Algebra of Phasors | 相量的代数运算

📚 Algebra of Phasors | 相量的代数运算

Phasor algebra extends ordinary complex-number arithmetic into the polar domain, making it an essential tool for analysing alternating current circuits, mechanical vibrations, and any system described by sinusoidal functions. By treating amplitude and phase as a single complex quantity, addition, subtraction, multiplication, division, and rotation become geometrically intuitive operations that underpin IB Mathematics: Analysis and Approaches and Applications and Interpretation alike.

相量代数将普通的复数运算推广到极坐标领域，使其成为分析交流电路、机械振动以及任何用正弦函数描述的系统的基本工具。通过将幅值和相位组合成一个复数，加法、减法、乘法、除法和旋转都变成了几何直观的运算，这正是 IB 数学分析与方法和应用与解释的重要基础。

1. Complex Numbers and Phasor Representation | 复数与相量表示

A phasor is a complex number whose magnitude equals the amplitude of a sinusoid and whose argument equals its phase angle. At a fixed angular frequency ω, the time‑varying signal A cos(ωt+φ) is represented by the stationary phasor A ∠ φ in the complex plane, removing the time dependence while preserving all amplitude and phase information.

相量是一个复数，其模等于正弦量的振幅，辐角等于其相位角。在固定角频率 ω 下，随时间变化的信号 A cos(ωt+φ) 用复平面上静止的相量 A ∠ φ 来表示，这样就去掉了时间依赖性，但保留了所有幅值和相位信息。

In Cartesian form, a phasor is expressed as z = a + b i, where a = A cos φ is the real part and b = A sin φ is the imaginary part. The modulus r = |z| = √(a² + b²) recovers the amplitude, while the argument φ = arctan(b/a) gives the phase shift, linking the rectangular and polar views directly.

在笛卡儿形式中，相量表示为 z = a + b i，其中 a = A cos φ 是实部，b = A sin φ 是虚部。模 r = |z| = √(a² + b²) 恢复了振幅，而辐角 φ = arctan(b/a) 给出相移，从而直接将直角坐标与极坐标视图联系起来。

---

2. Polar Form and Euler’s Formula | 极坐标形式与欧拉公式

Every phasor can be written in compact polar notation as z = r ∠ θ, where r ≥ 0 is the magnitude and θ is the phase angle measured from the positive real axis. Euler’s identity e^(iθ) = cos θ + i sin θ provides the exponential equivalent z = r e^(iθ), which simplifies multiplication and division drastically.

每个相量都可以用简洁的极坐标记号写作 z = r ∠ θ，其中 r ≥ 0 是幅值，θ 是从正实轴量起的相位角。欧拉恒等式 e^(iθ) = cos θ + i sin θ 给出了等价的指数形式 z = r e^(iθ)，它极大地简化了乘法和除法运算。

Conversion between Cartesian and polar forms is fundamental: given a + b i, the modulus is r = √(a² + b²) and the argument satisfies tan θ = b/a, with quadrant adjustment. Conversely, r ∠ θ expands to r cos θ + i r sin θ. Mastery of these transformations is essential for phasor algebra.

笛卡儿坐标与极坐标之间的转换是基础：给定 a + b i，模为 r = √(a² + b²)，辐角满足 tan θ = b/a，并需根据象限校正。反过来，r ∠ θ 展开为 r cos θ + i r sin θ。掌握这些转换对相量代数至关重要。

---

3. Addition and Subtraction of Phasors | 相量的加法与减法

Phasor addition is most efficiently performed in Cartesian coordinates. If z₁ = a₁ + b₁ i and z₂ = a₂ + b₂ i, then z₁ + z₂ = (a₁ + a₂) + (b₁ + b₂) i. The resultant phasor represents the superposition of two sinusoidal signals of the same frequency, and its magnitude and phase are obtained by converting the sum back to polar form.

相量加法在笛卡儿坐标下最为高效。若 z₁ = a₁ + b₁ i，z₂ = a₂ + b₂ i，则 z₁ + z₂ = (a₁ + a₂) + (b₁ + b₂) i。合成相量代表两个同频正弦信号的叠加，其幅值和相位可通过将和转换回极坐标获得。

Subtraction is analogous: z₁ − z₂ = (a₁ − a₂) + (b₁ − b₂) i. Geometrically, subtraction corresponds to adding the negative of the subtrahend, which is an inversion through the origin followed by a vector addition. The parallelogram law governs both operations.

