EGR 2201 Unit 11Sinusoids and Phasors

Read Alexander & Sadiku, Chapter 9 and Appendix B.

Homework #11 and Lab #11 due next week.

Quiz next week.

DC Versus AC

In a direct-current (DC) circuit, current flows in one direction only. The textbook’s Chapters 1 through 8

cover DC circuits. In an alternating-current (AC) circuit,

current periodically reverses direction. The book’s Chapters 9 through 11 cover

AC circuits.

The Math Used in AC Circuits

Our study of AC circuits will rely heavily on two areas of math: Sine and cosine functions Complex numbers

We’ll review the math after introducing some terminology used in discussing AC voltages and currents.

Waveforms

The graph of a current or voltage versus time is called a waveform. Example:

Note that this is an AC waveform: negative values of voltage mean the opposite polarity (and therefore opposite direction of current flow) from positive values.

Periodic Waveforms Often the graph of a voltage or current

versus time repeats itself. We call this a periodic waveform.

Common shapes for periodic waveforms include:

Sinusoid Square Triangle Sawtooth

Image from http://en.wikipedia.org/wiki/Sinusoid

Sinusoids are the most important of these.

Cycle

In a periodic signal, each repetition is called a cycle.

How many cycles are shown in the diagram below?

Waveform Parameters

Important parameters associated with periodic waveforms include:Period TFrequency fAngular Frequency Amplitude Vm (or Peak Value Vp)Peak-to-Peak Value Instantaneous Values

Period

The time required for one cycle is called the waveform’s period.

The symbol for period is T. Period is measured in seconds,

abbreviated s. Example: If a waveform repeats

itself every 3 seconds, we’d writeT = 3 s

Frequency

A waveform’s frequency is the number of cycles that occur in one second.

The symbol for frequency is f. Frequency is measured in hertz,

abbreviated Hz. Some old-timers say “cycles per second”

instead of “hertz.” Example: If a signal repeats itself 20

times every second, we’d writef = 20 Hz

Period and Frequency

Period and frequency are the reciprocal of each other:

f = 1 / T T = 1 / f

Radians

Recall that the radian (rad) is the SI unit for measuring angle.

It is related to degrees by radians = 180

We’ll often need to convert between radians and degrees: To convert radians to degrees, multiply

by . To convert degrees to radians, multiply

by .

Angular Frequency

The quantity 2f, which appears in many equations, is called the angular frequency.

Its symbol is , and its unit is rad/s: = 2f

One Question, Three Answers

So we have three ways of answering the question, “How fast is the voltage (or current) changing?”

1. Period, T, unit = seconds (s) Tells how many seconds for one cycle.

2. Frequency, f, unit = hertz (Hz) Tells how many cycles per second.

3. Angular frequency, , unit = rad/s Tells size of angle covered per second.

Relating T, f, and

If you know any one of these three (period, frequency, angular frequency), you can easily compute the other two.

The key equations that you must memorize are:

T = 1/f

= 2f = 2/T

Amplitude (or Peak Value)

The maximum value reached by an ac waveform is called its amplitude or peak value.

Peak-to-Peak Value

A waveform’s peak-to-peak value is its total height from its lowest value to its highest value.

Many waveforms are symmetric about the horizontal axis. In such cases, the peak-to-peak value is equal to twice the amplitude.

Instantaneous Value

The waveform’s instantaneous value is its value at a specific time.

A waveform’s instantaneous value constantly changes, unlike the previous parameters (period, frequency, angular frequency, amplitude, peak-to-peak value), which usually remain constant.

Lead and Lag

When two waveforms have the same frequency but are not “in phase” with each other, we say that the one shifted to left leads the other one.

And we say that the one shifted to the right lags the other one.

Phase Angle

To quantify the idea of how far a waveform is shifted left or right relative to a reference point, we assign each waveform a phase angle .

A positive phase angle causes the waveform to shift left along the x-axis.

A negative phase angle causes it to shift right.

Sinusoids A sinusoid is a sine wave or a cosine

wave or any wave with the same shape, shifted to the left or right.

Sinusoids arise in many areas of engineering and science. They are the waveform used most frequently in electrical circuit theory.

The waveform we’ve been looking at is a sinusoid.

