Circuit Theory - Wikimedia

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Circuit Theory/All Chapters 1

Circuit Theory/All Chapters

Circuit TheoryWikibooks: The Free Library

Preface

This wikibook is going to be an introductory text about electric circuits. It will cover some the basics of electric

circuit theory, circuit analysis, and will touch on circuit design. This book will serve as a companion reference for a

1st year of an Electrical Engineering undergraduate curriculum. Topics covered include AC and DC circuits, passive

circuit components, phasors, and RLC circuits. The focus is on students of an electrical engineering undergraduate

program. Hobbyists would benefit more from reading Electronics instead.

This book is not nearly completed, and could still be improved. People with knowledge of the subject are encouraged

to contribute.

The main editable text of this book is located at http:// en. wikibooks. org/ wiki/ Circuit_Theory. The wikibooks

version of this text is considered the most up-to-date version, and is the best place to edit this book and contribute to

it.

Electric Circuits IntroductionThe theory of electrical circuits can be a complex area of study. The chapters in this section will introduce the reader

to the world of electric circuits, introduce some of the basic terminology, and provide the first introduction to passive

circuit elements.

Introduction

Who is This Book For?

This book is intended to supplement a first year of electrical engineering exposition for college students. However,

any reader with an understanding of math and differential calculus can read this book and understand the material. A

knowledge of Integral Calculus, Differential Equations, or Physics (especially of forces, fields, and energy) will

provide extra insight into the material, but are not necessary. This book will expect the reader to have a firm

understanding of Calculus specifically, and will not stop to explain the fundamental topics in Calculus.

For information on Calculus, see the wikibook: Calculus.
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What Will This Book Cover?

This book will attempt to cover linear circuits, and linear circuit elements. We will start off discussing some of the

basic building blocks of circuits (wires and resistors), and we will discuss how to use them and how to analyze them.

We will then move into chapters on Capacitors and Inductors. The second half of this book will then start to talk

about AC power, and will go into some basic techniques for understanding and solving AC circuits. This book will

discuss the Laplace and Fourier Transforms, but will not cover them completely, opting instead to let later books inthe series (specifically the Signals and Systems book) cover them in depth.

Where to Go From Here

With a basic knowledge of electric circuits and electricity concepts under your belt, there are a number of different

paths available for study.

For a further discussion of related materials, see the Electronics wikibook.

To begin a course of study in Computer Engineering, see the Digital Circuits wikibook.

For the next step in Electrical Engineering theory, see the Signals and Systems wikibook.

There are certainly other paths to be taken from here, depending on interest; however, wikibooks currently lacksinformation in these fields of study.

Basic Terminology

Basic Terminology

There are a few key terms that need to be understood at the beginning of this book, before we can continue. This is

only a partial list of all terms that will be used throughout this book, but these key words are important to know

before we begin the main narrative of this text.

Time domain

The concept of a domain is very important in mathematics, and is also very important in engineering.

Depending on what domain you are in, there are different tools and techniques for analyzing circuits. The

"Time domain" is simply another way of saying that our circuits change with time, and that the major variable

used to describe the system is time. Another name is "Temporal".

Frequency domain

The frequency domain is a very commonly used method of describing the behavior of a circuit as functions of

the frequency of the signals within it. Another name is the "Fourier domain". Other domains that an engineer

might encounter are the "Laplace domain" (or the "s domain"), and the "Z domain". Sometimes is also

indicated with the word "Spectral"

Circuit Response

Circuits generally have inputs and outputs. In fact, it is safe to say that a circuit isn't useful if it doesn't have

one or the other (usually both). The response to a circuit is the relationship between the circuit's input to the

circuit's output. The circuit response may be a measure of either current or voltage.

Steady State

When something changes in a circuit, there is a certain amount of transition period before a circuit "settles

down", and reaches its final value. This final value, when all elements have a constant or periodic behaviour, is

known as the steady-state value of the circuit. The circuit response at steady state (when things aren't

changing) is also known as the "steady state response".

Transient Response
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Energy

Energy is measured most commonly in Joules, which are abbreviated with a "J" (upper-case J). The variable most

commonly used with energy is "w" (lower-case W). This book will not talk much about energy, although the Modern

Physics wikibook will. Refer to that book for more information.

Electric Circuit Basics

Circuits

Circuits (also known as "networks") are collections of circuit elements and wires. Wires are designated on a

schematic as being straight lines. Nodes are locations on a schematic where 2 or more wires connect, and are usually

marked with a dark black dot. Circuit Elements are "everything else" in a sense. Most basic circuit elements have

their own symbols so as to be easily recognizable, although some will be drawn as a simple box image, with the

specifications of the box written somewhere that is easy to find. We will discuss several types of basic circuit

components in this book.

Ideal Wires

For the purposes of this book, we will assume that an ideal wire has zero total resistance, no capacitance, and no

inductance. A consequence of these assumptions is that these ideal wires have infinite bandwidth, are immune to

interference, and are in essence completely uncomplicated. This is not the case in real wires, because all wires

have at least some amount of associated resistance. Also, placing multiple real wires together, or bending real wires

in certain patterns will produce small amounts of capacitance and inductance, which can play a role in circuit design

and analysis. This book will assume that all wires are ideal.

Ideal Nodes

Schematic representation of wire crossing with (a) and

without (b) connection between the wires. The third

style is preferred.

Like ideal wires, we assume that connecting nodes have zero

resistance, et al. Nodes connect two or more wires together. On a

schematic, nodes are frequently denoted with a small filled-in

black dot. When 2 wires cross on a schematic, but they do not

physically intersect (for instance if one wire lays on top of another

wire), there is no node drawn.

The diagram on the right shows three ways of drawing the

interesection of two wires with connection (a) and no connection

(b). The modern convention is to use the third style.In real life, nodes are often connected together, either by wire nuts,

or solder, or other connectors. These connectors can have a certain

amount of associated resistance, capacitance, or inductance

associated with them. This book will not, however, take this

interference into account, as it is usually negligible.
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Active vs Passive

The elements which are capable of delivering energy are called "Active elements".The elements which will receive

the energy and dissipate or store it are called "Passive elements".

Voltage and Current Generators are examples of active elements that can deliver the energy from one point to some

other point. These are generally considered independent generators of electric energy. From a system point of view,

this is not an accurate depiction since the energy output will be directly related to the energy put into the system or

stored in the system previously. Some examples of these generators are alternators, batteries etc...

A previous definition stated; "A dependent source will generate current or voltage but the energy output will depend

on some other individual parameter(may be voltage or current) in the same circuit, whereas an independent source

will generate regardless of the connections of the circuit."

From a localized perspective, this definition can still be useful. This definition can be used to differentiate a power

source ("independent source") from an active power control device, or amplifier ("dependent source"). It is probably

more useful to think of "dependent sources" as "energy amplifiers" or "active devices".

The three linear passive elements are the Resistor, the Capacitor and the Inductor. Examples of non-linear passive

devices would be diodes, switches and spark gaps. Examples of active devices are Transistors, Triacs, Varistors,Vacuum Tubes, relays, solenoids and piezo electric devices.

Open and Closed Circuits

A closed circuit is one in which a series of device(s) complete a connection between the terminals, and charge is

allowed to flow freely.

An open circuit is a section of a circuit for which there is no connection. Current does not flow between the

terminals of an open circuit, although a voltage may be applied between the terminals, and a capacitance may exist

between them. At steady state, there is no current flow in an open circuit, and most examples will assume that there

is no capacitance between nodes of an open circuit, for simplicity.

