EGR 2201

Circuit Analysis

Professor Nick Reeder

Reminders

Please turn off cell phones.

No food or soft drinks in the classroom.

Stow water bottles at floor level.

EGR 2201 Unit 1Basic Concepts

Read Alexander & Sadiku, Chapter 1. Homework #1 and Lab #1 due next

week. Quiz next week.

What This Course Is About In this course you’ll learn

mathematical techniques for studying electric circuits.

Our focus is not on practical circuits that do interesting things.

You’ll study those in later courses, using the techniques that you learn in this course.

The Math That We’ll Use

Calculator or Math Software Some of the math we’ll do is time-

consuming with a basic calculator. Examples: TI-30 or Casio fx-115

It’s faster with a powerful calculator that can solve systems of linear equations and manipulate complex numbers.

Examples: TI-86 or TI-89 You can use any calculator on exams, but no

cell phones. Recommendation: Learn one calculator and

use it for all homework and exams. Another option: Use MATLAB software,

which you may have used in other courses.

Calculator or Math Software (2) Be aware of the calculator policy for the

Fundamentals of Engineering exam and the Principles and Practice of Engineering exam, administered by the National Council of Examiners for Engineering and Surveying (NCEES).

In a few years you may decide to take these exams for professional advancement.

What is a Circuit? Our book’s definition (page 4):

An electric circuit is an interconnection of electrical elements.

Five electrical elements that we’ll focus on:

Resistors Capacitors Inductors Voltage Sources Current Sources

Example Circuit: A Power Supply from a Flat-Screen Television

Resistor

CapacitorsInductor

Schematic Diagrams To discuss circuits, we draw

schematic diagrams that represent those circuits.

Schematic diagrams do not show the parts of the circuit as they actually look. Instead, they contain standard symbols that represent electrical elements.

Example Schematic Diagram: A Radio Transmitter (from book’s page 4)

Resistor Symbol

Capacitor Symbol

Inductor Symbol

A Simpler Example Schematic Diagram: Flashlight

Light Bulb

Battery (Voltage Source)

Switch

When the switch is open (as drawn), no current flows, so the bulb is dark.

When the switch is closed, current flows, and the bulb lights.

Another Simple Example: A Voltage Source And Two Resistors

Polarity of a Battery

Positive terminal

Negative terminal

Note that the symbol for a battery is asymmetric. The end with the longer line represents the battery’s positive terminal, and the other end represents its negative terminal.

+

Direction of Current Flow For historical reasons, we say that in our

simple circuit current flows out of the battery’s positive terminal and into its negative terminal.

Modern science tells us that electrons actually move in the opposite direction, but we’ll follow the standard convention shown above.

The schematic diagrams so far have been incomplete. They show what kind of elements are in

the circuit and how those elements are connected to each other.

But they do not show numerical ratings that let us quantify the circuit’s behavior.

Every voltage source has a numerical rating in volts (V).

Every resistor has a numerical rating in ohms ().

Element Ratings

Examples of Voltage Sources What is the rating of these sources?

Flashlight battery ____ V

Wall outlet ____ V

But the battery is a DC voltage source, while the outlet is an AC voltage source.

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.

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 voltage source

Schematic Symbols for Independent Voltage Sources

Several different symbols are commonly used for voltage sources:

V or v?

Some authors use uppercase letters for constant quantities, such as V for the voltage of a constant DC voltage source.

And they use lowercase letters for time-varying quantities, such as v for the voltage of an AC voltage source.

Our textbook mentions this convention on pages 7 and 10, but usually uses lowercase letters for both constant and time-varying quantities.

DC Voltage Sources on Our Trainer

Fixed +5 V voltage source

Variable positive voltage source, controlled by the left-hand knob. We’ll usually use this one.

No matter which redsocket you use, youmust also use the GROUND socket.

Fixed -5 V voltage source

Variable negative voltage source, controlled by the right-hand knob.

Using a Digital Multimeter to Measure Voltage

We’ll use a digital multimeter, like the Fluke 45 shown, to measure voltage.

