Circuit Theory/All Chapters

From Wikibooks, the open-content textbooks collection

< Circuit Theory

Jump to: navigation, search

This is the print version of Circuit Theory

If you print this page, choose "print preview" in your browser, or click printable version, you will see the page without this notice, without navigational elements to the left or top, and without the TOC boxes on each page.
Refresh this page to see the latest changes.
For more information about the print version, including how to create a truly printable PDF File, see Wikibooks:Print versions.

Circuit Theory
Wikibooks: The Free Library

Circuit Theory

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 Introduction

The 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

Circuit Theory

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.

Basic Terminology

Circuit Theory

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. Other domains that an engineer might encounter are the frequency domain, the Laplace Domain, and the Z domain.

Temporal: Temporal is basically another way of saying "Time Domain". This is opposed to the word "Spectral", which is analogous to the term "Frequency Domain."

Circuit Response: Circuits generally have inputs and outputs. In fact, it is safe to say that a circuit isn't useful if it doesnt have one or the other (preferably 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 everything is done getting ready, 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: The response that a circuit has before it settles on its steady-state response is known as the transient response.

Variables and Standard Units

Circuit Theory

Electric Charge (Coulombs)

Note:
An electron has a charge of
-1.602×10E-19 C.

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 6.24150962915265×10E18 electrons, thus a single electron has a charge of −1.602 × 10E−19 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:

$i = \frac{dq}{dt}$

For the rest of this book, the lower-case J ( j ) will be used to denote an imaginary number, and the lower-case I ( i ) will be used to denote current.

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).

$V = \frac{W}{q}$

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.

Electric Circuit Basics

Circuit Theory

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 stored 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 explode.

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.

Voltmeters should never be placed in-line with other circuit elements.

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.

Ammeters should never be placed parallel to other circuit elements, without some resistance.

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 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.

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

There is 0 voltage difference between the terminals x and y.
There is 0 current flowing on terminals x and y.
There is 0 current flowing on terminal z.

For instance, if we know that x has a voltage with respect to ground of +5V, then we know that y also has a voltage with respect to ground of +5V.

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:

Dependent sources cannot be turned off.
Current Sources become an Open Circuit when turned off.
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.

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 $t < T 0$ , and acts like a closed-circuit for all time $t \ge T_0$ .

Unit Step Function

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

[Unit Step Function]

$u(t) = \left\{ \begin{matrix} 0, & \mbox{if }t < 0 \\ 1, & \mbox{if }t \ge 0 \end{matrix} \right.$

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.

Resistors and Resistance

Circuit Theory

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 could be defined as

R t = R o (1 + a T)

Where:

R_t is Final resistance,

R_o is Initial resistance,

a is Temperature coefficient,

T is 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

Wikipedia has related information at Electrical resistance.

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

$r = {\rho L \over A}$

where ρ 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 Ω). The associated variable is "G":

$r = \frac{1}{G}$

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 this wikibook will probably never mention it again.

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]

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.

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.

Resistive Circuit Analysis Techniques

Circuit Theory

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 resitor 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:

$v=iR \,$

The resistance is given the symbol "R" and is the unit of ohm which is denoted by a capital omega, Ω. 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:

$\Omega = \dfrac{\mbox{V}}{\mbox{A}} = \dfrac{\mbox{m}^2 \cdot \mbox{kg}}{\mbox{s}^{3} \cdot \mbox{A}^2}$

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

$P=vi \,$

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

$P=i^2 R \,$

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:

$v=iR \,$

$i=\frac{1}{R}v$

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

$i=Gv \,$

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:

$S = \dfrac{\mbox{A}}{\mbox{V}} = \dfrac{\mbox{s}^{3} \cdot \mbox{A}^2}{\mbox{m}^2 \cdot \mbox{kg}}$

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).

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.

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 R_eq.

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

$R_{eq}=\frac{v}{i}$

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:

$v=iR \,$

If the resistance is zero, we have

$v=i \times 0$

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:

$i=\frac{1}{R}v$

If the resistance tends to infinity, we have

$i =\lim \limits_{R \to \infty } \left( {{1 \over R}v} \right) = 0$

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.

$x = \sum_{n}^{} x_n$
$x = \left( \sum_{n}^{} \frac{1}{x_n} \right)^{-1}$

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.

Resistors

Resistors appearing in series can be converted into a single resistor, $r s e r i e s$ according to the following equation:

Resistors in a series configuration

[Resistors In Series]

$r_{tot} = \sum_{n}^{} r_n$

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, $v s e r i e s$ given by the equation:

$v_{series} = \sum_{n}^{} v_n$

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 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:

A diagram of several resistors appearing in parallel with each other.

