basic ohms law

8/17/2019 Basic Ohms Law

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Basic Ohm's Law

Here we'll attempt to explain Ohm's law basics!

Ohm's law can be very difficult to understand by anyone who has

never had any basic understanding or training in basic electricity. We'll assume that you have some knowledge of basic electricity. We'llexplain it in terms of water flow! DON' " W"!

What is Ohm's Law:

Ohm's Law is made from 3 mathematical equations that shows therelationship between electric voltage, current and resistance.

What is voltage? An anology would be a huge water tank filled with

thousands of gallons of water high on a hill.The difference between the pressure of water in the tank and the water thatcomes out of a pipe connected at the bottom leading to a faucet isdetermined by the size of the pipe and the size of the outlet of the faucet.This difference of pressure between the two can be thought of as potentialoltage.

What is current? An analogy would be the amount of flow determined bythe pressure !"oltage# of the water thru the pipes leading to a faucet. The

term current refers to the $uantity, "olume or intensity of electrical flow, asopposed to "oltage, which refers to the force or %pressure% causing thecurrent flow.

What is resistance? An analogy would be the size of the water pipes andthe size of the faucet. The larger the pipe and the faucet !less resistance#,the more water that comes out& The smaller the pipe and faucet, !moreresistance#, the less water that comes out& This can be thought of asresistance to the flow of the water current.

All three of these "oltage, current and resistance directly interact in Ohm's

law.(hange any two of them and you effect the third.

Info: Ohm's Law was named after Bavarian mathematician and physicistGeorg Ohm.

Ohm's Law can be stated as mathematical equations, all deri"ed from thesame principle.)n the following e$uations,V is "oltage measured in volts the sie o" the water tan#$%


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& is current measured in amperes relate to the pressure Voltage$ o"water thru the pipes an "aucet$ and

( is resistance measured in ohms as relate to the sie o" the pipes an"aucet:

V * & x ( Voltage ) *urrent multiplie b+ (esistance$

( * V , & (esistance ) Voltage ivie b+ *urrent$

& * V , ( *urrent ) Voltage -ivie b+ (esistance$

.nowing an+ two o" the values o" a circuit, one can determine!calculate# the thir% using Ohm's Law.

/or example% to "in the Voltage in a circuit:

&" the circuit has a current o" 0 amperes% an a resistance o" 1 ohm% 2these are the two 3#nowns3$% then accoring to Ohms Law an the"ormulas above% voltage equals current multiplie b+ resistance:

V ) 0 amperes x 1 ohm ) 0 volts$4 5o "in the current in the same circuit above assuming we did notknow it but we #now the voltage an resistance:

& ) 0 volts ivie b+ the resistance 1 ohm ) 0 amperes4

&n this thir example we #now the current 0 amperes$ an the voltage0 volts$4444what is the resistance?6ubstituting the "ormula:( ) Volts ivie b+ the current 0 volts ivie b+ 0 amperes ) 1 ohm

6ometimes it's ver+ help"ul to associate these "ormulas Visuall+4 5heOhms Law 3wheels3 an graphics below can be a ver+ use"ul tool to

7og +our memor+ an help +ou to unerstan their relationship4


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5he wheel above is ivie into three sections:

Volts V on top o" the iviing line$8mps amperes$ & lower le"t below the iviing line$(esistance ( lower right below the iviing line$9 represents the multipl+ b+ sign$emorie this wheel

5o use% 7ust cover the un#nown quantit+ +ou nee with +our mins e+ean what is le"t is the "ormula to "in the un#nown4

;xample:

5o "in the current o" a circuit &$% 7ust cover the & or 8mps section in+our mines e+e an what remains is the V volts above the iviing linean the ( ohms resistance$ below it4 our answer will be the current in the circuit45he same proceure is use to "in the volts or resistance o" a circuit!

Here is another example:

>ou #now the current an the resistance in a circuit but +ou want to"in out the voltage4

=ust cover the voltage section with +our mins e+e444what's le"t is the &9 ( sections4 =ust multipl+ the & value times the ( value to get +ouranswer! ractice with the wheel an +ou'll be surprise at how well itwor#s to help +ou remember the "ormulas without tr+ing!

