Complex numbers and phasor algebra

230 V, and a phase

• A mains voltage has a size, 230 V, and a timing, its phase. One ordinary number cannot hold both.
• A point on a plane can: a complex number. With it, phase shifts become rotations and gains become scalings.

Rectangular form, modulus and argument

• A complex number 复数 $z = a + jb$, with $j^{2} = -1$, has a real part $a$ and an imaginary part $b$. Engineers write $j$ because $i$ is already the current.
• Its modulus 模 is the length $|z| = \sqrt{a^{2} + b^{2}}$; its argument 辐角 is the angle $\theta = \text{atan2}(b, a)$, measured from the positive real axis.

Euler's identity joins the two forms

• Euler's identity 欧拉公式: $a + jb = r(\cos\theta + j\sin\theta) = r\,e^{j\theta}$. The polar form $r\angle\theta$ is shorthand for $r\,e^{j\theta}$.
• Use rectangular form to add and subtract (add the parts), and polar form to multiply and divide.

Multiplying rotates and scales

• $r_{1}e^{j\theta_{1}} \cdot r_{2}e^{j\theta_{2}} = r_{1}r_{2}\,e^{j(\theta_{1}+\theta_{2})}$: moduli multiply, angles add. That is exactly what a gain and a phase shift do to a sinusoid.
• Multiplying by $j = 1\angle 90^{\circ}$ rotates a number by $+90^{\circ}$; multiplying by $-j$ rotates it by $-90^{\circ}$.

Dividing in polar form

• Moduli divide, angles subtract: $\dfrac{6\angle 30^{\circ}}{2\angle 80^{\circ}} = 3\angle -50^{\circ}$, in one line. In rectangular form the same sum needs $\dfrac{5.196 + j3}{0.347 + j1.970}$ and a rationalised denominator.
• Check with the inverse: $3\angle -50^{\circ} \times 2\angle 80^{\circ} = 6\angle 30^{\circ}$. The round trip must return the start.

Complex roots carry decay and ringing

• A damped system's equation $s^{2} + 2s + 5 = 0$ has roots $s = \dfrac{-2 \pm \sqrt{4 - 20}}{2} = -1 \pm j2$: a conjugate pair 共轭对, at $2.236\angle \pm 116.6^{\circ}$.
• The real part, −1, is the decay rate; the imaginary part, 2 rad/s, is the ringing frequency. One complex number carries both behaviours.

Three common slips

Recap