[Resistors in Parallel]

$r_{tot} = \left(\sum_{n}^{} \frac{1}{r_n}\right)^{-1}$

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

$r_1 \| r_2 = \frac{r_1 r_2}{r_1 + r_2}$ .

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:

$i_{parallel} = \sum_{n}^{} i_n$

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 Current: current flowing into a node
Negative Current: current 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:

1A->           <-i
o--------o-------o
         N

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

1 A + 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:

1A->          1A->
o--------o-------o
         N

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 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:

Forward current on a source creates positive voltage, or a voltage increase.
Forward current on a load element creates a voltage drop, or a negative voltage.
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:

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:

v = - (1 A)(5Ω) = - 5 V

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      |
-|     <-i  |
 +----------+

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:

$12V + vr = 0 \to vr = -12V$

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:

v r = 12 V

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:

$12V = i(5\Omega) \to i = \frac{12V}{5\Omega} = 2.4A$

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

$I_i = I_{in} \frac{\frac{1}{Ri}}{\sum_{k=1}^n \frac{1}{R_k}}$

or equivalently, using conductances:

$I_i = I_{in} \frac{G_i}{\sum_{k=1}^n G_k}$

Construction

Voltage Divider

Definition

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

$V_i = V_{in} \frac{Ri}{\sum_{k=1}^n R_k}$

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

Circuit Theory

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.

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 resistance R_s and the voltage source, v_s by attaching and 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 v_s and R_s can be found using the following equation:

$v=v_s+iR_s \,$

The are two variables, so two values of i 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, 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 Thévenin's Theorem after French telegraph engineer Léon Charles Thévenin, and the circuit consisting of the voltage source and resistance is called the Thévenin Equivalent Circuit.

Norton's Theorem

Recall from above that the output voltage, v, of a Thévenin equivalent circuit can be expressed as

$v=v_s+iR_s \,$

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

$i=-\frac{v_s}{R_s}+\frac{v}{R_s}$

This is equivalent to a KCL description of the following circuit. We can call the constant term v_s/R_s the source current, i_s.

The equivalent current source and the equivalent resitance 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 i_sc (which is exactly the same as the source current i_s).

Circuit Transforms

We have just shown turns out that the Thévenin 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 Thévenin source and a Norton source from outside the black box, and that we can directly equate the two as below:

$\equiv$

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, R_eq:

$R_{s_n}=R_{s_t}=R_s=R_{eq}$

The value of a Thévenin voltage source is the value of the Norton current source times the equivalent resistance (Ohm's law):

$v_s=i_sr\,$

If these rules are followed, the circuts will behave identically. Using these few rules, we can transform a Norton circuit into a Thévenin circuit, and vice versa. This method is called source transformation. See Example 3.

Open Circuit Voltage and Short Circuit Current

The open-circuit voltage, v_oc of a circuit is the voltage across the terminals when the current is zero, and the short-circuit current i_sc is the current when the voltage across the terminals in zero:


The open circuit voltage	The short circuit current

We can also observe the following:

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

$v_s=v_{oc}\,$

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

$i_s=i_{sc}\,$

We can say that, generally,

$R_{eq}=\frac{v_{oc}}{i_{sc}}$

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.

Resistive Circuit Analysis Methods

Circuit Theory

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:

$I=\frac{\Delta V }{R} = \frac{V_{high}-V_{low}}{R}$ .

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

Identify the nodes. These are places where one device ends and another begins (ie a wire connects to a resistor).
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.
Label all the nodes, usually written as V_n, where n is the number of the node
Use Kirchoff's Current Law to set up an equation for each node. This will leave you with a System of Equations.
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: $V_0 = 0V\,$ (defined as ground node)
node 1: $V_1 = 9V\,$ (free node voltage)
node 2: $\frac{V_1 - V_2}{1k} = \frac{V_2 - V_0}{3k} + \frac{V_2 - V_3}{2k}$
node 3: $\frac{V_2 - V_3}{2k} = \frac{V_3 - V_0}{2k}$

which results in the following system of linear equations:

$\left\{\begin{matrix} +11 V_2 & -3 V_3 & = & 54 \\ +1 V_2 & -2 V_3 & = & 0\end{matrix}\right.$

Therefore, the solution is:

$\left\{\begin{matrix} V_0 =0.00V \\ V_1 =9.00V \\ V_2 =5.68V \\ V_3 =2.84V\end{matrix}\right.$

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. For M meshes, there will be M equations.