5his Ohm's Law 5riangle graphic is also help"ul tolearn the "ormulas4

=ust cover the un#nown value an "ollow the graphic as in the +ellowwheel examples above4

>ou'll have to insert the 9 between the & an ( in the graphic animagine the horiontal ivie line but the principal is 7ust the same4


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&n the above Ohm's law wheel +ou'll notice that is has an ae section$ "or ower an the letter ;@ has been use instea o" the letter V "orvoltage45his wheel is use in the exact same "ashion as the other wheels angraphics above4

>ou will also notice in the blue,green areas there are onl+ two #nownvalues with the un#nown value in the +ellow sections4 5he re barsseparate the "our units o" interest4

8n example o" the use o" this wheel is:Let's sa+ that +ou #now the power an the current in a circuit an wantto #now the voltage4

/in +our un#nown value in the +ellow areas V or ;@ in this wheel$an 7ust loo# outwar an pic# the values that +ou o #now4 5hesewoul be the an the &4 6ubstitute +our values in the "ormula% ivie b+ &$ o the math an +ou have +our answer!

#nfo$ ypically% Ohm's &aw is only applied to D circuits and not (circuits4 ) he letter *"* is sometimes used in representations of Ohm's &aw for voltage instead of the *+* as in the wheel above.

How voltage, current, and resistance relate

An electric circuit is formed when a conductive path is created to allow free

electrons to continuously move. This continuous movement of free electrons through theconductors of a circuit is called a current , and it is often referred to in terms of "flow," just like

the flow of a liquid through a hollow pipe.

The force motivating electrons to "flow" in a circuit is called voltage. Voltage is a specificmeasure of potential energy that is always relative between two points. hen we speak of a


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certain amount of voltage being present in a circuit, we are referring to the measurement ofhow much potential energy e!ists to move electrons from one particular point in that circuit to

another particular point. ithout reference to two particular points, the term "voltage" has nomeaning.

ree electrons tend to move through conductors with some degree of friction, or opposition tomotion. This opposition to motion is more properly called resistance. The amount of current in

a circuit depends on the amount of voltage available to motivate the electrons, and also theamount of resistance in the circuit to oppose electron flow. #ust like voltage, resistance is a

quantity relative between two points. or this reason, the quantities of voltage and resistanceare often stated as being "between" or "across" two points in a circuit.

To be able to make meaningful statements about these quantities in circuits, we need to beable to describe their quantities in the same way that we might quantify mass, temperature,

volume, length, or any other kind of physical quantity. or mass we might use the units of"kilogram" or "gram." or temperature we might use degrees ahrenheit or degrees $elsius.

%ere are the standard units of measurement for electrical current, voltage, and resistance&

The "symbol" given for each quantity is the standard alphabetical letter used to represent that

quantity in an algebraic equation. 'tandardi(ed letters like these are common in the disciplines

of physics and engineering, and are internationally recogni(ed. The "unit abbreviation" for eachquantity represents the alphabetical symbol used as a shorthand notation for its particular unitof measurement. And, yes, that strange)looking "horseshoe" symbol is the capital *reek letter

+, just a character in a foreign alphabet apologies to any *reek readers here-.

ach unit of measurement is named after a famous e!perimenter in electricity& The amp after

the renchman Andre /. Ampere, the volt after the 0talian Alessandro Volta, and the ohm afterthe *erman *eorg 'imon 1hm.

The mathematical symbol for each quantity is meaningful as well. The "2" for resistance andthe "V" for voltage are both self)e!planatory, whereas "0" for current seems a bit weird. The "0"

is thought to have been meant to represent "0ntensity" of electron flow-, and the other symbol

for voltage, "," stands for "lectromotive force." rom what research 03ve been able to do,there seems to be some dispute over the meaning of "0." The symbols "" and "V" areinterchangeable for the most part, although some te!ts reserve "" to represent voltage across

a source such as a battery or generator- and "V" to represent voltage across anything else.

All of these symbols are e!pressed using capital letters, e!cept in cases where a quantity

especially voltage or current- is described in terms of a brief period of time called an"instantaneous" value-. or e!ample, the voltage of a battery, which is stable over a long

period of time, will be symboli(ed with a capital letter "," while the voltage peak of a lightningstrike at the very instant it hits a power line would most likely be symboli(ed with a lower)case

letter "e" or lower)case "v"- to designate that value as being at a single moment in time. Thissame lower)case convention holds true for current as well, the lower)case letter "i"

representing current at some instant in time. /ost direct)current 4$- measurements,however, being stable over time, will be symboli(ed with capital letters.


