Mathematical methods are integral to the study of electronics.

Mathematics in electronics

Electronics engineering Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current f ...

careers usually include courses in

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

(single and multivariable),

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

(both ordinary and

partial Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...

and

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...

and

Z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...

s are also subjects which are usually included in

electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

programs.

Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...

can simplify computing RLC network behaviour.

Basic applications

A number of electrical laws apply to all electrical networks. These include *

Faraday's law of induction Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic in ...

: Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. *

Gauss's Law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...

: The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity. * Kirchhoff's current law: the sum of all currents entering a node is equal to the sum of all currents leaving the node or the sum of total current at a junction is zero * Kirchhoff's voltage law: the directed sum of the electrical potential differences around a circuit must be zero. *

Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...

: the voltage across a resistor is the product of its resistance and the current flowing through it.at constant temperature. *

Norton's theorem In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of ...

: any two-terminal collection of voltage sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor. *

Thévenin's theorem As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that ''"For any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B ...

: any two-terminal combination of voltage sources and resistors is electrically equivalent to a single voltage source in series with a single resistor. *

Millman's theorem In electrical engineering, Millman's theorem (or the parallel generator theorem) is a method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute the voltage at the ends of a circuit made up of only branches in ...

: the voltage on the ends of branches in parallel is equal to the sum of the currents flowing in every branch divided by the total equivalent conductance. * See also Analysis of resistive circuits. Circuit analysis is the study of methods to solve linear systems for an unknown variable. *

Circuit analysis A network, in the context of electrical engineering and electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many ...

Components

There are many electronic components currently used and they all have their own uses and particular rules and methods for use. *

Electronic components An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not ...

Complex numbers and Complex Analysis

If you apply a voltage across a capacitor, it 'charges up' by storing the electrical charge as an electrical field inside the device. This means that while the voltage across the capacitor remains initially small, a large current flows. Later, the current flow is smaller because the capacity is filled, and the voltage raises across the device. Complex Analysis methods are also important in electrical engineering in fields such as signal processing, power electronics, control systems, and others A similar though opposite situation occurs in an inductor; the applied voltage remains high with low current as a magnetic field is generated, and later becomes small with high current when the magnetic field is at maximum. The voltage and current of these two types of devices are therefore out of phase, they do not rise and fall together as simple resistor networks do. The mathematical model that matches this situation is that of

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

, using an imaginary component to describe the stored energy.

Signal analysis

. Deconstructing a periodic waveform into its constituent frequencies; see also:

Fourier theorem A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...

. *

Nyquist–Shannon sampling theorem The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that per ...

. *

Information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...

. Sets fundamental limits on how information can be transmitted or processed by any system. {{DEFAULTSORT:Mathematical Methods In Electronics Electronic engineering Applied mathematics