1. 1 EENG224 Eeng224 Circuit II, Course Information  Instructor: Huseyin Bilgekul, Room No: EE 207, Office Tel: 630 1333  Office Hours: Monday 10.30-12.30, Wednesday 8:30-10:30 (Any time that I am present in the office)  Course Webpage: http://www.ee.emu.edu.tr/eeng224 http://www.ee.emu.edu.tr/eeng224  Lab Assistant: Sevki Kandulu  Textbook: C. K. Alexander and M. N. O. Sadiku, Electric Circuits, 3rd Edition, McGraw-Hill.  Grading: Midterm 1 Exam: % 20 Midterm 2 Exam: % 20 Final Examination : % 30 HW & Quizzes : % 15 Lab Work : % 15  Prerequisite: EENG223 Circuit Theory I  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.  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

2. 2 EENG224 Chapter 9 Sinusoids and Phasors Huseyin Bilgekul EENG224 Circuit Theory II Department of Electrical and Electronic Engineering Eastern Mediterranean University 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.

3. 3 EENG224 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 (v 1, v 2 ).  Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value, denoted by the uppercase letters V m.  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  The sinusoidal waveform is the only alternating waveform whose shape is unaffected by the response characteristics of R, L, and C elements. T

4. 4 EENG224 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.  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:  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.

5. 5 EENG224 Sinusoids  The basic mathematical format for the sinusoidal waveform is: V m sinα  V m 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:

6. 6 EENG224  Sketch of V m sin  t. 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. (a) As a function of  t. (b) As a function of t. V m is the AMPLITUDE of the sinusoid.  is the ANGULAR FREQUENCY in radians/s. f is the FREQUENCY in Hertz. T is the period in seconds. T Period

7. 7 EENG224 Phase of Sinusoids  A periodic function is one that satisfies v(t) = v(t + nT), for all t and for all integers n.  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.

8. 8 EENG224 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:

9. 9 EENG224 Phase of Sinusoids  Consider the sinusoidal voltage having phase φ, v 2 LEADS v 1 by phase φ. v 1 LAGS v 2 by phase φ. v 1 and v 2 are out of phase.

10. 10 EENG224 (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.

11. 11 EENG224 Trigonometric Identities  Sine and cosine form conversions. Graphically relating sine and cosine functions.

12. 12 EENG224 Figure shows a pair of waveforms v 1 and v 2 on an oscilloscope. Each major vertical division represents 20 V and each major division on the horizontal (time) scale represents 20 ms. Voltage v 1 leads. Prepare a phasor diagram using v 1 as reference. Determine equations for both voltages.

13. 13 EENG224 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.