减法类似：z₁ − z₂ = (a₁ − a₂) + (b₁ − b₂) i。从几何上看，减法相当于加上减数的相反复数，即先关于原点反演再作矢量加法。平行四边形法则支配着这两种运算。

In phasor diagrams, the tail-to-head method visually constructs the sum, while the difference is seen as the vector connecting the heads when the tails coincide. Direct polar addition is unpractical, so conversion to Cartesian form is the standard procedure.

在相量图中，头尾相接的方法直观地构造和，而差则可看作当尾部重合时连接两个头部的矢量。直接的极坐标加法并不实用，因此转换为笛卡儿形式是标准步骤。

---

4. Multiplication of Phasors | 相量的乘法

Multiplication in polar form is elegantly simple: if z₁ = r₁ ∠ θ₁ and z₂ = r₂ ∠ θ₂, then z₁ z₂ = (r₁ r₂) ∠ (θ₁ + θ₂). The magnitudes multiply and the arguments add. Using exponential notation, r₁ e^(iθ₁) × r₂ e^(iθ₂) = r₁ r₂ e^(i(θ₁+θ₂)), which directly illustrates this rule.

极坐标形式下的乘法异常简洁：若 z₁ = r₁ ∠ θ₁，z₂ = r₂ ∠ θ₂，则 z₁ z₂ = (r₁ r₂) ∠ (θ₁ + θ₂)。模相乘，辐角相加。使用指数记法，r₁ e^(iθ₁) × r₂ e^(iθ₂) = r₁ r₂ e^(i(θ₁+θ₂))，直接展示了这条法则。

When one of the multipliers has unit magnitude, multiplication becomes a pure rotation. Multiplying any phasor by 1 ∠ α rotates it counter‑clockwise by angle α without changing its amplitude; this property is vital for phase‑shifting operations in signal processing.

当其中一个乘数的模为 1 时，乘法就变为纯粹的旋转。将任意相量乘以 1 ∠ α 可将其逆时针旋转角度 α 而不改变振幅；这一性质在信号处理中的移相操作里至关重要。

---

5. Division of Phasors | 相量的除法

Division follows naturally: z₁ ÷ z₂ = (r₁ / r₂) ∠ (θ₁ − θ₂), provided r₂ ≠ 0. The magnitudes divide and the arguments subtract. In exponential form, r₁ e^(iθ₁) / r₂ e^(iθ₂) = (r₁ / r₂) e^(i(θ₁−θ₂)), clearly showing the phase‑difference interpretation.

除法自然遵循：z₁ ÷ z₂ = (r₁ / r₂) ∠ (θ₁ − θ₂)，前提是 r₂ ≠ 0。模相除，辐角相减。在指数形式中，r₁ e^(iθ₁) / r₂ e^(iθ₂) = (r₁ / r₂) e^(i(θ₁−θ₂))，清楚地显示出相位差的含义。

In Cartesian form, division is performed by multiplying numerator and denominator by the complex conjugate of the denominator: (a₁ + b₁ i)/(a₂ + b₂ i) = [(a₁ + b₁ i)(a₂ − b₂ i)] / (a₂² + b₂²). This routine is necessary when phasors are given in rectangular coordinates.

在笛卡儿形式下，除法通过将分子和分母同时乘以分母的共轭复数来实现：(a₁ + b₁ i)/(a₂ + b₂ i) = [(a₁ + b₁ i)(a₂ − b₂ i)] / (a₂² + b₂²)。当相量以直角坐标给出时，这一常规步骤必不可少。

---

6. Complex Conjugate and Phasor Conjugate | 共轭复数与相量共轭

The complex conjugate of a phasor z = a + b i is z* = a − b i, and in polar form z* = r ∠ (−θ) = r e^(−iθ). Conjugation reflects the phasor across the real axis, useful for computing real power and for manipulating expressions involving magnitudes.

相量 z = a + b i 的共轭复数为 z* = a − b i，极坐标形式中 z* = r ∠ (−θ) = r e^(−iθ)。共轭运算将相量关于实轴反射，这在计算有功功率和处理包含模的表达式时非常有用。

A key identity is z z* = |z|² = a² + b² = r², which yields a purely real number equal to the squared amplitude. This identity underpins the Cartesian division algorithm and is constantly used when rationalising denominators.

一个关键的恒等式是 z z* = |z|² = a² + b² = r²，它得到一个纯实数，等于幅值的平方。这个恒等式支撑了笛卡儿除法算法，并在分母有理化时频繁使用。

---

7. Phasor Rotation and Phase Shift | 相量旋转与相位移动

Multiplication by the unit phasor e^(iΔ) = 1 ∠ Δ rotates a phasor by an angle Δ counter‑clockwise. Successive rotations multiply: rotating by α and then by β results in a total rotation of α + β, consistent with the addition‑of‑arguments rule.

