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Specific Objectives Understand how a sine wave of alternating voltage is generated. Explain the three ways to express the amplitude of a sinusoidal waveform and the relationship between them. Calculate the RMS, average, and peak-to-peak values of a sine wave when the peak value is known. Calculate the instantaneous value of a sine wave. Convert peak, peak-to-peak, average, and RMS voltage and current values from one value to another. Copyright © Texas Education Agency, All rights reserved.

Specific Objectives (Continued)
Explain the importance of the .707 constant and how it is derived. Define frequency and period and list the units of each. Calculate the period when the frequency is known and frequency when period is known. Explain the sine, cosine, and tangent trigonometric functions. Calculate the value of the sine of any angle between 0° and 360°. Copyright © Texas Education Agency, All rights reserved.

Introduction AC and the characteristics of a sinusoidal waveform Time and frequency measurement of a waveform Trigonometric functions This lesson discusses AC and the characteristics of a sinusoidal waveform in detail . The time and frequency measurement of a waveform are reviewed, and an introduction to some trigonometric functions will be presented. Copyright © Texas Education Agency, All rights reserved.

What is AC? Alternating Current (AC) is a useful form of voltage. AC is a flow of electric charge that periodically changes direction. Direct Current (DC) is another useful form of voltage. DC is the unidirectional flow of electric charge. AC and DC are both useful power sources for heating, lights, and motors. Copyright © Texas Education Agency, All rights reserved.

AC vs DC DC can power the same things AC can power. The devices may have to be constructed differently, but they would work the same. AC cannot power some of the things that DC can power. Example: electronic devices AC motors were not invented at the time Thomas Edison built his first DC power station in 1882. This means that DC is actually more useful than AC as a power source. In fact, the first type of commercial electricity used in the U.S. was DC and was championed by Thomas Edison in the late 1800s. However, the advantages of AC soon made it more popular than DC. Copyright © Texas Education Agency, All rights reserved.

Why Use AC Instead of DC? AC is relatively easy to produce. It is created through rotational motion. AC generation can produce large amounts of power economically. AC voltage can be changed from one value to another relatively easily. DC voltage cannot be changed from one value to another easily. Copyright © Texas Education Agency, All rights reserved.

Why Change Voltage? AC voltage can be stepped up to a higher value using a transformer (up to 760 kV). The basic concept is that “power in” equals “power out” of a transformer. Power is voltage times current, so if voltage is made to increase, then current will decrease. After the voltage is increased, AC is much more efficient to transmit over long distances. Voltages above 120,000 volts are used for electrical power transmission. Above 760 KV, AC starts radiating power, losing efficiency. There is no advantage for using AC over DC, but there is a huge advantage in the efficiency of transmitting high voltage AC over long distances. Copyright © Texas Education Agency, All rights reserved.

Why is AC More Efficient?
Electrical power (voltage and current) is sent over transmission wires from a generator to users. Lower current sent through those transmission wires will produce less heat (Joule’s Law). Fewer moving electrons reduce the amount of friction. Voltage does not create heat. Less heat means less power is lost during delivery. We can use thinner wires for power transmission. Thin wires are more economical. No one wants a large electrical generating plant in their backyards, so power plants are typically a long distance from the users of that power. Copyright © Texas Education Agency, All rights reserved.

How Do You Make AC? Most electricity is produced by induction. In a generator, induction occurs when a conductor moves through a magnetic field. AC is the type of electricity generated by a conductor moving in a circle through a static magnetic field. Circular motion is easy to produce. Like a wheel going around, it is a simple and efficient process. Static means the magnetic field itself is not changing. Unlike in a bar magnet where magnetic field lines are circular, the magnetic field in a generator goes in a straight line from one pole to the other. The magnetic poles are on opposite sides of the stator, and the rotor will turn between the two poles. Copyright © Texas Education Agency, All rights reserved.

Rotary Motion The process of using a water wheel as a mechanical power source for milling and sawing has been used for thousands of years. A water wheel produces circular motion. This is also called a water-driven turbine. This process was applied to creating electricity in the late 1800s. Two more things were necessary to make AC power viable: the alternator and the transformer. Copyright © Texas Education Agency, All rights reserved.