"Shorting" an element

We will often hear the term "shorting an element" in later chapters of circuit analysis. Shorting a circuit is

equivalent to placing a wire across the terminals of the element. Because current takes all paths available, shorting an

element allows some current to bypass the element. The amount of current taking the short path is inverse to that

path's resistance. If an ideal wire with effectively no resistance is used, all the current flows through the short and no

current flows through the device . This practice must be done with care, because reducing the resistance of a certain

portion of a circuit to zero can theoretically raise the current to infinity, and the circuit will become damaged.


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Measuring instruments

Voltmeters and Ammeters are devices that are used to measure the voltage across an element, and the current

flowing through a wire, respectively.

Ideal Voltmeters

An ideal voltmeter has an infinite resistance (in reality, several megaohms), and acts like an open circuit. A

voltmeter is placed across the terminals of a circuit element, to determine the voltage across that element.

Ideal Ammeters

An ideal ammeter has zero resistance (practically, a few ohms or less), and acts like a short circuit. Ammeters are

placed in-line in a circuit, so that all the current from one terminal flows through to the other terminal. By

convention, current into the + terminal is displayed as positive.

Sources

Sources come in 2 basic flavors: Current sources, and Voltage sources. These sources may be further broken down

into independent sources, and dependent sources.

Current Sources

Current sources are sources that output a specified amount of current. The voltage produced by the current source

will be dependent on the current output, and the resistance of the load (ohm's law).

Voltage Sources

Voltage sources produce a specified amount of voltage. The amount of current that flows out of the source is

dependent on the voltage and the resistance of the load (again, ohm's law). This can be dangerous because if avoltage source is shorted (a resistance-less wire is placed across its terminals), the resulting current output

approaches infinity! No voltage source in existence can output infinity current, so the source will usually melt or

explode long before it reaches that value. This is an important point to keep in mind, however.

An example of a voltage source is a battery, which is specified as being "9V" or "6V" or something similar. The

amount of current that the circuit draws from the battery determines how long the "battery life" is.

Ideal Op Amps

Op amps, (short for operational amplifiers) is an active circuit component. We will not discuss the internals of op

amps in this wikibook, but will instead only consider the ideal case. Op amps have 2 input terminals and 1 output

terminal.

Ideal op amps are governed by some very simple rules that allow an engineer to solve a circuit without having to

know exactly how an op amp does what it does. These rules are enumerated as follows.

1. Op Amp has infinite impedance

2. Op Amp has infinite bandwidth

3. Op Amp has infinite voltage gain

4. Op Amp has Zero output impedance

5. Op Amp has Zero offset (error)

We will consider an op amp with 2 inputs (x and y), and an output (z).

1. There is 0 voltage difference between the terminals x and y.

2. There is 0 current flowing on terminals x and y.


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3. There is 0 current flowing on terminal z.

Independent Sources

Independent sources produce current/voltage at a particular rate that is dependent only on time. These sources may

output a constant current/voltage, or they may output current/voltage that varies with time.

Dependent Sources

Dependent sources are current or voltage sources whose output value is based on time or another value from the

circuit. A dependent source may be based on the voltage over a resistor for example, or even the current flowing

through a given wire. The following sources are possible:

Current-controlled current source

Current-controlled voltage source

Voltage-controlled current source

Voltage-controlled voltage source

Dependent sources are useful for modelling transistors or vacuum tubes.

Turning Sources "Off"

Occasionally (specifically in Superposition) it is necessary to turn a source "off". To do this, we follow some general

rules:

1. Dependent sources cannot be turned off.

2. Current Sources become an Open Circuit when turned off.

3. Voltage Sources become a Closed Circuit when turned off.

Occasionally it is written that the source is "removed" from the circuit, because often it is physically possible to

remove the source component (be it a battery, or a plug, or any other source component) physically from the circuit.

We can't inactivate these sources.


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Source Warnings

The following image shows some configurations of current and voltage sources that are not permissible, and will

cause a problem in your circuit:

Switches

A switch then is a circuit element that is an open-circuit for all time , and acts like a closed-circuit for all

time .

Unit Step Function

Before talking about switches, we will introduce the Heaviside step function (also known as the unit step

function). The step function is defined piecewise as such:

[Unit Step Function]

This function provides a mathematical model for electrical engineers to describe circuit elements that change

between boolean states (on/off, high/low, etc.).
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Transducers

A transducer is a circuit component that transforms electrical energy into another type of energy. Some examples of

transducers are actuators and motors.

Resistors and Resistance

Resistors

Resistors are circuit elements that allow current to pass through them, but restrict the flow according to a specific

ratio called "Resistance". Flow that is restricted by resistors is said to be "lost to the resistor". Resistors are

commonly used as heating elements, because energy lost to the resistor is frequently dispersed into the surroundings

as heat. Every resistor has a given resistance. Resistors that have a variable resistance as a function of position are

known as "potentiometers". Resistors that have a variable resistance as a function of temperature are called

"thermisters".

Function of Temperature

Resistance also depends on surrounding temperature. It is defined as:

Where:

Rtis Final resistance,

Ro

is Initial resistance,

a is Temperature coefficient,

Tis Temperature

For most cases, especially in this book, we will treat resistance as being a constant, and not a function of time and

temperature.

Resistance

Resistance is measured in terms of units called "Ohms" (volts per ampere), which is commonly abbreviated with the

Greek letter . Ohms are also used to measure the quantities of impedance and reactance, as described in a later

chapter. The variable most commonly used to represent resistance is "r" or "R".

Resistance is defined as:

where is the resistivity of the material, L is the length of the resistor, and A is the cross-sectional area of the

resistor.
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Conductance

Conductance is the inverse of resistance. Conductance has units of "Siemens" (S), sometimes referred to as mhos

(ohms backwards, abbreviated as an upside-down ). The associated variable is "G":

Conductance can be useful to describe resistors in parallel, since the sum of conductances is equal to the equivalent

conductance. However, conductance is rarely used in practice, and it is outside of the scope of this textbook.

Ohm's Law

Image 1

A simple circuit diagram relating current,

voltage, and resistance

Ohm's law is a fundamental tenet of Electrical Engineering, and it is a

building block of circuit analysis techniques. Without a knowledge of Ohm's

law, the remainder of this wikibook will not be possible.

From image 1, we can relate the values R, V, and I with the following

equation:

[Ohm's Law]

In plain English, Ohm's law relates the voltage drop across a resistive

element to the current flowing through the element and its resistance. It is

important to note that across a resistive element, the voltage drops, whereas

across a voltage source, the voltage increases. Sometimes it is important to

denote that the voltage in ohm's law is a negative voltage to correspond to the voltage drop, although frequently it is

sufficient to remember that this is a drop, and not an increase.

Ohm's law is fundamental and axiomatic. We can accept it without proof.

Resistive CircuitsWe've been introduced to passive circuit elements such as resistors, sources, and wires. Now, we are going to

explore how complicated circuits using these components can be analyzed.

Resistive Circuit Analysis Techniques

Resistive Circuit Analysis

There are certain established rules that can be used to examine a circuit that is comprised entirely of resistors and

sources. Among these are tools for combining resistors in certain configurations into a single conceptual resistor that

has an equivalent resistance to the two that have been replaced. In this manner, very complicated circuits can be

reduced to be a very simple circuit with few components.

