Transcript No Slide Title - faraday - Eastern Mediterranean University

Eeng224 Circuit II, Course Information
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Instructor: Huseyin Bilgekul, Room No: EE 207, Office Tel: 630 1333
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Office Hours: Monday 10.30-12.30, Wednesday 8:30-10:30 (Any time that I am present in the office)
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Course Webpage: http://www.ee.emu.edu.tr/eeng224
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Lab Assistant: Sevki Kandulu
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Textbook: C. K. Alexander and M. N. O. Sadiku, Electric Circuits, 3rd Edition, McGraw-Hill.
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Grading: Midterm 1 Exam: % 20 Midterm 2 Exam: % 20
Final Examination : % 30
HW & Quizzes
: % 15
Lab Work
: % 15
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Prerequisite: EENG223 Circuit Theory I
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NG Policy: NG grade will be given to students who do not attend more than 50% of the course lecture
hours, miss the exams and fail.
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Makeup Exams: Makeup exams will NOT be granted to students with less than 50% attendance.
Huseyin Bilgekul
EENG224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
EENG224
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Chapter 9
Sinusoids and Phasors
Chapter Objectives:
 Understand the concepts of sinusoids and phasors.
 Apply phasors to circuit elements.
 Introduce the concepts of impedance and admittance.
 Learn about impedance combinations.
 Apply what is learnt to phase-shifters and AC
bridges.
Huseyin Bilgekul
EENG224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
EENG224
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Alternating (AC) Waveforms
 The term alternating indicates only that the waveform alternates between two prescribed levels in a set
time sequence.
 Instantaneous value: The magnitude of a waveform at any instant of time; denoted by the lowercase
letters (v1, v2).
 Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value,
denoted by the uppercase letters Vm.
 Period (T): The time interval between successive repetitions of a periodic waveform.
 Cycle: The portion of a waveform contained in one period of time.
 Frequency: (Hertz) the number of cycles that occur in 1 s
f 1
T
 The
sinusoidal waveform is the only alternating waveform whose shape is
unaffected by the response characteristics of R, L, and C elements.
T
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Sinusoids
 The sinusoidal wave form can be derived from the length of the vertical projection of a radius vector
rotating in a uniform circular motion about a fixed point.
Vm sin 
Vm cos 
 The velocity with which the radius vector rotates about the center, called the angular velocity, can be
determined from the following equation:
 The angular velocity () is:    t
 Since () is typically provided in radians per second, the angle α obtained using α = t is usually in
radians.
 The time required to complete one revolution is equal to the period (T) of the sinusoidal waveform. The
radians subtended in this time interval are 2π.

2
T
or   2 f
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Sinusoids
 The basic mathematical format for the sinusoidal waveform is:
Vmsinα
 Vm is the peak value of the waveform and α is the unit of measure for the horizontal axis.
 The equation α = t states that the angle α through which the rotating vector will pass is determined by
the angular velocity of the rotating vector and the length of time the vector rotates.
 For a particular angular velocity (fixed ), the longer the radius vector is permitted to rotate (that is, the
greater the value of t ), the greater will be the number of degrees or radians through which the vector will
pass. The general format of a sine wave can also be as:
Vm sin(t )
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Sinusoids
 A SINUSOID is a signal that has the form of the sine or cosine function.
 The sinusoidal current is referred to as AC. Circuits driven by AC sources are referred to as AC Circuits.
 Sketch of Vmsint.
T Period
(a) As a function of t.
(b) As a function of t .
• Vm is the AMPLITUDE of the sinusoid.
•  is the ANGULAR FREQUENCY in radians/s.
• f is the FREQUENCY in Hertz.   2 f
and
f 1
T
• T is the period in seconds.
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Phase of Sinusoids
 A periodic function is one that satisfies v(t) = v(t + nT),
for all t and for all integers n.
f 
1
Hz   2 f
T
 Only two sinusoidal values with the same frequency can be
compared by their amplitude and phase difference.
 If phase difference is zero, they are in phase; if phase difference is
not zero, they are out of phase.
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Phase of Sinusoids
 The terms lead and lag are used to indicate the relationship between two
sinusoidal waveforms of the same frequency plotted on the same set of axes.
 The cosine curve is said to lead the sine curve by 90°.
 The sine curve is said to lag the cosine curve by 90°.
 90 is referred to as the phase angle between the two waveforms.
When determining the phase measurement we first note that each sinusoidal
function has the same frequency, permitting the use of either waveform to determine
the period.
 Since the full period represents a cycle of 360°, the following ratio can be formed:
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Phase of Sinusoids
 Consider the sinusoidal voltage having phase φ,
• v2 LEADS v1 by
v(t )  Vm sin(t   )
phase φ.
• v1 LAGS v2 by phase φ.
• v1 and v2 are out of phase.
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(120 V at 60 Hz) versus (220 V at 50 Hz) AC
 In North and South America the most common available ac supply is 120 V at 60 Hz, while
in Europe and the Eastern countries it is 220 V at 50 Hz.
 Technically there is no noticeable difference between 50 and 60 cycles per second (Hz).
 The effect of frequency on the size of transformers and the role it plays in the generation and
distribution of power was also a factor.
 The fundamental equation for transformer design is that the size of the transformer is
inversely proportional to frequency.
 A 50 HZ transformer must be larger than a 60 Hz (17% larger) sinusoidal voltage having
phase φ.
 Higher frequencies result in concerns about arcing, increased losses in the transformer core
due to eddy current and hysteresis losses, and skin effect phenomena.
 Larger voltages (such as 220 V) raise safety issues beyond those of 120 V.
 Higher voltages result in lower current for the same demand, permitting the use of smaller
conductors.
 Motors and power supplies, found in common home appliances and throughout the
industrial community, can be smaller in size if supplied with a higher voltage.
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Trigonometric Identities
 Sine and cosine form conversions.
sin( A  B )  sin A cos B  cos A sin B
cos( A  B )  cos A cos B sin A sin B
sin(t  180)   sin t
cos(t  180)   cos t
sin(t  90)   cos t
cos(t  90)  sin t
Graphically relating sine
and cosine functions.
cos(t  90)  sin t
A cos t  B sin t  C cos(t   )
Where
B
C= A 2  B 2 and  =tan -1
A
sin(t  180)   sin t
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Figure shows a pair of waveforms v1
and v2 on an oscilloscope. Each
major vertical division represents
20 V and each major
division on the horizontal (time)
scale represents 20 ms. Voltage v1
leads. Prepare a phasor diagram
using v1 as reference. Determine
equations for both voltages.
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EXERCISE
 Voltage and current are out of phase by 40°, and voltage lags. Using
current as the reference, sketch the phasor diagram and the
corresponding waveforms.
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