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Circuit Theory
Wikibooks: The Free Library

Preface

This wikibook is going to be an introductory text about electric circuits. It will cover some the basics of electric circuit theory, circuit analysis, and will touch on circuit design. This book will serve as a companion reference for a 1st year of an Electrical Engineering undergraduate curriculum. Topics covered include AC and DC circuits, passive circuit components, phasors, and RLC circuits. The focus is on students of an electrical engineering undergraduate program. Hobbyists would benefit more from reading Electronics instead.

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

The main editable text of this book is located at http://en.wikibooks.org/wiki/Circuit_Theory. The wikibooks version of this text is considered the most up-to-date version, and is the best place to edit this book and contribute to it.

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.

Basic Terminology

Basic Terminology

There are a few key terms that need to be understood at the beginning of this book, before we can continue. This is only a partial list of all terms that will be used throughout this book, but these key words are important to know before we begin the main narrative of this text.

Time domain 

The concept of a domain is very important in mathematics, and is also very important in engineering. Depending on what domain you are in, there are different tools and techniques for analyzing circuits. The "Time domain" is simply another way of saying that our circuits change with time, and that the major variable used to describe the system is time.

Frequency domain

The frequency domain is a very commonly used method of describing the behavior of a circuit as functions of the frequency of the signals within it. Another name is the "Fourier domain". Other domains that an engineer might encounter are the the "Laplace domain" (or the "s 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 doesn't have one or the other (usually both). The response to a circuit is the relationship between the circuit's input to the circuit's output. The circuit response may be a measure of either current or voltage.

Steady State 

When something changes in a circuit, there is a certain amount of transition period before a circuit "settles down", and reaches its final value. This final value, when all elements have a constant or periodic behaviour, is known as the steady-state value of the circuit. The circuit response at steady state (when things aren't changing) is also known as the "steady state response".

Transient Response 

The response that a circuit has before it settles on its steady-state response is known as the transient response.

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 6.24150962915265×10E18 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:

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.

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.

The diagram on the right shows three ways of drawing the interesection of two wires with connection (a) and no connection (b). The modern convention is to use the third style.

In real life, nodes are often connected together, either by wire nuts, or solder, or other connectors. These connectors can have a certain amount of associated resistance, capacitance, or inductance associated with them. This book will not, however, take this interference into account, as it is usually negligible.

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 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 all the current from one terminal flows through to the other terminal. By convention, current into the + terminal is displayed as positive.

Sources

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

Current Sources

Current sources are sources that output a specified amount of current. The voltage produced by the current source will be dependent on the current output, and the resistance of the load (ohm's law).

Voltage Sources

Voltage sources produce a specified amount of voltage. The amount of current that flows out of the source is dependent on the voltage and the resistance of the load (again, ohm's law). This can be dangerous because if 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.

1. Op Amp has infinite impedance
2. Op Amp has infinite bandwidth
3. Op Amp has infinite voltage gain
4. Op Amp has Zero output impedance
5. Op Amp has Zero offset (error)

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

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

Independent Sources

Independent sources produce current/voltage at a particular rate that is dependent only on time. These sources may output a constant current/voltage, or they may output current/voltage that varies with time.

Dependent Sources

Dependent sources are current or voltage sources whose output value is based on time or another value from the circuit. A dependent source may be based on the voltage over a resistor for example, or even the current flowing through a given wire. The following sources are possible:

• Current-controlled current source
• Current-controlled voltage source
• Voltage-controlled current source
• Voltage-controlled voltage source

Dependent sources are useful for modelling transistors or vacuum tubes.

Turning Sources "Off"

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

1. Dependent sources cannot be turned off.
2. Current Sources become an Open Circuit when turned off.
3. Voltage Sources become a Closed Circuit when turned off.

Occasionally it is written that the source is "removed" from the circuit, because often it is physically possible to remove the source component (be it a battery, or a plug, or any other source component) physically from the circuit. we can't 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 function u(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.

Resistors and Resistance

Resistors

Resistors are circuit elements that allow current to pass through them, but restrict the flow according to a specific ratio called "Resistance". Flow that is restricted by resistors is said to be "lost to the resistor". Resistors are commonly used as heating elements, because energy lost to the resistor is frequently dispersed into the surroundings as heat. Every resistor has a given resistance. Resistors that have a variable resistance as a function of position are known as "potentiometers". Resistors that have a variable resistance as a function of temperature are called "thermisters".

Function of Temperature

Resistance also depends on surrounding temperature. It is defined as

R_t = R_o(1 + aT)

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:

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":

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.

Resistive Circuit Analysis Techniques

Resistive Circuit Analysis

There are certain established rules that can be used to examine a circuit that is comprised entirely of resistors and sources. Among these are tools for combining resistors in certain configurations into a single conceptual resistor that has an equivalent resistance to the two that have been replaced. In this manner, very complicated circuits can be reduced to be a very simple circuit with few components.

The Resistor

The resistor is an electrical component that limits the flow of electrical current. The physical characteristics of a resistor are described in the Electronics Wikibooks and information on resistors as found in real life are in Practical Electronics.

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

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 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:

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'',

Similarly, since i = v / 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:

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

Conductance has the unit siemens, named after the German scientist Ernst Werner von Siemens. The symbol for siemens is S. Conductance is expressed in base units as:

A siemens is therefore the current in the resistor per unit voltage across it. As with all SI units named after people, the word "siemens" is lowercase, but the symbol is uppercase (S).

In older texts, and often times in handwritten calculations, the older symbol Mho (: an upside-down capital Greek letter Omega) is used because an uppercase (S) is too easily confused with the following: lowercase (s) for "seconds"; a variable "S"; and lastly, the numeral (5).

Real Resistors

Resistors tend to heat up as they pass more current, due to an increase in dissipated power. Since usually the resistance of a material changes with temperature, this produces a distorted v-i graph:

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

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.

Resistors

Resistors appearing in series can be converted into a single resistor, r_series 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, v_series given by the equation:

It is to be noted that the magnitude of voltage should be a signed integer depending on the polarity of the voltage source. The convention for polarity can vary from one reference to another. But usually the voltage source 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. This is merely a restatement of the law of conservation of energy - we cannot get current out where no current went in.

For a node with n connections to other nodes, where i_k is the current flowing into this node from node k, Kirchhoff's Current Law states:

This is a vector sum in that the direction of the current matters. In fact the common convention is to define positive and negative currents as follows:

• Positive Current is current flowing into a node
• Negative Current is current flowing out 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 look at a few simple examples.

KCL Example 1

Kirchoff's Voltage Law (KVL)

Kirchoff's voltage law (KVL) states that the sum of the voltages around a closed loop must all equal zero. Again, we should come up with a convention, although this one will be a little bit more complicated.

Forward Current

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

Backwards Current

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

Once we have our notions of forward and backward current flow decided upon, we can then write out our convention for voltage increases and decreases:

1. Forward current on a source creates positive voltage, or a voltage increase.
2. Forward current on a load element creates a voltage drop, or a negative voltage.
3. Backwards current on a passive load element also creates a voltage 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 = − (1A)(5Ω) = − 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:

     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:

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 divider is 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 divider is 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

Source Transformations

Source Transformations

Independent current sources can be turned into independent voltage sources, and vice-versa, by methods called "Source Transformations." These transformations are useful for solving circuits. We will explain the two most important source transformations, Thevenin's Source, and Norton's Source, and we will explain how to use these conceptual tools for solving circuits.

Black Boxes

A circuit (or any system, for that matter) may be considered a black box if we don't know what is inside the system. For instance, most people treat their computers like a black box because they don't know what is inside the computer (most don't even care), all they know is what goes in to the system (keyboard and mouse input), and what comes out of the system (monitor and printer output).

Black boxes, by definition, are systems whose internals aren't known to an outside observer. The only methods that an outside observer has to examine a black box is to send input into the systems, and gauge the output.

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 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 v_s and R_s can be found using the following equation:

There 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

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 v_s/R_s the 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 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:

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:

• The value of a Thévenin 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 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:

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

We can say that, generally,

Why Transform Circuits?

Why would we ever bother transforming our circuits? Let's say that we have a resistor in series with a Norton circuit. If we transform the circuit to a Thevenin circuit, we can add the resistor values together! Likewise, let's say that we have a resistor in parallel to a Thevenin circuit: if we transform to a norton circuit, the resistors will be in parallel, and we can combine them! Many circuits can be completely simplified down into a circuit with a single resistor and a single source.