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1ne foundational unit of electrical measurement, often taught in the beginnings of electronicscourses but used infrequently afterwards, is the unit of the coulomb, which is a measure of

electric charge proportional to the number of electrons in an imbalanced state. 1ne coulomb ofcharge is equal to 5,678,888,888,888,888,888 electrons. The symbol for electric charge

quantity is the capital letter "9," with the unit of coulombs abbreviated by the capital letter "$."0t so happens that the unit for electron flow, the amp, is equal to : coulomb of electrons

passing by a given point in a circuit in : second of time. $ast in these terms, current is the rateof electric charge motion through a conductor.

As stated before, voltage is the measure of potential energy per unit charge available tomotivate electrons from one point to another. ;efore we can precisely define what a "volt" is,

we must understand how to measure this quantity we call "potential energy." The general

metric unit for energy of any kind is the joule, equal to the amount of work performed by a

force of : newton e!erted through a motion of : meter in the same direction-. 0n ;ritish units,this is slightly less than pound of force e!erted over a distance of : foot. ?ut in common

terms, it takes about : joule of energy to lift a pound weight : foot off the ground, or todrag something a distance of : foot using a parallel pulling force of pound. 4efined in these

scientific terms, : volt is equal to : joule of electric potential energy per divided by- : coulombof charge. Thus, a @ volt battery releases @ joules of energy for every coulomb of electrons

moved through a circuit.

These units and symbols for electrical quantities will become very important to know as we

begin to e!plore the relationships between them in circuits. The first, and perhaps mostimportant, relationship between current, voltage, and resistance is called 1hm3s aw,

discovered by *eorg 'imon 1hm and published in his :B6C paper, The Galvanic Circuit

Investigated Mathematically . 1hm3s principal discovery was that the amount of electric current

through a metal conductor in a circuit is directly proportional to the voltage impressed acrossit, for any given temperature. 1hm e!pressed his discovery in the form of a simple equation,

describing how voltage, current, and resistance interrelate&

0n this algebraic e!pression, voltage - is equal to current 0- multiplied by resistance 2-.Dsing algebra techniques, we can manipulate this equation into two variations, solving for 0 and

for 2, respectively&

et3s see how these equations might work to help us analy(e simple circuits&


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0n the above circuit, there is only one source of voltage the battery, on the left- and only onesource of resistance to current the lamp, on the right-. This makes it very easy to apply 1hm3s

aw. 0f we know the values of any two of the three quantities voltage, current, and resistance-in this circuit, we can use 1hm3s aw to determine the third.

0n this first e!ample, we will calculate the amount of current 0- in a circuit, given values ofvoltage - and resistance 2-&

hat is the amount of current 0- in this circuitE

0n this second e!ample, we will calculate the amount of resistance 2- in a circuit, given valuesof voltage - and current 0-&


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hat is the amount of resistance 2- offered by the lampE

0n the last e!ample, we will calculate the amount of voltage supplied by a battery, given values

of current 0- and resistance 2-&

hat is the amount of voltage provided by the batteryE

1hm3s aw is a very simple and useful tool for analy(ing electric circuits. 0t is used so often inthe study of electricity and electronics that it needs to be committed to memory by the serious

student. or those who are not yet comfortable with algebra, there3s a trick to rememberinghow to solve for any one quantity, given the other two. irst, arrange the letters , 0, and 2 in a

triangle like this&


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0f you know and 0, and wish to determine 2, just eliminate 2 from the picture and see what3sleft&

0f you know and 2, and wish to determine 0, eliminate 0 and see what3s left&

astly, if you know 0 and 2, and wish to determine , eliminate and see what3s left&

ventually, you3ll have to be familiar with algebra to seriously study electricity and electronics,

but this tip can make your first calculations a little easier to remember. 0f you are comfortable

with algebra, all you need to do is commit F02 to memory and derive the other two formulaefrom that when you need themG

• REVIEW:

• Voltage measured in volts, symboli(ed by the letters "" or "V".

• $urrent measured in amps, symboli(ed by the letter "0".

