Phasors | 相量

📚 Phasors | 相量

A phasor is a complex number that represents a sinusoidal function whose amplitude, angular frequency and initial phase are time-invariant. In the context of IB Mathematics, the concept of phasors bridges complex numbers and trigonometry, providing a powerful tool for analysing oscillatory phenomena, particularly in physics and electrical engineering. Understanding phasors allows you to treat alternating currents and voltages as rotating vectors in the complex plane, simplifying addition, subtraction, differentiation and integration of sinusoidal signals.

相量是一个复数，用来表示振幅、角频率和初相均不随时间改变的正弦函数。在IB数学中，相量的概念将复数与三角学连接在一起，为分析振动现象——尤其是在物理和电气工程中——提供了一种强有力的工具。掌握相量后，你可以把交流电流和电压视为复平面上的旋转向量，从而大幅简化正弦信号的加减、微分与积分运算。

1. What is a Phasor? | 什么是相量？

A phasor is essentially a ‘frozen’ snapshot of a rotating vector. Given a sinusoidal waveform v(t) = Vₘ sin(ωt + φ), we can associate it with a complex number whose magnitude is Vₘ and whose argument is the phase angle φ. By convention, phasors are expressed in polar form or as complex exponentials. Although the actual signal oscillates at frequency ω, the phasor discards the time-varying part e^(iωt) or cos(ωt) and retains only the amplitude and phase offset, which carry all the information needed for linear operations.

相量本质上是旋转向量在某一时刻的“定格快照”。给定正弦波形 v(t) = Vₘ sin(ωt + φ)，我们可以将它对应到一个幅值为Vₘ、辐角为初相φ的复数。按惯例，相量用极坐标形式或复指数形式表示。虽然真实信号以角频率ω振荡，相量却丢弃了含时部分 e^(iωt) 或 cos(ωt)，只保留幅值和相位偏移，而这些信息已足够完成所有线性运算。

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2. Complex Numbers in Polar Form | 极坐标形式的复数

The foundation of phasors lies in the polar representation of complex numbers. Any complex number z = a + bi can be written as z = r (cosθ + i sinθ) or in shorthand as r cis θ, where r = |z| = √(a² + b²) and θ = arg(z) = arctan(b/a) (adjusted for the correct quadrant). The angle θ is typically given in radians. For phasors, r represents the peak amplitude of the sinusoid, and θ represents its phase shift relative to a reference sinusoid sin(ωt).

相量的基础在于复数的极坐标表示。任何复数 z = a + bi 都可以写成 z = r (cosθ + i sinθ)，或简写为 r cis θ，其中 r = |z| = √(a² + b²) 是模，θ = arg(z) = arctan(b/a)（需根据象限调整）是辐角，通常以弧度为单位。对于相量来说，r 代表正弦量的峰值，θ 则代表它相对于参考正弦 sin(ωt) 的相位移。

Using Euler’s formula, we can also express this as z = r e^(iθ). In IB notation, cis θ = cosθ + i sinθ. This compact form makes multiplication and division particularly straightforward: multiplying two complex numbers multiplies their moduli and adds their arguments; dividing them divides moduli and subtracts arguments.

利用欧拉公式，我们还可将其写作 z = r e^(iθ)。在IB记法中，cis θ = cosθ + i sinθ。这种紧凑形式让乘除运算格外直接：两个复数相乘，模相乘、辐角相加；相除时模相除、辐角相减。

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3. From Sine Waves to Phasors | 从正弦波到相量

Consider a general sinusoidal voltage: v(t) = Vₘ sin(ωt + φ). We can also represent it as v(t) = Im(Vₘ e^(i(ωt+φ))), where Im denotes the imaginary part. If we factor out the rotating term e^(iωt), we are left with a complex constant V = Vₘ e^(iφ) = Vₘ∠φ. This complex constant is the phasor. Switching between time-domain representation and phasor representation is a central skill: time function v(t) ⇌ phasor V.

考虑一般的正弦电压：v(t) = Vₘ sin(ωt + φ)。我们也可以将其表示为 v(t) = Im(Vₘ e^(i(ωt+φ)))，其中Im表示取虚部。若提取出旋转因子 e^(iωt)，剩下的是一个复常数 V = Vₘ e^(iφ) = Vₘ∠φ。这个复常数就是相量。在时域表示与相量表示之间切换是一项核心技能：时间函数 v(t) ⇌ 相量 V。

It is crucial to note that a phasor does not include the frequency ω; the frequency is assumed to be the same for all phasors in a given linear circuit or system. Phasors thus reduce differential equations describing sinusoids to algebraic equations in complex numbers.