Resistive Circuit Analysis Methods

Analysis Methods

When circuits get large and complicated, it is useful to have various methods for simplifying and analyzing the circuit. There is no perfect formula for solving a circuit. Depending on the type of circuit, there are different methods that can be employed to solve the circuit. Some methods might not work, and some methods may be very difficult in terms of long math problems. Two of the most important methods for solving circuits are Nodal Analysis, and Mesh Current Analysis. These will be explained below.

Nodal Analysis

Nodal analysis is the application of Kirchoff's Current Law (KCL) to solve for the voltages at each node in an equation. A node voltage is defined as the potential difference between the given node and a designed reference node (ground). Since one node is defined as ground, a circuit with N nodes will require N-1 equations to solve completely.

If all sources are current sources, all N-1 equations will be KCL equations: The sum of the current into the node is equal to the sum of the current out of the node. Currents not connected to current sources can be found using:

.

Given M voltage sources (for M less than or equal to N-1), there will be M KVL equations and (N-1)-M KCL equations. A supernode may be formed if necessary.

Steps

1. Identify the nodes. These are places where one device ends and another begins (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 node
4. Use Kirchoff's Current Law to set up an equation for each node. This will leave you with a System of Equations.
5. Solve the System of Equations for each unknown variable

Example

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

• 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 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

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:

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 of superposition. 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 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:

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:

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

Step by Step

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

1. Isolate a source - Select a source, and set all of the remaining sources to zero. The consequences of "turning off" these sources are explained in Open and Closed Circuits. In summary, turning off a voltage source results in a short circuit, and turning off a current source results in an open circuit. (Reasoning - no current can flow through a open circuit and there can be no voltage drop across a short circuit.)
2. Find the output from the isolated source - Once a source has been isolated, the response from the source in question can be found using any of the techniques we've learned thus far.
3. Repeat steps 1 and 2 for each source - Continue to choose a source, set the remaining sources to zero, and find the response. Repeat this procedure until every source has been accounted for.
4. Sum the Outputs - Once the output due to each source has been found, add them together to find the total response.

Impulse Response

An impulse response of a circuit can be used to determine the output of the circuit:

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

.

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 definite integral, and as such, solving convolution equations will require knowledge of integral calculus. This wikibook will not require a prior knowledge of integral calculus, and therefore will not go into more depth on this subject then a simple definition, and some light explanation.

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:

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.

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

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

Capacitance is defined as the capability of a capacitor to store charge of a voltage . Capacitance is measured in units called "Farads", abbreviated with an "F" (capital F).

The ratio of Charge over Voltage gives a value of Capacitance

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

Z_C = R_C + jX_C

Where:

R_C Resistance of the Capacitor

X_C Reactance of the Capacitor = 1 / jωC

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_XC lags V_RC by 90°
• V_RC 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 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.

Z_C = R_C + 1 / jωC = 0 or

jω = 1 / CR_C

• ω = 0 . 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

For capacitors in parallel:

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

Capacitors can "Pop"

Many capacitors are polarized in a particular way. If you apply voltage across the terminals of a polarized capacitor, the capacitor itself might pop. This is made more dangerous by the fact that many capacitors have chlorine gas inside, because the chlorine raises the capacitance of the capacitor. Popping a chlorine-filled capacitor will be very unpleasant (if not down-right dangerous).

Capacitors can Kill

Strong capacitors, 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

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

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 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_XL leads V_RL by 90^ο
• V_RC 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 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

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 + infinity . Inductor is Open circuited V_o ≈ 0

Inductors in Series

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

Inductors in Parallel

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.

First-Order Circuits

First Order Circuits

First order circuits are circuits that contain only one energy storage element (capacitor or inductor), and that can therefore be described using only a first order differential equation. The two possible types of first-order circuits are:

1. RC (resistor and capacitor)
2. RL (resistor and inductor)

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

RL Circuits

An RL 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:

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:

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

RC Circuits

Wikipedia has related information at 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:

First Order Solution

Series RL

The differtial equation of the series RL circuit

. A = e^C

t	I(t)
0	A
1	36% A
2	A
3	A
4	A
5	1% A

Series RC

The differtial equation of the series RC circuit

. A = e^C

t	V(t)
0	A
1	36% A
2	A
3	A
4	A
5	1% A

Time Constant

The series RL and RC has a Time Constant

In general, from an engineering standpoint, we say that the system is at steady state ( Voltage or Current is almost at Ground Level ) after a time period of five Time Constants.

RLC Circuits

Wikipedia has related information at RLC Circuit

Series RLC Circuit

Second Order Differential Equation

Let then the above equation becomes quadratic equation with two roots

s = − α ±

Where

When

α² = β² . R =

The equation only has one real root .

The solution for

The I - t curve would look like

When

α² > β² . R >

The equation only has two real root . s = − α ±

The solution for

The I - t curve would look like

When

α² < β² . R <

The equation has two complex root . s = − α ± j

The solution for

The I - t curve would look like

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:

Circuit Type	Series RLC	Parallel RLC
Damping Factor
Resonance Frequency

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

Bandwidth

Δω = 2ζ

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.

Resonance

With R = 0

When R = 0 , the circuit reduce to series LC circuit . When circuit in resonance , circuit will vibrate at resonance frequency

Z_L = Z_C

The circuit vibrate and has the capability of producing Standing Wave when R = 0 , L = C

With R ≠ 0

When R ≠ 0 and the circuit operates in resonance .

The frequency dependent components L , C cancel out ie Z_L - Z_C = 0 so that the total impedance of the circuit is Z_R + Z_L + Z_C = R + [Z_L − Z_C] = R + 0 = R

The current of the circuit is

The Operating Frequency is

If the current is halved by doubling the value of resistance then

Circuit will be stable over the range of frquencies from ω₁ − ω₂

The circuit has the capability to select bandwidth where the circuit is stable . Therefore, it is best suited for Tuned Resonance Select Bandwidth Filter

Once using L or C to tune circuit into resonance at resonance frequency The current is at its maximum value . Reduce current above circuit will respond to narrower bandwidth than ω₁ − ω₂. Reduce current below circuit will respond to wider bandwidth than ω₁ − ω₂.

Conclusion

Circuit	General	Series RLC	Parallel RLC
Circuit			200
Impedance	Z
Roots	λ	λ =	λ =
I(t)	Aeλ1t + Beλ2t	Aeλ1t + Beλ2t	Aeλ1t + Beλ2t
Damping Factor	ζ
Resonant Frequency	ωo
Band Width	Δω = 2ζ
Quality factor

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 circuit
2. Determine the resonant frequency and the damping ratio
3. Obtain the characteristic equations of the circuit
4. Find the roots of the characteristic equation
5. Find the natural response
6. Find the forced response
7. 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.

Note

Parallel 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 single 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 first 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:

If either of these two values are used for s in the assumed solution x = Ae^st and 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 of damping 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:

α > ω₀

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:

α = 1 / (2RC)

ω₀ = 1 / sqrt(LC)

In a series circuit:

α = R / (2L)

ω₀ = 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.

ζ < ω₀

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:

λ₁ = − ζ + iω_c

and

λ₂ = − ζ − iω_c

where

The solutions are:

for arbitrary constants A and B. Using Euler's formula, we can simplify the solution as:

for arbitrary constants C and D. 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:

ζ = ω₀

In this case, the solutions to the characteristic equation is a double root. The two roots are identical (λ₁ = λ₂ = λ), the solutions are:

I(t) = (A + Bt)e^λt

for arbitrary constants A and B. 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 resistor R, a capacitor C, and an inductor L is as follows:

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

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:

Where f(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 as x_n(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:

x_n = A₁e^st + A₂e^st

Where s are the roots of the characteristic equation of the circuit. The reasons for choosing this specific solution for x_n is based in differential equations theory, and we will just accept it without proof for the time being. We can solve for the constant values, by using a system of two equations:

x(0) = A₁ + A₂

Where x 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 as x_f(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:

Input Form	Output Form
K (constant)	A (constant)
Msin(ωt)	Asin(ωt) + Bcos(ωt)
Me − at	Ae − at

Complete Response

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

x_c(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.

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 the primary 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.

