ELECTRICAL CIRCUIT ET 201 Define and explain characteristics of sinusoidal wave, phase relationships and phase shifting

SINUSOIDAL ALTERNATING WAVEFORMS (CHAPTER 1.1 ~ 1.4)

Understand Alternating Current DIRECT CURRENT (DC) – IS WHEN THE CURRENT FLOWS IN ONLY ONE DIRECTION. Constant flow of electric charge EX: BATTERY ALTERNATING CURRENT AC) – THE CURRENT FLOWS IN ONE DIRECTION THEN THE OTHER. Electrical current whose magnitude and direction vary cyclically, as opposed to direct current whose direction remains constant. EX: OUTLETS

Sources of alternating current By rotating a magnetic field within a stationary coil By rotating a coil in a magnetic field

Generation of Alternating Current A voltage supplied by a battery or other DC source has a certain polarity and remains constant. Alternating Current (AC) varies in polarity and amplitude. AC is an important part of electrical and electronic systems.

Faraday’s and Lenz’s Law involved in generating a.c current Faraday’s Laws of electromagnetic Induction.   Induced electromotive field Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. e.m.f, e = -N d N = Number of turn dt  = Magnetic Flux Lenz’s law An electromagnetic field interacting with a conductor will generate electrical current that induces a counter magnetic field that opposes the magnetic field generating the current.

Sine Wave Characteristics The basis of an AC alternator is a loop of wire rotated in a magnetic field. Slip rings and brushes make continuous electrical connections to the rotating conductor. The magnitude and polarity of the generated voltage is shown on the following slide.

Sine Wave Characteristics

Sine Wave Characteristics The sine wave at the right consists of two, opposite polarity, alternations. Each alternation is called a half cycle. Each half cycle has a maximum value called the peak value.

Sine Wave Characteristics Sine waves may represent voltage, current, or some other parameter. The period of a sine wave is the time from any given point on the cycle to the same point on the following cycle. The period is measured in time (t), and in most cases is measured in seconds or fractions thereof.

Frequency The frequency of a sine wave is the number of complete cycles that occur in one second. Frequency is measured in hertz (Hz). One hertz corresponds to one cycle per second. Frequency and period have an inverse relationship. t = 1/f, and f = 1/t. Frequency-to-period and period-to-frequency conversions are common in electronic calculations.

Peak Value The peak value of a sine wave is the maximum voltage (or current) it reaches. Peak voltages occur at two different points in the cycle. One peak is positive, the other is negative. The positive peak occurs at 90º and the negative peak at 270º. The positive and negative have equal amplitudes.

Average Values The average value of any measured quantity is the sum of all of the intermediate values. The average value of a full sine wave is zero. The average value of one-half cycle of a sine wave is: Vavg = 0.637Vp or Iavg = 0.637Ip Chapter 6 - Alternating Current

rms Value One of the most important characteristics of a sine wave is its rms or effective value. The rms value describes the sine wave in terms of an equivalent dc voltage. The rms value of a sine wave produces the same heating effect in a resistance as an equal value of dc. The abbreviation rms stands for root-mean-square, and is determined by: Vrms = 0.707Vp or Irms = 0.707Ip Chapter 6 - Alternating Current

Peak-to-Peak Value Another measurement used to describe sine waves are their peak-to-peak values. The peak-to-peak value is the difference between the two peak values.

Form Factor Form Factor is defined as the ratio of r.m.s value to the average value. Form factor = r.m.s value = 0.707  peak value average value 0.637  peak valur = 1.11

Peak Factor Crest or Peak or Amplitude Factor Peak factor is defined as the ratio of peak voltage to r.m.s value.

13.1 Introduction Alternating waveforms Alternating signal is a signal that varies with respect to time. Alternating signal can be categories into ac voltage and ac current. This voltage and current have positive and negative value.

