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EGR 1101 Unit 5 Complex Numbers in Engineering (Chapter 5 of Rattan/Klingbeil text)

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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

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Two Uses of i and j Dont confuse this use of i and j with the use of and as unit vectors in the x- and y-directions (from previous week).

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A Unique Property of j j is the only number whose reciprocal is equal to its negation: Therefore, for example,

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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.

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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

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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).

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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.

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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 and its angle is given by

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Converting 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 by

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Example: 3 /6 3e j /6 Exponential Form 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|e j

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Eulers Identity The exponential form is based on Eulers identity, which says that, for any,

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Mathematical Operations Well need to know how to perform the following operations on complex numbers: Addition Subtraction Multiplication Division Complex Conjugate

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Addition Adding complex numbers is easiest if the numbers are in rectangular form. Suppose z 1 = a 1 +jb 1 and z 2 = a 2 +jb 2 Then z 1 + z 2 = (a 1 +a 2 ) + j(b 1 +b 2 ) 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.

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Subtraction Subtracting complex numbers is also easiest if the numbers are in rectangular form. Suppose z 1 = a 1 +jb 1 and z 2 = a 2 +jb 2 Then z 1 z 2 = (a 1 a 2 ) + j(b 1 b 2 ) 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.

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Multiplication Multiplying complex numbers is easiest if the numbers are in polar form. Suppose z 1 = |z 1 | 1 and z 2 = |z 2 | 2 Then z 1 z 2 = (|z 1 | |z 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.

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Division Dividing complex numbers is also easiest if the numbers are in polar form. Suppose z 1 = |z 1 | 1 and z 2 = |z 2 | 2 Then z 1 ÷ z 2 = (|z 1 |÷|z 2 |) ( 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.

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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|

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Entering Complex Numbers in MATLAB Entering a number in rectangular form: >>z1 = 2+i 3 Entering a number in polar (actually, exponential) form: >>z3 = 5 exp(i pi/6) You must give the angle in radians, not degrees.

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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.

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Built-In Complex Functions in MATLAB Useful MATLAB functions: real() gives a numbers real part imag() gives a numbers imaginary part abs() gives a numbers magnitude angle() gives a numbers angle conj() gives a numbers complex conjugate

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This Weeks Examples 1. Impedance of an inductor 2. Impedance of a capacitor 3. Total impedance of a series RLC circuit 4. Current in a series RL circuit 5. Voltage in a series RL circuit

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Review: Resistors A resistor has a constant resistance (R), measured in ohms ().

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Review: Inductors An inductor has a constant inductance (L), measured in henries (H). It also has a variable inductive reactance (X L ), measured in ohms. Well see in a minute how to compute X L.

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A New Electrical Component: The Capacitor A capacitor has a constant capacitance (C), measured in farads (F). It also has a variable capacitive reactance (X C ), measured in ohms.

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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 (X L )Capacitive Reactance (X C )

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Reactance Depends on Frequency A resistors resistance is a constant and does not change. But an inductors reactance or a capacitors reactance depends on the frequency of the current thats passing through it.

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Formulas for Reactance For inductance L and frequency f, inductive reactance X L is given by: X L = 2 fL For capacitance C and frequency f, capacitive reactance X C is given by: X C = 1 (2 fC) As frequency increases, inductive reactance increases, but capacitive reactance decreases.

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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). Theyre related by the following: = 2 f

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Formulas for Reactance (Again) Using = 2 f, we can rewrite the earlier formulas for reactance. For inductance L and frequency f, inductive reactance X L is given by: X L = 2 fL = L For capacitance C and frequency f, capacitive reactance X C is given by: X C = 1 (2 fC) = 1 ( C)

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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 ). Z R = R Inductive reactance is positive imaginary (angle = +90 ). Z L = j X L = j 2 fL = j L Capacitive reactance is negative imaginary (angle = 90 ). Z C = j X C = j (2 fC) = j ( C)

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