The concept of electrical elements is used in the analysis of electrical networks. Any electrical network can be modeled by decomposing it down to multiple, interconnected electrical elements in a schematic diagram or circuit diagram. Each electrical element affects the voltage in the network or current through the network in a particular way. By analyzing the way a network is affected by its individual elements, it is possible to estimate how a real network will behave on a macro scale.
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Elements vs. components
There is a distinction between real, physical electrical or electronic components and the ideal electrical elements by which they are represented.
- Electrical elements do not exist physically, and are assumed to have ideal properties according to a lumped element model.
- Conversely, components do exist, have less than ideal properties, their values always have a degree of uncertainty, they always include some degree of nonlinearity and typically require a combination of multiple electrical elements to approximate their functions.
Circuit analysis using electric elements is useful for understanding many practical electrical networks using components.
The elements
The four fundamental circuit variables are current, I; voltage, V, charge, Q; and magnetic flux, Φm. Only 5 elements, produced by manipulating these four variables, are required to represent any component or network in a linear system:
- Two sources:
- Current source, measured in amperes - produces a current in a conductor. Affects charge according to the relation dQ = − Idt.
- Voltage source, measured in volts - produces a potential difference between two points. Affects magnetic flux according to the relation dΦm = Vdt.
- Three passive elements:
- 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 = RdI.
- 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 = CdV.
- Inductance L, measured in henries - produces a voltage proportional to the rate of change of current through the element. Relates flux and current according to the relation dΦm = LdI.
- 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 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).
Non-linear elements
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 clear 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 memristance. It only has any meaning as a non-linear element; as a linear element it reduces to a regular resistor. The constitutive relations of the passive elements are given by;[1]
- Resistance: constitutive relation defined as f(V,I) = 0.
- Capacitance: constitutive relation defined as f(V,Q) = 0.
- Inductance: constitutive relation defined as f(Φm,I) = 0.
- Memristance: constitutive relation defined as f(Φm,Q) = 0.
- where f() is an arbitrary function.
The fourth passive element, the memristor, was theorized 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.[2][3][4][5] With the advent of the memristor, each pairing of the four variables can now be related. Memristors 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.
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 voltage.
References
- ^ Wai-Kai Chen, The electrical engineering handbook, pp.75-77, Academic Press, 2005 ISBN 0121709604
- ^ Strukov, Dmitri B; Snider, Gregory S; Stewart, Duncan R; Williams, Stanley R (2008), "The missing memristor found", Nature 453: 80–83, doi:, http://www.nature.com/nature/journal/v453/n7191/full/nature06932.html
- ^ EETimes, 04/30/2008, 'Missing link' memristor created, EETimes, 04/30/2008
- ^ Engineers find 'missing link' of electronics - 04/30/2008
- ^ Researchers Prove Existence of New Basic Element for Electronic Circuits -- 'Memristor' - 04/30/2008
See also
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