Note that the meter has a red lead and a black lead. See next slide ….

Meter’s Red and Black Leads

When you measure a voltage, the order of the red and black leads determines whether the value is displayed as positive or negative.

Meter will display 5.00 V Meter will display 5.00 V

Resistance

Resistance is opposition to the flow of electrons.

Resistance’s unit of measure is the ohm ().

A perfect conductor would have zero resistance and a perfect insulator would have infinite resistance.

A resistor is a device manufactured to have a specific amount of resistance.

Resistor Ratings

The resistors in our labs range in value from 10 to 10,000,000 .

Instead of having the value printed in numbers on the case, our resistors are marked with a four-band color code to indicate the value.

Resistor Color Code

The first three color bands specify the resistance’s nominal value.

Digit Color

0 Black

1 Brown

2 Red

3 Orange

4 Yellow

5 Green

6 Blue

7 Violet

8 Gray

9 White

Resistor Color Code (2)

The fourth band (“tolerance band”) gives the percent variation from the nominal value that the actual resistance may have.

Many websites have color-code charts and calculators, such as this one.

Tolerance Color

5% Gold

10% Silver

20% None

Tolerance Calculations

To find a resistor’s tolerance in ohms, multiply its nominal value by the percentage tolerance.

Example: For a 220 resistor with 5% tolerance, the tolerance in ohms is

220 0.05 = 11 .

Then…

Tolerance Calculations (2)

To find the minimum value that a resistor can have, subtract its tolerance in ohms from its nominal value.

In example above, the nominal value was 220 and the tolerance was 11 . So the minimum value is

220 11 = 209 . To find the maximum value that a resistor

can have, add its tolerance in ohms to its nominal value.

In example above, the maximum value is 220 + 11 = 231 .

Using a Digital Multimeter to Measure Resistance

Digital multimeters can measure resistance as well as voltage.

When measuring a resistor’s resistance, the resistor must be out of circuit: definitely no power applied and disconnected from other elements.

Selecting the Measurement Type on the Digital Multimeter

DC Voltage DC Current Resistance

AC Voltage AC Current

Plugging the Meter’s Leads into the Jacks

Black lead always in this

jack.

Red lead here to measure voltage or resistance.

Red lead here to measure current.

These two circuits will perform differently. In particular, the different element ratings will result in: Different current values Different voltage values

Same Circuit Layout, but Different Element Ratings

Current

Current is the flow of electric charge through a circuit.

We use the symbol I or i to represent current.

Current’s unit of measure is the ampere, or amp (A).

For example, To say that a current is 2.5 amperes, we

write

I = 2.5 A or i = 2.5 A

Voltage

Voltage is a measure of how forcefully charge is being pushed through a circuit.

We use the symbol V or v to represent voltage.

Voltage’s unit of measure is the volt (V).

For example, To say that a voltage is 5 volts, we write

V = 5 V or v = 5 V

Summary of Some Electrical Quantities, Units, and Symbols

Quantity Symbol SI Unit Symbol for the Unit

Current I or i ampere A

Voltage V or v volt V

Resistance R ohm

Plumbing Analogy

It may help to think of a circuit as being like a plumbing system, with water flowing through pipes.

On this analogy, voltage is like the water pressure in a pipe. Its value will be different at different points in the circuit.

Current is like the volumetric flow rate through a pipe.

See Wikipedia article on Hydraulic analogy.

Plumbing Analogy in Our Simple Circuit

A voltage source is like a water pump. Its voltage rating (in volts)tells you how strong it is.

Resistors are like partial blockages in the pipe. They restrict the amount of current that flows through the circuit.

A wire is like a water pipe. The amount of electricity per second flowing through a wire is the current, which is measured in amperes.

The voltage (pressure) at this point is greater than the voltage at this point.

This course’s main goal: to learn how, given the schematic diagram of a circuit, to compute the voltages and currents in the circuit.

