25961867-7076151-Basic-Electrical

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Circuit Theory/All Chapters

From Wikibooks, the open-content textbooks collection

Page 1 of 65Circuit Theory/All Chapters - Wikibooks, collection of open-content textbooks

9/25/2008http://en.wikibooks.org/wiki/Circuit_Theory/All_Chapters


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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.




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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 fundamental topics in Calculus.

For information on Calculus, see the wikibook: Calculus.

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 in the 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 lacks information in these fields of study.




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Variables and Standard Units

Electric Charge (Coulombs)

Electric charge is a property of some subatomic particles. Electric Charge is measured in SI units called "Coulombs", which are abbreviated with the letter

capital C. A Coulomb is the total charge of 6.2415096291526510E18 electrons, thus a single electron has a charge of 1.602 10E19 C. The variable used

to represent a quantity of charge is the "q" (lower-case Q). Electric charge is the subject of many fundamental laws, such as Coulomb's Law, and Gauss' Law.

However, Charge is not important for the study of electric circuits, and so this wikibook will not make much use of it after this.

For further information about electric charge, Coulomb's Law or Gauss' Law, see the wikibook Modern Physics.

Current (Amperes)

Current is a measurement of the flow of electricity. Current is measured in units called Amperes (or "Amps"). Technically, an ampere is measured in terms of "coulombs per second" although in

reality, the coulomb is actually defined in terms of the ampere. Amperes are abbreviated with an "A" (upper-case A), and the variable most often associated with current is the letter "i" (lower-case I). In

terms of coulombs, an ampere is:

Because of the wide-spread use of complex numbers in Electrical Engineering, it is common for electrical engineering texts to use the letter "j" (lower-case J)

as the imaginary number, instead of the "i" (lower-case I) commonly used in math texts. This wikibook will adopt the "j" as the imaginary number, to avoid

confusion.

Voltage (Volts)

Voltage is a measure of the work required to move a charge from one point to another in a electric field. Thus the unit "volt" is defined as a Joules (J) per Coloumb (C).

W represents work, q represents an amount of charge.

Voltage is sometimes called "electric potential", because voltage represents the potential of a forcing function to produce current in a circuit. More voltage means more potential for current. Voltage also

can be called "Electric Pressure", although this is far less common. Voltage is not measured in absolutes but in relative terms compared to a reference point. The reference point against which all voltages

are measured in a given circuit is known as ground.

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.

Note:An electron has a charge of

-1.60210E-19 C.

For the rest of this book, the lower-case J ( j ) will be used to denote an

imaginary number, and the lower-caseI ( i ) will be used to denote current.




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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

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.

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.

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 s tored in the system previously. Some examples of these

generators are alternators, batteries etc...

A previous definition stated; "A dependant 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 ("independant source") from an active power control device, or amplifier

("dependant source"). It is probably more useful to think of "dependant 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 an ideal wire across the terminals of the element. Because current will

take the path of least resistance, shorting an element redirects all current around the element. Because there is no current, the element also has no voltage across its terminals. This practice must be done

with care, because reducing the resistance of a certain portion of a circuit to zero can potentially raise the current to infinity, and the circuit will become damaged.

Ideal Voltmeters

Voltmeters and Ammeters are devices that are used to measure the voltage across an element, and the current flowing through a wire, respectively.

An ideal voltmeter has an infinite resistance (in reality, several megohms), 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 that all the current from one terminal flows through 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 a voltage source is shorted (a resistance-less wire is placed across its terminals), the resulting current output approaches infinity! No voltage source in existance 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 f rom 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 s imple 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 impedance2. Op Amp has infinite bandwidth3. Op Amp has infinite voltage gain4. Op Amp has Zero output impedance5. Op Amp has Zero offset (error)

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




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1. There is 0 voltage difference between the terminals x and y.2. There is 0 current flowing on terminals x and y.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 wr itten 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 inactive these sources

Source Warnings

The following image shows some configurations of current and voltage sources that are not permissable, and will cause a problem in your circuit:

Switches

A switch then is a circuit element that is an open-circuit for all time t< T0, and acts like a closed-circuit for all time .