乘以单位相量 e^(iΔ) = 1 ∠ Δ 可将相量逆时针旋转角度 Δ。相继旋转相乘：先旋转 α 再旋转 β 的总旋转量为 α + β，这与辐角相加法则一致。

A rotation of 90° corresponds to multiplication by i, since i = 0 + 1 i = 1 ∠ 90°. Thus multiplying a phasor by i advances its phase by a quarter cycle, a fact heavily exploited in AC circuit analysis for inductive and capacitive reactances.

旋转 90° 相当于乘以 i，因为 i = 0 + 1 i = 1 ∠ 90°。因此将相量乘以 i 会使相位超前四分之一个周期，这一事实在交流电路分析中广泛应用于电感和电容电抗的计算。

---

8. Powers and Roots of Phasors | 相量的乘幂与开方

De Moivre’s theorem states that for any integer n, (r ∠ θ)ⁿ = rⁿ ∠ (nθ). This formula extends to rational exponents, allowing the calculation of roots. The n distinct nth roots of a phasor are given by r^(1/n) ∠ ((θ + 2kπ)/n) for k = 0, 1, …, n−1, forming a symmetric star in the complex plane.

棣莫弗定理指出，对任意整数 n，(r ∠ θ)ⁿ = rⁿ ∠ (nθ)。该公式可推广到有理指数，从而能够计算方根。一个相量的 n 个不同的 n 次方根由 r^(1/n) ∠ ((θ + 2kπ)/n) 给出，其中 k = 0, 1, …, n−1，在复平面上构成对称的星形。

For example, the square roots of 4 ∠ 60° are 2 ∠ 30° and 2 ∠ 210°. Phasor powers and roots frequently appear in filter design and in solving differential equations where characteristic roots are complex exponentials.

例如，4 ∠ 60° 的平方根为 2 ∠ 30° 和 2 ∠ 210°。相量的乘幂与开方经常出现在滤波器设计以及求解特征根为复指数的微分方程中。

---

9. Solving Equations with Phasors | 用相量解方程

Phasor methods convert differential equations governing linear AC circuits into algebraic equations. For a steady‑state sinusoidal input, derivatives become multiplications by iω, and the unknown phasor is solved by ordinary complex algebra, after which the time‑domain signal is recovered by taking the real part and multiplying by e^(iωt).

相量方法将支配线性交流电路的微分方程转换为代数方程。对于稳态正弦输入，求导变为乘以 iω，未知相量通过普通的复数代数解出，之后通过取实部并乘以 e^(iωt) 恢复时域信号。

Quadratic and higher-order phasor equations are solved using the quadratic formula or factorisation, just as with ordinary complex numbers. The principle of superposition permits separate handling of multiple sources, each represented by its own phasor.

二次及更高次的相量方程可使用二次公式或因式分解求解，与普通复数完全相同。叠加原理允许分别处理多个源，每个源用各自的相量表示。

---

10. Summary of Key Phasor Operations | 相量关键运算总结

The following table summarises the essential phasor algebra operations, highlighting the most efficient coordinate system for each.

下表总结了基本的相量代数运算，并标出每种运算最有效的坐标系统。

Operation | 运算	Polar Form | 极坐标	Cartesian Form | 笛卡儿坐标
Addition / Subtraction 加法/减法	Not convenient 不便	(a₁±a₂) + (b₁±b₂)i
Multiplication 乘法	r₁r₂ ∠ (θ₁+θ₂)	(a₁a₂−b₁b₂)+(a₁b₂+a₂b₁)i
Division 除法	(r₁/r₂) ∠ (θ₁−θ₂)	Multiply by conjugate 乘以共轭
Conjugation 共轭	r ∠ (−θ)	a − b i
Power n 乘幂	rⁿ ∠ (nθ)	Use polar 用极坐标
Roots 方根	r^(1/n) ∠ ((θ+2kπ)/n)	Use polar 用极坐标

Choosing the right representation—polar for multiplication, division, powers and phase shifts; Cartesian for superposition sums—transforms phasor problems into manageable algebraic tasks, forming a core technique for IB Higher Level and Standard Level complex number topics.

选择合适的表示法——极坐标用于乘法、除法、乘幂和相移；笛卡儿坐标用于叠加求和——能将相量问题转化为易处理的代数任务，这构成了 IB 高水平与标准水平复数主题的核心技术。

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

更多咨询请联系16621398022（同微信）