Transcript Unit 5 PowerPoint Slides

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 = (a1a2) + j(b1b2)

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
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
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
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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.
Operating on Complex Numbers in
MATLAB
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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:
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




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
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An inductor has a constant inductance
(L), measured in henries (H).
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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
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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
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For inductance L and frequency f,
inductive reactance XL is given by:
XL = 2fL
For capacitance C and frequency f,
capacitive reactance XC is given by:
XC = 1  (2fC)
As frequency increases, inductive
reactance increases, but capacitive
reactance decreases.
Frequency & Angular Frequency
Two common ways of specifying a
frequency:
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


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:
 = 2f
Formulas for Reactance (Again)
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
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Using  = 2f, we can rewrite the
earlier formulas for reactance.
For inductance L and frequency f,
inductive reactance XL is given by:
XL = 2fL = L
For capacitance C and frequency f,
capacitive reactance XC is given by:
XC = 1  (2fC) = 1  (C)
Total Impedance
To find total impedance of combined
resistances and reactances, treat them
as complex numbers (or as vectors).
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

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Resistance is positive real (angle = 0).
ZR = R
Inductive reactance is positive imaginary
(angle = +90).
ZL = j XL = j 2fL = j L
Capacitive reactance is negative imaginary
(angle = −90).
ZC = −j XC = −j  (2fC) = −j  (C)