Unit Step Function

Before talking about switches, we will introduce the Heaviside step functionu(t) (also known as the unit step function). The step function is defined piecewise as such:

This function provides a mathematical model for electrical engineers to describe circuit elements that change between boolean states (on/off, high/low, etc).

Transducers

A transducer is a circuit component that transforms electrical energy into another type of energy. Some examples of transducers are actuators and motors.

[Unit Step Function]




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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 r esistance. 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

Rt=R

o(1 + aT)

Where:

Rtis Final r esistance,

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 ofimpedance andreactance, 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.

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

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:

i = v /r

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.

Image 1A simple circuitdiagram relating

current, voltage, andresistance

[Ohm's Law]




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Resistive Circuits

We'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.




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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 r esistors 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 s imple 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 foundin real life are in Practical Electronics.

The v-i characteristic of a resistor is simply a straight line, passing through the origin:

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 v=iR,

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.

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 r esistor per unit voltage across it. As with all SI units named after people, the word "siemens" is lowercase, but the symbol is uppercase (S).

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:

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.

Equivalent Resistances

Non-ideal resistor value with nonlinear

resistance characteristic




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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.

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:

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. For capacitors, the formula have to be interchanged.




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Resistors

Resistors appearing in series can be converted into a single resistor, rseries

according to the following equation:

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.

Voltage Sources

Voltage sources given in series can be added together to form a single source, vseries

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 f rom one reference to another. But usually

the voltage source whose positive 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:

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.)

Current Sources

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

Kirchoff's Laws

Kirchoff 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. To come to this conclusion, we need to work out a current convention:

Positive Currentcurrent flowing into a node

Negative Currentcurrent flowing out of a node

The 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 examine some circuits:

in the above diagram, 1 Amp of current is moving into node N. We can then perform the summation, and solve for current i:

Resistors in a series configuration

[Resistors In Series]

A diagram of several resistors appearing in parallelwith each other.

[Resistors in Parallel]

1A->


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1A + i = 0 and therefore: i = 1

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

Granted, this is a simple example, but it gets the point across.

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

Backwards CurrentCurrent 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 drop

This convention is more tricky, so we will examine a few small examples first:

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:

v = (1A)(5) = 5V

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:

To figure out the voltage drop across the resistor using only Ohm's law, we would need to know the current in the loop, i. This circuit has 2 unknowns now: the voltage across the resistor (vr), and the

current, i. Using KVL, we can 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:

vr= 12V

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.

Definition

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

or equivalently, using conductances:

Construction

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

1A-> 1A->

o--------o-------o

N

1A-> 5ohm

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

+ v -

5ohm

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

+| + vr - |

( )12V |

-|


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A simplevoltage divider

diagram




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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'tknow 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.

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).

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 isequal to the voltage source, v

s. This is also called the open-circuit voltage, v

oc.

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 Theorem

Recall 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, i

s.

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, v

oc. 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 the short-circuit current isc

(which is exactly the same as the source current is).

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:

(a) (b)




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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:

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, g enerally,

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 r esistor 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.

The open circuit voltage The short circuit current




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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 withNnodes will requireN-1 equations to solve completely.

If all sources are current sources, allN-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:

.

GivenMvoltage sources (forMless than or equal to N-1), there will beMKVL equations and (N-1)-MKCL equations. A supernode may be formed if necessary.

Steps

1. Identify the nodes. These are places where one device ends and another begins (ie 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 multiple other 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 node4. 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.

node 0: (defined as ground node)

node 1: (free node voltage)

node 2:

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 currentis 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

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

Loop1: 0 = 9 - 1000I1 - 3000(I1 -I2)

Loop2: 0 = 3000(I1 -I2) - 2000I2 - 2000I2

Simplifying we get the simultaneous equations:

example circuit for nodal analysisexample.

Circuit diagram for use with the MeshCurrent example problem.