Ideal Transformers

Modern Inductors

When the coils of number of turns N₁ conducts current . There exists a Magnetic Field B on the coil . Changes of B will generates an Induced Voltage on the turns of coil N₁ and N₂ as shown

-ξ_p =

-ξ_s =

The ratio of -ξ₂ over -ξ₁

-ξ_p / -ξ_s =

If Input voltage at coil of turn N_p = -ξ_p and the Output voltage will be

= -ξ_s / -ξ_p =

Thus, this device is capable of Increase, Decrease and Conduct Voltage just by changing the turn ratio of the coils

Therefore, the output voltage can be

• Increased or Step Up by increasing number of turns of coil N_s greater than N_p

• Decreased or Step Down by Decreasing number of turns of coil N_s less than N_p

• Buffered by setting number of turns of coil N_s equal to N_p

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

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.

further reading

• Electronics/Transformers

(This section has not yet been written)

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 first-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

Sinusoidal Sources

Wikipedia has related information at AC power

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(ωt + φ)

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 ω is what is known as the "Radial Frequency". The term φ 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:

Period 

The 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 be confused with a lower-case t, which is used as the independant variable for time. A sinusoid moving through an entire

Frequency 

Frequency is the reciprocal of the period, and is the number of times, per second, that the sinusoid completes an entire cycle. Frequency is measured in Hertz (Hz). The relationship between frequency and the Period is as follows:

Where f is the variable most commonly used to express the frequency.

Radian Frequency 

Radian frequency is the value of the frequency expressed in terms of Radians Per Second, instead of Hertz. Radian Frequency is denoted with the variable ω. The relationship between the Frequency, and the Radian Frequency is as follows:

ω = 2πf

Phase 

The phase is a quantity, expressed in radians, of the time shift of a sinusoid. A sinusoid phase-shifted φ = + 2π is moved forward by 1 whole period, and looks exactly the same. An important fact to remember is this:

Phase is often expressed with many different variables, including φ,ψ,θ,γ etc... This wikibook will try to stick with the symbol φ, to prevent confusion.

Lead and Lag

A circuit element may have both a voltage across its terminals, and a current flowing through it. If one of the two (current or voltage) is a sinusoid, then the other must be a sinusoid. (remember, voltage is the derivative of the current, and the derivative of a sinusoid is always a sinusoid). However, the sinusoids of the voltage and the current may differ by quantities of magnitude and phase.

If the current has a lower phase angle then the voltage, the current is said to lag the voltage. If the current has a higher phase angle then the voltage, it is said to lead the voltage. Many circuits can be classified and examined using lag and lead ideas.

Fundamental Principle

There is a fundamental principle in engineering that states: you get out what you put in. While this might apply metaphorically to scholastic effort, it definately applies concretely to electric circuits. For instance, if we have a polynomial forcing function, our circuit response will be a polynomial output. If our input is exponential, our output will similarly be exponential. And (most important for our present uses) Sinusoidal input will produce sinusoidal output. This principle only holds for passive circuit elements, because active circuit elements (particularly a rectifier) can convert a sinusoid into a constant value, but we don't worry about that for this book.

Sinusoidal Response

In general, we say that given an input sinusoid, we will produce an output sinusoid:

Of course, as we can see from all the terms in the above equations, there is plenty of room for the input signal to be altered by the circuit. The fundamental tenet holds throughout this book that you get out what you put in. For that reason, if the input is a polynomial, the output will be a polynomial. If the input is a sinusoid, the output will be a sinusoid. If the input is an exponential, the output will be an exponential, et cetera.

For ease of use, we simply say that if the input sinusoid is a sine function, then the output will also be a sine function. If the input sinusoid is a cosine function, the output will also be a sinusoid. Sometimes it might be easier to write one or the other, but we like to stick to this convention.

Sinusoidal Conventions

For the purposes of this book we will generally use cosine functions, as opposed to sine functions. If we absolutely need to use a sine, we can remember the following trigonometric identity:

cos(ωt) = sin(ωt + π / 2)

We can express all sine functions as cosine functions. This way, we dont have to compare apples to oranges per se. This is simply a convention that this wikibook chooses to use to keep things simple. We could easily choose to use all sin( ) functions, but further down the road it is often more convenient to use cosine functions instead by default.

RMS

Wikipedia has related information at Root mean square

Sinusoidal functions for current and voltage can be best considered as single values, through the root mean square (RMS) calculation. RMS is essentially an averaging operation of all the magnitude values of the sinusoid. The RMS formula for a continuous function f(t) defined over the interval (for a periodic function the interval should be a whole number of complete cycles) is:

Since we are only considering sinusoids, we can reduce this calculation to:

Where f_max is the peak value of the sinusoid. RMS values for current and voltage, I_rms and V_rms can both be used in the standard power equation like normal:

P_avg = I_rmsV_rms

Phasor Representation

This book contains mathematical formulae that look better rendered as PNG.

Input and Output

If the input to a linear circuit is a sinusoid, then the output from the circuit will be a sinusoid. Specifically, if we have a voltage sinusoid as such:

v(t) = M_vcos(ωt + φ_v)

Then the current through the linear circuit will also be a sinusoid, although its magnitude and phase may be different quantities:

i(t) = M_icos(ωt + φ_i)

Note that both the voltage and the current are sinusoids with the same radial frequency, but different magnitudes, and different phase angles. Passive circuit elements cannot change the frequency of a sinusoid, only the magnitude and the phase. Why then do we need to write ω in every equation, when it doesnt change? For that matter, why do we need to write out the cos( ) function, if that never changes either? The answers to these questions is that we don't need to write these things every time. Instead, engineers have produced a short-hand way of writing these functions, called "phasors".

Euler's Equation

An important mathematical rule that needs to be understood for this discussion to continue is Euler's Equation. This rule is very difficult to understand at first, and it is often useful to derive the rule for better understanding, but in this wikibook we will take Euler's Equation as being axiomatic:

Me^{j(ωt + φ)} = Mcos(ωt + φ) + jMsin(ωt + φ)

This equation allows us to view sinusoids as complex exponential functions. Specifically, this allows us to consider the magnitude and phase angles directly, without having to worry about annoying trigonometry. Also, by viewing these quantities in terms of the complex quantity , We can graph the point (X, Y) on the complex plane. On the complex plane, the Imaginary axis is the vertical axis, and the horizontal axis is the real axis, and points can be plotted exactly the same as they are plotted on a regular Cartesian plane.

Using this fact, we can get the angle from the origin of the complex plane to out point (X, Y) with the function:

And using the pythagorean theorem, we can find the magnitude of C -- the distance from the origin to the point (X, Y) -- as:

.

The system of using an angle and a distance to describe a point, instead of an X and Y coordinate system is known as Polar Graphing. Using X and Y coordinates is known as Rectangular Graphing.

Phasors

If the radial frequency for all sources is the same then we can employ a representation called "Phasors". Phasors are similar to vectors, but they represent a sinusoid instead of a location in space. This is an important fact, because at any time a phasor can be converted into a cosine function, and any cosine function can be converted into a phasor. However, the user has to remember to write down the value for the radial frequency, because phasor notation does not include the radial frequency.

If we have the following complex exponential:

v(t) = M_ve^{j(ωt + φ)}

We know what the value of ω is because we have already figured it out and written it down someplace safe. The only important pieces of information then are the phase angle φ, and the magnitude M. We can then separate out these quantities into a new representation called a phasor:

Phasors will always be written out either with a large bold letter (as above), or will be written out with a vector notation, such as the letter This wikibook prefers the former notation for the simple reason that phasors are not vectors (or else presumably, we would call them "vectors").

Cosine Convention

It is important to remember which trigonometric function your phasors are mapping to. Since a phasor only includes information on magnitude and phase angle, it is impossible to know whether a given phasor maps to a sin( ) function, or a cos( ) function instead. By convention, this wikibook will say that all phasors map to cosine functions. This way, there is no ambiguity when we are moving from one notation to another.

Granted, the other convention could just as well be used, but it is important to pick a single convention and to stick with it. In this book, we assume that all phasors map to cosine functions, just as an arbitrary convention.

Any Sinusoid Can Be a Phasor

It is also important to remember that phasors can represent any quantity that can be represented by either a sinusoid, or a complex exponential function (as per Eulers Equation, above). For this reason, Current and Voltage can both be written as phasors, using the notation and , respectively. Other complex values can also be expressed as phasors, including Impedance, and Complex Power, which are discussed in later chapters.