The Resistor

The resistor is an electrical component that limits the flow of electrical current. The physical characteristics of a

resistor are described in the Electronics Wikibooks and information on resistors as found in real life are in Practical

Electronics.

The v-i characteristic of a resistor is simply a straight line, passing through the origin:
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We call the gradient of this line the resistance of the element. It is defined as the voltage developed across the

element per ampere of current through it. An ideal resistor therefore obeys Ohm's Law:

The resistance is given the symbol "R" and is the unit ofohm which is denoted by a capital omega, . It is named

after Georg Ohm so the word "ohm" is always lowercase (unless it begins the sentence) - this is the standard for SI

units. The ohm is given in base units by:

In real life, resistors are generally made of carbon or metal films or ceramic, depending on the application and

desired accuracy. Resistance is always positive for such devices. Negative resistances are possible, but they are not

found in resistors.

Power in a Resistor

The power,P, dissipated in a resistor is given by

And from Ohm's law it can be seen that, since ,

Similarly, since ,

This means that the power is the product of a square and a resistance, both of which must be positive. Therefore, the

power dissipated in a resistor is always positive.
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Conductance

The resistor is an invertible component. This means that we can rearrange the v-i characteristic into an i-v

characteristic:

The constant of proportionality (1/R) is called the conductance and is generally given the symbol "G":

Conductance has the unit siemens, named after the German scientist Ernst Werner von Siemens. The symbol for

siemens is S. Conductance is expressed in base units as:

A siemens is therefore the current in the resistor per unit voltage across it. As with all SI units named after people,

the word "siemens" is lowercase, but the symbol is uppercase (S).

In older texts, and oftentimes in handwritten calculations, the older symbol Mho ( : an upside-down capital

Greek letter Omega) is used because an uppercase (S) is too easily confused with the following: lowercase (s) for

"seconds"; a variable "S"; and lastly, the numeral (5).

Real Resistors

Resistors tend to heat up as they pass more current, due to an increase in dissipated power. Since usually the

resistance of a material changes with temperature, this produces a distorted v-i graph:

Non-ideal resistor value with nonlinear resistance

characteristic

This generally has a very complicated v-i characteristic, dependent of the thermal coefficient (how the resistance

changes with temperature), the ability of the resistor to dissipate heat into its surroundings (which is itself dependent

on many things) as well as the nominal resistance and the applied voltage or current.
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Equivalent Resistances

An unknown circuit element can be modeled as a single resistance if it has a directly proportional v-i relationship.

This resistance is called the equivalent resistance, and is often written as Req

.

In order to determine the value of the equivalent resistance, either a voltage can be applied to the terminals of the

unknown element, and the resulting current measured, or a constant current can be applied, and the resulting voltage

measured.

The equivalent resistance is then given by

Since a resistive element has a straight v-i characteristic, only one measurement is needed. Note that if the element is

not purely resistive, this method will give an erroneous result.
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Degenerate Resistors

Short Circuit

Consider the v-i characteristic for a resistor:

If the resistance is zero,

This limiting case is called a short-circuit. We can consider a short circuit to be a voltage source with a zero value.

Just as current in a voltage source is arbitrary (depends completely on the rest of the circuit), so is the current in a

short circuit (whatever i is, when multiplied by the zero in the v-i characteristic, it will be zero). We therefore have

the following equivalence:

Open Circuit

Consider the i-v characteristic for a resistor:

If the resistance tends to infinity, we have

This limiting case is called an open-circuit. We can consider an open circuit to be a current source with a zero value.

Just as voltage across a current source is arbitrary (depends completely on the rest of the circuit), so is the voltage

across an open circuit. We therefore have the following equivalence:
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Elements in Series

Circuit elements are said to occur in series when they are directly connected end-to-end with no branching nodes in

between them. Circuit elements are in parallel if they share a common starting and ending node.

There are two formulas that apply to series and parallel combinations of passive circuit elements. Which formula

applies depends on the element and the combination.

1.

2.

Where x is the quantity being considered. The first equation applies to series resistors, while the second applies to

parallel resistors. The same formula are applicable for inductors when their mutual inductance is neglected. Forcapacitors, the formula have to be interchanged.

Resistors

Resistors appearing in series can be converted into a single resistor, according to the following equation:

Resistors in a series configuration

[Resistors In Series]

Where n is the number of resistors in series. Multiple resistors appearing in series then, can be converted

conceptually into a single resistor whose resistance is simply the sum of the parts.
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Voltage Sources

Voltage sources given in series can be added together to form a single source, given by the equation:

It is to be noted that the magnitude of voltage should be a signed integer depending on the polarity of the voltage

source. The convention for polarity can vary from one reference to another. But usually the voltage source whosepositive terminal is connected to other element in the assumed direction of the circuit is considered as positive. If the

aforementioned condition is with the negative terminal, the polarity is considered to be negative

Current Sources

Current sources may not appear in series, as doing so would violate Kirchoff's Current Law (explained below).

Elements in Parallel

This section will talk about how to condense circuit elements that exist in parallel. "Parallel" is defined as elements

that share common endpoints.

Resistors

If multiple resistors are parallel to each other, we can calculate out the conceptual resultant resistor as follows:

A diagram of several resistors appearing in

parallel with each other.

[Resistors in Parallel]

In the special case of 2 resistors in parallel the following notation is used:

.

Voltage Sources

Placing voltage sources in parallel has no effect on their voltage, and, theoretically, has no effect at all, as a proper

voltage source is capable of producing infinite current. However, as the Sources subheading of Section 1 notes, no

voltage source can offer unlimited current, and the most common voltage sources, batteries, generally have fairly

low current limits.

So, for practical purposes, voltage sources are placed in parallel to offer more current. Assuming two identical

voltage sources, such as a pair of "AA" batteries, two cells in parallel offer twice the current of one of the cells. This

can be used to either power devices with larger current draws or to extend battery life. (Doubled current potential

means double battery life with a given load.)
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Current Sources

When Current Sources are in parallel, they may be replaced with a single source with the output:

Kirchoff's LawsKirchoff has two important laws that govern electrical circuits: the current law (KCL) and the voltage law (KVL).

These laws, along with Ohm's law are the three fundamental formulas that are needed to analyze circuits. Without

these three laws, many of the more advanced techniques and situations that we are going to discuss in this book

would not be possible.

Kirchoff's Current Law (KCL)

Kirchoff's current law (KCL) states that the sum of all the currents entering into a single node must equal zero.

This is merely a restatement of the law of conservation of charge - we cannot get current out where no current went

in.

For a node with n connections to other nodes, where ik

is the current flowing into this node from node k, Kirchhoff's

Current Law states:

[Kirchhoff's Current Law]

This is a vector sum in that the direction of the current matters. In fact the common convention is to define positive

and negative currents as follows:

Positive Current is current flowing into a node

Negative Current is current flowing out ofa nodeThe opposite convention may also be used, but the user needs to make sure that they use the same current flow

convention throughout an entire problem, or the answers will be wrong. The point cannot be stressed enough that

when doing circuit work, conventions must be specified and they must be followed exactly. Failure to do so will

cause all the calculations to be wrong from that point forward.

Now, let us look at a few simple examples.

KCL Example 1

Problem: In the diagram below, find i.

In this setup, 1 Amp of current is moving into node N2

from node N1. We can then perform the summation, and

solve for current i, which is the current into N2 from N 3:

and therefore:
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A negative current flows away from the node, as per our convention, and therefore we can update our schematic:

This example seems very simple, but the principle underpins all of electronics and it is vital that it is understood.