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0 = 9 - 4000I1

+ 3000I2

0 = 0 + 3000I1

- 7000I2

Solving to get:

I1

= 3.32mA

I2

= 1.42mA

Superposition

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

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 s ource, 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 f ew 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, 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 avoltage 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 ashort 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:

The output y is the convolution h * x of the input x and the impulse response:

.

If the input, x(t), was an impulse ((t)), 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 s ubject then a

simple definition, and some light explanation.

Definition

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

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 a * b = b * a. Convolution is also distributive over addition, i.e. a * (b + c) = a * b + a * c, and associative, i.e. a * (b * c) = (a * b) * c.

Systems, and convolution

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

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

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.

[Convolution]

Remember:Asterisks mean convolution, not

multiplication

x(t) = system input h(t) = impulse response y(t) = system output

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.




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Capacitors and Inductors

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




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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 put across the capacitor. Capacitors can transfer voltage and current across the dielectric, until the electric field inside the capacitor reaches

its maximum capacity. At which point, the field is saturated, and no more charges can travel from one place to the other. With a constant charge applied to the capacitor therefore, the capacitor

eventually becomes an open circuit.

With a constant voltage across the capacitor, the steady-state current becomes zero. Stored energy can be discharged from a capacitor by removing an external forcing voltage, and by shorting the

capacitor with a load resistance.

Capacitance

The ability of a capacitor to store energy is called Capacitance, and is measured in units called "Farads", abbreviated with an "F" (capital F). The variable most commonly associated with capacitance is

"C" (upper-case C). The capacitance associated with a capacitor can be found as a ratio of the amount of charge that the element can hold, divided by the voltage across the capacitor:

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

Energy Storage

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

Capacitors in Series

The total capacitance of a series of capacitors is given by the following formula:

Impedance

Impedance is the characteristic of Capacitor resist current flows when a Voltage is applied on the Capacitor

Impedance of Capacitor is defined as the sum of its Resistance and Reactance

ZC

=RC

+XC

Where:

RC

Resistance of the Capacitor

XC

Reactance of the Capacitor = 1 / jC

j =- 1 = 2fC= Capacitance of the Capacitor

Direct Current

Capacitor acts as Open Circuit . At the load would see zero Voltage .

Alternating Current

When apply a Voltage on the Capacitor . The Voltage of the Reactance is lagging The Voltage of the Resistance one angle equals to 90 . Voltage of the Resistance is in the same phase with the AppliedVoltage . The Load Voltage is at an angle with the total Voltage of Resistance and Reactance

VX

Clags V

RC

by 90

VR

Cis the same phase with V

i

A capacitor is a frequency dependent element. There is one frequency at which the capacitor react or start to conduct current and this frequency is called Response Frequency denoted as o

= 1 / RCand

[Capacitor Relation]

A set of N capacitors in series

the time that it takes to reach this frequency is t = RC

But how do you arrive at the frequency response? Ideally, when there is no voltage apply on capacitor there will be no current flows. Therefore, The impedance of the capacitor is equal to 0.

ZC

= RC

+ 1 / jC = 0 or

j = 1 / CRC

= 0 . Open circuit Vo

= 0

= o . Starts to react or conduct current. Vo Vi = infinityZ

C=R

C+ 0 . V

o V

i.

Capacitors in Parallel

For capacitors in parallel:




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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, specifically the capacitors in microwave ovens and CRT screens can remain charged when 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. Be careful when

dealing with old capacitors.

Inductors

An 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 short-circuit. 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.

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". The variable

associated with inductance is "L".