• 2esistance measured in ohms, symboli(ed by the letter "2".

• 1hm3s aw& F 02 H 0 F =2 H 2 F =0

An analogy for Ohm's Law


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1hm3s aw also makes intuitive sense if you apply it to the water)and)pipe

analogy. 0f we have a water pump that e!erts pressure voltage- to push water around a"circuit" current- through a restriction resistance-, we can model how the three variables

interrelate. 0f the resistance to water flow stays the same and the pump pressure increases,the flow rate must also increase.

0f the pressure stays the same and the resistance increases making it more difficult for thewater to flow-, then the flow rate must decrease&

0f the flow rate were to stay the same while the resistance to flow decreased, the required

pressure from the pump would necessarily decrease&

As odd as it may seem, the actual mathematical relationship between pressure, flow, and

resistance is actually more comple! for fluids like water than it is for electrons. 0f you pursuefurther studies in physics, you will discover this for yourself. Thankfully for the electronicsstudent, the mathematics of 1hm3s aw is very straightforward and simple.


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• REVIEW:

• ith resistance steady, current follows voltage an increase in voltage means an

increase in current, and vice versa-.

• ith voltage steady, changes in current and resistance are opposite an increase in

current means a decrease in resistance, and vice versa-.

• ith current steady, voltage follows resistance an increase in resistance means an

increase in voltage-.

Calculating electric power

e3ve seen the formula for determining the power in an electric circuit& by

multiplying the voltage in "volts" by the current in "amps" we arrive at an answer in "watts."et3s apply this to a circuit e!ample&

0n the above circuit, we know we have a battery voltage of :B volts and a lamp resistance of <

+. Dsing 1hm3s aw to determine current, we get&

Iow that we know the current, we can take that value and multiply it by the voltage to

determine power&

Answer& the lamp is dissipating releasing- :8B watts of power, most likely in the form of both

light and heat.

et3s try taking that same circuit and increasing the battery voltage to see what happens.

0ntuition should tell us that the circuit current will increase as the voltage increases and thelamp resistance stays the same. ikewise, the power will increase as well&


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Iow, the battery voltage is . Jou can check this by dividing >.

Dsing algebra again to manipulate the formulae, we can take our original power formula andmodify it for applications where we don3t know both voltage and current&

0f we only know voltage - and resistance 2-&

0f we only know current 0- and resistance 2-&


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An historical note& it was #ames ?rescott #oule, not *eorg 'imon 1hm, who first discovered the

mathematical relationship between power dissipation and current through a resistance. Thisdiscovery, published in :B>:, followed the form of the last equation ? F 062-, and is properly

known as #oule3s aw. %owever, these power equations are so commonly associated with the1hm3s aw equations relating voltage, current, and resistance F02 H 0F=2 H and 2F=0- that

they are frequently credited to 1hm.

• REVIEW:

• ?ower measured in watts, symboli(ed by the letter "".

• #oule3s aw& ? F 062 H ? F 0 H ? F 6 =2

Resistors

;ecause the relationship between voltage, current, and resistance in any circuit

is so regular, we can reliably control any variable in a circuit simply by controlling the othertwo. ?erhaps the easiest variable in any circuit to control is its resistance. This can be done by

changing the material, si(e, and shape of its conductive components remember how the thinmetal filament of a lamp created more electrical resistance than a thick wireE-.

'pecial components called resistors are made for the e!press purpose of creating a precise

quantity of resistance for insertion into a circuit. They are typically constructed of metal wire or

carbon, and engineered to maintain a stable resistance value over a wide range ofenvironmental conditions. Dnlike lamps, they do not produce light, but they do produce heat as

electric power is dissipated by them in a working circuit. Typically, though, the purpose of aresistor is not to produce usable heat, but simply to provide a precise quantity of electrical

resistance.

