In practice, real memristors are modeled using a xx that captures this memory, where memristance is M(x)M(x) and the state evolves according to dxdt=f(i,x)\frac{dx}{dt} = f(i, x), with ff representing the internal dynamics, such as migration in the material. The concept of the was theoretically predicted in 1971 by Leon Chua as part of a systematic classification of two-terminal nonlinear circuit elements, completing the set of fundamental relations between voltage, current, charge, and flux. However, no physical realization existed until 2008, when researchers at Laboratories demonstrated the first practical using a thin film of (TiO₂) with a doped , exhibiting the predicted pinched and effects at the nanoscale. Memristors have emerged as key enablers for technologies, such as (ReRAM), where their multi-level resistance states allow for high-density data storage with low power consumption compared to traditional . In neuromorphic computing, memristor arrays mimic by adjusting conductance based on input patterns, facilitating efficient brain-inspired architectures for and tasks. While ideal memristors assume perfect flux-charge coupling without additional parasitics, real implementations often incorporate the state-variable model to account for material-specific dynamics, enabling more accurate simulations. As of 2025, prototypes face ongoing challenges in scalability, including difficulties in achieving uniform fabrication across large arrays due to variations in material doping and interfaces, which can lead to inconsistent switching behavior. Volatility remains a concern in some designs, where the stored state may degrade over time without continuous power, limiting reliability for long-term applications despite advances in oxide-based and 2D material variants. Recent developments as of October 2025 include unified memristor-ferroelectric memories for energy-efficient AI computing and techniques to transform device noises into advantages for computational innovations like .

Transmission Lines

Transmission lines represent distributed electrical elements that model the propagation of electromagnetic waves along extended conductors, particularly at high frequencies where the signal wavelength is comparable to or smaller than the physical length of the line, in contrast to lumped elements that assume instantaneous signal propagation. These structures are essential for transmitting power and signals in applications such as and high-speed , where wave effects like phase shifts and reflections must be accounted for. Unlike lumped components, transmission lines are characterized by distributed per-unit-length parameters: series resistance RR (in ohms per meter, accounting for conductor losses), series inductance LL (in henries per meter, due to ), shunt conductance GG (in siemens per meter, representing dielectric losses), and shunt capacitance CC (in farads per meter, from between conductors). The voltage V(z)V(z) and current I(z)I(z) along the line satisfy the in the frequency domain: dV(z)dz=−(R+jωL)I(z)\frac{d V(z)}{d z} = -(R + j \omega L) I(z) dI(z)dz=−(G+jωC)V(z)\frac{d I(z)}{d z} = -(G + j \omega C) V(z) These equations lead to the characteristic impedance Z0Z_0, which is the ratio of voltage to current for a wave traveling in one direction without reflection, given by Z0=[R+j](/page/RandJ)ωLG+jωCZ_0 = \sqrt{\frac{[R + j](/page/R_and_J) \omega L}{G + j \omega C}}
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