The relationship between inductance, current, and voltage through an inductor is given by the formula: v(t) = Ldi/dt

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

ZL

=RL

+XL

RL

Resistance of the Inductor

XL

Reactance of the Inductor = jL

j = -1 = 2fL = Inductance of the Inductor

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 AppliedVoltage

VX

Lleads V

RL

by 90

VR

Cis the same phase with V

i

Inductor is frequency dependent element . There is one frequency at which the Inductor r eact 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 ZL

= RL

+ 0 . Inductor is Short circuited Vo V

i

= o

Inductor Starts to React or conduct current . Vo V

i

= infinityZL

=RL

+ infinity . Inductor is Open circuited Vo 0

Inductors in Series

Like resistors, inductors appearing in series can be conceptually converted into a single inductor, with a total inductance,Lseries given as follows:

Inductors in Parallel

A set of N capacitors in parallel

[Inductor Relation]

A set of N inductors in series




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If multiple inductors are in parallel, we can calculate out the resultant inductance of the circuit as follows:

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.

A set of N inductors in parallel




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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). These circuits are known as "First Order" circuits,because the solution to the circuit can be written as a f irst-order differential equation.

RL Circuits

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.

RC Circuits

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 s ide of the circuit, and combine them into a thevenin source. Then if we apply KVL around the

resulting loop, we get the following equation:

We will talk about the general solutions to these equations below.

First Order Solution

RL and RC circuits will both produce a first-order differential equation. The reader does not, however, require a prior knowledge of differential equations to read this topic, because we work through to

the general solution of the equation. To understand the material fully, you would need a knowledge of derivatives and integrals. We will replace the capacitor voltages and the inductor currents in the

previous equations with anx to signify that this will be a general solution to either type of problem. Here, we will consider a general equation of the form:

Where k is a constant value that corresponds to the source value (current for RL and voltage for RC circuits), possibly scaled by a certain factor based on the resistance, inductance, and/or capacitance ofthe circuit, when we divide through. T

cis a value known as the "Time Constant".

If we separate out the variables, we can get all thex terms on one side of the equation, and all the tterms on the other:

We can integrate both sides of this equation. The left side can be integrated with respect tox, and the left side can be integrated with respect to t. Performing the integrations gives us the following

equation:

WhereD is an arbitrary constant of integration. If we raise both sides to e (to get rid of the natural log function) we will get the following final result:

A = eD

It turns out that A is also the value of the initial condition of the circuit,x(0). Also kTc

is equal to the value of the steady-state value of the function. Combining this knowledge, we get the following

equation:

Where:

Is the steady state value ofx. This is our general result. Remember thatx gets replaced by the function for either the capacitor voltage or the inductor current, to get the solution to an RC or an RL circuit,

respectively.

Time Constant

The Time Constant, Tc, is an indicator of the amount of time it takes for a system to react to an input. The Time Constant is based on the amount of total resistance, capacitance, and inductance of a

circuit. In general, the Time Constant for an RL circuit is:

and the time constant for an RC circuit is:

An RL parallel circuit

A parallel RC Circuit

[First Order Solution]




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27/65

RLC Circuits

Series RLC Circuit

Circuit Impedance

Zt= Z

R+ Z

L+ Z

C

Zt= R + jL + 1 / jC

Zt= jRC + (j)2LC + 1

Zt= (j)2LC + jRC + 1

Zt= (j)2 + j(R/L) + 1/LC

Frequency Response of the circuit

1. There is one frequency at which ZC

= ZL

jL = 1 / jC

The Frequency is called Resonant Frequency

2. When there is no applied voltage, there will be no current. Therefore, the impedance of the circuit is zero.

Zt= (j)2 + j(R / L) + (1 / LC) = 0

Take the derivative of both sides:

2j + R / L = 0

The Frequency is called Damping Factor. The Damping Factor of the series RLC is . In other words, the damping factor of the series RLC is equal to half of the Response Frequency of the

inductor.

Compare the resonant frequency with the damping factor. We have three situations depending on the relationship between these values:

1. The Resonant Frequency = the Damping Factor (o

= ). The circuit is Critically Damped

2. The Resonant Frequency > the Damping Factor (o

> ). The circuit is Under Damped

3. The Resonant Frequency < the Damping Factor (o

< ). The circuit is Over Damped

Parallel RLC Circuit

Circuit Impedance

1/Z = 1/ZR

+ 1/ZL

+ 1/ZC

1/Z = 1/R + 1/jL + jC

1/Z = jL + R + (j)2RLC / jLR

1/Z = (j)2RLC + jL + R / jLR

Frequency Response of the Circuit

1. There is one frequency at whichZC

= ZL

jL = 1 / jC

The Frequency is called The Resonant Frequency.

