Phasors and Complex Impedance | 相量与复阻抗

📚 Phasors and Complex Impedance | 相量与复阻抗

In AC circuit analysis, phasors and complex impedance provide a powerful bridge between trigonometric functions and algebraic complex numbers. By representing sinusoidally varying voltages and currents as rotating vectors (phasors) in the complex plane, differential equations are reduced to simple algebraic equations, making steady-state analysis straightforward and elegant.

在交流电路分析中，相量与复阻抗在三角函数和代数复数之间架起了一座强有力的桥梁。将正弦变化的电压和电流表示为复平面上的旋转矢量（相量）后，微分方程被简化为简单的代数方程，使稳态分析变得直接而优雅。

1. Review of Complex Numbers | 复数回顾

A complex number can be written in rectangular form as a + jb, where a is the real part, b is the imaginary part, and j is the imaginary unit (j² = −1). In polar form, the same number is expressed as r∠θ or r e^jθ, with magnitude r = √(a² + b²) and phase angle θ = tan⁻¹(b/a). Euler’s formula e^jθ = cosθ + j sinθ links the two representations and is fundamental to phasor analysis.

复数可以写成直角坐标形式 a + jb，其中 a 为实部，b 为虚部，j 为虚数单位（j² = −1）。在极坐标形式下，同一个数可以表示为 r∠θ 或 r e^jθ，幅值 r = √(a² + b²)，相位角 θ = tan⁻¹(b/a)。欧拉公式 e^jθ = cosθ + j sinθ 将两种表示联系起来，是相量分析的基础。

2. Sinusoidal Signals and Phasor Representation | 正弦信号与相量表示

A sinusoidal voltage v(t) = V_m cos(ωt + φ) can be viewed as the real part of the complex exponential V_m e^j(ωt+φ). The phasor V is defined by extracting the constant complex amplitude: V = V_m e^jφ = V_m∠φ. The time-dependent rotation factor e^jωt is omitted in phasor domain because all quantities share the same angular frequency ω, allowing us to work only with magnitudes and phase differences.

正弦电压 v(t) = V_m cos(ωt + φ) 可视为复指数 V_m e^j(ωt+φ) 的实部。相量 V 的定义是提取其恒定的复振幅：V = V_m e^jφ = V_m∠φ。在相量域中，时间旋转因子 e^jωt 被省略，因为所有量都具有相同的角频率 ω，这使得我们可以只处理幅值和相位差。

3. Complex Impedance Definition | 复阻抗的定义

Complex impedance Z is defined as the ratio of the voltage phasor to the current phasor for a two-terminal element: Z = V / I. Its real part is the resistance R, and its imaginary part is the reactance X, so Z = R + jX. The magnitude |Z| = √(R² + X²) gives the ratio of voltage amplitude to current amplitude, while the angle φ = tan⁻¹(X/R) represents the phase shift between

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

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