必须注意，相量并不包含频率 ω；在线性电路或系统中，所有相量默认具有相同的频率。这样一来，相量就把描述正弦量的微分方程简化成了复数的代数方程。

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4. Euler’s Formula and the Rotating Vector | 欧拉公式与旋转向量

Euler’s formula e^(iθ) = cosθ + i sinθ is the mathematical engine behind phasors. In the complex plane, e^(iωt) traces out a unit circle at a constant angular speed ω. Multiplying this rotating unit vector by a constant complex number V stretches the radius to |V| and shifts the initial angle by arg(V). Therefore, v(t) = Im(V e^(iωt)) describes a sinusoidal projection onto the imaginary axis.

欧拉公式 e^(iθ) = cosθ + i sinθ 是相量背后的数学引擎。在复平面上，e^(iωt) 以恒定角速度 ω 画出一个单位圆。将这一旋转单位向量乘以一个常复数 V，会使其半径拉伸为 |V|，初始角度偏移 arg(V)。因此，v(t) = Im(V e^(iωt)) 描述的就是向虚轴的正弦投影。

When we work with phasors, we commonly adopt the convention that the instantaneous value is the projection onto either the imaginary axis (sine) or the real axis (cosine). Consistency is key: if we start with sine-based time functions, we must consistently take the imaginary part.

当使用相量时，我们通常约定瞬时值是向虚轴（正弦）或实轴（余弦）的投影。关键在于保持一致：若时间函数基于正弦，就必须始终提取虚部。

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5. Phasor Representation of Sinusoidal Functions | 正弦函数的相量表示

Given a sinusoid x(t) = A cos(ωt + φ), the corresponding phasor is X = A∠φ (or A cis φ). If the signal is expressed as a sine, x(t) = A sin(ωt + φ), we can either convert it to cosine using sin(θ) = cos(θ – π/2), or simply adopt the sine-reference phasor with the understanding that the time-domain reconstruction uses the imaginary part. A safe approach is to always convert to a cosine reference: A cos(ωt + φ) → A∠φ.

给定正弦量 x(t) = A cos(ωt + φ)，对应相量为 X = A∠φ（或 A cis φ）。如果信号表示为正弦 x(t) = A sin(ωt + φ)，可以利用 sin(θ) = cos(θ – π/2) 将其转换为余弦，或直接采用正弦参考相量（重构时提取虚部）。一个稳妥的做法是始终转换为余弦参考：A cos(ωt + φ) → A∠φ。

Thus, the phasor transform is a mapping from the set of equal-frequency sinusoids to the set of complex numbers. Inverse transform: given phasor X = |X|∠θ, the time-domain signal (cosine-based) is x(t) = |X| cos(ωt + θ).

由此，相量变换就是同频正弦量集合到复数集合的一个映射。逆变换：给定相量 X = |X|∠θ，其时域信号（基于余弦）为 x(t) = |X| cos(ωt + θ)。

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6. Phasor Addition and Subtraction | 相量加减法

Adding two sinusoids of the same frequency directly in the time domain requires trigonometric identities and can be clumsy. With phasors, addition reduces to vector addition in the complex plane. If v₁(t) = A₁ cos(ωt + φ₁) and v₂(t) = A₂ cos(ωt + φ₂), their phasors are V₁ = A₁∠φ₁ and V₂ = A₂∠φ₂. The resultant phasor is V_sum = V₁ + V₂, which we compute by converting to rectangular form, adding real and imaginary parts separately, and then converting back to polar form.

在时域直接叠加两个同频正弦量需要三角恒等式，过程较为繁琐。使用相量后，加法简化为复平面上的向量加法。若 v₁(t) = A₁ cos(ωt + φ₁)、v₂(t) = A₂ cos(ωt + φ₂)，它们对应相量 V₁ = A₁∠φ₁、V₂ = A₂∠φ₂。合成相量 V_sum = V₁ + V₂，我们通过转换为直角坐标形式，分别相加实部和虚部，再转回极坐标形式来计算。

Subtraction operates identically, taking care to reverse the direction of the subtrahend phasor. The resultant phasor then maps back to a single sinusoid, proving that any linear combination of equal-frequency sinusoids yields another sinusoid of the same frequency.

减法操作相同，只需将减相量反向即可。合成相量映回时域后就是一个单一的正弦波，这证明任何同频正弦量的线性组合仍然是同频的正弦波。

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7. Phasor Multiplication and Division | 相量乘除法

Multiplying two phasors corresponds to multiplying their moduli and adding their angles. However, the product of two phasors does not directly correspond to a time-domain product of two sinusoids at the same frequency—it would rather yield a signal of double frequency plus a DC component. Therefore, in linear AC circuit analysis, multiplication by a constant complex number (such as impedance) is common, but phasor–phasor multiplication is used mainly when analysing power or when dealing with complex scaling.