Mechanical Power Source
An electrical power plant has a capacity, but the actual amount of power produced is a function of user demand. Higher user demand creates a larger load on the generator. The generator then needs to draw more mechanical power from the prime mover. If the prime mover cannot provide the additional mechanical power, the plant will shut down. User demand is actually a function of resistance. Because electrical circuits are created as parallel circuits, more devices turned on will create additional parallel circuits. This will reduce total resistance, and with a constant source voltage, these circuits will require additional current. This current is the load on the generator. The amount of mechanical power a plant is able to produce determines the plant capacity. Copyright © Texas Education Agency, All rights reserved.

Hydroelectric Power The Ames Hydroelectric Generating Plant was the world's first commercial system to produce and transmit AC electricity for industrial use. In 1890, Westinghouse Electric supplied the station's generator and motor. The AC was proven to be effective as it was transmitted two miles (3 km) at a loss of less than 5%. The maximum distance for DC was about a mile. The Ames Hydroelectric Generating Plant was located near Ophir, Colorado. Copyright © Texas Education Agency, All rights reserved.

AC Generation In an AC generator, one rotation of the rotor shaft creates one cycle of voltage. This voltage is not steady over the cycle, it changes and reverses polarity depending on the direction of motion of the conductor through the magnetic field. The magnetic field goes in a line from the north pole to the south pole. The rotor that contains the conductors move in a circle through the linear magnetic field. Copyright © Texas Education Agency, All rights reserved.

AC Generation Electricity is produced by induction. Induction occurs when a conductor cuts through a magnetic field line. A conductor must move perpendicular to the magnetic field line to cut through it. Conductor motion parallel to the magnetic field does not cut through any magnetic field lines. Copyright © Texas Education Agency, All rights reserved.

Sine Wave Generation During a rotation, the motion of a conductor changes from perpendicular to parallel. Perpendicular is a 90° angle. Parallel is a 0° (or 360°) angle. Between these examples, the conductor motion is at some other angle relative to the direction of the magnetic field. Rotation goes through 0 to 360 degrees and then repeats. Copyright © Texas Education Agency, All rights reserved.

Voltage Amplitude The amount of voltage produced at any point is proportional to the sine of the angle of motion relative to the direction of the magnetic field lines. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle

Voltage is a Function of Angle
A conductor is placed in the magnetic field. This is actually a single conductor that loops back through the magnetic field. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
Arrows show the direction of motion for the conductor in this position. This motion is perpendicular to the direction of the field lines (at 90°). Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle

Voltage is a Function of Angle

Voltage is a Function of Angle
Here is the position of a conductor after a rotation of 45° (actual angle = 135°). The motion is at an angle of 45° to the direction of the magnetic field lines. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
The sine of 45° (or 135°) is We are here on the sine wave. The amount of voltage produced at this angle is of the peak voltage. Note that at an angle of either 45 degrees or 135 degrees, we are at the same amplitude of voltage. One point is before the peak and rising; the other point is after the peak and falling. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
This shows a conductor after another rotation of 45° from the previous example. This motion is parallel to the magnetic field lines and represents an angle of 180°. Since we started off with an angle of 90 degrees and rotated through another 90 degrees, the new angle is 180. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
The sine of 180° (or 0°) is zero. We are here on the sine wave. At this instant the voltage produced by the generator is zero volts. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
The conductor rotates another 45°. Polarity starts to reverse. This motion is now down through the magnetic field lines (the opposite direction). Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
The Sine of 225° is We are here on the sine wave. Polarity is opposite because the direction of motion is going the opposite way. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
Here is the position of the conductor after another 45° rotation. This more clearly shows that the direction of motion (red circle) is down. Copyright © Texas Education Agency, All rights reserved.

Voltage is a Function of Angle
The sine of 270° is negative one. We are here on the sine wave. This is the negative peak with equal but opposite amplitude of the positive peak. Copyright © Texas Education Agency, All rights reserved.

Waveform Plot At each point in the rotating cycle, the amount of voltage produced is equal to the sine of the angle created by the direction of motion of the conductor relative to the direction of the magnetic field. This produces the changing voltage used for electricity known as the sine wave. As the angle between the conductor and the magnetic field changes, the voltage changes. Peak RMS Copyright © Texas Education Agency, All rights reserved.

Common Terms A plot of the voltage vs time produces a wave form in the shape of a sine wave. This wave form has many terms we need to learn. There are terms for the amplitude or size of the wave: Peak, peak to peak, RMS, instantaneous, and average voltage There are terms for how fast the wave changes: Frequency, period Copyright © Texas Education Agency, All rights reserved.