2. When there is no applied voltage . There will be no current . Therefore, the circuit Impedance is zero

3. Z = (j)2 + j(1/RC) x + (1/LC) = 0Take derivative of both sides2j + 1 / RC = 0

=

The Frequency (1 / RC) is called the Damping Factor.The Damping Factor of the parallel RLC is (1 / RC) . In other words, the damping factor of the parallel RLC is equal half of the Response Frequency of the capacitor.

Compare the resonance frequency with the damping f actor, we have the following three situations:

1. The Resonant Frequency = the Damping Factor . The circuit is Critical Damp1 /(LC) = (R / L)

Z = (j)2 + j(1/RC) x + (1/LC) = 0 . This equation only has one root

Because the resonance frequency is equal to the damping factor. Therefore, the square root is zero.

= 0

I(t) = (A +Bt)etFor arbitrary constantsA andB. Critically damped circuits typically have low overshoot, no oscillations, and quick settling time.

2. The Resonant Frequency < the Damping Factor . The circuit is Under Damp(L / C) > (R / L)

Z = (j)2 + j(1/RC) x + (1/LC) = 0 . This equation has 2 imaginaries Roots

Because The Resonance Frequency is greater than the Damping Factor . Therefore , The Square Roots is negative . And Square Root of a negative number is an imaginary number

is a negative number, and square root of a negative number is an imaginary number.

1

= + ic

and2 = ic

Where:

The solutions are:

200

RLC series Critically Damped




28/65

For arbitrary constantsA andB. Using Euler's formula, we can simplify the solution as:

For arbitrary constants CandD. These solutions are characterized by exponentially decaying sinusoidal response. The higher the Quality Factor (below), the longer it takes for theoscillations to decay.

3. The Resonant Frequency > the Damping Factor . The circuit is Over Damp(L / C) < (R / L)

Z = (j)2 + j(1/RC) x + (1/LC) = 0 . This Equation has 2 Real Roots

Because The Resonant Frequency is greater than the Damping Factor. Therefore , The Square Root is a positive number

Overdamped circuits are characterized as having a very large settling time, and possibly a large steady-state error.

Resonance

A circuit containing resistors, capacitors, and inductors is said to be in resonance when the reactance of the inductor cancels that of the capacitor to

leave the resulting total resistance of the circuit to be equal to the value of the component resistor. The resonance state is achieved by fine tuning the frequency of the circuit to a value where the resulting

impedance of the capacitor cancels that of the inductor, resulting in a circuit that appears entirely resistive.

In other words, the resonant frequency of the circuit is the f requency where the circuit impedance is the lowest, and the circuit produces the highest output gain. We define the resonant frequency of an

RLC circuit as:

Damping Factor

The damping factor is the amount by which the oscillations of a circuit gradually decrease over time. We define the damping ratio to be:

Compare The Damping factor with The Resonance Frequency give rise to different types of circuits: Overdamped, Underdamped, and Critically Damped.

Bandwidth

= 2

For series RLC circuit:

For Parallel RLC circuit:

Quality Factor

For Series RLC circuit:

For Parallel RLC circuit:

Stability

Because inductors and capacitors act differently to different inputs, there is some potential for the circuit response to approach infinity when subjected to certain types and amplitudes of inputs. When the

output of a circuit approaches infinity, the circuit is said to be unstable. Unstable circuits can actually be dangerous, as unstable elements overheat, and potentially rupture.

A circuit is considered to be stable when a "well-behaved" input produces a "well-behaved" output response. We use the term "Well-Behaved" differently for each application, but generally, we mean

"Well-Behaved" to mean a finite and controllable quantity.

Second-Order Solution

The solution proceedures for the second-order circuit is a sufficiently complex-enough task that we are going to devote the entire next page to it. See: Second-Order Solution.