When dealing with phasors, we must transform every element in the circuit into a phasor representation. This means that every source, and every load element (resistor, capacitor, or inductor) can be transformed into a phasor quantity. We will discuss all these transformations later.

Phasor Arithmetic

Phasors can be mapped into a variety of different representations, each of which is good for different calculations. Since phasors map neatly to exponential functions, the multiplication of 2 phasors is incredibly easy. If we have a phasor A and a phasor B, we can multiply them together, as such:

Keep in mind that as an exponential function, we are multiplying the coefficients (the magnitudes), and we are adding the exponents (the phase angles). As an exercise, the reader should convert the above phasors A and B into exponential representation, and do the math by yourself to prove this point.

Likewise, division is done extremely easily with phasors in their exponential notation, because the magnitudes can be divided into each other, and the phase angles can be subtracted from one another:

It is important to remember that the phasor on the bottom of the division (the denominator) is subtracted in the phase angles, and is the denominator of the magnitude division.

However, phasors are not good for the operations of addition and subtraction. For this, we need to convert the phasors into rectangular notation:

X = Mcos(φ), Y = Msin(φ)

This notation makes it very easy to add and subtract, so long as we follow some very simple rules:

1. Real parts get added together.
2. Imaginary parts get added together.

So, to add together 2 phasors, in rectangular form, we just add the real parts and imaginary parts, as such:

We often include subscripts on the different quantities, so we can remember which phasor each value belongs to. If we want to convert a phasor in rectangular form back into its exponential form, we can use the following relations to make the transformation:

The first part of the equation (the magnitudes) many readers will recognize as being the pythagorean theorem, and the inverse tangent part some readers will recognize as being used to find the angle from the origin to a point on the cartesian plane. This makes intuitive sense when we consider that complex values can be graphed on the complex plane. On the complex plane, the value of the magnitude is simply the distance from the origin to the point, and the phase angle is the angle the line makes with the positive Real axis.

Once we have a phasor in exponential form, we can convert it back into a sinusoid (remember, we are using cosine functions by convention) by following 2 simple steps:

1. Use Eulers equation to separate the exponential function out into sin and cos functions.
2. Take the Real value as the sinusoid result.

For instance:

Variable Frequency

If the radial frequency of a circuit is variable, we can include a notation to show that the phasors are dependent on the radial frequency:

This does not, however change the representation of the phasor, we still only write it in terms of the magnitude and the phase angle. However, different circuit components respond differently to different frequencies (we will see this in detail later), so when the frequency is variable we need to keep that information handy so we don't get confused. The downside to this is that for every different value for the frequency, conceptually we need to recompute all our phasors. Also, phasors of different frequencies cannot be combined, or used in the same equation. All the phasors need to be in relation to the same radial frequency before they can be used.

Using Phasors

It is important to remember a few details when working with phasors:

1. Phasors may represent Current, Voltage, Power, or Impedance, so it is important to remember which quantity you are describing.
2. Phasors do not include information about the radial frequency, so this information needs to be written down somewhere.
3. Phasors are a short-hand way of writing a complex exponential, so phasors are manipulated in the same way that exponentials are manipulated.
4. Phasors can be used to represent either Sine or Cosine functions, so the exact relationship being used needs to be considered. By convention, this wikibook maps all phasors to cosine functions.

Different Representations, Same Values

Phasors are simply a different way to write values that are equivalent to the "normal" time-domain way we have been writing them. However, phasors make many operations and methods of analysis much easier then they have been. Specifically, phasors allow us to ignore differential equations (good), they allow us to combine resistors, capacitors, and inductors together (good), and they allow us to focus our attention on the quantities that change in a circuit (the phase and the magnitude), without having to write out alot of other repetitive things (very good). Here are a few of the representations that we will be using throughout the rest of our discussion of phasors:

"Polar Notation" or "Phasor Notation"

"Exponential Notation"

"Rectangular Notation"

"sinusoid notation"

These 4 notations are all just different ways of writing the same exact thing.

Note on Notation

There are other ways to denote a phasor. Some common notations are:

• (the large bold block-letters we use in this wikibook)
• ("bar" notation, used by Wikipedia)
• (vector arrow notation)
• 

further reading

• Wikipedia:Euler's_Identity uses Euler's equation to prove "the most beautiful theorem in mathematics".

Phasor Circuit Analysis

Phasor Analysis

The mathematical representations of individual circuit elements can be converted into phasor notation, and then the circuit can be solved using phasors.

Resistance, Impedance and Admittance

In phasor notation, resistance, capacitance, and inductance can all be lumped together into a single term called "impedance". The phasor used for impedance is . The inverse of Impedance is called "Admittance" and is denoted with a

And the Ohm's law for phasors becomes:

It is important to note at this point that Ohm's Law still holds true even when we switch from the time domain to the phasor domain. This is made all the more amazing by the fact that the new term, impedance, is no longer a property only of resistors, but now encompasses all load elements on a circuit (capacitors and inductors too!).

Impedance is still measured in units of Ohms, and admittance (like Conductance, its DC-counterpart) is still measured in units of Siemens.

Let's take a closer look at this equation:

If we break this up into polar notation, we get the following result:

Wikipedia has related information at Electrical impedance

This is important, because it shows that not only are the magnitude values of voltage and current related to each other, but also the phase angle of their respective waves are also related. Different circuit elements will have different effects on both the magnitude and the phase angle of the voltage given a certain current. We will explore those relationships below.

Resistors

Resistors do not affect the phase of the voltage or current, only the magnitude. Therefore, the impedance of a resistor with resistance R is:

Through a resistor, the phase difference between current and voltage will not change. This is important to remember when analyzing circuits.

Capacitors

A capacitor with a capacitance of C has a phasor value:

To write this in terms of degrees, we can say:

We can accept this for now as being axiomatic. If we consider the fact that phasors can be graphed on the imaginary plane, we can easily see that the angle of − π / 2 points directly downward, along the negative imaginary axis. We then come to an important conclusion: The impedance of a capacitor is imaginary, in a sense. Since the angle follows directly along the imaginary axis, there is no real part to the phasor at all. Because there is no real part to the impedance, we can see that capacitors have no resistance (because resistance is a real value, as stated above).

Reactance

A capacitor with a capacitance of C in an AC circuit with an angular velocity ω has a reactance given by

Reactance is the impedance specific to an AC circuit with angular velocity ω.

Inductors

Inductors have a phasor value:

Where L is the inductance of the inductor. We can also write this using degrees:

Like capacitors, we can see that the phasor for inductor shows that the value of the impedance is located directly on the imaginary axis. However, the phasor value for inductance points in exactly the opposite direction from the capacitance phasor. We notice here also that inductors have no resistance, because the resistance is a real value, and inductors have only an imaginary value.

Reactance

In an AC circuit with a source angular velocity of ω, and inductor with inductance L.

Impedances Connected in Series

If there are several impedances connected in series, the equivalent impedance is simply a sum of the impedance values:

----[ Z1 ]----[ Z2 ]--- ... ---[ Zn ]---   ==> ---[ Zseries ]---

Notice how much easier this is than having to differentiate between the formulas for combining capacitors, resistors, and inductors in series. Notice also that resistors, capacitors, and inductors can all be mixed without caring which type of element they are. This is valuable, because we can now combine different elements into a single impedance value, as opposed to different values of inductance, capacitance, and resistance.

Keep in mind however, that phasors need to be converted to rectangular coordinates before they can be added together. If you know the formulas, you can write a small computer program, or even a small application on a programmable calculator to make the conversion for you.

Impedances in Parallel

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

Where N is the total number of impedances connected in parallel with each other. Impedances may be multplied in the polar representation, but they must be converted to rectangular coordinates for the summation. This calculation can be a little bit time consuming, but when you consider the alternative (having to deal with each type of element separately), we can see that this is much easier.

Steps For Solving a Circuit With Phasors

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

1. Convert all elements to phasor notation
2. Combine impedances, if possible
3. Combine Sources, if possible
4. Use Ohm's Law, and Kirchoff's laws to solve the circuit
5. Convert back into time-domain representation

Unfortunately, phasors can only be used with sinusoidal input functions. We cannot employ phasors when examining a DC circuit, nor can we employ phasors when our input function is any non-sinusoidal periodic function. To handle these cases, we will look at more general methods in later chapters

Network Function

The network function is a phasor, that is a ratio of the circuit's input to its output. This is important, because if we can solve a circuit down to find the network function, we can find the response to any sinusoidal input, by simply multiplying by the network function. With time-domain analysis, we would have to solve the circuit for every new input, and this would be very time consuming indeed.