Amplitude, Frequency, Phase Angle

Any two sinusoids must have the same shape, but can vary in three ways:

Amplitude (maximum value) Angular frequency (how fast the values change) Phase angle (how far shifted to the left or right)

We’ll use mathematical expressions for sinusoids that specify these three things….

Mathematical Expression For a Sinusoid

The mathematical expression for a sinusoid looks like this:

v(t) = Vmcos(t + )

where Vm is the amplitude, is the angular frequency, and is the phase angle.

Example:

v(t) = 20 cos(180t + 30) V

Calculator’s Radian Mode and Degree Mode

Recall that when using your calculator’s trig buttons (such as cos), you must pay attention to whether the calculator is in radian mode or degree mode. Example: If the calculator is in radian

mode, then cos(90) returns 0.448, which is the cosine of 90 radians.

But if the calculator is in degree mode, then cos(90) returns 0, which is the cosine of 90 degrees.

Caution: Radians and Degrees

In the expression for a sinusoid,

v(t) = Vmcos(t + )

is usually given in degrees, but is always given in radians per second.

Recommendation: To compute a sinusoid’s instantaneous value, leave your calculator in radian mode, and convert to radians.

Type of Voltage Source

Symbol Used in Our Textbook

Symbol Used in Multisim Software

Generic voltage source (may be DC or AC)

DC voltage source

AC sinusoidal voltage source

Schematic Symbols for Independent Voltage Sources

Several different symbols are commonly used for voltage sources:

Function Generator

We use a function generator to produce periodic waveforms.

Trainer’s Function Generator

TTL Mode Output, controlled only by the FREQUENCY and RANGE knobs. Used in Digital Electronics courses.

Regular Output, controlled by all four knobs. In this course we’ll always use this one.

No matter which oneof these you use, youmust also use the GROUND connection.

Oscilloscope

We use an oscilloscope to display waveforms.

Using it, wecan measure amplitude, period, and phase angle of ac waveforms, as well as dc values, transients, and more.

Oscilloscope Challenge Game

The oscilloscope is a complex instrument that you must learn to use.

To learn the basics, play my Oscilloscope Challenge game at http://people.sinclair.edu/nickreeder/flashgames.htm.

Sine or Cosine?

A sinusoidal waveform can be expressed mathematically using either the sine function or the cosine function.

Example: these two expressions describe the same waveform: v(t) = 20 sin(300t + 30) v(t) = 20 cos(300t 60)

In a problem where you’re given a mixture of sines and cosines, your first step should be to convert all of the sines to cosines.

Trigonometric Identities Relating Sine and Cosine

You can convert from sine to cosine (or vice versa) using the trig identities sin(x + 90) = cos(x) sin(x 90) = cos(x) cos(x + 90) = sin(x) cos (x 90) = sin(x)

These identities reflect the fact that the cosine function leads the sine function by 90.

A Graphical Method Instead of Trig Identities

Remembering and applying trig identities may be difficult.

The book describes a graphical method that relies on the following diagram:

To use it, recall that we measure positive angles counterclockwise, and negative angles clockwise.

Mathematical Review: Complex Numbers

The system of complex numbers is based on the so-called imaginary unit, which is equal to the square root of 1.

Mathematicians use the symbol i for this number, but electrical engineers use j:

or 1i 1j

Rectangular versus Polar Form

Any complex number can be expressed in three forms:

Rectangular form Example: 3 + j 4

Polar form Example: 5 53.1

Exponential form Example: 5e j 53.1 or 5e j 0.927

Rectangular Form

In rectangular form, a complex number z is written as the sum of a real part x and an imaginary part y:

z = x + jy

The Complex Plane

We often represent complex numbers as points in the complex plane, with the real part plotted along the horizontal axis (or “real axis”) and the imaginary part plotted along the vertical axis (or “imaginary axis”).

Polar Form

In polar form, a complex number z is written as a magnitude r at an angle :

z = r

The angle is measured from the positive real axis.

Converting Between Rectangular and Polar Forms

We will very often have to convert from rectangular form to polar form, or vice versa.

This is easy to do if you remember a bit of right-angle trigonometry.

Given a complex number z with real part x and imaginary part y, its magnitude is given by

and its angle is given by

Converting from Rectangular Form to Polar Form

22 yxr

x

y1tan

Inverse Tangent Button on Your Calculator

When using your calculator’s tan1 (inverse tangent) button, pay attention to whether the calculator is in degree mode or radian mode.