If all sources are voltage sources, all M equations will be KVL.

If the circuit has N current sources, there will be N KCLs and M-N KVLs.

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: $0 = 9 - 1000 I 1 - 3000(I 1 - I 2)$
Loop2: $0 = 3000(I 1 - I 2) - 2000 I 2 - 2000 I 2$

Simplifying we get the simultaneous equations:

0 = 9 - 4000 I 1 + 3000 I 2

0 = 0 + 3000 I 1 - 7000 I 2

Solving to get:

I 1 = 3.32 m A

I 2 = 1.42 m A

Superposition

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

The superposition 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. If f is taken to be the response, and a and b are constant, then:

$f(ax_1+bx_2)= f(nx_1) + f(jx_2) \,$

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 response y is equal to the square of the input, i.e. y=x². If a and b are constant, then:

$y=(ax_1+bx_2)^2 \ne (ax_1)^2 + (bx_2)^2 = y_1+y_2\,$

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:

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.)
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.
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.
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:

[Convolution]

$y(t) = (h*x)(t) = \int_{-\infty}^{+\infty} h(t-s)x(s)ds$ .

If the input, x(t), was an impulse ( $δ(t)$ ), 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 definate 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.

Definition

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

$(a * b)(t) = \int_{-\infty}^\infty a(\tau)b(t - \tau)d\tau$

Remember:
Asterisks mean convolution, not multiplcation

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:

	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 ω 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

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

Circuit Theory

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

Wikipedia has related information at Capacitance.

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:

$C = \frac{q}{v}$

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

[Capacitor Relation]

$i = C\frac{dv}{dt}$

Energy Storage

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

$E = \frac{1}{2}Cv^2$

Capacitors in Series

A set of N capacitors in series

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

$C_{series} = \frac{{\prod_{n}^{} C_{n}}}{{\sum_{n}^{} C_{n}}} = \left ( \sum_{n}^{} \frac{1}{x_n} \right )^{-1}$

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

Z C = R C + X C

Where:

R_C Resistance of the Capacitor

X_C Reactance of the Capacitor = 1 / jωC

j =ν - 1

ω = 2πf

C = 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^ο . Voltage of the Resistance is in the same phase with the Applied Voltage . The Load Voltage is at an angle θ with the total Voltage of Resistance and Reactance

V_{X_C} lags V_{R_C} by 90°
V_{R_C} is 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 ω_o = 1 / RC and the time that it takes to reach this frequency is t = RC

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

Z_C = R_C + 1 / jωC = 0 or

jω = 1 / CR_C

ω = 0 $Z_C = R_C + \infty$ . Open circuit V_o = 0
ω = ω_o . Starts to react or conduct current. V_o ≈ V_i
ω = infinity $Z C = R C + 0$ . V_o ≈ V_i.

Capacitors in Parallel

A set of N capacitors in parallel

For capacitors in parallel:

$C_{parallel} = {\sum_{n}^{} C_{n}}$

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. When the magnetic field has reached it's maximum capacity, the inductor no longer stores energy, and the inductor becomes a short-circuit.

Inductance

Wikipedia has related information at 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

[Inductor Relation]

$v = L\frac{di}{dt}$

Energy Storage

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

$w = \frac{1}{2}Li^2$

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 of transformers, 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

Z L = R L + X L

R_L Resistance of the Inductor

X_L Reactance of the Inductor = jωL

j = γ -1

ω = 2πf

L = 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^ο . Voltage of the Resistance is in the same phase with the Applied Voltage

V_{X_L} leads V_{R_L} by 90^ο
V_{R_C} is the same phase with V_i

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 ω_o = R / L and the time that it takes to reach this frequency is t = L / R

How do you arrived the Frequency reponse ? Idealy, When there is no Voltage apply on Inductor . There will be no current flows . Therefore, The Impedance of the Inductor is equal to 0

Z_L = R_L + jωL = 0 or

jω = R_L / L

ω = 0 Z_L = R_L + 0 . Inductor is Short circuited V_o ≈ V_i
ω = ω_o Inductor Starts to React or conduct current . V_o ≈ V_i
ω = infinity $Z L = R L + i n f i n i t y$ . Inductor is Open circuited V_o ≈ 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, $L s e r i e s$ given as follows:

$L_{series} = \sum_{n}^{} L_n$

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:

$L_{parallel} = \frac{\Pi L_n}{\Sigma L_n}$

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

Circuit Theory

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:

RC (resistor and capacitor)
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 first-order differential equation.