Network functions are defined in the following way:

Where is the phasor representation of the circuit's output, and is the representation of the circuit's input. In the time domain, to find the output, we would need to convolute the input with the impulse response. With the network function, however, it becomes a simple matter of multiplying the input phasor with the network function, to get the output phasor. Using this method, we have converted an entire circuit to become a simple function that changes magnitude and phase angle.

Gain

Gain is the amount by which the magnitude of the sinusoid is amplified or attenuated by the circuit. Gain can be computed from the Network function as such:

Where the bars around the phasors are the "magnitude" of the phasor, and not the "absolute value" as they are in other math texts. Again, gain may be a measure of the magnitude change in either current or voltage. Most frequently, however, it is used to describe voltage.

Phase Shift

The phase shift of a function is the amount of phase change between the input signal and the output signal. This can be calculated from the network function as such:

Where the denotes the phase of the phasor.

Again, the phase change may represent current or voltage.

Phasor Theorems

Circuit Theorems

Phasors would be absolutely useless if they didn't make the analysis of a circuit easier. Luckily for us, all our old circuit analysis tools work with values in the phasor domain. Here is a quick list of tools that we have already discussed, that continue to work with phasors:

• Ohm's Law
• Kirchoff's Laws
• Superposition
• Thevenin and Norton Sources
• Maximum Power Transfer

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

Ohm's Law

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

Separating this out, we get:

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

Kirchoff's Laws

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

Kirchoff's Current Law

Kirchoff's current law states that the amount of current entering a particular node must equal the amount of current leaving that node. Notice that KCL never specifies what form the current must be in: any type of current works, and KCL always holds true.

Kirchoff's Voltage Law

KVL states: The sum of the voltages around a closed loop must always equal zero. Again, the form of the voltage forcing function is never considered: KVL holds true for any input function.

Superposition

Superposition may be applied to a circuit if all the sources have the same frequency. However, superposition must be used as the only possible method to solve a circuit with sources that have different frequencies. The important part to remember is that impedance values in a circuit are based on the frequency. Different reactive elements react to different frequencies differently. Therefore, the circuit must be solved once for every source frequency. This can be a long process, but it is the only good method to solve these circuits.

Thevenin and Norton Circuits

Thevenin Circuits and Norton Circuits can be manipulated in a similar manner to their DC counterparts: Using the phasor-domain implementation of Ohm's Law.

It is important to remember that the does not change in the calculations, although the phase and the magnitude of both the current and the voltage sources might change as a result of the calculation.

Maximum Power Transfer

The maximum power transfer theorem in phasors is slightly different then the theorem for DC circuits. To obtain maximum power transfer from a thevenin source to a load, the internal thevenin impedance () must be the complex conjugate of the load impedance ():

Complex Power

Complex Power

Just like the other values of voltage, current, and resistance, power also has a complex phasor quantity that we are going to become familiar with. Complex power is denoted with a symbol. It is calculated as such:

Where the quantity denotes the complex conjugate of the phasor current. To get the complex conjugate of , we have two formulas:

Given:

• (rectangular)
• (polar)

There is more information on complex conjugation of phasors in the Appendix.

Apparent Power

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

Where is called the apparent power. It is this quantity that we can measure, because it makes no sense to measure an imaginary number or a complex value.

Average and Reactive Power

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

, and

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

We can then convert this quantity into rectangular form where:

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

Units

Unfortunately, Power is not as simple a quantity as impedance. Unlike Impedance and resistance, The different power quantities do not all share the same units. We list the units for each type of power, below:

Time-Domain Power 

Watts (w)

Average Power

Watts (w)

Complex Power 

Volt-Amps (VA)

Reactive Power

Volt-Amps Reactive (VAR)

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

Power and Impedance

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

If the element in question is a resistor, the reactive power delivered will be 0. Likewise, if the element is a capacitor or an inductor, the average power delivered will be zero. If the impedance is complex, then the delivered power will be complex.

Conservation of Power

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

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

Power Factor

The relationship between the average power, and the apparent power is called the power factor. Power factor is given the variable p_f, and is calculated as such:

p_f = cos(φ_v − φ_i)

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

p_fangle = φ_v − φ_i

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

cos(φ_v − φ_i) = cos(φ_i − φ_v)

This means that to be able to accurately calculate the phase angles of the current and the voltage from the power factor, we need an additional specifier of either leading or lagging.

Lagging 

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

φ_V > φ_I

Leading 

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

φ_V < φ_I

Maximum Transfer Theorem

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

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

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

This book contains mathematical formulae that look better rendered as PNG.

Laplace Transform

The Laplace Transform is a powerful tool for an Electrical Engineer to use. The transform allows equations in the "time domain" to be transformed into an equivalent equation in the Complex S Domain. The laplace transform is an integral transform, although the reader does not need to have a knowledge of integral calculus because all results will be provided. This page will discuss the Laplace transform as being simply a tool for solving and manipulating ordinary differential equations.

Laplace transformations of circuit elements are similar to phasor representations, but they are not the same. Laplace transformations are more general than phasors, and can be easier to use in some instances. Also, do not confuse the term "Complex S Domain" with the complex power ideas that we have been talking about earlier. Complex power uses the variable , while the Laplace transform uses the variable s. The Laplace variable s has nothing to do with power.

The transform is named after the mathematician Pierre Simon Laplace, who lived in the 18th century. The transform itself did not become popular until Oliver Heaviside, a famous electrical engineer, began using a variation of it to solve electrical circuits.

The Transform

The mathematical definition of the Laplace transform is as follows:

The transform, by virtue of the definite integral, removes all t from the resulting equation, leaving instead the new variable s. In essence, this transform takes the function f(t), and "transforms it" into a function in terms of s, F(s). As a general rule the transform of a function f(t) is written as F(s). Time-domain functions are written in lower-case, and the resultant S-domain functions are written in upper-case.

we will use the following notation to show the transform of a function:

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

The Inverse Transform

The inverse laplace transform converts a function in the complex S-domain to its counterpart in the time-domain. Its mathematical definition is as follows:

where c is a real constant such that all of the poles s₁,s₂,...,s_n of F(s) fall in the region . In other words, c is chosen so that all of the poles of F(s) are to the left of the vertical line intersecting the real axis at s = c.

The inverse transform is more difficult mathematically than the transform itself is. However, luckily for us, extensive tables of laplace transforms and their inverses have been computed, and are available for easy browsing.

Laplace Domain

The Laplace domain, or the "Complex s Domain" is the domain into which the Laplace transform transforms a time-domain equation. s is a complex variable, composed of real parts:

s = σ + jω

The Laplace domain graphs the real part (σ) as the horizontal axis, and the imaginary part (ω) as the vertical axis. The real and imaginary parts of s can be considered as independent quantities.

Transform Properties

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

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

Or for an arbitrary derivative:

where the notation f⁽ⁿ⁾(t) means the n^th derivative of the function f at the point t, and f⁽⁰⁾(t) means f(t).

In plain english, the laplace transform converts differentiation into polynomials. The only important thing to remember is that we must add in the initial conditions of the time domain function, but for most circuits, the initial condition is 0, leaving us with nothing to add.

For integrals, we get the following:

Initial Value Theorem

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

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

Final Value Theorem

Similar to the Initial Value Theorem, the Final Value Theorem states that we can find the value of a function f, as t approaches infinity, in the laplace domain, as such:

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

Transfer Function

If we have a circuit with impulse-response h(t) in the time domain, with input x(t) and output y(t), we can find the Transfer Function of the circuit, in the laplace domain, by transforming all three elements:

In this situation, H(s) is known as the "Transfer Function" of the circuit. It can be defined as both the transform of the impulse response, or the ratio of the circuit output to it's input in the Laplace domain:

Transfer functions are powerful tools for analyzing circuits. If we know the transfer function of a circuit, we have all the information we need to understand the circuit, and we have it in a form that is easy to work with. When we have obtained the transfer function, we can say that the circuit has been "solved" completely.