Also recall that the calculator’s answer may be in the wrong quadrant, and that you may need to adjust the answer by 180.

The tan1 button always returns an angle in Quadrants I or IV, even if you want an answer in Quadrants II or III.

Given a complex number z with magnitude r and angle , its real part is given by

and its imaginary part is given by

Converting from Polar Form to Rectangular Form

cosrx

sinry

Complex numbers may also be written in exponential form. Think of this as a mathematically respectable version of polar form.

In exponential form, should be in radians.

Example: 330 3ej/6

Exponential Form

Polar form Exponential Form

r rej

Euler’s Identity

The exponential form is based on Euler’s identity, which says that, for any ,

sincos je j

Mathematical Operations

You must be able to perform the following operations on complex numbers:

Addition Subtraction Multiplication Division Complex Conjugate

Addition

Adding complex numbers is easiest if the numbers are in rectangular form.

Suppose z1 = x1+jy1 and z2 = x2+jy2

Then z1 + z2 = (x1+x2) + j(y1+y2)

In words: to add two complex numbers in rectangular form, add their real parts to get the real part of the sum, and add their imaginary parts to get the imaginary part of the sum.

Subtraction

Subtracting complex numbers is also easiest if the numbers are in rectangular form.

Suppose z1 = x1+jy1 and z2 = x2+jy2

Then z1 z2 = (x1x2) + j(y1y2)

In words: to subtract two complex numbers in rectangular form, subtract their real parts to get the real part of the result, and subtract their imaginary parts to get the imaginary part of the result.

Multiplication

Multiplying complex numbers is easiest if the numbers are in polar form.

Suppose z1 = r1 1 and z2 = r2 2

Then z1 z2 = (r1r2) (1+ 2)

In words: to multiply two complex numbers in polar form, multiply their magnitudes to get the magnitude of the result, and add their angles to get the angle of the result.

Division

Dividing complex numbers is also easiest if the numbers are in polar form.

Suppose z1 = r1 1 and z2 = r2 2

Then z1 ÷ z2 = (r1 ÷ r2) (1 2)

In words: to divide two complex numbers in polar form, divide their magnitudes to get the magnitude of the result, and subtract their angles to get the angle of the result.

Complex Conjugate

Given a complex number in rectangular form, z = x + jyits complex conjugate is simply z* = x jy

Given a complex number in polar form, z = r its complex conjugate is simply z* = r

Performing Complicated Operations on Complex Numbers Solving a problem may require us to

perform many operations on complex numbers.

Example: With a powerful calculator such as the TI-

89, you can do this quickly and easily. With other calculators it’s more tedious, since you must repeatedly convert between rectangular and polar forms.

Another option is to use MATLAB.

Entering Complex Numbers in MATLAB

Entering a number in rectangular form:>>z1 = 2+j3

Entering a number in polar (actually, exponential) form:>>z3 = 5exp(jpi/6)

You must give the angle in radians, not degrees.

MATLAB always displays complex numbers in rectangular form.

Operating on Complex Numbers in MATLAB

Use the usual mathematical operators for addition, subtraction, multiplication, division:

>>z5 = z1+z2

>>z6 = z1*z2

and so on.

Built-In Complex Functions in MATLAB

Useful MATLAB functions: real() gives a number’s real part imag() gives a number’s imaginary part abs() gives a number’s magnitude angle() gives a number’s angle in radians conj() gives a number’s complex conjugate

Reminder About Calculators

In this course I’ll let you use any calculator or MATLAB on the exams.

But the Fundamentals of Engineering exam and the Principles and Practice of Engineering exam have a restrictive calculator policy.

To succeed on those exams, you must be able to do complex-number math with a “bare-bones” calculator.

Useful Properties of j j is the only number whose reciprocal is

equal to its negation:

Therefore, for example,

Also, Therefore multiplication by j is equivalent to a

counterclockwise rotation of 90 in the complex plane.

jj

1

C

j

Cj

1

Sinusoids Everywhere

If you connect a sinusoidal voltage source or current source to a circuit made up of resistors, capacitors, and inductors, then all voltages and currents in the circuit will be sinusoids.

You can’t make the same statement for triangle waves, square waves, sawtooth waves, or other waveshapes.

Kirchhoff’s Laws in AC Circuits

KCL and KVL hold in AC circuits. But to apply these laws, we must add

(or subtract) sinusoids instead of adding (or subtracting) numbers.