Conclusion

RLC series Over-Damped Response

[Resonant Frequency]

Circuit Type Series RLC Parallel RLC

Damping Factor

Resonance Frequency

[Bandwidth]

[Quality Factor]

Cicuit General Series RLC Parallel RLC

Cicuit

200




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Impedance Z

Roots = =

I(t) Ae1t + Be2t Ae1t + Be2t Ae1t + Be2t

DampingFactor

ResonantFrequency

o

BandWidth

= 2

Qualityfactor




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The Second-Order Circuit Solution

Second-Order Solution

This page is going to talk about the solutions to a second-order, RLC circuit. The second-order solution is resonably complicated, and a complete understanding of it will require an understanding of

differential equations. This book will not require you to know about differential equations, so we will describe the solutions without showing how to derive them. The derivations may be put into another

chapter, eventually.

The aim of this chapter is to develop the complete response of the second-order circuit. There are a number of steps involved in determining the complete response:

1. Obtain the differential equations of the circuit2. Determine the resonant frequency and the damping ratio3. Obtain the characteristic equations of the circuit4. Find the roots of the characteristic equation5. Find the natural response6. Find the forced response7. Find the complete response

We will discuss all these steps one at a time.

Finding Differential Equations

A Second-order circuit cannot possibly be solved until we obtain the second-order differential equation that describes the circuit. We will discuss here some of the techniques used for obtaining the

second-order differential equation for an RLC Circuit.

NoteParallel RLC Circuits are easier to solve in terms of current. Series RLC circuits are easier to solve in terms of voltage.

The Direct Method

The most direct method for finding the differential equations of a circuit is to perform a nodal analysis, or a mesh current analysis on the circuit, and then solve the equation for the input function. The

final equation should only contain derivatives, no integrals.

The Variable Method

If we create 2 variables, g and h, we can use them to create a second-order differential equation. First, we set g and h to be either inductor currents, capacitor voltages, or both. Next, we create a s ingle

first order differential equation that has g = f(g, h). Then, we write another first-order differential equation that has the form:

or

Next, we substitute in our second equation into our f irst equation, and we have a second-order equation.

Zero-Input Response

The zero-input response of a circuit is the state of the circuit when there is no forcing function (no current input, and no voltage input). We can set the differential equation as such:

This gives rise to the characteristic equation of the circuit, which is explained below.

Characteristic Equation

The characteristic equation of an RLC circuit is obtained using the "Operator Method" described below, with zero input. The characteristic equation of an RLC circuit (series or parallel) will be:

The roots to the characteristic equation are the "solutions" that we are looking for.

Finding the Characteristic Equation

This method of obtaining the characteristic equation requires a little trickery. First, we create an operator s such that:

Also, we can show higher-order operators as such:

Where x is the voltage (in a series circuit) or the current (in a parellel circuit) of the circuit source. We write 2 first order differential equations for the inductor currents and/or the capacitor voltages in

our circuit. We convert all the differentiations to s, and all the integrations (if any) into (1/s). We can then use Cramer's rule to solve for a solution.

Solutions

The solutions of the characteristic equation are given in terms of the resonant frequency and the damping ratio:

Ifeitherof these two values are used for s in the assumed solutionx =Aestand that solution completes the differential equation then it can be considered a valid solution. We will discuss this more,below.

Damping

The solutions to a circuit are dependant on the type ofdamping that the circuit exhibits, as determined by the relationship between the damping ratio and the resonant frequency. The different types of

damping are Overdamping, Underdamping, and Critical Damping.

Overdamped

A circuit is called Overdamped when the following condition is true:

[Characteristic Equation Solution]




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> 0

In this case, the solutions to the characteristic equation are two distinct, positive numbers, and are given by the equation:

, where

In a parallel circuit:

= 1 / (2RC)

0= 1 /sqrt(LC)

In a series circuit:

=R / (2L)

0= 1 /sqrt(LC)

Overdamped circuits are characterized as having a very large settling time, and possibly a large steady-state error.

