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Complex Numbers Introduction: Let’s say you have the following quadratic equation: 5 x 2 x 0.25 0 (1) The solution to this equation is: (see quadratic formula) x 1 12 4 5 0.25 1 4 25 10 (2) If you didn’t know any better, you would say that x does not exist, because the 4 does not exist. However, math gives us an elegant solution. Given the imaginary unit j 1 , the solution to (2) is: x 0.1 j 0.2 (3) Equation 3 gives the value of x as a complex number. It is important to note that the two components of this complex number are real numbers. The real part of x is -0.1, whereas the imaginary part of x is ±0.2. In general, a complex number is written as: C a j b , where the real part of C is a and the imaginary part of C is b. Note that both a and b are real numbers. Forms: The complex number, C, can be written in rectangular (or Cartesian) form or in the polar form. Both forms have their uses. Cartesian form The complex plane: Imaginary(C) C=a+jb b r J a Real(C) Figure 1: A vector in the complex plane. A complex number can be used to denote a vector in the x-y plane. This is indicated in Figure 1, which shows the complex plane in terms of the x-axis (which corresponds to the real part of the complex number), and the y-axis (which corresponds to the imaginary part of the complex number.) The projection of the vector C onto the x-axis is equal to a, while the projection of the vector C onto the y-axis is equal to b. Polar form You can also think of the same vector in terms of its magnitude r (the length of the red arrow) and the angle, J, of the vector away from the x-axis (this is called the phase.) Thought in those terms, a rcosJ and b rsin J . So, the complex number, C, can be written as C rcosJ j rsin J . jJ jJ Another way towrite the same thing is C r e since e cos J j sin J Think Trigonometry! C a jb rcosJ jrsin J Let’s take the sum of the square of the real part of C and the square of the imaginary part of C: a2 b2 r sin J r cosJ 2 2 r 2 sin 2 J r 2 cos 2 J r 2 sin 2 J cos 2 J But, sin2J + cos2Jso: a2 b2 r 2 r is the magnitude of the complex number C, and is calculated as a2+b2 when C is given in Cartesian form. Let’s divide the imaginary part of C (in other words, b) by the real part of C (in other words, a). Remember, a=rcosJ and b=rsinJ. So: Re(C) b r sin J sin J tan J Im( C) a r cosJ cosJ So, if the complex number C is given in terms of a and b (real and imaginary), the phase, J, can be calculated as J arctan b a This sounds straightforward, and it is, as long as you take into account the properties of the arctan function. A calculator will always return a value for the arctan between the angles -90°to 90°, that is in the first or fourth quadrant of the complex plane (see Figure 2 – the areas denoted I and IV). For example, in Figure 2, C=a+jb, and C2=-a-jb. The magnitudes of the complex numbers are the same, but their phases differ by 180°, or . b b A calculator, however, would yield the same result for arctan and arctan . It is a a your responsibility to observe that, in the case of C2, the complex number has a<0 and b<0, and add to the final result. It is also a good practice to always calculate the phase in radians. Similarly, a complex number in quadrant II might yield a result that “looks like” a quadrant IV number. Follow the rules in Table 1 to calculate the phase correctly in each case. Imaginary(C) II I C=a+jb b r J J a Real(C) r C2=-a-jb III IV Figure 2: Complex plane showing the quadrants. C=a+jb a>0 and b>0 a<0 and b>0 a<0 and b<0 a>0 and b<0 Quadrant Result in rads I arctan(b/a) II arctan(b/a)+ III arctan(b/a)+ IV arctan(b/a) Table 1: Calculating the correct phase. Result in degs arctan(b/a) arctan(b/a)+180° arctan(b/a)+180° arctan(b/a) J is the phase of the complex number, C, and is calculated as arctan(b/a) when C is given in Cartesian form. So, given C in terms of the real and imaginary parts (Cartesian form – C=a+jb), one can calculate the magnitude and phase from: r a2 b2 J arctan b a Conversely, given C in polar form ( C r e jJ ), one can calculate the real and imaginary parts from: a r cosJ b r sin J This result is VERY IMPORTANT. Depending on the application, one would want to use one or the other form to simplify one’s calculations. Some Math Addition and Subtraction: For example, if one wants to add (or subtract) two complex numbers, then the preferred form would be the Cartesian: Given: C1 a1 j b1 C1 C 2 a1 a 2 j b1 b2 C 2 a 2 j b2 However, if one wants to multiply or divide two complex numbers, the Cartesian form proves cumbersome. Multiplication and Division: If one wants to multiply two complex numbers, one need only remember the formula for multiplying exponents: e x e y e x y Let’s see why this simplifies complex number multiplication: Given: C1 r1 e jJ1 C1 C2 r1 r2 e j J1 J2 jJ2 C2 r2 e This yields the additional result that: Mag C1 C2 r1 r2 PhC1 C2 J1 J2 Similarly, for division, one need only remember that: ex e x y y e to see that given: C1 r1 e jJ1 C1 r1 j J1 J2 e C2 r2 e jJ2 C2 r2 with the additional result that: C r Mag 1 1 C 2 r2 C Ph 1 J1 J2 C2 These basic principles of complex numbers math are essential for this class. Review: Complex numbers: Cartesian form: Real part: Imaginary part: Magnitude: Phase: Polar form: Real part: Imaginary part: Magnitude: Phase: C a j b ReC a ImC b Mag C a 2 b 2 b PhC arctan a C r e jJ ReC a r cosJ ImC b r sin J Mag (C ) r Ph(C ) J Addition and Subtraction: C1 C2 a1 a2 j b1 b2 Multiplication: New magnitude: New phase: C1 C2 r1 r2 e j J1 J2 Mag C1 C2 r1 r2 PhC1 C2 J1 J2 Division: New magnitude: New phase: C1 r1 j J1 J2 e C2 r2 C r Mag 1 1 C 2 r2 C Ph 1 J1 J2 C2