강의노트 복소수

복소수

• 실수(a)와 허수(b)의 합으로 구성
• 직교좌표식(retangular form)

$$Z = a + jb$$

$$j = \sqrt {-1}$$

$$\vert Z \vert = \sqrt{a^2 + b^2}$$

$$\theta = \tan^{-1} \left( \dfrac{b}{a} \right)$$

• 극좌표식(polar form)

$$Z = \vert Z \vert \angle \theta = \vert Z \vert e^{j \theta}$$

복소수의 사칙연산

$$Z_1 = a_1 + jb_1 = \vert Z_1 \vert \angle \theta_1 = \vert Z_1 \vert e^{j\theta_1}$$

$$Z_2 = a_2 + jb_2 = \vert Z_2 \vert \angle \theta_2 = \vert Z_2 \vert e^{j\theta_2}$$

• 덧셈

$$Z_1 + Z_2 = (a_1 + a_2) +j(b_1 + b_2)$$

• 뺄셈

$$Z_1 - Z_2 = (a_1 - a_2) +j(b_1 - b_2)$$

• 곱셈

$$Z_1 * Z_2 = \vert Z_1 \vert * \vert Z_2 \vert \angle (\theta_1 + \theta_2) = \vert Z_1 \vert * \vert Z_2 \vert e^{ j(\theta_1 + \theta_2)}$$

• 나눗셈

$$\dfrac{Z_1}{Z_2} = \dfrac{\vert Z_1 \vert}{\vert Z_2 \vert} \angle (\theta_1 - \theta_2) = \dfrac{\vert Z_1 \vert}{\vert Z_2 \vert} e^{j(\theta_1 - \theta_2)}$$