Kirchoff's Voltage Law (KVL)

Kirchoff's voltage law (KVL) states that the sum of the voltages around a closed loop must all equal zero. Again,

we should come up with a convention, although this one will be a little bit more complicated.

Forward Current

Current is flowing from the negative terminal of an element to the positive terminal.

Backwards Current

Current is flowing from the positive terminal of an element to the negative terminal.

Once we have our notions of forward and backward current flow decided upon, we can then write out our convention

for voltage increases and decreases:

1. Forward current on a source creates positive voltage, or a voltage increase.

2. Forward current on a load element creates a voltage drop, or a negative voltage.

3. Backwards current on a passive load element also creates a voltage dropThis convention is more tricky, so we will examine a few small examples first:

1A-> 5ohm

o----/\/\/\----o

+ v -

Here, the current is flowing from the positive terminal of the resistor to the negative terminal. By our convention,

this is a "Backwards Current" flow, and therefore over the resistor we have a voltage drop:

This voltage drop corresponds to the fact that on the left side of our schematic there is more electrical potential then

on the right side of the schematic.

now, let's look at a whole circuit:

5ohm

+--/\/\/\--+

+| + vr - |

( )12V |

-|


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sum the voltage contributions of both the source and the resistor, as such:

Keeping in mind that this is a resistor, and therefore it is a voltage drop across the resistor, we can reverse the sign to

show the voltage drop of the resistor:

Using Ohm's law now, we can calculate i because we know the resistance of the resistor, and the voltage across the

terminals of the resistor:

Current Divider

Current dividers and voltage dividers are two types of circuits with similar intentions: to decrease current or voltage

by a certain factor, using only resistors. This page will talk about current dividers and voltage dividers.

DefinitionA current divideris formed by connecting resistors in parallel. The current through any single resistor can be found

by:

or equivalently, using conductances:

Construction

A simple current divider

diagram
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Voltage Divider

Definition

A voltage divideris created by connecting resistors in series. The voltage of resistor i in an n-resistor voltage divider

is:

Voltage division can be used to adapt 220-240V AC to 110-120V AC (to allow 120V US devices to run on 220V).

However, voltage division can be inefficient since the resistors have to dissipate large amounts of heat. More

efficient adapters use transformers.

Construction

A simple voltage divider

diagram

Source Transformations

Source Transformations

Independent current sources can be turned into independent voltage sources, and vice-versa, by methods called

"Source Transformations." These transformations are useful for solving circuits. We will explain the two most

important source transformations, Thevenin's Source, and Norton's Source, and we will explain how to use these

conceptual tools for solving circuits.

Black Boxes

A circuit (or any system, for that matter) may be considered a black box if we don't know what is inside the system.

For instance, most people treat their computers like a black box because they don't know what is inside the computer

(most don't even care), all they know is what goes in to the system (keyboard and mouse input), and what comes out

of the system (monitor and printer output).

Black boxes, by definition, are systems whose internals aren't known to an outside observer. The only methods that

an outside observer has to examine a black box is to send input into the systems, and gauge the output.
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Thevenin's Theorem

Let's start by drawing a general circuit consisting of a source and a load, as a block diagram:

Let's say that the source is a collection of voltage sources, current sources and resistances, while the load is a

collection of resistances only. Both the source and the load can be arbitrarily complex, but we can conceptually say

that the source is directly equivalent to a single voltage source and resistance (figure (a) below).

(a) (b)

We can determine the value of the resistanceRs

and the voltage source, vs

by attaching an independent source to the

output of the circuit, as in figure (b) above. In this case we are using a current source, but a voltage source could also

be used. By varying i and measuring v, both vs

andRs

can be found using the following equation:

There are two variables, so two values ofi will be needed. See Example 1 for more details. We can easily see from

this that if the current source is set to zero (equivalent to an open circuit), then v is equal to the voltage source, vs.

This is also called the open-circuit voltage, voc

.
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This is an important concept, because it allows us to model what is inside a unknown (linear) circuit, just by

knowing what is coming out of the circuit. This concept is known as Thvenin's Theorem after French telegraph

engineer Lon Charles Thvenin, and the circuit consisting of the voltage source and resistance is called the

Thvenin Equivalent Circuit.

Norton's TheoremRecall from above that the output voltage, v, of a Thvenin equivalent circuit can be expressed as

Now, let's rearrange it for the output current, i:

This is equivalent to a KCL description of the following circuit. We can call the constant term vs/R

sthe source

current, is.

The equivalent current source and the equivalent resistance can be found with an independent source as before (see

Example 2).

When the above circuit (the Norton Equivalent Circuit, after Bell Labs engineer E.L. Norton) is disconnected from

the external load, the current from the source all flows through the resistor, producing the requisite voltage across the

terminals, voc

. Also, if we were to short the two terminals of our circuit, the current would all flow through the wire,

and none of it would flow through the resistor (current divider rule). In this way, the circuit would produce theshort-circuit current i

sc(which is exactly the same as the source current i

s).
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Circuit Transforms

We have just shown turns out that the Thvenin and Norton circuits are just different representations of the same

black box circuit, with the same Ohm's Law/KCL equations. This means that we cannot distinguish between

Thvenin source and a Norton source from outside the black box, and that we can directly equate the two as below:

We can draw up some rules to convert between the two:

The values of the resistors in each circuit are conceptually identical, and can be called the equivalent resistance,

Req

:

The value of a Thvenin voltage source is the value of the Norton current source times the equivalent resistance

(Ohm's law):

If these rules are followed, the circuits will behave identically. Using these few rules, we can transform a Norton

circuit into a Thvenin circuit, and vice versa. This method is called source transformation. See Example 3.

Open Circuit Voltage and Short Circuit Current

The open-circuit voltage, voc

of a circuit is the voltage across the terminals when the current is zero, and the

short-circuit current isc

is the current when the voltage across the terminals in zero:
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The open circuit voltage The short circuit current

We can also observe the following:

The value of the Thvenin voltage source is the open-circuit voltage:

The value of the Norton current source is the short-circuit current:

We can say that, generally,

Why Transform Circuits?

Why would we ever bother transforming our circuits? Let's say that we have a resistor in series with a Norton circuit.

If we transform the circuit to a Thevenin circuit, we can add the resistor values together! Likewise, let's say that we

have a resistor in parallel to a Thevenin circuit: if we transform to a norton circuit, the resistors will be in parallel,and we can combine them! Many circuits can be completely simplified down into a circuit with a single resistor and

a single source.

Maximum Power Transfer


Often we would like to transfer the most power from a source to a load placed across the terminals as possible. How

can we determine the optimum resistance of the load for this to occur?

Let us consider a source modelled by a Thvenin equivalent (a Norton equivalent will lead to the same result, as the

two are directly equivalent), with a load resistance, RL

. The source resistance isRs

and the open circuit voltage of the

source is vs:


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The current in this circuit is found using Ohm's Law:

The voltage across the load resistor, vL

, is found using the voltage divider rule:

We can now find the power dissipated in the load,PL

as follows:

We can now rewrite this to get rid of theRL

on the top:

Assuming the source resistance is not changeable, then we obtain maximum power by minimising the bracketed part

of the denominator in the above equation. It is an elementary mathematical result that is at a minimum

whenx=1. In this case, it is equal to 2. Therefore, the above expression is minimum under the following condition:

This leads to the condition that:

We will get maximum power out of the source if the load resistance is identical to the internal source resistance. Thisis the Maximum Power Transfer Theorem.
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Efficiency

The efficiency, of the circuit is the proportion of all the energy dissipated in the circuit that is dissipated in the

load. We can immediately see that at maximum power transfer to the load, the efficiency is 0.5, as the source resistor

has half the voltage across it. We can also see that efficiency will increase as the load resistance increases, even

though the power transferred will fall.