两个相量相乘，等于模相乘、辐角相加。然而，两个相量的乘积并不直接对应于两个同频正弦量在时域的乘积——后者的结果会包含二倍频分量和直流分量。因此，在线性交流电路分析中，常乘以一个常复数（如阻抗），而相量与相量相乘主要用于功率分析或复比例缩放。

Division of phasors involves dividing their magnitudes and subtracting their angles: V/I = (|V|/|I|)∠(θ_V – θ_I). This division is fundamental in defining impedance Z = V/I in AC circuits, where both V and I are phasors representing voltage and current.

相量相除则是模相除、辐角相减：V/I = (|V|/|I|)∠(θ_V – θ_I)。这一除法在定义交流电路的阻抗 Z = V/I 时至关重要，此时V和I都是表示电压和电流的相量。

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8. Phasor Diagrams | 相量图

A phasor diagram is a graphical representation of several phasors on the complex plane, typically drawn as arrows from the origin. The length of each arrow corresponds to the magnitude, and the counterclockwise angle from the positive real axis corresponds to the phase. Phasor diagrams are especially useful for qualitative analysis: you can visualise leading and lagging relationships, perform graphical addition, and check the relative phase between voltage and current in reactive components.

相量图是在复平面上以箭头形式表示多个相量的图形。箭头的长度对应幅值，从正实轴起逆时针度量的角度对应相位。相量图在定性分析中尤其有用：你可以直观看出超前与滞后关系，进行图解加法，并验证电抗元件上电压与电流的相对相位。

To draw a phasor diagram, choose a reference phasor (often the one with phase 0) and place it along the positive real axis. Other phasors are then drawn with angles relative to this reference. The resultant is obtained by the parallelogram law or tip-to-tail method.

绘制相量图时，先选定一个参考相量（通常取相位为0），将其放在正实轴方向。其他相量则按其相对角度画出。合成相量可通过平行四边形法则或首尾相接法求得。

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9. Differentiation and Integration in Phasor Form | 相量形式的微分与积分

One of the most powerful aspects of phasor analysis is how it handles differentiation and integration of sinusoids. In the time domain, if i(t) = I cos(ωt + φ), then di/dt = –ωI sin(ωt + φ), which is equivalent to ωI cos(ωt + φ + π/2). In the phasor domain, this transformation becomes multiplication by iω (or jω): the phasor I corresponds to di/dt via the phasor iω I. Similarly, integration corresponds to division by iω.

相量分析最强大的特性之一，是它如何处理正弦量的微分与积分。在时域中，若 i(t) = I cos(ωt + φ)，则 di/dt = –ωI sin(ωt + φ)，这等价于 ωI cos(ωt + φ + π/2)。而在相量域中，这一变换变成了乘以 iω：微分后的相量为 iω I。同理，积分则对应除以 iω。

Thus, a differential equation such as L di/dt + Ri = v(t) becomes an algebraic equation iωL I + R I = V, where I and V are phasors. This is the basis of AC steady-state analysis, turning calculus into complex algebra.

因此，像 L di/dt + Ri = v(t) 这样的微分方程，变成了代数方程 iωL I + R I = V，其中I和V为相量。这便是交流稳态分析的基础，它将微积分转化为复数代数。

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10. Application to AC Circuit Analysis | 交流电路分析应用

In AC circuit theory, resistors, inductors and capacitors are described by complex impedances: Z_R = R, Z_L = iωL, Z_C = 1/(iωC) = –i/(ωC). Ohm’s law in phasor form is V = Z I. Kirchhof’s laws apply directly to phasor currents and voltages. This allows students to solve AC networks using the same techniques as DC circuits—series, parallel combinations, node-voltage and mesh-current methods—only with complex arithmetic.

在交流电路理论中，电阻、电感和电容用复阻抗描述：Z_R = R、Z_L = iωL、Z_C = 1/(iωC) = –i/(ωC)。相量形式的欧姆定律为 V = Z I。基尔霍夫定律直接适用于相量电流和电压。这让学生能够使用与直流电路相同的技巧——串并联等效、节点电压法、网孔电流法——只是计算变为复数运算。

Although circuit analysis may be more emphasized in physics, IB Mathematics often includes these applications to illustrate the power of complex numbers. Understanding phasors thus builds a vital bridge between pure mathematics and real-world engineering problems.