13.2 Sinusoidal AC Voltage Characteristics and Definitions Voltage and current value is represent by vertical axis and time represent by horizontal axis. In the first half, current or voltage will increase into maximum positive value and come back to zero. Then in second half, current or voltage will increase into negative maximum voltage and come back to zero. One complete waveform is called one cycle. volts or amperes units of time

13.2 Sinusoidal AC Voltage Characteristics and Definitions Defined Polarities and Direction The voltage polarity and current direction will be for an instant in time in the positive portion of the sinusoidal waveform. In the figure, a lowercase letter is employed for polarity and current direction to indicate that the quantity is time dependent; that is, its magnitude will change with time. 20

13.2 Sinusoidal AC Voltage Characteristics and Definitions Defined Polarities and Direction For a period of time, a voltage has one polarity, while for the next equal period it reverses. A positive sign is applied if the voltage is above the axis. For a current source, the direction in the symbol corresponds with the positive region of the waveform. 21

13.2 Sinusoidal AC Voltage Characteristics and Definitions There are several specification in sinusoidal waveform: 1. period 2. frequency 3. instantaneous value 4. peak value 5. peak to peak value 6. angular velocity 7. average value 8. effective value

13.2 Sinusoidal AC Voltage Characteristics and Definitions Period (T) Period is defines as the amount of time is take to go through one cycle. Period for sinusoidal waveform is equal for each cycle. Cycle The portion of a waveform contained in one period of time. Frequency (f) Frequency is defines as number of cycles in one seconds. It can derives as f = Hz T = seconds (s)

13.2 Sinusoidal AC Voltage Characteristics and Definitions The cycles within T1, T2, and T3 may appear different in the figure above, but they are all bounded by one period of time and therefore satisfy the definition of a cycle.

13.2 Sinusoidal AC Voltage Characteristics and Definitions Signal with lower frequency Signal with higher frequency Frequency = 1 cycle per second Frequency = 21/2 cycles per second Frequency = 2 cycles per second 1 hertz (Hz) = 1 cycle per second (cps)

13.2 Sinusoidal AC Voltage Characteristics and Definitions Instantaneous value Instantaneous value is magnitude value of waveform at one specific time. Symbol for instantaneous value of voltage is v(t) and current is i(t).

13.2 Sinusoidal AC Voltage Characteristics and Definitions Peak Value The maximum instantaneous value of a function as measured from zero-volt level. For one complete cycle, there are two peak value that is positive peak value and negative peak value. Symbol for peak value of voltage is Em or Vm and current is Im . Peak value, Vm = 8 V

13.2 Sinusoidal AC Voltage Characteristics and Definitions Peak to peak value The full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks. Symbol for peak to peak value of voltage is Ep-p or Vp-p and current is Ip-p Peak to peak value, Vp-p = 16 V

13.2 Sinusoidal AC Voltage Characteristics and Definitions Angular velocity Angular velocity is the velocity with which the radius vector rotates about the center. Symbol of angular speed is and units is radians/seconds (rad/s) Horizontal axis of waveform can be represent by time and angular speed.

13.2 Sinusoidal AC Voltage Characteristics and Definitions Angular velocity Degree Radian 90° (π/180°) x ( 90°) = π/2 rad 60° (π/180°) x ( 60°) = π/3 rad 30° (π/180°) x (30°) = π/6 rad Radian Degree π /3 (180° /π) x (π /3) = 60° π (180° /π) x (π ) = 180° 3π /2 (180°/π) x (3π /2) = 270°

13.2 Sinusoidal AC Voltage Characteristics and Definitions Plotting a sine wave versus (a) degrees and (b) radians.

13.2 Sinusoidal AC Voltage Characteristics and Definitions 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. Waveform picture with respect to angular velocity

13.2 Sinusoidal AC Voltage Characteristics and Definitions Angular velocity Formula of angular velocity Since (ω) is typically provided in radians/second, the angle ϴ obtained using ϴ = ωt is usually in radians.

13.2 Sinusoidal AC Voltage Characteristics and Definitions Angular velocity The time required to complete one cycle is equal to the period (T) of the sinusoidal waveform. One cycle in radian is equal to 2π (360o). (rad/s) or

13.2 Sinusoidal AC Voltage Characteristics and Definitions Angular velocity Demonstrating the effect of  on the frequency f and period T.

13.2 Sinusoidal AC Voltage Characteristics and Definitions Example 13.6 Given  = 200 rad/s, determine how long it will take the sinusoidal waveform to pass through an angle of 90 Solution

13.2 Sinusoidal AC Voltage Characteristics and Definitions Example 13.7 Find the angle through which a sinusoidal waveform of 60 Hz will pass in a period of 5 ms. Solution

13.2 Sinusoidal AC Voltage Characteristics and Definitions Average value Average value is average value for all instantaneous value in half or one complete waveform cycle. It can be calculate in two ways: Calculate the area under the graph: Average value = area under the function in a period period 2. Use integral method For a symmetry waveform, area upper section equal to area under the section, so just take half of the period only.

