Complex Numbers What if we want to add

Complex Numbers • What if we want to add two sinusoids? – V(t) = 5 cos(377 t + 45 ) + 7 sin(377 t – 60) – Trig Identity: – Not very pleasant! • To deal more effectively with sinusoidal signals we will learn a new tool: – Complex Numbers 1

Complex Numbers, Intro (video) 2

Complex Nums – Rectangular 3

Complex Nums – Rectangular adding and subtracting (video) 4

Complex Nums – Rectangular Dividing (video) 5

Complex Arithmetic (Rectangular) Given: a = 3 + 4 j b= -5 + 6 j Find a+b a–b a*b a/b 6

Complex Numbers – Polar (video) 7

Conversion: Rect to Polar (video) 8

Conversion: Rect to Polar (video) 9

Conversion: Polar to Rect (video) 10

Euler’s Formula (video) 11

Polar to Rect (and back) • Convert a = 2 – 3 j and b = -4 + 5 j – to polar form: • Convert c = 2<30 o and d = 4<200 o – To rectangular form 12

Complex: Add/Multiply in Polar Form (video) 13

Complex: Divide in Polar (video) 14

Summary: Mathematic operation of complex numbers http: //www. 1728. com/compnumb. htm 1. Addition 2. Subtraction 3. Multiplication 4. Division 5. Reciprocal 6. Square root 7. Complex conjugate 8. Euler’s identity 15

Polar Complex Mult/Div Given: c = 8<45 d = 12<250 Find c*d c/d c* (complex conj of c) 16

Example 3 Calculator Demo! • Evaluate the following complex numbers: a. b. Solution: a. – 15. 5 + j 13. 67 b. 8. 293 + j 2. 2 17