The efficiency can be calculated using the following equation:

wherePs

is the power in the source resistor. This can be found using a simple modification to the equation forPL

:

The graph below shows the power in the load (as a proportion of the maximum power,Pmax

) and the efficiency for

values ofRL

between 0 and 5 timesRs.

It is important to note that under conditions of maximum power transfer as much power is dissipated in the source asin the load. This is not a desirable condition if, for example, the source is the electricity supply system and the load is

your electric heater. This would mean that the electricity supply company would be wasting half the power it

generates. In this case, the generators, power lines, etc. are designed to give the lowest source resistance possible,

giving high efficiency. The maximum power transfer condition is used in (usually high-frequency) communications

systems where the source resistance can not be made low, the power levels are relatively low and it is paramount to

get as much signal power as possible to the receiving end of the system (the load).
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Resistive Circuit Analysis Methods

Analysis Methods

When circuits get large and complicated, it is useful to have various methods for simplifying and analyzing the

circuit. There is no perfect formula for solving a circuit. Depending on the type of circuit, there are different methods

that can be employed to solve the circuit. Some methods might not work, and some methods may be very difficult in

terms of long math problems. Two of the most important methods for solving circuits are Nodal Analysis, and Mesh

Current Analysis. These will be explained below.

Nodal Analysis

Nodal analysis is the application of Kirchoff's Current Law (KCL) to solve for the voltages at each node in an

equation. A node voltage is defined as the potential difference between the given node and a designed reference

node (ground). Since one node is defined as ground, a circuit with N nodes will require N-1 equations to solve

completely.

If all sources are current sources, all N-1 equations will be KCL equations: The sum of the current into the node is

equal to the sum of the current out of the node. Currents not connected to current sources can be found using:

.

Given M voltage sources (for M less than or equal to N-1), there will be M KVL equations and (N-1)-M KCL

equations. A supernode may be formed if necessary.

Steps

1. Identify the nodes. These are places where one device ends and another begins (i.e. a wire connects to a resistor).

2. Choose one node to be the reference node, and identify it with a ground symbol. Nodes which connect to multipleother nodes, or which are near a voltage source are the easiest.

3. Label all the nodes, usually written as V_n, where n is the number of the node

4. Use Kirchoff's Current Law to set up an equation for each node. This will leave you with a System of Equations.

5. Solve the System of Equations for each unknown variable

Example

Given the Circuit below, find the voltages at all nodes.

example circuit for nodal analysis example.

node 0: (defined as ground node)

node 1: (free node voltage)

node 2:
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node 3:

which results in the following system of linear equations:

Therefore, the solution is:

Mesh Current Analysis

Mesh analysis is the application of Kirchoff's Voltage Law (KVL) to solve for mesh currents. A mesh current is

defined as the current in a mesh: a loop not containing any other loops. ForMmeshes, there will beMequations.

If all sources are voltage sources, allMequations will be KVL.

If the circuit hasNcurrent sources, there will beNKCLs andM-NKVLs.

Mesh analysis is often easier as it requires fewer unknowns; however, it can only be used on planar circuits.

Example

Circuit diagram for use with the Mesh Current

example problem.

The circuit has 2 loops indicated on the diagram. Using KVL we get:

Loop1:

Loop2:

Simplifying we get the simultaneous equations:

Solving to get:
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Superposition

One of the most important principals in the field of circuit analysis is the principal ofsuperposition. It is valid only

in linear circuits.

Thesuperposition principle states that the total effect of multiple contributing sources on a linear circuit is equal to

the sum of the individual effects of the sources, taken one at a time.

What does this mean? In plain English, it means that if we have a circuit with multiple sources, we can "turn off" all

but one source at a time, and then investigate the circuit with only one source active at a time. We do this with every

source, in turn, and then add together the effects of each source to get the total effect. Before we put this principle to

use, we must be aware of the underlying mathematics.

Necessary Conditions

Superposition can only be applied to linear circuits; that is, all of a circuit's sources hold a linear relationship with

the circuit's responses. Using only a few algebraic rules, we can build a mathematical understanding of

superposition. Iffis taken to be the response, and a and b are constant, then:

In terms of a circuit, it clearly explains the concept of superposition; each input can be considered individually and

then summed to obtain the output. With just a few more algebraic properties, we can see that superposition cannot be

applied to non-linear circuits. In this example, the responsey is equal to the square of the input x, i.e. y=x2. Ifa and b

are constant, then:

Note that this is only one of an infinite number of counter-examples...

Step by Step

Using superposition to find a given output can be broken down into four steps:

1. Isolate a source - Select a source, and set all of the remaining sources to zero. The consequences of "turning off"

these sources are explained in Open and Closed Circuits. In summary, turning off a voltage source results in a

short circuit, and turning off a current source results in an open circuit. (Reasoning - no current can flow through a

open circuit and there can be no voltage drop across a short circuit.)

2. Find the output from the isolated source - Once a source has been isolated, the response from the source in

question can be found using any of the techniques we've learned thus far.

3. Repeat steps 1 and 2 for each source - Continue to choose a source, set the remaining sources to zero, and find the

response. Repeat this procedure until every source has been accounted for.

4. Sum the Outputs - Once the output due to each source has been found, add them together to find the total

response.

Impulse Response

An impulse response of a circuit can be used to determine the output of the circuit:
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The output y is the convolution h * x of the input x and the impulse response:

[Convolution]

.

If the input, x(t), was an impulse ( ), the output y(t) would be equal to h(t).

By knowing the impulse response of a circuit, any source can be plugged-in to the circuit, and the output can be

calculated by convolution.

Convolution

The convolution operation is a very difficult, involved operation that combines two equations into a single resulting

equation. Convolution is defined in terms of a definite integral, and as such, solving convolution equations will

require knowledge of integral calculus. This wikibook will not require a prior knowledge of integral calculus, and

therefore will not go into more depth on this subject then a simple definition, and some light explanation.

DefinitionThe convolution a * b of two functions a and b is defined as:

Remember:

Asterisks mean convolution, not multiplication

The asterisk operator is used to denote convolution. Many computer systems, and people who frequently write

mathematics on a computer will often use an asterisk to denote simple multiplication (the asterisk is the

multiplication operator in many programming languages), however an important distinction must be made here: The

asterisk operator means convolution.

Properties

Convolution is commutative, in the sense that . Convolution is also distributive over addition, i.e.

, and associative, i.e. .

Systems, and convolution

Let us say that we have the following block-diagram system:

x(t) = system input

h(t) = impulse response

y(t) = system output

Where x(t) is the input to the circuit, h(t) is the circuit's impulse response, and y(t) is the output. Here, we can find

the output by convoluting the impulse response with the input to the circuit. Hence we see that the impulse response

of a circuit is not just the ratio of the output over the input. In the frequency domain however, component in the

output with frequency is the product of the input component with the same frequency and the transition function at

that frequency. The moral of the story is this: the output to a circuit is the input convolved with the impulse response.

Capacitors and Inductors
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Resistors, wires, and sources are not the only passive circuit elements. Capacitors and Inductors are also common,

passive elements that can be used to store and release electrical energy in a circuit. We will use the analysis methods

that we learned previously to make sense of these complicated circuit elements.

