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AC Circuit Generalization of Ohm’s Law

1890

Oliver Heaviside
Charles Proteus Steinmetz

For alternating current (AC) circuits, Ohm’s law is generalized using complex numbers to [latex]\mathbf{V} = \mathbf{I} \mathbf{Z}[/latex]. Here, [latex]\mathbf{V}[/latex] and [latex]\mathbf{I}[/latex] are complex phasors representing the sinusoidally varying voltage and current, capturing both magnitude and phase. [latex]\mathbf{Z}[/latex] is the complex impedance, which extends the concept of resistance to include the effects of capacitors and inductors.

In AC circuits, capacitors and inductors introduce a phase shift between the voltage and current. Simple resistance is insufficient to describe this behavior. Impedance, [latex]\mathbf{Z}[/latex], is a complex quantity defined as [latex]\mathbf{Z} = R + jX[/latex], where [latex]R[/latex] is the resistance (the real part) and [latex]X[/latex] is the reactance (the imaginary part). Reactance is the opposition to current flow from capacitors ([latex]X_C[/latex]) and inductors ([latex]X_L[/latex]), and it is frequency-dependent.

This phasor form of Ohm’s law allows engineers to use algebraic methods, similar to those for DC circuits, to solve for magnitudes and phase angles in complex AC systems. The magnitude of the impedance, [latex]|\mathbf{Z}| = \sqrt{R^2 + X^2}[/latex], determines the ratio of voltage amplitude to current amplitude. The phase angle of the impedance, [latex]\phi = \arctan(X/R)[/latex], represents the phase difference between the voltage and current. This generalization is indispensable for modern electrical engineering, underpinning the design of everything from power supplies to wireless 沟通 devices.

电容器, 电气工程, 电力电子, 信号处理

UNESCO Nomenclature: 3308

– Electrical engineering

类型

抽象系统

中断

实质性

使用方法

广泛使用

前体

Ohm’s law for dc circuits
Michael Faraday’s law of induction, describing how changing magnetic fields create voltage
Joseph Henry’s work on self-inductance
James Clerk Maxwell’s unified theory of electromagnetism
The mathematical 框架 of complex numbers and phasor analysis

应用

音频滤波器的设计（高通、低通、带通）
输配电网分析
通信系统的射频 (rf) 电路设计
天线阻抗匹配以最大化功率传输
电子学中 RLC 电路的建模与分析

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Related to: AC circuits, impedance, reactance, phasor, complex numbers, alternating current, circuit theory, RLC circuit, phase shift, electrical engineering.