Underdamped

A Circuit is called Underdamped when the damping ratio is less than the resonant frequency.

< 0

In this case, the characteristic polynomial's solutions are complex conjugates. This results in oscillations or ringing in the circuit. The solution consists of two conjugate roots:

1

= + ic

and

2

= ic

where

The solutions are:

for arbitrary constantsA andB. Using Euler's formula, we can simplify the solution as:

for arbitrary constants CandD. These solutions are characterized by exponentially decaying sinusoidal response. The higher the Quality Factor (below), the longer it takes for the oscillations to decay.

Critically Damped

A circuit is called Critically Damped if the damping factor is equal to the resonant frequency:

= 0

In this case, the solutions to the characteristic equation is a double root. The two roots are identical (1

= 2

= ), the solutions are:

I(t) = (A +Bt)et

for arbitrary constantsA andB. Critically damped circuits typically have low overshoot, no oscillations, and quick settling time.

Series RLC

The differential equation to a simple series circuit with a voltage source V, and a resistor R, a capacitor C, and an inductor L is:

Where v is the voltage across the circuit. The characteristic equation then, is as follows:

With the two roots:

and

Parallel RLC

The differential equation to a parallel RLC circuit with a resistorR, a capacitor C, and an inductorL is as follows:

Where v is the voltage across the circuit. The characteristic equation then, is as follows:

RLC series Over-Damped Response

RLC series Critically Damped

A series RLC circuit.




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With the two roots:

and

Circuit Response

Once we have our differential equations, and our characteristic equations, we are ready to assemble the mathematical form of our circuit response. RLC Circuits have differential equations in the form:

Wheref(t) is the forcing function of the RLC circuit.

Natural Response

The natural response of a circuit is the response of a given circuit to zero input (i.e. depending only upon the initial condition values). The natural Response to a circuit will be denoted asxn(t). The

natural response of the system must satisfy the unforced differential equation of the circuit:

We remember this equation as being the "zero input response", that we discussed above. We now define the natural response to be an exponential function:

xn

=A1est+A

2est

Where s are the roots of the characteristic equation of the circuit. The reasons for choosing this specific solution forxn

is based in differential equations theory, and we will just accept it w ithout proof for

the time being. We can solve for the constant values, by using a system of two equations:

x(0) =A1

+A2

Wherex is the voltage (of the elements in a parallel circuit) or the current (through the elements in a series circuit).

Forced Response

The forced response of a circuit is the way the circuit responds to an input forcing function. The Forced response is denoted asxf(t).

Where the forced response must satisfy the forced differential equation:

The forced response is based on the input function, so we can't give a general solution to it. However, we can provide a set of solutions for different inputs:

Complete Response

The Complete response of a circuit is the sum of the forced response, and the natural response of the system:

xc(t) =x

t(t) +x

s(t)

Once we have derived the complete response of the circuit, we can say that we have "solved" the circuit, and are finished working.

A parallel RLC Circuit.

[Unforced function]

[Forced function]

Input Form Out put Form

K(constant) A (constant)

Msin(t) Asin(t) +Bcos(t)

Me at Ae at

[Complete Response]




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Mutual Inductance

Magnetic Fields

Inductors store energy in the form of a magnetic field. The magnetic field of an inductor actually extends outside of the inductor, and can be affected (or can affect) another inductor close by. The image

above shows a magnetic field (red lines) extending around an inductor.

Mutual Inductance

If we accidentally or purposefully put two inductors close together, we can actually transfer voltage and current from one inductor to another. This property is called

Mutual Inductance. A device which utilizes mutual inductance to alter the voltage or current output is called a transformer.

The inductor that creates the magnetic field is called theprimary coil, and the inductor that picks up the magnetic field is called the secondary coil. Transformers are

designed to have the greatest mutual inductance possible by winding both coils on the same core. (In calculations for inductance, we need to know which materials

form the path for magnetic flux.Air core coils have low inductance; Cores of iron or other magnetic materials are better 'conductors' of magnetic flux.)