The most common schematic symbol for a resistor is a (ig)(ag line&

2esistor values in ohms are usually shown as an adjacent number, and if several resistors are

present in a circuit, they will be labeled with a unique identifier number such as 2:, 26, 2


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2eal resistors look nothing like the (ig)(ag symbol. 0nstead, they look like small tubes orcylinders with two wires protruding for connection to a circuit. %ere is a sampling of different

kinds and si(es of resistors&

0n keeping more with their physical appearance, an alternative schematic symbol for a resistor

looks like a small, rectangular bo!&

2esistors can also be shown to have varying rather than fi!ed resistances. This might be for the

purpose of describing an actual physical device designed for the purpose of providing anadjustable resistance, or it could be to show some component that just happens to have an

unstable resistance&


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0n fact, any time you see a component symbol drawn with a diagonal arrow through it, that

component has a variable rather than a fi!ed value. This symbol "modifier" the diagonalarrow- is standard electronic symbol convention.

Variable resistors must have some physical means of adjustment, either a rotating shaft orlever that can be moved to vary the amount of electrical resistance. %ere is a photograph

showing some devices called potentiometers, which can be used as variable resistors&

;ecause resistors dissipate heat energy as the electric currents through them overcome the"friction" of their resistance, resistors are also rated in terms of how much heat energy they

can dissipate without overheating and sustaining damage. Iaturally, this power rating is

specified in the physical unit of "watts." /ost resistors found in small electronic devices such asportable radios are rated at :=> 8.67- watt or less. The power rating of any resistor is roughly

proportional to its physical si(e. Iote in the first resistor photograph how the power ratingsrelate with si(e& the bigger the resistor, the higher its power dissipation rating. Also note how

resistances in ohms- have nothing to do with si(eG

Although it may seem pointless now to have a device doing nothing but resisting electriccurrent, resistors are e!tremely useful devices in circuits. ;ecause they are simple and so

commonly used throughout the world of electricity and electronics, we3ll spend a considerableamount of time analy(ing circuits composed of nothing but resistors and batteries.

or a practical illustration of resistors3 usefulness, e!amine the photograph below. 0t is a picture

of a printed circuit board , or PCB& an assembly made of sandwiched layers of insulatingphenolic fiber)board and conductive copper strips, into which components may be inserted and


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secured by a low)temperature welding process called "soldering." The various components onthis circuit board are identified by printed labels. 2esistors are denoted by any label beginning

with the letter "2".

This particular circuit board is a computer accessory called a "modem," which allows digitalinformation transfer over telephone lines. There are at least a do(en resistors all rated at :=>

watt power dissipation- that can be seen on this modem3s board. very one of the blackrectangles called "integrated circuits" or "chips"- contain their own array of resistors for their

internal functions, as well.

Another circuit board e!ample shows resistors packaged in even smaller units, called "surface

mount devices." This particular circuit board is the underside of a personal computer hard disk


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drive, and once again the resistors soldered onto it are designated with labels beginning withthe letter "2"&

There are over one hundred surface)mount resistors on this circuit board, and this count ofcourse does not include the number of resistors internal to the black "chips." These two

photographs should convince anyone that resistors )) devices that "merely" oppose the flow of

electrons )) are very important components in the realm of electronicsG

0n schematic diagrams, resistor symbols are sometimes used to illustrate any general type of

device in a circuit doing something useful with electrical energy. Any non)specific electricaldevice is generally called a load , so if you see a schematic diagram showing a resistor symbol

labeled "load," especially in a tutorial circuit diagram e!plaining some concept unrelated to theactual use of electrical power, that symbol may just be a kind of shorthand representation of

something else more practical than a resistor.


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To summari(e what we3ve learned in this lesson, let3s analy(e the following circuit, determiningall that we can from the information given&

All we3ve been given here to start with is the battery voltage :8 volts- and the circuit current

6 amps-. e don3t know the resistor3s resistance in ohms or the power dissipated by it inwatts. 'urveying our array of 1hm3s aw equations, we find two equations that give us answers

from known quantities of voltage and current&

0nserting the known quantities of voltage - and current 0- into these two equations, we candetermine circuit resistance 2- and power dissipation ?-&

or the circuit conditions of :8 volts and 6 amps, the resistor3s resistance must be 7 +. 0f wewere designing a circuit to operate at these values, we would have to specify a resistor with a

minimum power rating of 68 watts, or else it would overheat and fail.

• REVIEW:

• 4evices called resistors are built to provide precise amounts of resistance in electric

circuits. 2esistors are rated both in terms of their resistance ohms- and their abilityto dissipate heat energy watts-.

• 2esistor resistance ratings cannot be determined from the physical si(e of the

resistors- in question, although appro!imate power ratings can. The larger theresistor is, the more power it can safely dissipate without suffering damage.

