Download Trigonometric Functions Applied to AC Circuits

Trigonometric Functions Applied
to AC Circuits
Adapted material to accompany
Lesson 2315-0
IDENTIFYING THE SIDES OF A RIGHT
TRIANGLE
• The sides of a right triangle are named the opposite side,
adjacent side, and hypotenuse
• The hypotenuse is the longest side of a right triangle and
is always opposite the right angle
• The positions of the opposite and adjacent sides depend
on the reference angle
• The opposite side is opposite the reference angle
• The adjacent side is next to the reference angle
2
EXAMPLES OF IDENTIFYING SIDES
• The two right triangles below each have their sides labeled
according to a given reference angle
Adjacent
•
A
Adjacent
B
Opposite
Note: The opposite and adjacent sides vary depending on the reference angle
while the hypotenuse stays in the same position in both cases.
3
TRIGONOMETRIC FUNCTIONS
• Three trigonometric functions are defined in the table below
FUNCTION SYMBOL
•
DEFINITION
sine θ
sin θ
opposite side
hypotenuse
cosine θ
cos θ
adjacent side
hypotenuse
tangent θ
tan θ
opposite side
adjacent side
Note: The symbol  denotes the reference angle
4
RATIO EXAMPLE
• The sides of the triangle below are labeled with different
letters, then each of the listed trigonometric functions are
given using 1 as the reference angle
a
sin 1 =
c
c
a
1
b
b
cos 1 =
c
a
tan 1 =
b
5
DEFINITIONS
• A vector is a quantity that has both magnitude and
direction
• Vectors are shown as directed line segments. The length
of the segment represents the magnitude and the
arrowhead represents the direction of the quantity
• Vectors have an initial point and a terminal point. An
arrowhead represents the terminal point
• A vector is named by its two end points or by a single
lowercase letter. Arrows are placed above the line
segments
6
DEFINITIONS
• Equal vectors have identical magnitudes and directions. A
vector can be repositioned provided its magnitude and
direction remain the same
• Vectors are usually shown on the rectangular coordinate (x,y)
system. A vector is in standard position when the initial point
is at the origin of the rectangular coordinate (x,y) system and
its angle is measured counterclockwise from the positive xaxis
7
EXAMPLES
• The vector shown below could be named either
AB or f .
B
A
f
• Vectors a and b below are not equal vectors because
they do not have identical magnitudes and directions.
a
b
8
VECTOR NOTATION
• A vector can be represented on the
rectangular coordinate system using ordered
pair notation (x,y), or vectors can be shown
and solved as lengths and angles. Note: Vector
angles are often represented as  (theta)
• The vector shown on the following slide could
either be read as 3x + 4y [(3,4)cm] in the
rectangular coordinate system or as 5cm at
53.1° ( 32 + 42 = 5)
9
VECTOR NOTATION
10
Circuit for example problem
Problem, part 1:
• In the previous figure,
how much current is
flowing
a) Through the 30Ω
resistor?
b) Through the 40Ω
capacitive reactance,
XC?
c) To and from the
terminals of the
applied voltage, VT?
Answer:
Given that this is a series
circuit, all answers should
be 2A regardless of the
component used. The one
thing we should be able to
count on is the current is
the same through all
components.
Problem, part 2:
• In the previous figure,
what is the phase
relationship between
a) I and VR?
b) I and VC?
c) VR and VC?
Answers:
a) Should be zero degrees
as resistor is passive
component
b) VC should lag by 90
degrees
c) VC should lag by 90
degrees
Problem, part 3:
• In the previous figure,
how much is the
applied voltage, VT?
• How much is the total
impedance, ZT?
• Answers:
VT = 602 + 802 =
3600 + 6400 =
10000 = 100 volts
ZT = 302 + 402 =
900 + 1600 = 2500
= 50 ohms
Problem, part 4:
• Draw the phasor voltage triangle for the
previous figure (use VR as the reference
phasor; show two ways).
Vector representations
VR
VC
VT
VR
VC
VT