Energy Storage Elements

Energy Storage Elements

Resistors are not the only available circuit element. Far from it: There are many different types of elements that can

be found in circuits. Among passive elements, there are 2 more types besides resistors: capacitors and inductors.

Both capacitors and inductors can store energy, to be released back into the circuit under certain conditions.

Capacitors store energy in an electric field, while inductors store energy in a magnetic field.

Capacitors

Capacitors are passive circuit elements that can be used to store energy in the form of an electric field. In the

simplest case, a capacitor is a set of parallel metal plates separated by a dielectric substance.

Electric charges build up on the opposite plates as a voltage is applied to the capacitor.

With a constant voltage across the capacitor there will be no change in electric charge and thus the steady-state

current becomes zero. Stored energy can be discharged from a capacitor by removing an external forcing voltage andby shorting the capacitor with a load resistance.

Capacitance

Capacitance is defined as the capability of a capacitor to store the charge of a voltage. Capacitance is measured in

units called "Farads," abbreviated by an "F".

The ratio of charge over voltage gives a value of capacitance

The relationship between the current and the voltage of a capacitor is:

[Capacitor Relation]
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Energy Storage

The amount of energy that is storable in a capacitor is determined by:

Capacitors in Series

A set of N capacitors in series

The total capacitance of a series of capacitors is given by the

following formula:

Impedance

Impedance is the characteristic of capacitor to resist current flows when a voltage is applied on the capacitor.

Impedance of a capacitor is defined as the sum of its resistance and reactance:

Where:

RC

Resistance of the Capacitor

XC

Reactance of the Capacitor = 1 / jC

= 2f

C= Capacitance of the Capacitor

Direct Current

In a steady-state DC circuit, capacitors act as an open circuit.

Alternating Current

When applying a voltage to the capacitor the voltage of the reactance is lagging the voltage of its resistance an angle

equal to 90. Voltage of the resistance is in the same phase with the applied voltage: The load voltage is at an angle

with the total voltage of resistance and reactance:

VX

Clags VR

Cby 90

VR

Cis the same phase with Vi

A capacitor is a frequency dependent element. There is one frequency at which the capacitor reactance equals the

value of resistor. This frequency is called Response Frequency denoted as o

= 1 / RCor in time domain is called

Time Constant t = RC

But how do you arrive at the frequency response? Ideally, when there is no voltage applied to the capacitor there will

be no current flow. Therefore, The impedance of the capacitor is equal to 0.

ZC

= RC

+ 1 / jC = 0 or

j = 1 / CRC

= 0 . Open circuit Vo

= 0
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= o

. Vo

= Vi.

= infinity . Vo V

i.

Capacitors in Parallel

A set of N capacitors in parallel

For capacitors in parallel:

For assistance remembering this formula remember the construction of a capacitor, that capacitance increases with

the area of the plates.

Capacitors can "Pop"

Many capacitors are polarized in a particular way. If you apply voltage across the terminals of a polarized capacitor,

the capacitor itself might pop. This is made more dangerous by the fact that many capacitors have chlorine gas

inside, because the chlorine raises the capacitance of the capacitor. Popping a chlorine-filled capacitor will be very

unpleasant (if not down-right dangerous).

Capacitors can Kill

Strong capacitors, such as those in microwave ovens and CRT screens can remain charged even after the device is

turned off. Remember that a capacitor maintains its charge until a load is placed across its terminals or the capacitor

is shorted. For this reason large capacitors with a high voltage across their terminals can hold a dangerous charge

even if the device is turned off or has been out of use for a long time. Large capacitors can produce enough voltage

to create a 4 amp current across the hands of a person who grabs it. 4 amps is a fatal amount of current, so be careful

when dealing with old capacitors.

InductorsAn inductor is a coil of wire that stores energy in the form of a magnetic field. With a forcing voltage applied to the

inductor, the magnetic field charges up as the current passing through the inductor increases. When the magnetic

field has reached its maximum capacity, the inductor ceases storing more energy, and the inductor behaves as a

resistor. As a wire has the potential of creating a magnetic field around itself, inversely a magnetic field as the

potential of creating a current through a wire. If the current flowing through the inductor drops, the magnetic field

will also collapse, regenerating the lost current. Thinking of energy storage in a magnetic field can be unintuitive, so

it may be helpful to consider that an inductor is analogous to a flywheel, where the applied voltage is like a torque

applied to the flywheel. The faster the flywheel spins, the more kinetic energy it contains, just as the higher the

current through the inductor, the more "magnetic" energy it contains.
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Inductance

Inductance is the capacity of an inductor to store energy in the form of a magnetic field. Inductance is measured by

units called "Henries" which is abbreviated with a capital "H," and the variable associated with inductance is "L".

The relationship between inductance, current, and voltage through an inductor is given by the formula:

[Inductor Relation]

Energy Storage

The energy stored in an inductor is given with the formula:

Inductors are generally used in applications such as for limiting current through dc-dc converters, either for step-up

operations, or step-down operations. Also, because inductors convert electrical energy into a magnetic field, they are

the primary components oftransformers, which we will discuss later.

When the forcing voltage is removed from an inductor, the energy from an inductor is discharged.

Impedance

Impedance is the characteristic of inductor resist current flows when a voltage is applied on the inductor. Impedance

of inductor is defined as the sum of its resistance and reactance:

RL

Resistance of the Inductor

XL

Reactance of the Inductor = jL

= 2f

L = Inductance of the Inductor
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Direct Current

Inductor acts as Short Circuit. At the load would see the applied voltage.

Alternate Current

When apply a voltage on the inductor. The voltage of the reactance is leading the voltage of its resistance one angle

equals to 90

. Voltage of the resistance is in the same phase with the applied voltage: V

XL leads V

RL by 90

VR

C is the same phase with Vi

Inductor is frequency dependent element. There is one frequency at which the inductor react or start to conduct

current and this frequency is called response frequency, denoted as o

= R / L, and the time that it takes to reach this

frequency is t = L / R.

How do you arrived the frequency response? Ideally, when there is no voltage apply on inductor there will be no

current flows. Therefore, the impedance of the inductor is equal to 0:

ZL

= RL

+ jL = 0 or

j = RL/ L = 0 Z

L= R

L+ 0. Inductor is short circuited V

o V

i

= o

Inductor starts to react or conduct current. Vo V

i

= infinity . Inductor is open circuited Vo 0

Inductors in Series

A set of N inductors in series

Like resistors, inductors appearing in series can be

conceptually converted into a single inductor, with a

total inductance, given as follows:

Inductors in Parallel

A set of N inductors in parallel

If multiple inductors are in parallel, we can calculate out the

resultant inductance of the circuit as follows:
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Warnings

Inductors and capacitors have different associated dangers. For inductors, when the current flowing is interrupted a

high voltage pulse resulting from the consequent collapse in the inductor's magnetic field can be dangerous. Using

makeshift setups to conduct current through a large value inductance can be very dangerous especially when the

circuit is disconnected. Give adequate planning to how your circuit and apparatus will dissipate the voltages created

when power is removed from an inductor.

It is important to note also that the magnetic field of an inductor can cause magnetic interference with other electric

devices, and can damage sensitive digital circuits.

