Mathematical methods in electronics: Difference between revisions

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'''Mathematical methods''' are integral to the study of '''electronics'''.

== Mathematics in Electronicselectronics engineering ==

Mathematical Methods in Electronics Engineering involves applying mathematical principles to analyze, design, and optimize electronic circuits and systems. Key areas include:<ref>{{Citation |title=Preface |date=1986-01-31 |work=Mathematical Methods in Electrical Engineering |pages=vii–viii |url=http://dx.doi.org/10.1017/cbo9781139165945.001 |access-date=2024-05-26 |publisher=Cambridge University Press|doi=10.1017/cbo9781139165945.001 |isbn=978-0-521-30661-4 |url-access=subscription }}</ref><ref>{{Cite web |title=Signals and Systems {{!}} Supplemental Resources |url=https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/ |access-date=2024-05-26 |website=MIT OpenCourseWare |language=en}}</ref>

Electrical Engineering careers usually include courses in [[Calculus]] (single and [[Multivariable Calculus|multivariable]]), [[Complex analysis|Complex Analysis]], [[Differential Equations]] (both [[Ordinary differential equation|ordinary]] and [[Partial differential equation|partial]]), [[Linear Algebra]] and [[Probability]]. [[Fourier Analysis]] and [[Z-transform|Z-Transforms]] are also subjects which are usually included in electrical engineering programs.

* [[Linear algebra|Linear Algebra]]: Used to solve systems of linear equations that arise in circuit analysis. Applications include network theory and the analysis of electrical circuits using matrices and vector spaces

==Basic applications==

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

*[[Faraday's law of induction]]: 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|Gauss's Law]]: The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.

*[[Kirchhoff's circuit laws#Kirchhoff's current law|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 circuit laws#Kirchhoff's voltage law|Kirchhoff's voltage law]]: the directed sum of the electrical potential differences around a circuit must be zero.

*[[Ohm's law]]: the voltage across a resistor is the product of its resistance and the current flowing through it.at constant temperature.

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

*[[Thevenin's theorem]]: 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|Millman's theorem]]: 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]].

* [[Calculus]]: Essential for understanding changes in electronic signals. Used in the analysis of dynamic systems and control systems. Integral calculus is used in analyzing waveforms and signals.

Circuit analysis is the study of methods to solve linear systems for an unknown variable.

* [[Differential equation|Differential Equations]]: Applied to model and analyze the behavior of circuits over time. Used in the study of filters, oscillators, and transient responses of circuits.

*[[Circuit analysis]]

* [[Complex number|Complex Numbers]] and [[Complex analysis|Complex Analysis]]: Important for circuit analysis and impedance calculations. Used in signal processing and to solve problems involving sinusoidal signals.

==Components==

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

* [[Probability]] and [[Statistics]]: Used in signal processing and communication systems to handle noise and random signals. Reliability analysis of electronic components.

*[[Electronic components]]

* [[Fourier transform|Fourier]] and [[Laplace transform|Laplace Transforms]]: Crucial for analyzing signals and systems. Fourier transforms are used for frequency analysis and signal processing. Laplace transforms are used for solving differential equations and analyzing system stability.

==Complex numbers==

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.

* [[Numerical methods|Numerical Methods]]: Employed for simulating and solving complex circuits that cannot be solved analytically. Used in computer-aided design tools for electronic circuit design.

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.

* [[Vector calculus|Vector Calculus]]: Applied in electromagnetic field theory. Important for understanding the behavior of electromagnetic waves and fields in electronic devices.

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, using an imaginary component to describe the stored energy.

* [[Optimization]]: Techniques used to design efficient circuits and systems. Applications include minimizing power consumption and maximizing signal integrity.

==Signal analysis==

* [[Fourier analysis]]. Deconstructing a [[Wave|periodic]] waveform into its constituent frequencies; see also: [[Fourier theorem]], [[Fourier transform]].

* [[Nyquist-Shannon sampling theorem]].

*[[Information theory]]. Sets fundamental limits on how information can be transmitted or processed by any system.

These methods are integral to systematically analyzing and improving the performance and functionality of electronic devices and systems.

{{DEFAULTSORT:Mathematical Methods In Electronics}}

== Mathematical methods applied in foundational electrical laws and theorems ==

A number of fundamental electrical laws and theorems apply to all electrical networks. These include:<ref>{{Cite book |last=Kreyszig |first=Erwin |title=Advanced Engineering Mathematics |publisher=Wiley |year=2015 |isbn=978-0470458365}}</ref>

* [[Faraday's law of induction]]: 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|Gauss's Law]]: The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.

* [[Kirchhoff's circuitCurrent laws#Kirchhoff's current law|Kirchhoff's current lawLaw]]: theThe 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 circuit laws#Kirchhoff's voltage law|Kirchhoff's voltage law]]: theThe directed sum of the electrical potential differences around a circuit must be zero.

* [[Ohm's law|Ohm's Law]]: theThe voltage across a resistor is the product of its resistance and the current flowing through it., at constant temperature.

* [[Norton's theorem|Norton's Theorem]]: anyAny two-terminal collection of voltage sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor.

* [[TheveninThévenin's theorem|Thévenin's Theorem]]: anyAny two-terminal combination of voltage sources and resistors is electrically equivalent to a single voltage source in series with a single resistor.

* [[Millman's Theoremtheorem|Millman's theoremTheorem]]: theThe 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.

== Analytical methods ==

In addition to the foundational principles and theorems, several analytical methods are integral to the study of electronics:<ref>{{Cite book |last=James W. Nilsson, Susan Riedel |title=Electric Circuits |publisher=Pearson |year=2021 |isbn=9780137477845}}</ref><ref>{{Cite web |title=Mathematical Methods for Electrical Engineering {{!}} Lehrstuhl für Bildverarbeitung der RWTH Aachen |url=https://www.lfb.rwth-aachen.de/en/academics/lectures/mathematical-methods-for-electrical-engineering/ |access-date=2024-05-26 |language=en-US}}</ref>

* [[Network analysis (electrical circuits)]]: Essential for comprehending capacitor and inductor behavior under changing voltage inputs, particularly significant in fields such as signal processing, power electronics, and control systems. This entails solving intricate networks of resistors through techniques like [[Nodal analysis|node-voltage]] and [[Mesh analysis|mesh-current methods]].

* [[Signal analysis]]: Involves [[Fourier analysis]], [[Nyquist–Shannon sampling theorem]], and [[information theory]], essential for understanding and manipulating signals in various systems.

These methods build on the foundational laws and theorems provide insights and tools for the analysis and design of complex electronic systems.

== See also ==

* [https://pe.gatech.edu/courses/introduction-electronics Introduction to Electronics Georgia Tech]

* [https://catalog.ucsc.edu/en/current/general-catalog/academic-units/baskin-engineering/electrical-and-computer-engineering/electrical-engineering-bs/ University of California, Santa Cruz Electrical Engineering curriculum]

* [https://guide.berkeley.edu/courses/el_eng/ University of California, Berkeley Electrical Engineering curriculum (UCSC Catalog) (Berkeley Academic Guide)]

== References ==

{{Reflist}}{{DEFAULTSORT:Mathematical Methods In Electronics}}

[[Category:Electronic engineering]]

[[Category:Applied mathematics]]