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EGR 1101 Unit 5 Complex Numbers in Engineering (Chapter 5 of Rattan/Klingbeil text) Mathematical Review: Complex Numbers The system of complex numbers is based on the so-called imaginary unit, which is equal to the square root of 1. Mathematicians use the symbol i for this number, while electrical engineers use j: or i 1 j 1 Two Uses of i and j Don’t confuse this use of i and j with the use of iˆ and ˆ j as unit vectors in the xand y-directions (from previous week). A Unique Property of j j is the only number whose reciprocal is equal to its negation: 1 j j Therefore, for example, 1 j3 j 3 Rectangular versus Polar Form Just as vectors can be expressed in component form or polar form, complex numbers can be expressed in rectangular form or polar form. Rectangular Form In rectangular form, a complex number z is written as the sum of a real part a and an imaginary part b: z = a + ib or z = a + jb The Complex Plane We often represent complex numbers as points in the complex plane, with the real part plotted along the horizontal axis (or “real axis”) and the imaginary part plotted along the vertical axis (or “imaginary axis”). Polar Form In polar form, a complex number z is written as a magnitude |z| at an angle : z = |z| The angle is measured from the positive real axis. Converting from Rectangular Form to Polar Form Given a complex number z with real part a and imaginary part b, its magnitude is given by z a b 2 2 and its angle is given by b tan a 1 Converting from Polar Form to Rectangular Form Given a complex number z with magnitude |z| and angle , its real part is given by a z cos and its imaginary part is given by b z sin Exponential Form Complex numbers may also be written in exponential form. Think of this as a mathematically respectable version of polar form. Polar form |z| Example: 3/6 Exponential Form |z|ej 3ej/6 In exponential form, should be in radians. Euler’s Identity The exponential form is based on Euler’s identity, which says that, for any , e j cos j sin Mathematical Operations We’ll need to know how to perform the following operations on complex numbers: Addition Subtraction Multiplication Division Complex Conjugate Addition Adding complex numbers is easiest if the numbers are in rectangular form. Suppose z1 = a1+jb1 and z2 = a2+jb2 Then z1 + z2 = (a1+a2) + j(b1+b2) In words: to add two complex numbers in rectangular form, add their real parts to get the real part of the sum, and add their imaginary parts to get the imaginary part of the sum. Subtraction Subtracting complex numbers is also easiest if the numbers are in rectangular form. Suppose z1 = a1+jb1 and z2 = a2+jb2 Then z1 z2 = (a1a2) + j(b1b2) In words: to subtract two complex numbers in rectangular form, subtract their real parts to get the real part of the result, and subtract their imaginary parts to get the imaginary part of the result. Multiplication Multiplying complex numbers is easiest if the numbers are in polar form. Suppose z1 = |z1| 1 and z2 = |z2| 2 Then z1 z2 = (|z1||z2|) (1+ 2) In words: to multiply two complex numbers in polar form, multiply their magnitudes to get the magnitude of the result, and add their angles to get the angle of the result. Division Dividing complex numbers is also easiest if the numbers are in polar form. Suppose z1 = |z1| 1 and z2 = |z2| 2 Then z1 ÷ z2 = (|z1|÷|z2|) (1 2) In words: to divide two complex numbers in polar form, divide their magnitudes to get the magnitude of the result, and subtract their angles to get the angle of the result. Complex Conjugate Given a complex number in rectangular form, z = a + ib its complex conjugate is simply z* = a ib Given a complex number in polar form, z = |z| its complex conjugate is simply z* = |z| Entering Complex Numbers in MATLAB Entering a number in rectangular form: >>z1 = 2+i3 Entering a number in polar (actually, exponential) form: >>z3 = 5exp(ipi/6) You must give the angle in radians, not degrees. Operating on Complex Numbers in MATLAB Use the usual mathematical operators for addition, subtraction, multiplication, division: >>z5 = z1+z2 >>z6 = z1*z2 and so on. Built-In Complex Functions in MATLAB Useful MATLAB functions: real() gives a number’s real part imag() gives a number’s imaginary part abs() gives a number’s magnitude angle() gives a number’s angle conj() gives a number’s complex conjugate This Week’s Examples 1. 2. 3. 4. 5. Impedance of an inductor Impedance of a capacitor Total impedance of a series RLC circuit Current in a series RL circuit Voltage in a series RL circuit Review: Resistors A resistor has a constant resistance (R), measured in ohms (Ω). Review: Inductors An inductor has a constant inductance (L), measured in henries (H). It also has a variable inductive reactance (XL), measured in ohms. We’ll see in a minute how to compute XL. A New Electrical Component: The Capacitor A capacitor has a constant capacitance (C), measured in farads (F). It also has a variable capacitive reactance (XC), measured in ohms. Review: Impedance Resistance (R) and reactance (X) are special cases of a quantity called impedance (Z), also measured in ohms. Impedance (Z) Resistance (R) Reactance (X) Inductive Reactance (XL) Capacitive Reactance (XC) Reactance Depends on Frequency A resistor’s resistance is a constant and does not change. But an inductor’s reactance or a capacitor’s reactance depends on the frequency of the current that’s passing through it. Formulas for Reactance For inductance L and frequency f, inductive reactance XL is given by: XL = 2fL For capacitance C and frequency f, capacitive reactance XC is given by: XC = 1 (2fC) As frequency increases, inductive reactance increases, but capacitive reactance decreases. Frequency & Angular Frequency Two common ways of specifying a frequency: Frequency f, measured in hertz (Hz); also called “cycles per second”. Angular frequency , measured in radians per second (rad/s). They’re related by the following: = 2f Formulas for Reactance (Again) Using = 2f, we can rewrite the earlier formulas for reactance. For inductance L and frequency f, inductive reactance XL is given by: XL = 2fL = L For capacitance C and frequency f, capacitive reactance XC is given by: XC = 1 (2fC) = 1 (C) Total Impedance To find total impedance of combined resistances and reactances, treat them as complex numbers (or as vectors). Resistance is positive real (angle = 0). ZR = R Inductive reactance is positive imaginary (angle = +90). ZL = j XL = j 2fL = j L Capacitive reactance is negative imaginary (angle = −90). ZC = −j XC = −j (2fC) = −j (C)