The Goal of Circuit Analysis

For some circuits, such as this one, the math is simple (basic algebra).

More complicated circuits require more powerful math (trig, complex numbers, calculus, differential equations…).

Large and Small Numbers

We must often deal with very large or very small numbers.

Example: a resistor might have a resistance of 680,000 and a current of 0.000145 A.

It’s not convenient to use so many zeroes when writing or discussing numbers. Instead we use SI prefixes (or engineering prefixes), which are abbreviations for certain powers of 10.

Table 1.2

We rarely use these.

1,000,000,000,0001,000,000,0001,000,0001,000

1 / 1,0001 / 1,000,0001 / 1,000,000,0001 / 1,000,000,000,000

Engineering Prefix Game

You must memorize these prefixes. To practice, play my

Metric Prefix matching game at http://people.sinclair.edu/nickreeder/flashgames.htm.

You must also be able to convert between numbers written with engineering prefixes and numbers written in everyday (floating-point) notation. To practice, play my

Engineering- Notation game.

Using Engineering Prefixes

Whenever you have a number that’s greater than 1000 or less than 1, you should use these prefixes.

Examples: Instead of writing 680,000 ,

write 680 k (pronounced “680 kilohms”).

Instead of writing 0.000145 A, write 145 A (pronounced “145 microamps”).

Calculator’s Exponent Key

Scientific calculators have an exponent key (usually labeled EE, EXP, or E) that lets you easily enter numbers with engineering prefixes.

Examples: To enter 680 k, press 680 EE 3. To enter 145 , press 145 EE −6.

Calculator’s Engineering Mode

Most scientific calculators also have an engineering mode, which forces the answer always to be displayed with one of the engineering powers of 10.

Learn how to use this feature of your calculator. It will save you from making mistakes.

Measuring Voltage

A voltmeter is an instrument designed to measure voltage (also called potential difference).

Voltage measurements are always made across elements.

To measure avoltage in a circuit, you don’t need to disconnect any elements.

Measuring the voltage across R1.

Positive or Negative Voltage?

When you measure a voltage, the displayed value may be positive or negative.

In the drawing, the meter’s + lead is connected to point a and its – lead to point b. To indicate this, wewould say that we’re measuring vab.

If we swapped the leads, we’d be measuring vba. These two voltages, vab and vba, have the same

magnitude but different signs. Example: If vab = 1.60 V, then vba must be 1.60 V.

a

b

Voltage Drops and Rises

If vab = 1.60 V, wesay that there’s a voltage drop of1.60 V from point a to point b.

Equivalently, we say that there’s a voltage rise of 1.60 V from point b to point a.

Though it may seem confusing, we could also say that there’s a voltage rise of 1.60 V from point a to point b, or that there’s a voltage drop of 1.60 V from point b to point a.

a

b

Measuring Current

An ammeter is an instrument designed to measure current.

To measure the current at a point, you must break the circuit at that point and insert the ammeter in such a way that the current flowsthrough the ammeter.

Measuring current.

Positive or Negative Current?

When you measure a current, the displayed value may be positive or negative.

Note that in thedrawing, the meter’s + lead is connected to the battery and its – lead to R1.

The displayedvalue is the current flowing into the + lead and out of the – lead.

Positive or Negative Current? (2)

As with voltage measurements, swapping the leads would give the same magnitude but opposite sign. Example: If the meter displays 34.0 mA when

connected as shown, then it would display 34.0 mA if you swapped the leads.

We could express this by saying either that a current of 34.0 mA flowsfrom V1 to R1 (clockwise),or that a current of 34.0 mA flows from R1 to V1 (counter-clockwise).

Measuring Resistance

An ohmmeter is an instrument designed to measure resistance.

To measurean element’s resistance, you must removethe element from the circuit.

When measuring resistance, the meter will never display a negative value.

Measuring R1’s resistance.

Multimeter

A multimeter can measure voltage, current, or resistance, depending on the setting of a selector switch.

A multimeter must not be set to measure current when it is connected as a voltmeter, or set to measure voltage when it is connected as an ammeter.