• Any device that performs some useful task with electric power is generally known

as a load . 'ometimes resistor symbols are used in schematic diagrams to designate

a non)specific load, rather than an actual resistor.

Nonlinear conduction


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"dvances are made by answering !uestions #iscoveries are made by

!uestioning answers"

Bernhard Haisch, Astrophysicist

1hm3s aw is a simple and powerful mathematical tool for helping us analy(e electric circuits,but it has limitations, and we must understand these limitations in order to properly apply it to

real circuits. or most conductors, resistance is a rather stable property, largely unaffected byvoltage or current. or this reason we can regard the resistance of many circuit components as

a constant, with voltage and current being directly related to each other.

or instance, our previous circuit e!ample with the < + lamp, we calculated current through the

circuit by dividing voltage by resistance 0F=2-. ith an :B volt battery, our circuit currentwas 5 amps. 4oubling the battery voltage to


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The phenomenon of resistance changing with variations in temperature is one shared by almostall metals, of which most wires are made. or most applications, these changes in resistance

are small enough to be ignored. 0n the application of metal lamp filaments, the changehappens to be quite large.

This is just one e!ample of "nonlinearity" in electric circuits. 0t is by no means the onlye!ample. A "linear" function in mathematics is one that tracks a straight line when plotted on a

graph. The simplified version of the lamp circuit with a constant filament resistance of < +generates a plot like this&

The straight)line plot of current over voltage indicates that resistance is a stable, unchanging

value for a wide range of circuit voltages and currents. 0n an "ideal" situation, this is the case.2esistors, which are manufactured to provide a definite, stable value of resistance, behave very

much like the plot of values seen above. A mathematician would call their behavior "linear."

A more realistic analysis of a lamp circuit, however, over several different values of battery

voltage would generate a plot of this shape&


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The plot is no longer a straight line. 0t rises sharply on the left, as voltage increases from (eroto a low level. As it progresses to the right we see the line flattening out, the circuit requiring

greater and greater increases in voltage to achieve equal increases in current.

0f we try to apply 1hm3s aw to find the resistance of this lamp circuit with the voltage and

current values plotted above, we arrive at several different values. e could say that theresistance here is nonlinear , increasing with increasing current and voltage. The nonlinearity is

caused by the effects of high temperature on the metal wire of the lamp filament.

Another e!ample of nonlinear current conduction is through gases such as air. At standard

temperatures and pressures, air is an effective insulator. %owever, if the voltage between twoconductors separated by an air gap is increased greatly enough, the air molecules between the

gap will become "ioni(ed," having their electrons stripped off by the force of the high voltagebetween the wires. 1nce ioni(ed, air and other gases- become good conductors of electricity,

allowing electron flow where none could e!ist prior to ioni(ation. 0f we were to plot current overvoltage on a graph as we did with the lamp circuit, the effect of ioni(ation would be clearly seen

as nonlinear&

The graph shown is appro!imate for a small air gap less than one inch-. A larger air gap wouldyield a higher ioni(ation potential, but the shape of the 0= curve would be very similar&

practically no current until the ioni(ation potential was reached, then substantial conductionafter that.

0ncidentally, this is the reason lightning bolts e!ist as momentary surges rather than

continuous flows of electrons. The voltage built up between the earth and clouds or between

different sets of clouds- must increase to the point where it overcomes the ioni(ation potentialof the air gap before the air ioni(es enough to support a substantial flow of electrons. 1nce it

does, the current will continue to conduct through the ioni(ed air until the static chargebetween the two points depletes. 1nce the charge depletes enough so that the voltage falls

below another threshold point, the air de)ioni(es and returns to its normal state of e!tremelyhigh resistance.

/any solid insulating materials e!hibit similar resistance properties& e!tremely high resistanceto electron flow below some critical threshold voltage, then a much lower resistance at voltages

beyond that threshold. 1nce a solid insulating material has been compromised by high)voltagebrea$down, as it is called, it often does not return to its former insulating state, unlike mostgases. 0t may insulate once again at low voltages, but its breakdown threshold voltage will


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have been decreased to some lower level, which may allow breakdown to occur more easily inthe future. This is a common mode of failure in high)voltage wiring& insulation damage due to

breakdown. 'uch failures may be detected through the use of special resistance metersemploying high voltage :888 volts or more-.