Convolution Theorem

Earlier it was mentioned that we could compute the output of a system from the input and the impulse response by using the convolution operation. As a reminder, given the following system:

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

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

y(t) = x(t) * h(t)

Where the asterisk denotes convolution, not multiplication. However, in the S domain, this operation becomes much easier, because of a property of the laplace transform:

Where the asterisk operator denotes the convolution operation. This leads us to an english statement of the convolution theorem:

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

X(s) = Input H(s) = Transfer Function Y(s) = Output

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

Y(s) = X(s)H(s)

Notice that this property is very similar to phasors, where the output can be determined by multiplying the input by the network function. The network function and the transfer function then, are very similar quantities.

Resistors

The laplace transform can be used independently on different circuit elements, and then the circuit can be solved entirely in the S Domain (Which is much easier). Let's take a look at some of the circuit elements:

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

R(s) = r

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

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

Ohm's Law

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

V(s) = I(s)R

Now, following ohms law, the resistance of the circuit element is a ratio of the voltage to the current. So, we will solve for the quantity , and the result will be the resistance of our circuit element:

This ratio, the input/output ratio of our resistor is an important quantity, and we will find this quantity for all of our circuit elements. We can say that the transform of a resistor with resistance r is given by:

Capacitors

Let us look at the relationship between voltage, current, and capacitance, in the time domain:

Solving for voltage, we get the following integral:

Then, transforming this equation into the laplace domain, we get the following:

Again, if we solve for the ratio , we get the following:

Therefore, the transform for a capacitor with capacitance C is given by:

Inductors

Let us look at our equation for inductance:

putting this into the laplace domain, we get the formula:

V(s) = sLI(s)

And solving for our ratio , we get the following:

Therefore, the transform of an inductor with inductance L is given by:

Impedance

Since all the load elements can be combined into a single format dependant on s, we call the effect of all load elements impedance, the same as we call it in phasor representation. We denote impedance values with a capital Z (but not a phasor ).

Laplace Circuit Solution

One of the most important uses of the Laplace transform is to solve linear differential equations, just like the type of equations that represent our first- and second-order circuits. This page will discuss the use of the Laplace Transform to find the complete response of a circuit.

Steps

Here are the general steps for solving a circuit using the Laplace Transform:

1. Determine the differential equation for the circuit.
2. Use the Laplace Transform on the differential equation.
3. Solve for the unknown variable in the laplace domain.
4. Use the inverse laplace transform to find the time domain solution.

Another method that we can use is:

1. Transform the individual circuit components into impedance values using the Laplace Transform.
2. Find the Transfer function that describes the circuit
3. Solve for the unknown variable in the laplace domain.
4. Use the inverse laplace transform to find the time domain solution.

Fourier Transform

The Fourier Transform is a specific case of the Laplace transform. If we separate s into its real and imaginary parts:

s = σ + jω

Where s is the complex laplace variable, σ is the real part of s, and ω is the imaginary part of s. Remember, in Electrical Engineering, j is the imaginary number, not i.

Now, if we set , we can get the following:

s = jω

Plugging into the Laplace transform, we get the following formula:

The variable ω is known as the "radial frequency" of the circuit. This term refers to the frequency of the circuit. The Fourier transform, in the respect that it accounts only for the response of the circuit to a given frequency is very similar to phasor notation. However the Fourier Transform produces an equation that can be used to analyze the circuit for all frequencies, not just a single frequency like phasors are limited to.

As with the Laplace transform, there is an inverse Fourier transform:

However, there are extensive tables of Fourier transforms and their inverses available, so we need not waste time computing individual transforms.

Radial Frequency

In the Fourier transform, the value ω is known as the Radial Frequency, and has units of radians/second (rad/s). People might be more familiar with the variable f, which is called the "Frequency", and is measured in units called Hertz (Hz). The conversion is done as such:

ω = 2πf

For instance, if a given AC source has a frequency of 60Hz, the resultant radial frequency is:

ω = 2πf = 2π(60) = 120π

Fourier Domain

The Laplace transform converts functions from the time domain to the complex s domain. s has real and imaginary parts, and these parts form the axes of the s domain: the real part is the horizontal axis, and the imaginary part is the vertical axis. However, in the Fourier transform, we have the relationship:

And therefore we don't have a real part of s. The Fourier domain then is broken up into two distinct parts: the magnitude graph, and the phase graph. The magnitude graph has jω as the horizontal axis, and the magnitude of the transform as the vertical axis. Remember, we can compute the magnitude of a complex value C as:

C = A + jB

The Phase graph has jω as the horizontal axis, and the phase value of the transform as the vertical axis. Remember, we can compute the phase of a complex value as such:

C = A + jB

The phase and magnitude values of the Fourier transform can be considered independant values, although some abstract relationships do apply. Every fourier transform must include a phase value and a magnitude value, or it cannot be uniquely transformed back into the time domain.

The combination of graphs of the magnitude and phase responses of a circuit, along with some special types of formatting and interpretation are called Bode Plots, and are discussed in more detail in the next chapter.

Impedance and Reactance

In the Fourier domain, the concepts of capacitance, inductance, and resistance can be generalized into a single complex term called "Impedance." Impedance in this sense is exactly the same as the impedance quantities from the Laplace domain and the phasor domain. In the fourier domain however, the impedance of a circuit element is defined in terms of the voltage frequency across that element, as such:

Z(jω) = R(jω) + jX(jω)

Where R is the fourier transform of resistance, and X is the transform of reactance, that we discussed earlier.

Frequency Domain Analysis

Individual circuit elements can be transformed into the Fourier frequency domain according to a few simple rules. These transformed circuit elements can then be used to find the Frequency Response of the circuit.

Resistors

Resistors are not reactive elements, and their resistance is not a function of time. Therefore, when transformed, the fourier impedance value of a resistor is given as such:

Z_resistor(jω) = r

Resistors act equally on all frequencies of input.

Capacitors

Capacitors are reactive elements, and therefore they have reactance, but no resistance, as such:

Inductors

Inductors are also reactive elements, and have the following fourier transform:

Z_inductor(jω) = jωL

Current and Voltage Sources

The frequency representation of a source is simply the transform of that source's input function.

Frequency Response

If we set , and plug this value into our transfer function:

The function H(jω) is called the "Frequency Response". The frequency response can be used to find the output of a circuit from the input, in exactly the same way that the Transfer function can be:

Y(jω) = X(jω)H(jω)

In addition, the Convolution Theorem holds for the ω domain the same way as it works for the S domain:

Decibel

Decibels (abbreviated "dB") are not units per se. Instead, a Decibel is simply a logarithmic ratio of the input to the output of a circuit. There are 2 ways to calculate decibels, depending on whether we want to analyze the voltage gain or the power gain. Considering that decibels are just a ratio between other quantities, decibels are generally followed by other units, such as "dBW" (decibels, watts), "dBmV" (decibels, millivolts), etc. In this book, if no units are supplied with the decibels, we will assume we mean "dBV" (decibels, volts).

Voltage

This is to describe the difference between the input voltage to a system, and the output voltage from the system. The terms "Vin" and "Vout" can either be the time-domain values of the voltage input and voltage output, or they can be the magnitudes of the respective phasors.

Power Gain

This is used to compare the input power of a system to the output power of a system. Notice that the only difference between the equations for power gain and voltage gain is the coefficient out front.

Notes on Decibels

Decibels are plotted on a "log10" graph, where each hash-mark on the axis is a successive power of 10. Also, the values on the X-axis are plotted on a log-10 scale as well. Each successive power-of-10 on the frequency axis is known as a "decade". It is important to mention that decibels are simply a convenient way to represent a scaling factor, and that decibels are not numbers: they are ratios. Any place where "gain" is used in an equation, decibels can not be used, because decibels are not numbers. Many people get this confused, so don't worry if it takes a while for this to sink in.

Bode Plots

Bode plots can be broken down into 2 separate graphs: the magnitude graph, and the phase graph. Both graphs represent the circuit response in each catagory to sinusoids of different frequencies.

Magnitude Graph

The Bode Magnitude Graph is a graph where the radial frequency is plotted along the X-axis, and the gain of the circuit at that frequency is plotted (in Decibels) on the Y-axis. The bode magnitude graph most frequently plots the power gain against the frequency, although they may also be used to graph the voltage gain against the frequency. Also, the frequency axis may be in terms of hertz or radians, so the person drawing a bode plot should make sure to label their axes correctly.

Phase Graph

The Bode Phase Plot is a graph where the radial frequency is plotted along the X axis, and phase shift of the circuit at that frequency is plotted on the Y-axis. The phase change is almost always represented in terms of radians, although it is not unheard of to express them in terms of degrees. Likewise, the frequency axis may be in units of hertz or radians per second, so the axes need to be labeled correctly.