Waveform Values Graphically
VAVG A graph of a sine wave voltage vs time in degrees. Time is a function of frequency, angle is independent of frequency. The angle represents time. An angle is used because a time measurement would change with a change in frequency. Copyright © Texas Education Agency, All rights reserved.

Angular Measurement We use an angle to measure instantaneous voltage values at an instant of time. At a given angle, the relative amplitude at that point is the same for any sine wave regardless of the frequency. A sine wave is created due to a conductor moving in a circle through a magnetic field. A circle always has 360 degrees. A sine wave always has 360 degrees for one cycle. Copyright © Texas Education Agency, All rights reserved.

Radians There is another way to measure the angle of a circle or sine wave. It is called the radian measurement because of the relationship between the radius and the circumference of a circle. There are 2π radians in 360 degrees. C = 2πr Copyright © Texas Education Agency, All rights reserved.

Time-based Waveform Terms
Wave- a disturbance traveling through a medium. For AC electricity, the movement of the electrons back and forth in the wire Waveform- a graphic representation of a wave. Waveform depends on both movement and time. Example: ripple on the surface of a pond A change in the vertical dimension of a signal is a result of a change in the amount of voltage. Copyright © Texas Education Agency, All rights reserved.

Time-based Waveform Terms
Frequency (f)- the number of cycles of the waveform that occur in one second of time. Measured in hertz (Hz) Period (P)- the time required to complete one cycle of a waveform. Measured in seconds, tenths of seconds, milliseconds, or microseconds Copyright © Texas Education Agency, All rights reserved.

Frequency and Period There is an inverse relationship between frequency and period. Example: A frequency of 100 Hz gives a period of 0.01 sec (10 ms) f= 1 P P= 1 f and Copyright © Texas Education Agency, All rights reserved.

Waveform Terminology Amplitude- height of a wave Expressed in one of the following methods Peak (PK, pk, or Pk) Peak-to-peak (P-P, PP, or pp) Root-mean-square (RMS) Average (AVG) There are no conventions regarding capitalization of these terms. Copyright © Texas Education Agency, All rights reserved.

Waveform Amplitude Specification
Peak- the maximum positive or negative deviation of a waveform from its zero reference level. Sinusoidal waveforms are symmetrical. The positive peak value of sinusoidal will be equal to the value of the negative peak. Measured at an instant of time Copyright © Texas Education Agency, All rights reserved.

Waveform Amplitude Specification (continued
Peak-to-peak is the measurement from the highest amplitude peak to the lowest peak. Sinusoidal waveform If the positive peak value is 10 volts in magnitude, then the negative peak is also 10; therefore, peak-to- peak is 20 volts. Non-sinusoidal waveform It is determined by adding magnitude of positive and negative peaks. Copyright © Texas Education Agency, All rights reserved.

Waveform Amplitude Specification (Cont’d.)
Root-mean-square (RMS) Measured over one (or more) cycles of the wave. Allows the comparison of AC and DC circuit values. RMS values of AC create the same heat as that same numerical voltage value of DC. RMS is most common method of specifying the value of sinusoidal waveforms. Almost all voltmeter and ammeters are calibrated so that they measure AC values in terms of RMS amplitude. VRMS = VDC = heat Copyright © Texas Education Agency, All rights reserved.

RMS Heating Effect VRMS = Vpeak IRMS = Ipeak A sinusoidal voltage with peak amplitude of 1 volt has the same heating effect as a DC voltage of volts. AC creates slightly more heat than DC. Comparing RMS value to average voltage. Due to this, the RMS value of voltage is also referred to as the effective value. The back and forth movement of electrons in AC creates more friction and more heat than the steady movement of electrons in DC. This compares RMS value to average voltage (not peak voltage). Copyright © Texas Education Agency, All rights reserved.

Determining the 0.707 Constant
To determine the constant, you must use the mathematical procedure suggested by the name, root-mean-square. This has nothing to do with the sine function even though this is the same value as sin 45°. VRMS = V pk 2 The square root of 2 is related to the fact that power creates heat, and power is proportional to voltage squared. When solving for the heating value of voltage, the 2 comes from taking the square root of the sum of the squares with unit values (meaning normalized to 1). Copyright © Texas Education Agency, All rights reserved.

Average Voltage Average voltage is the DC equivalent voltage. The average voltage of AC over a full sine wave equals zero. The positive half cycle is equal to the negative half cycle. Because of this we only look at half the wave (the positive half cycle). This means we are looking at an angle of π radians. Copyright © Texas Education Agency, All rights reserved.