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:

$v(t) = L\frac{di}{dt}$

Wikipedia has related information at RL circuit.

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:

$i_{source}(t) = \frac{L}{R_n}\frac{di_{inductor}(t)}{dt} + i_{inductor}(t)$

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

RC Circuits

Wikipedia has related information at RC circuit.

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 thevenin source. Then if we apply KVL around the resulting loop, we get the following equation:

$v_{source} = RC\frac{dv_{capacitor}(t)}{dt} + v_{capacitor}(t)$

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 an x to signify that this will be a general solution to either type of problem. Here, we will consider a general equation of the form:

$\frac{dx(t)}{dt} + \frac{x(t)}{T_c} = k$

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 of the circuit, when we divide through. T_c is a value known as the "Time Constant".

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

$\frac{dx}{x - kT_c} = -\frac{dt}{T_c}$

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

$\ln(x - kT_c) = -\frac{t}{T_c} + D$

Where D 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:

$x(t) = kT_c + Ae^{-\frac{t}{T_c}}$

A = e D

It turns out that A is also the value of the initial condition of the circuit, x(0). Also kT_c is equal to the value of the steady-state value of the function. Combining this knowledge, we get the following equation:

[First Order Solution]

$x(t) = x(\infty) + [x(0) - x(\infty)]e^{-\frac{t}{T_c}}$

Where:

$x(\infty)$

Is the steady state value of x. This is our general result. Remember that x 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, T_c, 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:

$T_c = \frac{L}{R}$

and the time constant for an RC circuit is:

T c = R C

In general, from an engineering standpoint, we say that the system is at steady state after a time period of five time constants.

RLC Circuits

Circuit Theory

Wikipedia has related information at RLC Circuit.

Series RLC Circuit

Circuit Impedance

Z_t = Z_R + Z_L + Z_C

Z_t = R + jωL + 1 / jωC

Z_t = jωRC + (jω)²LC + 1

Z_t = (jω)²LC + jωRC + 1

Z_t = (jω)² + jω(R/L) + 1/LC

Frequency Response of the circuit

There is one frequency at which Z_C = Z_L

jωL = 1 / jωC

$\omega =1 /\sqrt{LC}$

The Frequency $1 /\sqrt{LC}$ is called Resonance Frequency
When there is no applied voltage, there will be no current. Therefore, the impedance of the circuit is zero.

Z_t = (jω)² + jω(R / L) + (1 / LC) = 0

Take the derivative of both sides:

2jω + R / L = 0

$\omega = \frac{1}{2}$ $\frac{R}{L}$

The Frequency $\frac{1}{2}\frac{R}{L}$ is called Damping Factor. The Damping Factor of the series RLC is $\frac{1}{2}$ $\frac{R}{L}$ . 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:

The Resonant Frequency = the Damping Factor ( $ω o = ζ$ ). The circuit is Critical Damped
The Resonant Frequency > the Damping Factor ( $ω o > ζ$ ). The circuit is Under Damped
The Resonant Frequency < the Damping Factor ( $ω o < ζ$ ). The circuit is Over Damped

Parallel RLC Circuit

Circuit Impedance

200

1/Z = 1/Z_R + 1/Z_L + 1/Z_C

1/Z = 1/R + 1/jωL + jωC

1/Z = jωL + R + (jω)²RLC / jωLR

1/Z = (jω)²RLC + jωL + R / jωLR

$Z = (j\omega)^2 + j\omega\frac{1}{RC} + \frac{1}{LC}$

Frequency Response of the Circuit

There is one frequency at which Z_C = Z_L

jωL = 1 / jωC

$1 / \sqrt{LC}$

The Frequency $1 / \sqrt{LC}$ is called The Resonant Frequency.
When there is no applied voltage . There will be no current . Therefore, the circuit Impedance is zero
Z = (jω)² + jω(1/RC) x + (1/LC) = 0

Take derivative of both sides

2jω + 1 / RC = 0

ω = $\frac{1}{2}$ $\frac {1}{RC}$

The Frequency ½ (1 / RC) is called the Damping Factor.
The Damping Factor of the parallel RLC is ½ (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 factor, we have the following three situations:

The Resonant Frequency = the Damping Factor . The circuit is Critical Damp

RLC series Critically Damped

1 / γ(LC) = ½ (R / L)

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

$s = - \zeta \pm \sqrt{\zeta^2 - \omega_o^2}$

Because the resonance frequency