Bode Plots by Different Methods

Bode plots can be used both with Phasors (Network Functions), and with the Fourier Transform (Frequency Response). However, there are slightly different methods to doing it each way, and those methods will be examined in the following chapters. The Laplace Transform can be used to construct a bode plot by transposing from the s-domain to the fourier-domain. However, this is rarely done in practice and the Laplace Transform is instead used with other graphical methods that are unfortunately, outside the scope of this wikibook.

Bode Analysis

The pages in this section will talk about how to analyze a bode plot of a given circuit, and draw conclusions from that plot.

(This section has not yet been written)

Log Magnitude Graph

Using a network function (remember phasors?), we can find the log magnitude and the phase bode plots of a circuit. This page will discuss how to find the bode polt from the network function of a circuit.

Phase Graph

Poles and Zeros

The topic of "Poles and Zeros" are discussed in excruciating detail in advanced texts in Electrical Engineering. We will introduce the concepts of what a pole and a zero are in this chapter.

Transfer Polynomials

Let's say that we have a given frequency response:

Where both Z and P are polynomials. We then set each of these equations to zero, and solve:

Z(jω) = 0

P(jω) = 0

The solutions to the equation Z = 0 are called the "Zeros" of H. The solutions to the equation P = 0 are called the "Poles" of H.

Properties of poles and zeros

Let's say that we have a frequency response that has a zero at N, and a pole at M. We then plug in these values to our frequency response:

And:

Now, some of the purists will immediately say "but you arent allowed to divide by zero", and to those people I say: you can write in a limit, if you really want to.

Bode Equation Format

let us say that we have a generic transfer function with poles and zeros:

Each term, on top and bottom of the equation, is of the form (ω_N + jω). However, we can rearrange our numbers to look like the following:

Now, if we do this for every term in the equation, we get the following:

This is the format that we are calling "Bode Equations", although they are simply another way of writing an ordinary frequency response equation.

DC Gain

The constant term out front:

is called the "DC Gain" of the function. If we set , we can see that everything in the equation cancels out, and the value of H is simply our DC gain. DC then is simply the input with a frequency of zero.

Break Frequencies

in each term:

the quantity ω_N is called the "Break Frequency". When the radial frequency of the circuit equals a break frequency, that term becomes (1 + 1) = 2. When the radial frequency is much higher then the break frequency, the term becomes much greater then 1. When the radial Frequency is much smaller then the break frequency, the value of that term becomes approximately 1.

Much Greater and Much Less

We use the term "much" as a synonym for the term "At least 10 times". So "Much Greater" becomes "At least 10 times greater" and "Much less" becomes "At least 10 times less". We also use the symbol "<<" to mean "is much less than" and ">>" to mean "Is much greater than". Here are some examples:

• 1 << 10
• 10 << 1000
• 2 << 20 Right!
• 2 << 10 WRONG!

Engineering Approximations

For a number of reasons, Electrical Engineers find it appropriate to approximate and round some values very heavily. For instance, manufacturing technology will never create electrical circuits that perfectly conform to mathematical calculations. When we combine this with the << and >> operators, we can come to some important conclusions that help us to simplify our work:

If A << B:

• A + B = B
• A - B = -B
• A / B = 0

All other mathematical operations need to be performed, but these 3 forms can be approximated away. This point will come important for later work on bode plots.

Using our knowledge of the Bode Equation form, the DC gain value, Decibels, and the "much greater, much less" inequalities, we can come up with a fast way to approximate a bode magnitude plot. Also, it is important to remember that these gain values are not constants, but rely instead on changing frequency values. Therefore, the gains that we find are all slopes of the bode plot. Our slope values all have units of "decibel per decade", or "db/decade", for short.

At Zero Radial Frequency

At zero radial frequency, the value of the bode plot is simply the DC gain value in decibels. Remember, bode plots have a log-10 magnitude Y-axis, so we need to convert our gain to decibels:

Magnitude = 20log₁₀(DCGain)

At a Break Point

We can notice that each given term changes it's effect as the radial frequency goes from below the break point, to above the break point. Let's show an example:

Our breakpoint occurs at 5 radians per second. When our radial frequency is much less than the break point, we have the following:

Gain = (1 + 0) = 1

Magnitude = 20log₁₀(1) = 0db / decade

When our radial frequency is equal to our break point we have the following:

Gain = (1 + 1) = 2

Magnitude = 20log₁₀(2) = 3db / decade

And when our radial frequency is much higher (10 times) our break point we get:

Magnitude = 20log₁₀(10) = 20db / decade

However, we need to remember that some of our terms are "Poles" and some of them are "Zeros".

Zeros

Zeros have a positive effect on the magnitude plot. The contributions of a zero are all positive:

Radial Frequency << Break Point 

0db/decade gain.

Radial Frequency = Break Point 

3db/decade gain.

Radial Frequency >> Break Point 

20db/decade gain.

Poles

Poles have a negative effect on the magnitude plot. The contributions of the poles are as follows:

Radial Frequency << Break Point 

0db/decade gain.

Radial Frequency = Break Point 

-3db/decade gain.

Radial Frequency >> Break Point 

-20db/decade gain.

Conclusions

To draw a bode plot effectively, follow these simple steps:

1. Put the frequency response equation into bode equation form.
2. identify the DC gain value, and mark this as a horizontal line coming in from the far left (where the radial frequency conceptually is zero).
3. At every "zero" break point, increase the slope of the line upwards by 20db/decade.
4. At every "pole" break point, decrease the slope of the line downwards by 20db/decade.
5. at every breakpoint, note that the "actual value" is 3db off from the value graphed.

And then you are done!

See Also

• Bode Plots on ControlTheoryPro.com

Impedance

Let's recap: In the transform domain, the quantities of resistance, capacitance, and inductance can all be combined into a single complex value known as "Impedance". Impedance is denoted with the letter Z, and can be a function of s or jω, depending on the transform used (Laplace or Fourier). This impedance is very similar to the phasor concept of impedance, except that we are in the complex domain (laplace or fourier), and not the phasor domain.

Impedance is a complex quantity, and is therefore comprised of two components: The real component (resistance), and the complex component (reactance). Resistors, because they do not vary with time or frequency, have real values. Capacitors and inductors however, have imaginary values of impedance. The resistance is denoted (as always) with a capital R, and the reactance is denoted with an X (this is common, although it is confusing because X is also the most common input designator). We have therefore, the following relationship between resistance, reactance, and impedance:

Z = R + jX

Susceptance and Admittance

The inverse of resistance is a quantity called "Conductance". Similarly, the inverse of reactance is called "Susceptance". The inverse of impedance is called "Admittance". Conductance, Susceptance, and Admittance are all denoted by the variables Y or G, and are given the units Siemens. This book will not use any of these terms again, and they are just included here for completeness.

Parallel Components

Once in the transform domain, all circuit components act like basic resistors. Components in parallel are related as follows:

Series Components

Series components in the transform domain all act like resistors in the time domain as well. If we have two impedances in series with each other, we can combine them as follows:

Z₁ in series with Z₂ = Z₁ + Z₂

Solving Circuits

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Polyphase Systems

A polyphase system is an AC circuit system that uses multiple sinusoidal inputs, with equal frequencies but different phase values, to transmit power to loads. Polyphase systems have a number of advantages over "single phase" systems: equal amounts of power can be transmitted using waves of smaller amplitude, polyphase motors and appliances run smoother than single-phase appliances, etc. One of the most popular polyphase implementations is 3-phase systems.

3-Phase Systems

3-phase systems use 3 sinusoidal forcing functions, each with the same magnitude and frequency, but 120^o apart from each other. 3-phase systems have a curious mathematical property:

This means that if the waves have the same amplitude and frequency, those waves can cancel each other out entirely. This means that if a transmission system is appropriately balanced among 3 wires, no ground wire is required. Even if the loads aren't perfectly balanced, a ground wire can be supplied, with the expectation that most of the current will be cancelled out, and the ground wire will carry very little current back to the source.

In a 3-phase system, we define the 3 main transmission wires as a, b, and c. Then the respective voltages on these are defined as:

• line a: v_a(t) = V_mcos(ωt)
• line b:
• line c:

3-Phase Phasors

In phasor representation we can say:

• 
• 
• 

This is because all these functions have the same frequency, but differing phase values.

