# Electrical element

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Electrical element

Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. Any electrical network can be analysed as multiple, interconnected electrical elements in a schematic diagram or circuit diagram, each of which affects the voltage in the network or current through the network. These ideal electrical elements represent real, physical electrical or electronic components but they do not exist physically and they are assumed to have ideal properties according to a lumped element model, while components are objects with less than ideal properties, a degree of uncertainty in their values and some degree of nonlinearity, each of which may require a combination of multiple electrical elements in order to approximate its function.

Circuit analysis using electric elements is useful for understanding many practical electrical networks using components. By analyzing the way a network is affected by its individual elements it is possible to estimate how a real network will behave.

## One-port elements

Only nine types of two-terminal elements, five passive and four active, are required to model any electrical component or circuit. Each element is defined by a relation between the state variables of the network: current, I; voltage, V, charge, Q; and magnetic flux, Φ.

### Passive elements

• Resistors with resistance R, measured in ohms – produces a voltage proportional to the current flowing through the element. Relates voltage and current according to the relation $dV = R\,dI$.
• Capacitors with capacitance C, 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 $dQ = C\,dV$.
• linear (capacitor)
• nonlinear (varactor)
• time-dependent nonlinear (memcapacitor)
• Inductors with inductance L, 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 $d\Phi = L\,dI$.
• linear (inductor)
• nonlinear inductor
• time-dependent nonlinear (meminductor)

The passive electrical elements and their variations are shown below in a form of a table where the rows represent the electrical properties and the columns represent the variations of these properties.

 Linear Nonlinear Memory Resistance Resistor Varistor Memristor Capacitance Capacitor Varactor Memcapacitor Inductance Inductor NL inductor Meminductor

The fourth passive element

In reality, all circuit elements are non-linear and can only be approximated to linear over a certain range. To more exactly describe the passive elements, their constitutive relation is used instead of simple proportionality. From any two of the circuit variables there are six constitutive relations that can be formed. From this it is supposed that there is a theoretical fourth passive element since there are only five elements in total found in linear network analysis. This additional element is called memristor. It only has any meaning as a time-dependent non-linear element; as a time-independent linear element it reduces to a regular resistor. The constitutive relations of the passive elements are given by;

• Resistance: constitutive relation defined as f(V,I) = 0.
• Capacitance: constitutive relation defined as f(V,Q) = 0.
• Inductance: constitutive relation defined as f(Φ,I) = 0.
• Memristance: constitutive relation defined as f(Φ,Q) = 0.
where f(x,y) 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 V = f(I).

The fourth passive element, the memristor, is proposed by Leon Chua in a 1971 paper, but a physical component demonstrating memristance was not created until thirty-seven years later. It was reported on April 30, 2008, that a working memristor had been developed by a team at HP Labs led by scientist R. Stanley Williams. With the advent of the memristor, each pairing of the four variables can now be related. Although memristors are analog memory elements, they are able to store one bit of non-volatile memory. They may see application in programmable logic, signal processing, neural networks, and control systems, among other fields. Because memristors are time-variant by definition, they are not included in linear time-invariant (LTI) circuit models.

### Sources

• Current source, measured in amperes – produces a current in a conductor. Affects charge according to the relation $dQ = -I\,dt$.
• Voltage source, measured in volts – produces a potential difference between two points. Affects magnetic flux according to the relation $d\Phi = V\,dt$.
Φ in this relationship does not necessarily represent anything physically meaningful. In the case of the current generator, Q, the time integral of current, represents the quantity of electric charge physically delivered by the generator. Here Φ is the time integral of voltage but whether or not that represents a physical quantity depends on the nature of the voltage source. For a voltage generated by magnetic induction it is meaningful, but for an electrochemical source, or a voltage that is the output of another circuit, no physical meaning is attached to it.

### Controlled sources

• Voltage-controlled voltage source (VCVS) Generates a voltage based on another voltage with respect to a specified gain. (has infinite input impedance and zero output impedance).
• Voltage-controlled current source (VCCS) Generates a current based on a voltage with respect to a specified gain, used to model field-effect transistors and vacuum tubes (has infinite input impedance and infinite output impedance).
• Current-controlled voltage source (CCVS) Generates a voltage based on an input current with respect to a specified gain. (has zero input impedance and zero output impedance).
• Current-controlled current source (CCCS) Generates a current based on an input current and a specified gain. Used to model bipolar junction transistors. (Has zero input impedance and infinite output impedance).

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

• Nullator: defined as V = I = 0
• Norator: defined as an element which places no restrictions on voltage and current whatsoever.

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

## Two-port elements

All the above are two-terminal, or one-port elements. There are two lossless, passive, linear two-port elements that are normally introduced into network analysis. Their constitutive relations in matrix notation are;

Transformer $\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} 0 & n \\ -n & 0 \end{bmatrix}\begin{bmatrix} I_1 \\ V_2 \end{bmatrix}$
Gyrator $\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} 0 & -r \\ r & 0 \end{bmatrix}\begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$

The transformer maps a voltage at one port to a voltage at the other in a ratio of n. The current between the same two port is mapped by 1/n. The gyrator, on the other hand, maps a voltage at one port to a current at the other. Likewise, currents are mapped to voltages. The quantity r in the matrix is in units of resistance. The gyrator is a necessary element in analysis because it is not reciprocal. Networks built from the basic linear elements only are obliged to be reciprocal and so cannot be used by themselves to represent a non-reciprocal system. It is not essential, however, to have both the transformer and gyrator. Two gyrators in cascade are equivalent to a transformer but the transformer is usually retained for convenience. Introduction of the gyrator also makes either capacitance or inductance non-essential since a gyrator terminated with one of these at port 2 will be equivalent to the other at port 1.

## Examples

The following are examples of representation 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 of time).
• 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 more often used when representing semiconductors. For example, on a first degree of approximation, a bipolar transistor may be represented by a variable current source that is controlled by the input current.

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