Example: In the circuit shown, KVL tells us that v = v1 + v2. But suppose

v1 = 10 cos(200t + 30) V and v2 = 12 cos(200t + 45) V

How can we add those to find v?

Adding Sinusoids

We often need to find the sum of two or more sinusoids.

A unique property of sinusoids: the sum of sinusoids of the same frequency is always another sinusoid of that frequency.

You can’t make the same statement for triangle waves, square waves, sawtooth waves, or other waveshapes.

Adding Sinusoids (Continued)

For example, if we add

v1 = 10 cos(200t + 30) V and v2 = 12 cos(200t + 45) V

we’ll get another sinusoid of the same angular frequency, 200 rad/s:

v1 + v2 = Vm cos(200t + ) V

But how do we figure out the resulting sinusoid’s amplitude Vm and phase angle ?

Complex Numbers to the Rescue!

One method for adding sinusoids relies on trig identities.

But we’ll use a simpler method, which relies on complex numbers. In fact, the main reason we’re interested

in complex numbers (in this course) is that they give us a simple way to add sinusoids.

Phasors

A phasor is a complex number that represents the amplitude and phase angle of a sinusoidal voltage or current.

The phasor’s magnitude r is equal to the sinusoid’s amplitude.

The phasor’s angle is equal to the sinusoid’s phase angle. Example: We use the phasor

V = 1030 V to represent the sinusoid v(t) = 10 cos(200t + 30) V.

Time Domain and Phasor Domain

Some fancy terms: We call an expression like

10 cos(200t + 30) V the time-domain representation of a sinusoid.

We call 1030 V the phasor-domain representation of the same sinusoid. (It’s also called the frequency-domain representation.)

Using Phasors to Add Sinusoids

To add sinusoids of the same frequency:

1. If any of your sinusoids are expressed using sine, convert them all to cosine.

2. Write the phasor-domain version of each sinusoid.

3. Add the phasors (which are just complex numbers).

4. Write the time-domain version of the resulting phasor.

Example of Using Phasors to Add Sinusoids

v1 = 10 cos(200t + 30) V and v2 = 12 cos(200t + 45) V:

Transform from time domain to phasor domain: V1 = 1030 V and V2 = 1245 V .

Add the phasors:1030 V + 1245 V = 21.838.2 V

Transform from phasor domain back to time domain:v1 + v2 = 21.8 cos(200t + 38.2) V

Phasor Relationships for Circuit Elements

We’ve seen how we can use phasors to add sinusoids.

Next we’ll revisit the voltage-current relationships for resistors, inductors, and capacitors, assuming that their voltages and current are sinusoids.

Phasor Relationship for Resistors

For resistors we have, in the time domain:

v = iR Example:

If i = 2 cos(200t + 30) A and R = 5 , then v = 10 cos(200t + 30) V

For this same example, in the phasor domain we have:If I = 230 A and R = 5 , then V = 1030 V

So we can write V = IR.

What This Means

For resistors, if i is a sinusoid, then v will be a sinusoid with the same frequency and phase angle as i.

Therefore i and v reach their peak values at the same instant.

We say that a resistor’s voltage and current are in phase.

Summary for Resistors

In the time domain:

In the phasor domain:

Phasor Relationship for Inductors

For inductors we have, in the time domain:

Example: If i = 2 cos(200t + 30) A and L = 5 H, then v = 2000 cos(200t + 120) V

For this same example, in the phasor domain we have:If I = 230 A and L = 5 H, then V = 2000120 V

So we can write V = jLI.

What This Means

For inductors, if i is a sinusoid, then v will be a sinusoid with the same frequency as i, but i will lag v by 90.

Summary for Inductors

In the time domain:

In the phasor domain:

Phasor Relationship for Capacitors

For capacitors we have, in the time domain:

Example: If v = 2 cos(200t + 30) V and C = 5 F, then i = 2000 cos(200t + 120) A

For this same example, in the phasor domain we have:If V = 230 V and C = 5 F, then I = 2000120 A

So we can write I = jCV.

What This Means

For capacitors, if i is a sinusoid, then v will be a sinusoid with the same frequency as i, but i will lead v by 90.

