Complex numbers and phasor algebra
Introduced| English | Meaning |
|---|---|
| complex number/ˈkɒmpleks ˈnʌmbə/ | A number expressible as a real part plus an imaginary part, using i with i squared equal to minus one. |
| modulus/ˈmɒdjʊləs/ | The non-negative magnitude of a number; for a real number, its distance from zero. |
| argument/ˈɑːɡjuːmənt/ | An angle describing the direction of a nonzero complex number from the positive real axis in the complex plane. |
| Euler's identity/ˈɔɪləz aɪˈdentɪti/ | e to the power jθ equals cos θ plus j sin θ, which links exponentials and sinusoids. |
| conjugate pair | conjugate pair |
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.
$z = 3 + j4$: $|z| = \sqrt{9 + 16} = 5$ and $\theta = \text{atan2}(4, 3) \approx 53.1^{\circ}$. So $3 + j4 = 5\angle 53.1^{\circ}$.
What is the modulus of z = 5 − j12?
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$\sqrt{5^{2} + 12^{2}} = \sqrt{169} = 13$. The minus sign affects the angle, not the length.
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.
Write 10∠30° in rectangular form. What is its real part? Give two decimal places.
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$10\cos 30^{\circ} \approx 8.66$; the imaginary part is $10\sin 30^{\circ} = 5$. So $10\angle 30^{\circ} = 8.66 + j5$.
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 $-90^{\circ}$.
Multiplying a phasor by j rotates it by −90°.
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$j = 1\angle 90^{\circ}$, so multiplying by $j$ adds $+90^{\circ}$. Multiplying by $-j$ rotates 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.
What is (8∠15°) ÷ (2∠40°)?
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Divide the moduli (8 ÷ 2 = 4) and subtract the arguments (15° − 40° = −25°). Multiplying back must return 8∠15°.
What is 3∠−50° multiplied by 2∠80°?
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Moduli multiply, $3 \times 2 = 6$; arguments add, $-50^{\circ} + 80^{\circ} = 30^{\circ}$. This undoes the division that produced 3∠−50°.
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.
Check by substitution: $(-1 + j2)^{2} + 2(-1 + j2) + 5 = (-3 - j4) + (-2 + j4) + 5 = 0$ exactly.
What are the roots of s² + 2s + 5 = 0?
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$s = -1 \pm j2$: modulus $\sqrt{1 + 4} \approx 2.236$. The upper root is at $+116.6^{\circ}$ and its conjugate at $-116.6^{\circ}$.
Three common slips
- Adding moduli. $|z_{1} + z_{2}|$ is not $|z_{1}| + |z_{2}|$; add the rectangular parts, then find the modulus.
- Degrees inside $e^{j\theta}$. Formulas such as $\dfrac{d}{dt}e^{j\omega t}$ assume radians.
- The direction of $j$. Multiplying by $j$ turns anticlockwise by $90^{\circ}$; by $-j$, clockwise.
Match each operation to how it is done.
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Each form makes some operations easy. Rectangular suits adding; polar suits multiplying and dividing.
Recap
- $z = a + jb = r\angle\theta = r\,e^{j\theta}$, with $r = \sqrt{a^{2}+b^{2}}$.
- Add in rectangular form; multiply and divide in polar form.
- Complex roots come in conjugate pairs: real part = decay, imaginary part = frequency.