Multimeter Challenge Game

You must learn how to use a multimeter.

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

Three that we have discussed:

Four new ones:

Some Quantities and Their Units

Quantity Symbol SI Unit Symbol for the Unit

Current I or i ampere A

Voltage V or v volt V

Resistance R ohm

Quantity Symbol SI Unit Symbol for the Unit

Charge Q or q coulomb C

Time t second s

Energy W or w joule J

Power P or p watt W

Charge

All electrical phenomena are based on the movement or separation of electric charge.

We don’t often measure charge directly, but sometimes we need to calculate it.

The symbol for charge is Q or q. Charge’s unit of measure is the coulomb

(C). For example,

To indicate a charge of 450 microcoulombs, we write

Q = 450 µC or q = 450 µC

Basic Facts About Charge

There are two kinds of charge, which we call positive and negative.

Opposite charges attract. Like charges repel. The smallest known charge is the

charge on a proton or an electron, 1.602 × 10-19 C. Most practical charges that we deal with are much larger than this—for example, nanocoulombs (nC) or microcoulombs (µC).

Formal Definition of Current

We’ve seen that current can informally be thought of as being like the flow rate of water through a plumbing system.

More formally, current is defined as the rate of change of charge per time:

Thus, one ampere is equal to one coulomb per second (1 A = 1 C/s).

dt

dqi

Differentiation and Integration

Recall that differentiation and integration are inverse operations.

Therefore, any relationship between two quantities that can be expressed in terms of derivatives can also be expressed in terms of integrals.

Charge and Current We saw above that current is

the derivative with respect to time of charge:

Therefore charge is the integral with respect to time of current:

In typical problems, we know the initial charge at time t0 and wish to find the charge at later time t. In such cases we use the definite integral:

dt

dqi

dtiq

)()( 0

0

tqdtitqt

t

Calculus or Algebra? As we’ve seen, the equations relating charge and

current contain derivatives and integrals:

Some problems involving current and charge therefore require calculus. (For example, Problems 1.2 and 1.3 in the textbook.)

But for many problems—in particular, problems in which current is constant—these equations simplify to algebraic equations:

dt

dqi dtiq

t

qi tiq

Energy

Energy is perhaps the most fundamental physical concept, underlying all areas of physics.

We don’t often measure energy directly, but sometimes we need to calculate it.

The symbol for energy is W or w. Energy’s unit of measure is the joule (J). For example,

To indicate an energy of 780 nanojoules, we write

W = 780 nJ or w = 780 nJ

Formal Definition of Voltage

We’ve seen that voltage can informally be thought of as being like water pressure in a plumbing system.

More formally, the voltage between two points is defined as the amount of energy needed to move a unit charge from one point to the other:

Thus, one volt is equal to one joule per coulomb (1 V = 1 J/C).

dq

dwv

Power At any time, some elements in a circuit

supply energy, and some elements absorb energy.

An element’s power is the rate at which that element supplies or absorbs energy.

The symbol for power is P or p:

Power’s unit of measure is the watt (W). One watt is equal to one joule per second (1 W = 1 J/s).

dt

dwp

Supplies energy

Absorb energy

Energy and Power We saw above that power is

the derivative with respect to time of energy:

Therefore energy is the integral with respect to time of power:

In typical problems, we know the initial energy at time t0 and wish to find the energy at later time t. In such cases we use the definite integral:

dt

dwp

dtpw

)()( 0

0

twdtptwt

t

Calculus or Algebra? As we’ve seen, the equations relating energy and

power contain derivatives and integrals:

Some problems involving power and energy therefore require calculus.

But for many problems—in particular, problems in which power is constant—these equations simplify to algebraic equations:

dt

dwp dtpw

t

wp tpw

Positive or Negative Power?

By convention, we assign a positive sign to a power value if the element is absorbing energy, and we assign a negative sign if the element is supplying energy.