First-Order Circuits

First Order Circuits

First order circuits are circuits that contain only one energy storage element (capacitor or inductor), and that can

therefore be described using only a first order differential equation. The two possible types of first-order circuits are:

1. RC (resistor and capacitor)

2. RL (resistor and inductor)

RL and RC circuits is a term we will be using to describe a circuit that has either a) resistors and inductors (RL), or

b) resistors and capacitors (RC).

RL Circuits

An RL parallel circuit

An RL Circuit has at least one resistor (R) and one inductor (L). These

can be arranged in parallel, or in series. Inductors are best solved by

considering the current flowing through the inductor. Therefore, we

will combine the resistive element and the source into a Norton Source

Circuit. The Inductor then, will be the external load to the circuit. We

remember the equation for the inductor:

If we apply KCL on the node that forms the positive terminal of the voltage source, we can solve to get the following

differential equation:

We will show how to solve differential equations in a later chapter.
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RC Circuits

A parallel RC Circuit

No, RC does not stand for "Remote Control". An RC circuit is a circuit

that has both a resistor (R) and a capacitor (C). Like the RL Circuit, we

will combine the resistor and the source on one side of the circuit, and

combine them into a thevening source. Then if we apply KVL around

the resulting loop, we get the following equation:

First Order Solution

Series RL

The differential equation of the series RL circuit

. A = eC

t I(t)

0 A

136% A

2A

3A

4A

51% A
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Series RC

The differential equation of the series RC circuit

. A = eC

t V(t)

0 A

136% A

2A

3A

4A

51% A

Time Constant

The series RL and RC has a Time Constant

In general, from an engineering standpoint, we say that the system is at steady state ( Voltage or Current is almost at

Ground Level ) after a time period of five Time Constants.


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RLC Circuits

Series RLC Circuit

Second Order Differential Equation

The characteristic equation is

Where

When

The equation only has one real root .

The solution for

The I - t curve would look like

When

. R >

The equation only has two real root .
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The solution for

The I - t curve would look like

When

. R ---[ Zseries ]---

[Impedances in Series]

Notice how much easier this is than having to differentiate between the formulas for combining capacitors, resistors,

and inductors in series. Notice also that resistors, capacitors, and inductors can all be mixed without caring which

type of element they are. This is valuable, because we can now combine different elements into a single impedancevalue, as opposed to different values of inductance, capacitance, and resistance.

Keep in mind however, that phasors need to be converted to rectangular coordinates before they can be added

together. If you know the formulas, you can write a small computer program, or even a small application on a

programmable calculator to make the conversion for you.

Impedances in Parallel

Impedances connected in parallel can be combined in a slightly more complicated process:

[Impedances in Parallel]

Where N is the total number of impedances connected in parallel with each other. Impedances may be multiplied in

the polar representation, but they must be converted to rectangular coordinates for the summation. This calculation

can be a little bit time consuming, but when you consider the alternative (having to deal with each type of element

separately), we can see that this is much easier.

Steps For Solving a Circuit With Phasors

There are a few general steps for solving a circuit with phasors:

1. Convert all elements to phasor notation

2. Combine impedances, if possible

3. Combine Sources, if possible

4. Use Ohm's Law, and Kirchoff's laws to solve the circuit

5. Convert back into time-domain representation

Unfortunately, phasors can only be used with sinusoidal input functions. We cannot employ phasors when examining

a DC circuit, nor can we employ phasors when our input function is any non-sinusoidal periodic function. To handle

these cases, we will look at more general methods in later chapters


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Phasor Theorems

Circuit Theorems

Phasors would be absolutely useless if they didn't make the analysis of a circuit easier. Luckily for us, all our old

circuit analysis tools work with values in the phasor domain. Here is a quick list of tools that we have already

discussed, that continue to work with phasors:

Ohm's Law

Kirchoff's Laws

Superposition

Thevenin and Norton Sources


This page will describe how to use some of the tools we discussed for DC circuits in an AC circuit using phasors.

Ohm's Law

Ohm's law, as we have already seen, becomes the following equation when in the phasor domain:

Separating this out, we get:

Where we can clearly see the magnitude and phase relationships between the current, the impedance, and the voltage

phasors.

Kirchoff's Laws

Kirchoff's laws still hold true in phasors, with no alterations.

Kirchoff's Current Law

Kirchoff's current law states that the amount of current entering a particular node must equal the amount of current

leaving that node. Notice that KCL never specifies what form the current must be in: any type of current works, and

KCL always holds true.

[KCL With Phasors]

Kirchoff's Voltage Law

KVL states: The sum of the voltages around a closed loop must always equal zero. Again, the form of the voltage

forcing function is never considered: KVL holds true for any input function.

[KVL With Phasors]


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Superposition

Superposition may be applied to a circuit if all the sources have the same frequency. However, superposition mustbe

used as the only possible method to solve a circuit with sources that have different frequencies. The important part to

remember is that impedance values in a circuit are based on the frequency. Different reactive elements react to

different frequencies differently. Therefore, the circuit must be solved once for every source frequency. This can be a

long process, but it is the only good method to solve these circuits.

Thevenin and Norton Circuits

Thevenin Circuits and Norton Circuits can be manipulated in a similar manner to their DC counterparts: Using the

phasor-domain implementation of Ohm's Law.

It is important to remember that the does not change in the calculations, although the phase and the magnitude of

both the current and the voltage sources might change as a result of the calculation.


The maximum power transfer theorem in phasors is slightly different then the theorem for DC circuits. To obtain

maximum power transfer from a thevenin source to a load, the internal thevenin impedance ( ) must be the

complex conjugate of the load impedance ( ):

[Maximum Power Transfer, with Phasors]

Complex Power

Complex Power

Just like the other values of voltage, current, and resistance, power also has a complex phasor quantity that we are

going to become familiar with. Complex power is denoted with a symbol. It is calculated as such:

[Complex Power]

Where the quantity denotes the complex conjugate of the phasor current. To get the complex conjugate of , we

have two formulas:

Given: (rectangular)

(polar)

There is more information on complex conjugation of phasors in the Appendix.
http://en.wikibooks.org/w/index.php?title=Circuit_Theory/Phasor_Arithmetic


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Apparent Power

If we take the magnitude of our Complex power variable, we get the following:

[Apparent Power]

Where is called the apparent power. It is this quantity that we can measure.

Average and Reactive Power

Let us break up our voltage and current phasors for a moment:

, and

if we plug those two values into our equation for complex power, above, we get the following:

We can then convert this quantity into rectangular form where:

[Average Power]

[Reactive Power]

We call P the Average Power and Q the Reactive Power. We will discuss these quantities later.

Units

Unfortunately, Power is not as simple a quantity as impedance. Unlike Impedance and resistance, The different

power quantities do not all share the same units. We list the units for each type of power, below:

Time-Domain Power

Watts (w)

Average Power

Watts (w)

Complex Power

Volt-Amps (VA)

Reactive Power

Volt-Amps Reactive (VAR)

Technically, all these units are equatable, but they are called different things as a matter of common convention.


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Power and Impedance

Complex power can be expressed in terms of impedance and complex current, using the following formula:

If the element in question is a resistor, the reactive powerdelivered will be 0. Likewise, if the element is a capacitor

or an inductor, the average powerdelivered will be zero. If the impedance is complex, then the delivered power will

be complex.

Conservation of Power

Power in a circuit is conserved. Therefore, the following equation holds true:

[Conservation of Power]

Remember that sources supply power, and that impedance elements (resistors, capacitors and inductors) absorbpower.