Average Voltage Average voltage over a half cycle: Or, VAVG = .637 VPK VAVG is produced by rectifying then filtering the AC in a voltage regulator. This is the process used in a DC power supply. VAVG = 2 Vpk π = Vpp π Copyright © Texas Education Agency, All rights reserved.

Instantaneous Voltage
This is an important value when sampling a wave at several points. Example: analog to digital conversion Different types of waves might have the same peak or average voltage but different instantaneous voltages Examples: square wave, triangle wave Copyright © Texas Education Agency, All rights reserved.

Instantaneous Voltage
Instantaneous voltage is the voltage at a single point or instant of time. To find the angle, take the inverse sine. Usually a 2nd function on a calculator V INST = V pk sin θ θ=sin −1 ( V INST V pk ) Copyright © Texas Education Agency, All rights reserved.

Relationship Calculations
EXAMPLES 120 VAC = 170 Vpk Formula: PK = RMS  0.707 120  = (round off to 170 Vpk) 18 72  = 19 Vpk Formula: PK = Instantaneous  Sine 18  Sine(72°) = (round off to 19) 30 Vpk = 21.2 VAC Formula: RMS = X PK 0.707 X 30 = 21.2  = 30 Vpk Formula PK = Instantaneous  Sine 350  Sine(23.5°) = (round off to 878) 50 Vpp = 17.7 Vrms Step 1: Need to find PK Formula: PK = P-P  2 50  2 = 25m Step 2: Find RMS 0.707 X 25 = (round off to 17.7 Vrms) Find the angle with 454 V instantaneous and a PK of 908 V Formula: Sine (θ) = Instantaneous  PK 454  908 = 0.5 2nd Sine (.5) = 30 20 V Average = 22.2 Vrms = 31.4 Vpk = 62.8 Vp-p Step 1: Find PK, Formula: PK = Average  0.637 = 20 0.637 = (round off to 31.4) Step 2: Find RMS, Formula: RMS = X PK = X 31.4 = (round off to 22.2) Step 3: Find P-P, Formula: PK = 2 X PK = 2 X 31.4 = 62.8 Copyright © Texas Education Agency, All rights reserved.

Calculate peak voltage given RMS voltage
VRMS = 120 V Formula: PK = RMS  0.707 120  = (round up to 170 Vpk) 120 VRMS = 170 Vpk P – P RMS PK PK INST AVG. Sine° PK PK Copyright © Texas Education Agency, All rights reserved.

Calculate RMS voltage given peak voltage
Vpk = 30 V Formula: RMS = X PK 0.707 X 30 = 21.2 V 30 Vpk = 21.2 VRMS P – P RMS PK PK INST AVG. Sine° PK PK Copyright © Texas Education Agency, All rights reserved.

Calculate RMS given p-p
Vpp = 50 V Step 1: Find Vpk Formula: PK = p-p  2 50  2 = 25 V Step 2: Find RMS Formula: RMS = X PK 0.707 X 25 = (round off to 17.7 Vrms) 50 Vpp = 17.7 Vrms P – P RMS PK PK INST AVG. Sine° PK PK Copyright © Texas Education Agency, All rights reserved.

Calculate VRMS, Vpk, and Vpp given VAVG
VAVG = 20 V Step 1: Find Vpk, Formula: PK = Average  0.637 = 20  = (round up to 31.4) Step 2: Find RMS, Formula: RMS = X PK = X 31.4 = 22.19 (round up to 22.2) Step 3: Find P-P, Formula: P-P = 2 X PK = 2 X 31.4 = 62.8 P – P RMS PK PK INST AVG. Sine° PK PK Copyright © Texas Education Agency, All rights reserved.

Calculate peak voltage from instantaneous voltage
VINST = 18 72° Formula: PK = Instantaneous  Sine of the angle = 18  Sine(72°) = 18  .951 = 18.9 V (round up to 19) 18 72  = 19 Vpk P – P RMS PK PK INST AVG. Sine° PK PK Copyright © Texas Education Agency, All rights reserved.

Calculate peak voltage from instantaneous voltage
VINST = ° Formula: PK = Instantaneous  Sine(angle) = 350  Sine(23.5°) = 350  0.4 = (round up to 878)  = 878 Vpk P – P RMS PK PK INST AVG. Sine° PK PK Copyright © Texas Education Agency, All rights reserved.

