Presentation on theme: "Complex Numbers in Engineering (Chapter 5 of Rattan/Klingbeil text)"— Presentation transcript:
1Complex Numbers in Engineering (Chapter 5 of Rattan/Klingbeil text) EGR 1101 Unit 5Complex Numbers in Engineering(Chapter 5 of Rattan/Klingbeil text)
2Mathematical 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:orCheck whether your calculator recognizes sqrt of -1.
3Two Uses of i and jDon’t confuse this use of i and j with the use of and as unit vectors in the x- and y-directions (from previous week).
4A Unique Property of jj is the only number whose reciprocal is equal to its negation:Therefore, for example,
5Rectangular 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.
6Rectangular FormIn 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 + jbEx: z = 5 + j2
7The Complex PlaneWe 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”).Plot the previous number (z = 5 + j2)
8Polar FormIn 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.
9Converting from Rectangular Form to Polar Form Given a complex number z with real part a and imaginary part b, its magnitude is given by and its angle is given byConverting between the two forms is exactly the same as for vectors!Do the previous number (z = 5 + j2)
10Converting from Polar Form to Rectangular Form Given a complex number z with magnitude |z| and angle , its real part is given by and its imaginary part is given byConverting between the two forms is exactly the same as for vectors!
11Exponential Form |z| |z|ej 3/6 3ej/6 Complex numbers may also be written in exponential form. Think of this as a mathematically respectable version of polar form.In exponential form, should be in radians.Polar formExponential Form|z||z|ejOne reason this is important: to enter numbers in polar form in MATLAB, you actually have to enter them in exponential form.Continuing example, write z = 5 + j2 in exponential form.Example:3/63ej/6
12Euler’s IdentityThe exponential form is based on Euler’s identity, which says that, for any ,
13Mathematical Operations We’ll need to know how to perform the following operations on complex numbers:AdditionSubtractionMultiplicationDivisionComplex Conjugate
14AdditionAdding 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.
15SubtractionSubtracting 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.
16MultiplicationMultiplying 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.
17DivisionDividing 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.
18Complex ConjugateGiven a complex number in rectangular form, z = a + ib its complex conjugate is simply z* = a ibGiven a complex number in polar form, z = |z| its complex conjugate is simply z* = |z|
19Entering Complex Numbers in MATLAB Entering a number in rectangular form: >>z1 = 2+i3Entering a number in polar (actually, exponential) form: >>z3 = 5exp(ipi/6)You must give the angle in radians, not degrees.
20Operating on Complex Numbers in MATLAB Use the usual mathematical operators for addition, subtraction, multiplication, division: >>z5 = z1+z2>>z6 = z1*z2 and so on.
21Built-In Complex Functions in MATLAB Useful MATLAB functions:real() gives a number’s real partimag() gives a number’s imaginary partabs() gives a number’s magnitudeangle() gives a number’s angleconj() gives a number’s complex conjugate
22This Week’s Examples Impedance of an inductor Impedance of a capacitor Total impedance of a series RLC circuitCurrent in a series RL circuitVoltage in a series RL circuit
23Review: ResistorsA resistor has a constant resistance (R), measured in ohms (Ω).
24Review: InductorsAn 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.
25A 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.Build the first four columns of a table showing component, abbreviation, unit (& abbrev), and schematic symbol.
26Review: Impedance Impedance (Z) 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)
27Reactance 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.Start fifth column (formula for impedance) in table.
28Formulas for Reactance For inductance L and frequency f, inductive reactance XL is given by:XL = 2fLFor capacitance C and frequency f, capacitive reactance XC is given by:XC = 1 (2fC)As frequency increases, inductive reactance increases, but capacitive reactance decreases.Start fifth column (formula for impedance) in table.
29Frequency & 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: = 2fComplete fifth column (formula for impedance) in table.
30Formulas 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 = LFor capacitance C and frequency f, capacitive reactance XC is given by:XC = 1 (2fC) = 1 (C)Start fifth column (formula for impedance) in table.
31Total ImpedanceTo find total impedance of combined resistances and reactances, treat them as complex numbers (or as vectors).Resistance is positive real (angle = 0) ZR = RInductive reactance is positive imaginary (angle = +90) ZL = j XL = j 2fL = j LCapacitive reactance is negative imaginary (angle = −90) ZC = −j XC = −j (2fC) = −j (C)Add a sixth column (position in complex plane) to table.