Power Factor

The relationship between the average power, and the apparent power is called the power factor. Power factor is

given the variable , and is calculated as such:

[Power Factor]

There is also a quantity called the power-factor angle, which is equal to the differences in phase angle between the

current and the voltage:

Since the cosine is an even function, the following values are equal:

This means that to be able to accurately calculate the phase angles of the current and the voltage from the power

factor, we need an additional specifier of either leading or lagging.

Lagging

The phase angle of the voltage is greater than the phase angle for the current.

Leading

The phase angle of the current is greater than the phase angle for the voltage.

a visual depiction of the relationship between P,

Q, and the angle
http://en.wikibooks.org/w/index.php?title=File:Electric_power_factor.jpg


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Maximum Transfer Theorem

Similarly to DC power, AC power has its own maximum power transfer theorem that can be expressed in terms of

phasors.

Given a Thevenin equivalent source with an impedance , maximum power transfer is attained when the

load impedance is:

In plain English, the source impedance must be the complex conjugate of the load impedance to attain maximum

power transfer.

The Laplace TransformThe Laplace Transform is a useful tool borrowed from mathematics to quickly and easily analyze systems that are

represented by high-order linear differential equations. The Fourier Transform, which is closely related, can also

provide us with insight about the frequency response characteristics of a system.

Laplace TransformThe Laplace Transform is a powerful tool that is very useful in Electrical Engineering. The transform allows

equations in the "time domain" to be transformed into an equivalent equation in the Complex S Domain. The

laplace transform is an integral transform, although the reader does not need to have a knowledge of integral calculus

because all results will be provided. This page will discuss the Laplace transform as being simply a tool for solving

and manipulating ordinary differential equations.

Laplace transformations of circuit elements are similar to phasor representations, but they are not the same. Laplace

transformations are more general than phasors, and can be easier to use in some instances. Also, do not confuse the

term "Complex S Domain" with the complex power ideas that we have been talking about earlier. Complex power

uses the variable , while the Laplace transform uses the variable s. The Laplace variable s has nothing to do with

power.

The transform is named after the mathematician Pierre Simon Laplace (1749-1827). The transform itself did not

become popular until Oliver Heaviside, a famous electrical engineer, began using a variation of it to solve electrical

circuits.

The Transform

The mathematical definition of the Laplace transform is as follows:

[The Laplace Transform]

Note:

The letter s has no special significance, and is used with the Laplace Transform as a matter of common convention.

The transform, by virtue of the definite integral, removes all t from the resulting equation, leaving instead the new

variable s, a complex number that is normally written as . In essence, this transform takes the function

f(t), and "transforms it" into a function in terms of s, F(s). As a general rule the transform of a function f(t) is written

as F(s). Time-domain functions are written in lower-case, and the resultant s-domain functions are written in

upper-case.

There is a table of Laplace Transform pairs in

the Appendix

we will use the following notation to show the transform of a function:
http://en.wikibooks.org/w/index.php?title=Circuit_Theory/Transform_Tables


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We use this notation, because we can convert F(s) back into f(t) using the inverse Laplace transform.

The Inverse Transform

The inverse laplace transform converts a function in the complex S-domain to its counterpart in the time-domain.

Its mathematical definition is as follows:

[Inverse Laplace Transform]

where is a real constant such that all of the poles of fall in the region . In

other words, is chosen so that all of the poles of are to the left of the vertical line intersecting the real axis

at .The inverse transform is more difficult mathematically than the transform itself is. However, luckily for us, extensive

tables of laplace transforms and their inverses have been computed, and are available for easy browsing.

Laplace Domain

The Laplace domain, or the "Complex s Domain" is the domain into which the Laplace transform transforms a

time-domain equation. s is a complex variable, composed of real parts:

The Laplace domain graphs the real part () as the horizontal axis, and the imaginary part () as the vertical axis.

The real and imaginary parts of s can be considered as independent quantities.

The similarity of this notation with the notation used in Fourier transform theory is no coincidence; for , the

Laplace transform is the same as the Fourier transform if the signal is causal.

Transform Properties

There is a table of Laplace Transform properties in

The Appendix

The most important property of the Laplace Transform (for now) is as follows:

Likewise, we can express higher-order derivatives in a similar manner:

Or for an arbitrary derivative:

where the notation means the nth

derivative of the function at the point , and means .

In plain English, the laplace transform converts differentiation into polynomials. The only important thing to

remember is that we must add in the initial conditions of the time domain function, but for most circuits, the initial

condition is 0, leaving us with nothing to add.

For integrals, we get the following:
http://en.wikibooks.org/w/index.php?title=Circuit_Theory/Transform_Tables


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Initial Value Theorem

The Initial Value Theorem of the laplace transform states as follows:

[Initial Value Theorem]

This is useful for finding the initial conditions of a function needed when we perform the transform of adifferentiation operation (see above).

Final Value Theorem

Similar to the Initial Value Theorem, the Final Value Theorem states that we can find the value of a function f, as t

approaches infinity, in the laplace domain, as such:

[Final Value Theorem]

This is useful for finding the steady state response of a circuit. The final value theorem may only be applied to stable

systems.

Transfer Function

If we have a circuit with impulse-response h(t) in the time domain, with input x(t) and output y(t), we can find the

Transfer Function of the circuit, in the laplace domain, by transforming all three elements:

In this situation, H(s) is known as the "Transfer Function" of the circuit. It can be defined as both the transform ofthe impulse response, or the ratio of the circuit output to its input in the Laplace domain:
http://en.wikibooks.org/w/index.php?title=File:LTI.png


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[Transfer Function]

Transfer functions are powerful tools for analyzing circuits. If we know the transfer function of a circuit, we have all

the information we need to understand the circuit, and we have it in a form that is easy to work with. When we have

obtained the transfer function, we can say that the circuit has been "solved" completely.

Convolution Theorem

Earlier it was mentioned that we could compute the output of a system from the input and the impulse response by

using the convolution operation. As a reminder, given the following system:

x(t) = system input

h(t) = impulse response

y(t) = system output

We can calculate the output using the convolution operation, as such:

Where the asterisk denotes convolution, not multiplication. However, in the S domain, this operation becomes much

easier, because of a property of the laplace transform:

[Convolution Theorem]

Where the asterisk operator denotes the convolution operation. This leads us to an English statement of the

convolution theorem:

Convolution in the time domain becomes multiplication in the S domain, and Convolution in the S domain becomes

multiplication in the time domain.

Now, if we have a system in the Laplace S domain:

X(s) = Input

H(s) = Transfer Function

Y(s) = Output

We can compute the output Y(s) from the input X(s) and the Transfer Function H(s):

Notice that this property is very similar to phasors, where the output can be determined by multiplying the input by

the network function. The network function and the transfer function then, are very similar quantities.
http://en.wikibooks.org/w/index.php?title=File:Laplace_Block.svghttp://en.wikibooks.org/w/index.php?title=File:System_Block.svg


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Resistors

The laplace transform can be used independently on different circuit elements, and then the circuit can be solved

entirely in the S Domain (Which is much easier). Let's take a look at some of the circuit elements:

Resistors are time and frequency invariant. Therefore, the transform of a resistor is the same as the resistance of the

resistor:

[Transform of Resistors]

Compare this result to the phasor impedance value for a resistance r:

You can see very quickly that resistance values are very similar between phasors and laplace transforms.

Ohm's Law

If we transform Ohm's law, we get the following equation:

[Transform of Ohm's Law]

Now, following ohms law, the resistance of the circuit element is a ratio of the voltage to the current. So, we will

solve for the quantity , and the result

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