13.2 Sinusoidal AC Voltage Characteristics and Definitions Average value Example: Calculate the average value of the waveform below. Solution: For a sinus waveform , average value can be calculate by

13.2 Sinusoidal AC Voltage Characteristics and Definitions Effective value The most common method of specifying the amount of sine wave of voltage or current by relating it into dc voltage and current that will produce the same heat effect. Effective value is the equivalent dc value of a sinusoidal current or voltage, which is 1/√2 or 0.707 of its peak value. The equivalent dc value is called rms value or effective value. The formula of effective value for sine wave waveform is;

13.2 Sinusoidal AC Voltage Characteristics and Definitions Example 13.21 The 120 V dc source delivers 3.6 W to the load. Find Em and Im of the ac source, if the same power is to be delivered to the load.

13.2 Sinusoidal AC Voltage Characteristics and Definitions Example 13.21 – solution and

13.2 Sinusoidal AC Voltage Characteristics and Definitions Example 13.21 – solution

Basic sine wave for current or voltage 13.5 General Format for the Sinusoidal Voltage or Current The basic mathematical format for the sinusoidal waveform is: where: Am : peak value of the waveform  : angle from the horizontal axis volts or amperes Basic sine wave for current or voltage

13.5 General Format for the Sinusoidal Voltage or Current The general format of a sine wave can also be as: General format for electrical quantities such as current and voltage is: where: and is the peak value of current and voltage while i(t) and v(t) is the instantaneous value of current and voltage. α= ωt

13.5 General Format for the Sinusoidal Voltage or Current Example 13.8 Given e(t) = 5 sin, determine e(t) at  = 40 and  = 0.8. Solution For  = 40, For  = 0.8

13.5 General Format for the Sinusoidal Voltage or Current Example 13.9 (a) Determine the angle at which the magnitude of the sinusoidal function v(t) = 10 sin 377t is 4 V. Determine the time at which the magnitude is attained.

13.5 General Format for the Sinusoidal Voltage or Current Example 13.9 - solution Hence, When v(t) = 4 V,

13.5 General Format for the Sinusoidal Voltage or Current Example 13.9 – solution (cont’d) (a) But α is in radian, so α must be calculate in radian: (b) Given, , so

13.6 Phase Relationship Phase angle Phase angle is a shifted angle waveform from reference origin. Phase angle is been represent by symbol θ or Φ Units is degree ° or radian Two waveform is called in phase if its have a same phase degree or different phase is zero Two waveform is called out of phase if its have a different phase.

13.6 Phase Relationship The unshifted sinusoidal waveform is represented by the expression:

13.6 Phase Relationship Sinusoidal waveform which is shifted to the right or left of 0° is represented by the expression: where  is the angle (in degrees or radians) that the waveform has been shifted.

13.6 Phase Relationship If the wave form passes through the horizontal axis with a positive-going (increasing with the time) slope before 0°:

13.6 Phase Relationship If the waveform passes through the horizontal axis with a positive-going slope after 0°:

13.6 Phase Relationship

13.6 Phase Relationship The terms leading and lagging are used to indicate the relationship between two sinusoidal waveforms of the same frequency f （or angular velocity ω) 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.

13.6 Phase Relationship Note: sin (- α) = - sin α cos(- α) = cos α cos (α-90o) sin (α+90o) Note: sin (- α) = - sin α cos(- α) = cos α Start at + sin α position;

13.6 Phase Relationship If a sinusoidal expression should appear as the negative sign is associated with the sine portion of the expression, not the peak value Em , i.e. And, since;

13.6 Phase Relationship Example 13.2 Determine the phase relationship between the following waveforms

13.6 Phase Relationship i leads v by 40 v lags i by 40 Example 13.2 – solution i leads v by 40 or v lags i by 40

13.6 Phase Relationship i leads v by 80 v lags i by 80 Example 13.2 – solution (cont’d) i leads v by 80 or v lags i by 80

13.6 Phase Relationship i leads v by 110 v lags i by 110 Example 13.2 – solution (cont’d) i leads v by 110 or v lags i by 110

13.6 Phase Relationship Example 13.2 – solution (cont’d) OR v leads i by 160 Or i lags v by 160 i leads v by 200 Or v lags i by 200