The voltage that appears in the secondary is caused by the change in the shared magnetic field, each time the current through the primary changes. Thus, transformers work on A.C. power, since the

voltage and current change continuously.

Modern Inductors

The following photo shows several examples of the construction of inductors and transformers. At the upper right is a toroidal core type (toroid is the mathematical term for a donut s hape). This shape

very efficiently contains the magnetic flux, so less power (or signal) is lost to heating up the core.




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Ideal Transformers

Step Up and Step Down

The terms 'step-up' and 'step-down' are used to compare the secondary (output) voltage to the voltage supplied to the primary.

Many transformers are specially designed to operate exclusively as step-up or step-down. While an ideal transformer could simply be 'turned around', we find that many actual transformers are built to

perform best at certain ranges of voltage and current.

For example, a power transformer may be used to step down household AC (about 120 Volts) to 24V for home heating controls, etc. The output current is higher than the primary current in this example,

so the transformer is made with a heavier gauge of wire in its secondary windings.

In transformers that deal with very high voltages, special attention is paid to insulation. The windings that deal with thousands of volts must resist arcing and other problems we do not see at home.

Finally, some transformers in electronic equipment are designed for a task known as 'impedance matching', rather than for specific in/out voltages. This function is explained in literature covering audio

and radio topics.




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further reading

Electronics/Transformers

(This section has not yet been written)




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State-Variable Approach

State Variables

A more modern approach to circuit analysis is known as the state variable method, which we will attempt to describe here. We use variables called state variables to describe the current state of the

energy storage elements (capacitors and inductors). Here, we are using the word "state" to mean "condition" or "status" of the elements.

Aim of State Variables

What is the goal of using state variables? We have all sorts of other methods for solving and describing circuits, so why would we introduce another method for dealing with circuits?

An answer to both these questions lay in the theory of the state variable method: the state variable approach attempts to describe a circuit using a system of first-order differential equations instead of a

single, higher-order equation. A system of first-order equations can then be easily manipulated using linear algebra techniques, and can be solved through brute-force methods such as cramers rule.

State Variable Selection

We choose state variables to be either the voltage across a capacitor, or the current through an inductor. If our circuit has multiple irreducible capacitors or inductors, we assign a state variable to each

one.

System of Equations

Once we have assigned our state variables, we are tasked with finding a f irst-order differential equation that describes each one individually. Once we have all of these equations, we can set them up in

matrix form, and use Cramers rule, or another analysis method to solve the system.

Further Reading

Control Systems/State-Space Equations




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Sinusoidal Sources

The circuits that we have analyzed previously have been DC , where a constant

voltage or current is applied to the circuit. In the following chapters, we will

discuss the topic of alternating current (AC), which utilizes sinusoidal forcingfunctions to stimulate a circuit.




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Sinusoidal Sources

Sinusoidal Sources

With the advent of AC power, analysis of circuits has become a much more complicated task that requires a whole set of new analysis tools. Many of the mathematical concepts and tools used to work

with sinusoids are very different from the kinds of tools that people are used to working with.

Sinusoidal Forcing Functions

Let us consider a general AC forcing function:

v(t) =Msin(t+ )

In this equation, the term M is called the "Magnitude", and it acts like a scaling factor that allows the peaks of the sinusoid to be higher or lower then +/- 1. The term is what is known as the "Radial

Frequency". The term is an offset parameter known as the "Phase".

Sinusoidal sources can be current sources, but most often they are voltage sources.

Other Terms

There are a few other terms that are going to be used in many of the following sections, so we will introduce them here:

PeriodThe period of a sinusoidal function is the amount of time, in seconds, that the sinusoid takes to make a complete wave. The period of a sinusoid is always denoted with a capital T. This is not to beconfused with a lower-case t, which is used as the independant variable for time. A sinusoid moving through an entire

FrequencyFrequency is the reciprocal of the period, and is the number of times, per second, that the sinusoid completes an entire cycle. Frequency i

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