3-Phase Loads

Each load impedance should be equal to the others, to provide for a balanced system:

We can define the currents on each

So that the return voltage should equal zero:

Circuit Functions

Circuit Functions

This appendix page will list the various values of the variable H that have been used throughout the circuit theory textbooks. These values of H are all equivalent, but are represented in different domains. All of the H functions are a ratio of the circuit input over the circuit output.

The "Impulse Response"

The impulse response is the time-domain relationship between the circuit input and the circuit output, denoted with the following notation:

h(t)

The impulse response is, strictly speaking, the output that the circuit will produce when an ideal impulse function is the input. The impulse response can be used to determine the output from the input through the convolution operation:

y(t) = h(t) * x(t)

We can determine the impulse response from the input and output functions by division:

The "Network Function"

The network function is the phasor-domain representation of the impulse response. The network function is denoted as such:

The network function is related to the input and output of the circuit through the following relationships:

Similarly, the network function can be received by dividing the output by the input, in the phasor domain.

The "Transfer Function"

The transfer function is the laplace-transformed representation of the impulse response. It is denoted with the following notation:

H(s)

The transfer function can be obtained by one of two methods:

1. Transform the impulse response.
2. Transform the circuit, and solve.

The Transfer function is related to the input and output as follows:

Y(s) = H(s)X(s)

The "Frequency Response"

The Frequency Response is the fourier-domain representation of the impulse response. It is denoted as such:

H(jω)

The frequency response can be obtained in one of three ways:

1. Transform the impulse response
2. Transform the circuit and solve
3. Substitute s = jω into the transfer function

The frequency response has the following relationship to the circuit input and output:

Y(jω) = H(jω)X(jω)

The frequency response is particularly useful when discussing a sinusoidal input, or when constructing a bode diagram.

Phasor Arithmetic

Phasor Arithmetic

This page will review phasors and phasor arithmetic topics.

Forms

Phasors have two components, the magnitude (M) and the phase angle (φ). Phasors are related to sinusoids through our cosine convention:

Remember, there are 3 forms to phasors:

• "Phasor Form"
• "Rectangular Form"
• "Exponential Form"

Phasor and Exponential forms are identical and are also referred to as polar form.

Converting between Forms

When working with phasors it is often necessary to convert between rectangular and polar form. To convert from rectangular form to polar form:

To convert from polar to rectangular form:

A is the part of the phasor along the real axis

A = Mcosφ

B is the part of the phasor along the imaginary axis

B = Msinφ

Addition

To add two phasors together, we must convert them into rectangular form:

This is a well-known property of complex arithmetic.

Subtraction

Subtraction is similar to addition, except now we subtract

Multiplication

To multiply two phasors, we should first convert them to polar form to make things simpler. The product in polar form is simply the product of their magnitudes, and the phase is the sum of their phases.

Keep in mind that in polar form, phasors are exponential quantities with a magnitude (M), and an argument (φ). Multiplying two exponentials together forces us to multiply the magnitudes, and add the exponents.

Division

Division is similar to multiplication, except now we divide the magnitudes, and subtract the phases

Inversion

An important relationship that is worth understanding is the inversion property of phasors:

On the normal cartesian plane, for instance, the negative X axis is 180 degrees around from the positive X axis. By using that fact on an imaginary axis, we can see that the Negative Real axis is facing in the exact opposite direction from the Positive Real axis, and therefore is 180 degrees apart.

Complex Conjugation

Similar to the inversion property is the complex conjugation property of phasors. Complex conjugation is denoted with an asterisk above the phasor to be conjugated. Since phasors can be graphed on the Real-Imaginary plane, a 90 degree phasor is a purely imaginary number, and a -90 degree phasor is its complex conjugate:

Essentially, this holds true for phasors with all angles: the sign of the angle is reversed to produce the complex conjugate of the phasor in polar notation. In general, for polar notation, we have:

In rectangular form, we can express complex conjugation as:

Notice the only difference in the complex conjugate of C is the sign change of the imaginary part.

Decibels

This appendix page is going to take a deeper look at the units of decibels, it will describe some of the properties of decibels, and will demonstrate how to use them in calculations.

Definition

Decibels are, first and foremost, a power calculation. With that in mind, we will state the definition of a decibel:

The letters "dB" are used as the units for the result of this calculation. dB ratios are always in terms of watts, unless otherwise noted.

Voltage Calculation

now, another formula has been demonstrated that allows a decibel calculation to be made using voltages, instead of power measurements. We will derive that equation here:

First, we will use the power calculation and Ohm's law to produce a common identity:

Now, if we plug that result into the definition of a decibel, we can create a complicated equation:

Now, we can cancel out the resistance values (R) from the top and bottom of the fraction, and rearrange the exponent as such:

If we remember the properties of logarithms, we will remember that if we have an exponent inside a logarithm, we can move the exponent outside, as a coefficient. This rule gives us our desired result:

Inverse Calculation

It is a simple matter of arithmetic to find the inverse of the decibel calculation, so it will not be derived here, but stated simply:

P = 10^{dB / 10}

Reference Units

Now, this decible calculation has proven to be so useful, that occasionally they are applied to other units of measurement, instead of just watts. Specifically, the units "dBm" are used when the power unit being converted was in terms of milliwatts, not just watts. Let's say we have a value of 10dBm, we can go through the inverse calculation:

P = 10^{10dBm / 10} = 10mW

Likewise, let's say we want to apply the decibel calculation to a completely unrelated unit: hertz. If we have 100Hz, we can apply the decibel calculation:

dB = 10log100Hz = 20dBHz

If no letters follow the "dB" lable, the decibels are referenced to watts.

Decibel Arithmetic

Decibels are ratios, and are not real numbers. Therefore, specific care should be taken not to use decibel values in equations that call for gains, unless decibels are specifically called for (which they usually aren't). However, since decibels are calculated using logarithms, a few principles of logarithms can be used to make decibels usable in calculations.

Multiplication

Let's say that we have three values, a b and c, with their respective decibel equivalents denoted by the upper-case letters A B and C. We can show that for the following equation:

a = b c

That we can change all the quantities to decibels, and convert the multiplication operations to addition:

A = B + C

Division

Let's say that we have three values, a b and c, with their respective decibel equivalents denoted by the upper-case letters A B and C. We can show that for the following equation:

a = b / c

Then we can show through the principals of logarithms that we can convert all the values to decibels, and we can then convert the division operation to subtraction:

A = B - C

Transform Tables

Laplace Transform Appendix

The Laplace Transform is defined as such:

The result of the transform of a time-domain function f(t) is F(s).

Laplace Transform Table

	Time Domain	Laplace Domain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

Laplace Transform Properties

Property	Definition
Linearity
Differentiation
Frequency Division
Frequency Integration
Time Integration
Scaling
Initial value theorem
Final value theorem
Frequency Shifts
Time Shifts
Convolution Theorem

Where:

s = σ + jω

Fourier Transform Appendix

The Fourier Transform is defined as such:

The result of the transform of a time-domain function f(t) is F(jω).

Fourier Transform Table

	Time Domain	Frequency Domain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Notes:	sinc(x) = sin(x) / x pτ(t) is the rectangular pulse function of width τ
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Resources

Further Reading

Pages listed here are sources of further information on the topic of electric circuits, or are additional subjects that may be of interest for a reader of this book. Many of the resources listed here are sources of information, and this may be treated as a bibliography for this wikibook.

Wikibooks

• Wikibooks: Electronics
• Wikibooks: Signals and Systems
• Wikibooks: Digital Circuits
• Wikibooks: Circuit Idea

The following Wikibooks list Circuit Theory as a prerequisite:

• Analog and Digital Conversion
• Semiconductors

Other Resources

• Horowitz and Hill, The Art of Electronics, Second Edition, Cambridge University Press, 1989. ISBN 0521370957
• US Navy, Basic Electrity, Dover, 1970. ISBN 0486209733
• US Navy, Basic Electronics, Dover, 1973. ISBN 0486210766
• Comer and Comer, Fundamentals of Electronic Circuit Design, John Wiley & Sons, 2003. http://www.wiley.com/college/comer/ ISBN 0471410160
• Dorf and Svoboda, Introduction to Electric Circuits, Sixth Edition, John Wiley & Sons, 2004. ISBN 0471447951

License

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Version 1.2, November 2002

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