AC Electricity

Muhajir Ab. RahimSchool of Mechatronic Engineering

Universiti Malaysia Perlis

Contents

1. Introduction

2. AC Sinusoidal Waveform

3. Measurement of AC Sinusoidal Waveform

4. Nonsinusoidal Waveforms

Contents

1. Introduction

2. AC Sinusoidal Waveform

3. Measurement of AC Sinusoidal Waveform

4. Nonsinusoidal Waveforms

Learning Objectives in Introduction

1.Able to describe the difference between DC and AC voltage and current.

2.Able to represent the sources of DC and AC voltage and current.

3.Understand the concept of operation of AC generator

Introduction• In 1887 direct current (DC) was

king. At that time there were 121 Edison power stations scattered across the United States delivering DC electricity to its customers.

• But DC had a great limitation -- namely, that power plants could only send DC electricity about a mile before the electricity began to lose power.

• In the late 1880s, George Westinghouse introduced his system based on high-voltage alternating current (AC), which could carry electricity hundreds of miles with little loss of power.

DC Current vs. AC Current (1/2) 

• Direct current (DC) flows in one direction the circuit. 

• Alternating current (AC) flows first in one direction then in the opposite direction.

The same definitions apply to alternating voltage (AC voltage): 

• DC voltage has a fixed polarity. • AC voltage switches polarity back and forth.

DC Current vs. AC Current  (2/2)

• There are numerous sources of DC and AC current and voltage. However:  Sources of DC are commonly shown as a cell or battery:  

• Sources of AC are commonly shown as an AC generator:

Switch is off, circuit is open,no current flow,

lamp is off

Switch is on to DC,Circuit is closed,

Current is flowing in one direction,Lamp is on

Switch is on to AC,Circuit is closed,

Current is flowing back and forth,Lamp is on

The purpose of a generator is to convert motion into electricity.

The wire passing through a magnetic field causes electrons in that wire

to move together in one direction. When the loop is spinning,

it's moving across the field first in one direction and then in the other,

which means that the flow of electrons keeps changing

Comparison of DC and AC loops and rings

This is Brighton Electric Light Station in 1887.Stationary steam engines drive tiny

direct current (DC) generators by means of leather belts

Tesla's alternating current (AC) induction motor was simple in concept and could be made sufficiently small to power individual machines in factories

Contents

1. Introduction

2. AC Sinusoidal Waveform

3. Measurement of AC Sinusoidal Waveform

4. Nonsinusoidal Waveforms

Learning Objectives in AC Sinusoidal Waveform

1. Able to describe the shape and main features of a AC sinusoidal waveform.

2. Able to calculate the instantaneous value of a current or voltage sine waveform, given the maximum value and angular displacement

The Sinusoidal AC Waveform (1/2)

• The most common AC waveform is a sine (or sinusoidal) waveform. 

• The vertical axis represents the amplitude of the AC current or voltage, in amperes or volts. 

• The horizontal axis represents the angular displacement of the waveform. The units can be degrees or radians. 

The Sinusoidal AC Waveform (2/2)

• The sine waveform is accurately represented by the sine function of plane trigonometry:  y = rsinθ where:   y = the instantaneous amplitude

r = the maximum amplitude

θ = the horizontal displacement

Instantaneous Current

i = Ipsinθwhere,

i = instantaneous current in amperes 

Ip= the maximum, or peak, current in amperes 

θ = the angular displacement in degrees or radians

Instantaneous Voltage 

v = Vpsinθ

where 

v = instantaneous voltage in volts  Vp = the maximum, or peak, voltage in volts  θ = the angular displacement in degrees or radians

Contents

1. Introduction

2. AC Sinusoidal Waveform

3. Measurement of AC Sinusoidal Waveform

4. Nonsinusoidal Waveforms

Learning Objectives in Measurement of AC Sinusoidal Waveform (1/2)

Amplitude of AC Sinusoidal Waveform• Able to define the terms peak and peak-to-peak as they apply to AC sinusoidal

waveforms. • Able to convert between peak and peak-to-peak values. • Able to describe the meaning of root-mean-square (RMS) as it applies to AC

sinusoidal waveforms. • Able to convert between RMS values and  peak values. • Able to describe the meaning of average values of a AC sinusoidal waveform. • Able to convert between averages values and  peak values. • Able to describe how instantaneous values for voltage and current differ from peak,

RMS, and averages values.

Period and Frequency of AC Sinusoidal Waveform• Able to define the  period of a AC waveform. • Able to cite the units of measure for the period of a waveform. • Able to define the  frequency of a waveform. • Able to cite the units of measure for the frequency of a waveform. • Able to convert between values for the period and frequency of a waveform

Learning Objectives in Measurement of AC Sinusoidal Waveform (2/2)

Phase Angle• Able to describe the meaning of phase angle. • Able to expand the formula for instantaneous sine voltage and

current to include a phase angle, then apply the formula to sketch accurate sinusoidal waveforms.