Summary for Capacitors

In the time domain:

In the phasor domain:

Summary: Textbook’s Table 9.2

A Memory Aid

To remember whether current leads or lags voltage in a capacitor or inductor, remember the phrase

“ELI the ICEman”

(For this to make sense, you must know that E is sometimes used as the abbreviation for voltage.)

Impedance

The impedance Z of an element or a circuit is the ratio of its phasor voltage V to its phasor current I:

Impedance is measured in ohms. Like resistance, impedance

represents opposition to current: for a fixed voltage, greater impedance results in less current.

A Resistor’s Impedance

For resistors, V = IR, so a resistor’s impedance is:

So a resistor’s impedance is a pure real number (no imaginary part), and is simply equal to its resistance.

To emphasize this, we could write

or

Resistors and Frequency

A resistor’s impedance does not depend on frequency, since Z=R for a resistor.

Therefore, a resistor doesn’t oppose high-frequency current any more or less than it opposes low-frequency current.

An Inductor’s Impedance

For inductors, V = jLI, so an inductor’s impedance is:

So an inductor’s impedance is a pure imaginary number (no real part).

To emphasize this, we could write

or

Inductors and Frequency

The magnitude of an inductor’s impedance is directly proportional to frequency, since Z=jL for an inductor.

Therefore, an inductor opposes high-frequency current more than it opposes low-frequency current.

Also, as 0, Z0, which is why inductors act like short circuits in dc circuits.

A Capacitor’s Impedance

For capacitors, I = jCV, so an inductor’s impedance is:

So a capacitor’s impedance is a pure imaginary number (no real part).

To emphasize this, we could write

or

Capacitors and Frequency

The magnitude of a capacitor’s impedance is inversely proportional to frequency, since for a capacitor.

Therefore, a capacitor opposes low-frequency current more than it opposes high-frequency current.

Also, as 0, Z, which is why capacitors act like open circuits in dc circuits.

Impedance, Resistance, and Reactance

Since impedance Z is a complex number, we can write it in rectangular form:

We call the real part (R) the resistance.

We call the imaginary part (X) the reactance.

Impedance, resistance, and reactance are measured in ohms.

Summary

Element Impedance Resistance Reactance

Resistor R R

Inductor jL L

Capacitor

Inductors and capacitors are called reactive elements because they have reactance but no resistance.

Admittance

Recall that conductance, measured in siemens (S), is the reciprocal of resistance:

G = 1 / R

The reciprocal of impedance is called admittance, abbreviated Y:

Y = 1 / Z The unit of admittance is the

siemens.

Admittance, Conductance, and Susceptance

Since admittance Y is a complex number, we can write it in rectangular form:

We call the real part (G) the conductance.

We call the imaginary part (B) the susceptance.

Admittance, conductance, and susceptance are measured in siemens.

Summary

Element Admittance Conductance Susceptance

Resistor G G

Inductor

Capacitor jC

Combining Impedances in Series

The equivalent impedance of series-connected impedances is thesum of the individual impedances:

Thus, series-connected impedances combine like series-connected resistors.

Combining Impedances in Parallel The equivalent impedance

of parallel-connected impedances is given by the reciprocal formula:

For two impedances in parallel we can also use the product-over-sum formula:

Thus, parallel-connected impedances combine like parallel-connected resistors.

Voltage-Divider Rule

As in dc circuits, the voltage-divider rule lets us find the voltage across an element in a series combination if we know the voltage across the entire series combination.

Example: In the circuit shown, and

Current-Divider Rule

As in dc circuits, the current-divider rule lets us find the current through an element in a parallelcombination if we know the current through the entire parallel combination.

Example: In the circuit shown, and

Summary of Chapter 9

We’ve seen that we can apply these familiar techniques to sinusoidal ac circuits in the phasor domain: Ohm’s law () Kirchhoff’s laws (KVL and KCL) Series and parallel combinations Voltage-divider rule Current-divider rule

In each case, we must use complex numbers (phasors) instead of real numbers.

Steps to Analyze AC Circuits

1. Transform the circuit from the time domain to the phasor domain.

2. Solve the problem using circuit techniques (Ohm’s law, Kirchhoff’s laws, voltage-divider rule, etc.)

3. Transform the resulting phasor to the time domain.

What’s Next?

In Chapter 10 we’ll see that we can also apply these other familiar techniques in the phasor domain: Nodal analysis Mesh analysis Superposition Source transformation Thevenin’s theorem Norton’s theorem