ENGIN 211, Engineering Math. Complex Numbers

Transcription

1 ENGIN 211, Engineering Math Complex Numbers 1

2 Imaginary Number and the Symbol J Consider the solutions for this quadratic equation: x = 0 x = ± 1 1 is called the imaginary number, and we use the symbol j to represent it: j = 1. Thus, the solutions Obviously, x = ±j j 2 = 1 This notation allowed us to deal with a large number of quadratic equations of the form: ax 2 + bb + c = 0 with the solution x = b± b2 4aa 2a In case b 2 4aa < 0, so we can rewrite x = b ± j 4aa b2 2a 2

3 Complex Numbers Example: x 2 + 2x + 3 = 0 We have two roots x 1 = 1 + j 2 and x 2 = 1 j 2 Both roots consisting of a real number and an imaginary number are complex number. Re(x 1 ) = 1 real part of the complex number x 1 Im(x 1 ) = 2 imaginary part of the complex number x 1 Re(x 2 ) = 1 real part of the complex number x 2 Im x 2 = 2 imaginary part of the complex number x 2 Note: Both Re(x) and Im(x) are real numbers themselves. 3

4 Operations of Complex Numbers Addition: Subtraction: Powers of j, 2 + jj + 3 jj = 1 + jj 2 + jj 3 jj = 5 + jjj j 2 = 1, j 3 = j 2 j = j, j 4 = j 2 j 2 = 1 1 = 1, Multiplication: 2 + jj 3 jj = jj + jj 3 + (jj)( jj) = 6 + jj + jjj j 2 24 = 6 + jjj 1 24 = 18 + jjj 4

5 Complex Conjugates Complex conjugate: a + jj and a jj are complex conjugate pairs a + jj a jj = a 2 jj 2 = a 2 + b 2 always a real number Example: 3 + jj 3 jj = = 25 5

6 Division of Complex Numbers Division by a real number 3+jj 2 Division by another complex number = jj 3 + jj 3 + jj 2 + jj = 2 jj 2 jj 2 + jj j(8 + 9) = = jj.31 Division by itself = 6 + jjj jj 3 + jj = 3 + jj 3 jj 3 + jj 3 jj = = 1 6

7 Equal Complex Numbers If two complex number are equal, their corresponding real and imaginary parts must be equal: Implies: Because we can write a + jj = c + jj a = c, and b = d a c = j(d b) And a real number never equals an imaginary number. Example: Then x 1 = 6, and x 2 = 8. x = x 1 + jx 2 = 6 + jj 7

8 Creating Complex Numbers in Matlab Complex numbers consist of two separate parts: a real part and an imaginary part. The basic imaginary unit is equal to the square root of -1. This is represented in MATLAB by either of two letters: i or j. The variable x is assigned a complex number with a real part of 2 and an imaginary part of 3: x = 2 + 3i; Another way to create a complex number is using the complex function. This function combines two numeric inputs into a complex output, making the first input real and the second imaginary: x = rand(3) * 5; y = rand(3) * -8; z = complex(x, y) 8

9 Matlab Complex Number Functions Function complex i or j real imag Description Construct complex data from real and imaginary components. Return the imaginary unit used in constructing complex data. Return the real part of a complex number. Return the imaginary part of a complex number. isreal Determine if a number is real or imaginary. 9

10 Complex Plane (Argand Diagram) Imaginary axis z = a + jj b (a, b) a Real axis Graphical addition of two complex numbers

11 Polar Form Imaginary axis r θ z = a + jj b a Real axis Conversion between polar and rectangular forms r: modulus of the complex number z θ: argument of the complex number z r = a 2 + b 2, θ = tan 1 b a a = r cos θ, b = r sin θ, π θ π Example: z = 6 jj can be converted into polar form, r = = 10, θ = tan = 36o 52 11

12 Exponential Form Maclaurin Series Let x = jj, e x = 1 + x + x2 2! + x3 3! + x4 4! + x5 5! + e jj = 1 + jj + jj 2 2! + jj 3 3! + jj 4 4! + jj 5 5! + = 1 θ2 2! + θ4 θ3 + j θ 4! 3! + θ5 5! = cos θ + j sin θ EEEEE s iiiiiiii Thus, r cos θ + j sin θ = re jj exponential form Three ways: z = a + jj = r cos θ + j sin θ = re jj 12

13 Which Form Is Better? It depends on what you want to do: For addition and subtraction, it is better to work with the rectangular form z = a + jj For multiplication and division, it is better to work with exponential form, z = re jj, e.g., 4e j60o 2e j30o = 2e j 60o 30 o = 2e j30o To convert between rectangular and exponential form, we work thru polar form, z = r cos θ + j sin θ 3 + jj = 5 cos 36 o 52 + j sin 36 o 52 = 5e j36o 52 13

14 Powers Consider: z = re jj z 2 = re jj 2 = r 2 e jjj = r 2 cos 2θ + j sin 2θ, z n = re jj n = r n e jjj = r n cos nn + j sin nn DeMoivre s theorem If nn > π, we need to add or subtract multiples of 2π so that the equivalent angle φ stays within φ π. Example: z = 2e j120o = o, z 5 = 2 5 e j5 120o = o = o z 120 o 120 o z 5 14

15 Roots Roots of a complex number is somewhat complicated and requires careful attention in manipulation. Example, w = o = 32e j120o, let s find w 1/5 =? 1) We can take do the following 32e j120o 1/5 = 32 1/5 e j120o /5 = 2e j24o = o But we certainly expect to recover z = 2e j120o = o because z 5 = o from the previous example of powers, so what happened? 2) Of course, we can say 120 o is also 360 o 120 o = 240 o, so now 32e j120o 1/5 = 32e j240 o 1/5 = 32 1/5 e j240o /5 = 2e j48o = 2244 o But the solution z = 2e j120o = o is still not recovered. 15

16 Roots (Cont d) 3) Well, we can keep adding 720 o 120 o = 600 o, so now 32e j120o 1/5 = 32e j600 o 1/5 = 32 1/5 e j600o /5 = 2e j120o = o This is it, but how do we know we have gotten them all? 4) We can also subtract 360 o, so 360 o 120 o = 480 o, then 32e j120o 1/5 = 32e j480 o 1/5 = 32 1/5 e j480o /5 = 2e j96o = o 5) Of course, we can also obtain 32e j120o 1/5 = 32e j840 o 1/5 = 32 1/5 e j840o /5 = 2e j168o = o When do we stop? 16

17 Roots (Cont d) Let s plot these roots in the complex plane o o o 33e j111o o o We can see that these five roots are equally separated by 360o = 72 o. 5 It is generally true that the n-th order roots have n number of values that are equally separated by 333o. n So in practice, we only need to find one as we did in Step 1), and the remaining n 1 roots will be generated similar to the illustration in the complex plane plot. The one nearest to the positive real axis is called the principal root. 17

18 Summary Key points: Recognize j = 1 for imaginary number Three forms of the complex number: rectangular, polar, and exponential Conversions between the three forms Complex conjugate pairs Addition, subtraction, multiplication (powers), and division Roots of n-th order (equally spaced by 360 o /n on a circle) 18