• Able to define the terms leading and lagging as they apply to sinusoidal waveforms.

AC Sinusoidal Power Waveform• Able to sketch voltage, current, and power sine waveforms on the

same axis. • Able to explain why the power waveform is always positive as long

as current and voltage are in phase. • Able to cite the fact that average power is equal to the product of

RMS current and RMS voltage

Amplitude of AC Waveform

• Peak and peak-to-peak values are most often used when measuring the amplitude of ac waveforms directly from an oscilloscope display.

Peak and Peak-to-Peak Voltage

• Peak voltage is the voltage measured from the baseline of an ac waveform to its maximum, or peak, level. 

• Unit: Volts peak (Vp)  Symbol: Vp

• For a typical sinusoidal waveform, the positive peak voltage is equal to the negative peak voltage.  

• Peak-to-peak voltage is the voltage measured from the maximum positive level to the maximum negative level. 

• Unit: Volts peak-to-peak  (Vp-p)  Symbol: Vp-p

• For a typical sinusoidal waveform, the peak-to-peak voltage is equal to 2 times the peak voltage. 

Conversion between Vp and Vp-p

• Convert Vp to Vp-p: 

Vp-p = 2 Vp

• Convert Vp-p to Vp : 

Vp =0.5 Vp-p

Exercise

1. What is the peak-to-peak value of a sinusoidal waveform that has a peak value of 12 V? 

2. What is the peak value of a sine wave that has a peak-to-peak value of 440 V?   

Ans: 24 Vp-p

Ans: 220 Vp

Vrms = 0.707Vp

RMS Voltage (1/2)• AC levels are assumed to be expressed as Root Mean Square, (RMS)

values unless clearly specified otherwise. • The RMS value is also referred to as the effective value. • Unit: Volts (V) 

Symbol: Vrms  • The RMS voltage of a sinusoidal waveform is equal to 0.707 times its peak

value. 

RMS Voltage (2/2)• The RMS value of a sinusoidal voltage is actually a measure of the heating effect of

the sine wave. • For example, when a resistor is connected across an AC (sinusoidal) voltage source,

as shown in Figure (a), a certain amount of heat is generated by the power in the resistor. Figure (b) shows the same resistor connected across a DC voltage source.

• The value of the AC voltage can be adjusted so that the resistor gives off the same amount of heat as it does when connected to the DC source.

• In a DC circuit, applying 2 V to a 1 Ω resistance produces 4 W of power.• In an AC circuit, applying 2 Vrms to a 1Ω resistance produces 4 W of power• RMS voltages are expressed without a + or - sign. 

Sinusoidal AC Source

DC Source

Conversion between Vp to Vrms

• Convert Vp to Vrms: 

Vrms = 0.707Vp  

• Convert Vrms to Vp : 

Vp =1.414Vrms  

Exercise

1. Determine the RMS value of a waveform that measures 15 Vp. 

2. Determine the peak value of 120 V. (Hint: Assume 120 V is in RMS) 

Ans: 10.6 V

Ans: 170 Vp

Average Voltage• Average voltage is the average value of all the values for one half-cycle of

the waveform.  • Unit: Volts average (Vave) 

Symbol: Vave  • The average voltage of a sinusoidal waveform is equal to 0.637 times its peak

value.  Vave = 0.637Vp

• The average voltage is determined from just one half-cycle of the waveform because the average value of a full cycle is zero. 

• Average voltages are expressed without a + or - sign 

Conversion between Vp to Vave

• Convert Vp to Vave: 

Vave = 0.637Vp  

• Convert  Vave to Vp: 

Vp =1.57Vave

Exercise

1. Determine the average value of a waveform that measured 16 Vp. 

2. What is the peak value of a waveform that has an average value of 22.4 V? 

Ans: 10.2 Vave

Ans: 35.2 Vp

Period of a Waveform

• The period of a waveform is the time required for completing one full cycle.  

• Math symbol: T  Unit of measure: seconds (s)

• One period occupies exactly 360º of a sine waveform. 

Frequency of a Waveform

• The frequency of a waveform is the number of cycles that is completed each second. 

• Math symbol: f  Unit of measure: hertz (Hz)

• This example shows four cycles per second, or a waveform that has a frequency of 4 Hz

Conversion betweenPeriod and Frequency

• Period to Frequency  f = 1/T   

• Frequency to Period  T = 1/f  

Frequency (f) is in hertz (Hz)  Period (T) is in seconds (s)

Exercise

1. A certain sine waveform has a frequency of 100 Hz. What is the period of this waveform? 

2. What is the frequency of a waveform that has a period of 200 ms? 

Ans: 10 ms

Ans: 5 kHz

Phase Angle

• The phase angle of a waveform is angular difference between two waveforms of the same frequency. 

• Math symbol: θ  Unit of measure: degrees or radians

• Two waveforms are said to be in phase when they have the same frequency and there is no phase difference between them. 

• Two waveforms are said to be out of phase when they have the same frequency and there is some amount of phase shift between them.