For example, To say that an element is absorbing 50

milliwatts, we could write

P = 50 mW or p = 50 mW

To say that an element is supplying 250 milliwatts, we could write

P = 250 mW or p = 250 mW

Kilowatt-hours

We’ve seen that in the SI system of units, energy is measured in joules (J) and power is measured in watts (W), with

1 J = 1 W 1 s But in the electrical power industry,

the unit of power most often used is the kilowatt (kW), and the unit of energy used is the kilowatt-hour (kWh).

1 kWh = 1 kW 1 hour

The Power Law

We now have the following definitions:

But the chain rule of calculus tells us that :

Therefore we can write: In words, an element’s power is equal to

its voltage times its current.

dt

dwp

dt

dqi

dq

dwv

dt

dq

dq

dw

dt

dw

vip

The Passive Sign Convention

To get the correct sign (+ or ) on the power value when we use the power law (p=vi), we must be careful with the signs of v and i.

We’ll always follow thepassive sign convention, which says that we regard the positive direction for current as being current into an element’s positive terminal.

Conservation of Energy

Any circuit must obey the law of conservation of energy.

Therefore the algebraic sum of the powers in a circuit must equal 0. Recall that an energy supplier’s power is

negative, while an energy absorber’s power is positive.

Example: In the circuit shown, if we know that the voltage source’s power is 100 mW, and R1’s power is 75 mW, then what must R2’s power be?

Supplies energy

Absorb energy

Review: Some Quantities and Their Units

Quantity Symbol SI Unit Symbol for the Unit

Current I or i ampere A

Voltage V or v volt V

Resistance R ohm

Charge Q or q coulomb C

Time t second s

Energy W or w joule J

Power P or p watt W

Active Elements

Circuit elements can be classified as active or passive, depending on whether they are capable of generating electric energy.

Active elements can generate electric energy. Examples:

Voltage sources Current sources

Passive Elements

Passive elements cannot generate electric energy. Examples:

Resistors Capacitors Inductors

An important difference among these is that capacitors and inductors can store energy for later use.

Resistors cannot store energy: they always dissipate energy as heat.

Ideal Sources

The most important active elements are voltage sources and current sources.

In each case the word “ideal” means that these are simplified models that ignore some of the effects present in real sources.

Ideal Independent Voltage Source

An ideal independent voltage source maintains a specified terminal voltage no matter what the rest of the circuit looks like.

We’ve discussed these previously.

The book’s Figure 1.11 shows two symbols for ideal independent voltage sources.

Ideal Independent Current Source

An ideal independent current source supplies a specified current no matter what the rest of the circuit looks like.

The arrow identifies itas a current source and shows the direction of positive currentflow.

Ideal Dependent Voltage Source

An ideal dependent voltage source maintains a terminal voltage whose value depends on a voltage or current somewhere else in the circuit.

The diamond-shaped body tells us that it’s a dependent source.

The +/- inside tells us that it’s a voltage source, and shows the voltage polarity.

Ideal Dependent Current Source

An ideal dependent current source supplies a current whose value depends on a voltage or current somewhere else in the circuit.

The diamond-shaped body tells us that it’s a dependent source.

The arrow inside tells us that it’s a current source and shows the direction of current flow.

Summary of Symbols for Ideal Sources

Ideal dependent

current source

Ideal dependent

voltagesource

Ideal independent

current source

Ideal independent

voltagesource

Four Kinds of Dependent Sources

A dependent source’s value depends on a voltage or current somewhere else in the circuit, giving rise to four kinds: A voltage-controlled voltage source. A current-controlled voltage source. A voltage-controlled current source. A current-controlled current source.

Text next to the symbol will let you tell exactly which kind it is….

Examples of Symbols for Controlled (Dependent) Sources

Current-controlledcurrent source

Voltage-controlledvoltagesource

Current-controlledvoltagesource

Voltage-controlledcurrent source

5v 5i 5v 5i

Example of a Controlled Source in a Schematic Diagram

If i in this circuit is equal to 2.5 A, then the dependent voltage source’s value is 25 V.