尽管电路分析在物理中更常见，IB数学常常引入这些应用以展示复数的威力。因此，理解相量在纯数学和实际工程问题之间架起了一座重要的桥梁。

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11. Worked Examples | 例题精讲

Example 1: Express the sinusoid v(t) = 10 sin(100πt + π/6) as a cosine-referenced phasor.
Solution: Rewrite as cosine: v(t) = 10 cos(100πt + π/6 – π/2) = 10 cos(100πt – π/3). Hence the phasor is V = 10∠(–π/3) or 10 cis(–π/3).

例题 1：将正弦量 v(t) = 10 sin(100πt + π/6) 表示为余弦参考相量。
解：改写为余弦：v(t) = 10 cos(100πt + π/6 – π/2) = 10 cos(100πt – π/3)。因此相量为 V = 10∠(–π/3) 或 10 cis(–π/3)。

Example 2: Given phasors I₁ = 4∠30° and I₂ = 3∠–60°, find the resultant I_total = I₁ + I₂.
Solution: Convert to rectangular form: I₁ = 4 cos30° + i 4 sin30° = 3.464 + i 2.000; I₂ = 3 cos(–60°) + i 3 sin(–60°) = 1.500 – i 2.598. Sum = (3.464+1.500) + i(2.000 – 2.598) = 4.964 – i 0.598. Magnitude = √(4.964² + (–0.598)²) ≈ 5.00; phase = arctan(–0.598/4.964) ≈ –6.87°. Thus I_total ≈ 5∠–6.87°.

例题 2：已知相量 I₁ = 4∠30°，I₂ = 3∠–60°，求合成相量 I_total = I₁ + I₂。
解：转换为直角坐标：I₁ = 4 cos30° + i 4 sin30° = 3.464 + i 2.000；I₂ = 3 cos(–60°) + i 3 sin(–60°) = 1.500 – i 2.598。和为 (3.464+1.500) + i(2.000 – 2.598) = 4.964 – i 0.598。模 ≈ √(4.964² + (–0.598)²) ≈ 5.00，辐角 ≈ arctan(–0.598/4.964) ≈ –6.87°。因此 I_total ≈ 5∠–6.87°。

Example 3: In a series AC circuit, R = 50 Ω, L = 0.2 H, ω = 100 rad/s. The phasor voltage across the series combination is V = 100∠0° V. Find the current phasor I.
Impedance Z = R + iωL = 50 + i(100)(0.2) = 50 + i20 Ω = √(50²+20²)∠arctan(20/50) = 53.85∠21.8° Ω. Then I = V / Z = (100∠0°) / (53.85∠21.8°) = 1.857∠–21.8° A.

例题 3：在串联交流电路中，R = 50 Ω，L = 0.2 H，ω = 100 rad/s。串联组合两端电压相量为 V = 100∠0° V。求电流相量 I。
阻抗 Z = R + iωL = 50 + i(100)(0.2) = 50 + i20 Ω，其模为 √(50²+20²) = 53.85，辐角为 arctan(20/50) = 21.8°，即 Z = 53.85∠21.8° Ω。于是 I = V / Z = (100∠0°) / (53.85∠21.8°) = 1.857∠–21.8° A。

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12. Summary and Key Formulae | 总结与关键公式

Phasors transform sinusoids into complex constants, enabling algebraic treatment of AC problems. The core conversions are: time domain x(t) = Xₘ cos(ωt + φ) ⇌ phasor X = Xₘ∠φ. Euler’s identity e^(iθ) = cosθ + i sinθ underpins the rotating vector interpretation. Addition of phasors is equivalent to vector addition; differentiation multiplies the phasor by iω; integration divides by iω. These principles provide a unified framework for solving linear systems driven by single-frequency sinusoidal sources.

相量将正弦量转化为复常数，使交流问题能用代数方法处理。核心转换：时域 x(t) = Xₘ cos(ωt + φ) ⇌ 相量 X = Xₘ∠φ。欧拉恒等式 e^(iθ) = cosθ + i sinθ 支撑着旋转向量的物理解释。相量加法等价于向量加法；微分相当于乘以 iω；积分相当于除以 iω。这些原理为求解单频正弦激励的线性系统提供了统一框架。

Key Formulae:    cis θ = cosθ + i sinθ   |   X = |X|∠φ   |   d/dt ⇌ iω   |   ∫ dt ⇌ 1/(iω)   |   Z_R = R, Z_L = iωL, Z_C = 1/(iωC)

掌握相量不仅能帮助IB学生在数学考试中游刃有余，也为未来学习物理、工程和信号处理打下坚实基础。

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