Leading Phase Angles  • A leading waveform is one that is

ahead of a reference waveform of the same frequency. 

• In this example, the blue waveform (sine wave A) is taken as the reference because it begins at 0 degrees on the horizontal axis.

• Sine wave B is shifted to the right by 90°.

• Thus, there is a phase angle of 90° between sine wave A and sine wave B.

• In terms of time, the positive peak of sine wave B occurs later than the positive peak of sine wave A because time increases to the right along the horizontal axis.

• In this case, sine wave B is said to lag sine wave A by 90°. In other words, sine wave A is said to be leading sine wave B by 90°

A leads B by 900, or B lags A by 900

Lagging Phase Angles  • A lagging waveform is one that is

behind a reference waveform of the same frequency.

• In this example, the blue waveform (sine wave A) is taken as the reference because it begins at 0 degrees on the horizontal axis.

• Sine wave B is shown shifted left by 90° .

• There is a phase angle of 90° between sine wave A and sine wave B.

• In this case, the positive peak of sine wave B occurs earlier in time than that of sine wave A.

• Therefore, sine wave B is said to lead sine wave A by 90°. In other words, sine wave A is said to be lagging sine wave B by 90°

B leads A by 900, or A lags B by 900

AC Power Waveform (1/2)

• The current and voltage waveforms are shown in phase. This is typical for a resistive load. The shaded green areas represent the corresponding levels of power. 

• The instantaneous value of power is equal to the instantaneous current times the instantaneous voltage. 

where, P = instantaneous value of power (Watt)I = instantaneous value of current (Amp)V = instantaneous value of voltage (V)

P = IV

AC Power Waveform (2/2)

• Notice that the power waveform is always positive.  

• A positive value of power indicates that the source is giving power to the load.

• A negative value of power would indicate that the circuit is returning power to the source (which will not happen in a resistor circuit).

• The power waveform is always positive because the values of current and voltage always have the same sign--both negative or both positive. In algebra, this means that the product of the two values is always a positive value. 

Average AC Power 

• When the current and voltage waveforms are in phase, the average power is equal to the RMS voltage times the RMS current: 

• Conventional use allows us to write this equation more simply as:  P = IV

• It is then assumed that P is an average value and the other two terms are RMS values. 

Pave = IRMS x VRMS

Contents

1. Introduction

2. AC Sinusoidal Waveform

3. Measurement of AC Sinusoidal Waveform

4. Nonsinusoidal Waveforms

Learning Objectives in Nonsinusoidal Waveforms

• Able to describe the features of a rectangular or square waveform.

• Able to calculate the total period and frequency of a rectangular waveform.

• Able to define duty cycle and average voltage as the terms apply to a rectangular waveform. 

• Able to describe the features of a triangular or sawtooth waveform

Rectangular Waveform (1/3)• A rectangular waveform is characterized

by flat maximum and minimum levels, fast-rising and fast-falling edges, and squared-off corners. Because of the squared corners, a rectangular waveform is also called a square waveform

• The amplitude of a rectangular waveform is a measure of the distance between the minimum and maximum levels--the peak-to-peak value. Amplitude is most often expressed in units of volts, although units of current and power can be useful at times. 

• The period of a rectangular waveform is the time required to complete one full cycle.  The period is measured in units of seconds. 

Rectangular Waveform (2/3)• The period of a rectangular waveform

can be further broken down into two phases:  Time High, TH -- The amount of time for the higher amplitude level.  Time Low, TL--The amount of time for the lower amplitude level.

• In these terms, the period of the waveform can be give by: 

T = total period of the waveform  TH = time high  TL = time low

T = TH + TL

Rectangular Waveform (3/3)• The duty cycle of a square waveform is the ratio of time high to the total period: 

• It is often expressed as a percentage where: 

• Rectangular waveforms are sometimes used for regulating the amount of power

applied to a load (such as a motor or lamp). The higher the duty cycle, the greater the amount of power applied to the load. 

• The frequency of a rectangular waveform is given by: 

• The average voltage of a square waveform is given by:   

duty cycle = TH / T

duty cycle (%) = (TH / T ) x 100

f = 1/T or   f = 1/(TH+TL) 

Vave = Vp x duty cycle 

Exercise

1. For a rectangular waveform, TL = 15 ms and TH = 10 ms. Calculate: 

(a) The total period of the waveform.  (b) The frequency of the waveform.  (c) The duty cycle of the waveform.

Ans: (a) 25 ms, (b) 40 Hz, (c) 0.4 or 40%

Sawtooth Waveform • A sawtooth waveform is

characterized by one sloping edge and one that instantaneously returns to the baseline.

• The slope of a sawtooth waveform is specified in terms of volts per second (V/s).

• If Vp is the amplitude and T is the period of sawtooth waveform, the slope is give by:

or since T = 1/f:

slope = Vp / T

slope = Vpf

Exercise

1. The amplitude of a sawtooth waveform is 12 V. If the frequency is 100 Hz, what is the slope?  slope = 1200 V/s