Calculate phase angle given instantaneous voltage and peak voltage
P – P RMS PK PK INST AVG. Sine° PK PK 454 VINST at unknown angle with a Vpk of 908 V Formula: Sine θ = Instantaneous  PK 454  908 = 0.5 2nd Sine (.5) = 30 Copyright © Texas Education Agency, All rights reserved.

Relationship Exercise

Sine Wave and Sine Trigonometric Function
The term sinusoidal has been used to describe a waveform produced by an AC generator. The term sinusoidal comes from a trigonometric function called the sine function. Sines, cosines, and tangents are numbers equal to the ratio of the lengths of the sides. The sine function is used in AC because the opposite side is the direction of motion of the conductor through a magnetic field relative to the angle. Copyright © Texas Education Agency, All rights reserved.

Right-Triangle: Side and Angle Relationships
A right triangle has a 90° angle. Each side is named with respect to the angle you are using (called the angle theta, or θ). The side of the triangle across from the angle theta is called the opposite side. The longest side of a right triangle is called the hypotenuse. The remaining side is called the adjacent. Each of these sides are commonly abbreviated to their initials, O, H, and A. Copyright © Texas Education Agency, All rights reserved.

Basic Trigonometric Functions
In trigonometry, there are three common ratios used to study right triangles. Sine The sine of the angle theta is equal to the ratio formed by the length of the opposite side divided by the length of the hypotenuse. Sine θ = opposite hypotenuse Copyright © Texas Education Agency, All rights reserved.

Basic Trigonometric Functions (Continued)
Cosine The cosine of the angle theta is equal to the ration formed by length of the adjacent side divided by the length of the hypotenuse. Cosine θ = adjacent hypotenuse Tangent The tangent of the angle theta is equal to the ratio formed by length of the opposite side divided by the length of the adjacent side. Tangent θ = opposite adjacent Copyright © Texas Education Agency, All rights reserved.

Basic Trigonometric Functions (Continued)

Trigonometric Exercise

Go to the Trigonometric Functions
Relationship Key # rms peak pk-to-pk average instantaneous 200mV 283mV 566mV 180mV 72º 80v 113V 226V 72V 90º 96.4 136V 272V 87V 235º 1.25V 1.77V 3.54V 1.13V 122º 764µV 1.08mV 2.16mV 689V 35º Go to the Trigonometric Functions Copyright © Texas Education Agency, All rights reserved.

Trigonometric Exercise Key
Triangle #1 Hypotenuse? 8’ Opposite 10’ Adjacent Hypotenuse = 12.8’ Triangle #2 5.3 Rods Opposite 6.8 Rods Adjacent Hypotenuse = 8.62 Rods Triangle #3 125 Miles Hypotenuse Opposite? 85 miles Adjacent Opposite = 91.7 miles Triangle #4 56’ 23.2’ Adjacent? Adjacent = 51’ Go to the Calculations End of Presentation Copyright © Texas Education Agency, All rights reserved.

Trigonometric Exercise Calculations

Trigonometric Exercise Calculations (Continued)
Triangle #2 Given Adjacent side = 6.8 rods and Opposite side = 5.3 rods What is the Hypotenuse? Step 1 - Find the degree angle Tangent = Opposite Adjacent = 5.3 rods 6.8 rod enter 2nd tangent on your calculator = 37.93° Step 2 - Change the degree angle to sine Hypotenuse = Opposite Sine sine (enter 37.93, enter sine on the calculator ) = 8.63 = Copyright © Texas Education Agency, All rights reserved.

Trigonometric Exercise Calculations (continued)
Triangle #3 Given Hypotenuse side = 125 miles and Adjacent side = 8.5 miles What is the Opposite side? Step 1 - Find the degree angle Cosine = = 0.68 enter 2nd function button on calculator enter cosine 0.68 = (round off to 47.16º) Step 2 - Change the degree angle to sine Opposite = Sine x Hypotenuse enter sine on calculator = = x 125 miles = (round off to 91.7) = 91.7 miles Adjacent Hypotenuse = Copyright © Texas Education Agency, All rights reserved.

Trigonometric Exercise Calculations (continued)
Triangle #4 Given Hypotenuse side = 56’ and Opposite side = 23.2’ What is Adjacent side? Step 1 - Find the degree angle Sine = = = enter 2nd sin = 24.47º Step 2 - Change the degree angle to cosine Adjacent = Cosine x Hypotenuse enter cosine on the calculator = x 56’ = (round off to 51’) = 51’ opposite hypotenuse = Copyright © Texas Education Agency, All rights reserved.