There are circuit components specifically engineered to provide nonlinear resistance curves,one of them being the varistor . $ommonly manufactured from compounds such as (inc o!ide

or silicon carbide, these devices maintain high resistance across their terminals until a certain"firing" or "breakdown" voltage equivalent to the "ioni(ation potential" of an air gap- is

reached, at which point their resistance decreases dramatically. Dnlike the breakdown of aninsulator, varistor breakdown is repeatable& that is, it is designed to withstand repeated

breakdowns without failure. A picture of a varistor is shown here&

There are also special gas)filled tubes designed to do much the same thing, e!ploiting the very

same principle at work in the ioni(ation of air by a lightning bolt.

1ther electrical components e!hibit even stranger current=voltage curves than this. 'ome

devices actually e!perience a decrease in current as the applied voltage increases. ;ecause the

slope of the current=voltage for this phenomenon is negative angling down instead of up as itprogresses from left to right-, it is known as negative resistance.


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/ost notably, high)vacuum electron tubes known as tetrodes and semiconductor diodes known

as %sa$i or tunnel diodes e!hibit negative resistance for certain ranges of applied voltage.

1hm3s aw is not very useful for analy(ing the behavior of components like these where

resistance varies with voltage and current. 'ome have even suggested that "1hm3s aw"

should be demoted from the status of a "aw" because it is not universal. 0t might be moreaccurate to call the equation 2F=0- a definition of resistance, befitting of a certain class of

materials under a narrow range of conditions.

or the benefit of the student, however, we will assume that resistances specified in e!amplecircuits are stable over a wide range of conditions unless otherwise specified. 0 just wanted to

e!pose you to a little bit of the comple!ity of the real world, lest 0 give you the false impressionthat the whole of electrical phenomena could be summari(ed in a few simple equations.

• REVIEW:

• The resistance of most conductive materials is stable over a wide range of

conditions, but this is not true of all materials.

• Any function that can be plotted on a graph as a straight line is called a linear

function. or circuits with stable resistances, the plot of current over voltage is linear0F=2-.

• 0n circuits where resistance varies with changes in either voltage or current, the

plot of current over voltage will be nonlinear not a straight line-.

• A varistor is a component that changes resistance with the amount of voltage

impressed across it. ith little voltage across it, its resistance is high. Then, at a

certain "breakdown" or "firing" voltage, its resistance decreases dramatically.

• &egative resistance is where the current through a component actually decreases

as the applied voltage across it is increased. 'ome electron tubes and semiconductor

diodes most notably, the tetrode tube and the %sa$i , or tunnel diode, respectively-e!hibit negative resistance over a certain range of voltages.

Circuit wiring

'o far, we3ve been analy(ing single)battery, single)resistor circuits with no

regard for the connecting wires between the components, so long as a complete circuit is


24/40

formed. 4oes the wire length or circuit "shape" matter to our calculationsE et3s look at acouple of circuit configurations and find out&

hen we draw wires connecting points in a circuit, we usually assume those wires havenegligible resistance. As such, they contribute no appreciable effect to the overall resistance of

the circuit, and so the only resistance we have to contend with is the resistance in the

components. 0n the above circuits, the only resistance comes from the 7 + resistors, so that isall we will consider in our calculations. 0n real life, metal wires actually do have resistance andso do power sourcesG-, but those resistances are generally so much smaller than the resistance

present in the other circuit components that they can be safely ignored. !ceptions to this rulee!ist in power system wiring, where even very small amounts of conductor resistance can

create significant voltage drops given normal high- levels of current.

0f connecting wire resistance is very little or none, we can regard the connected points in a

circuit as being electrically common. That is, points : and 6 in the above circuits may bephysically joined close together or far apart, and it doesn3t matter for any voltage or resistance

measurements relative to those points. The same goes for points < and >. 0t is as if the ends of the resistor were attached directly across the terminals of the battery, so far as our 1hm3s aw

calculations and voltage measurements are concerned. This is useful to know, because itmeans you can re)draw a circuit diagram or re)wire a circuit, shortening or lengthening the

wires as desired without appreciably impacting the circuit3s function. All that matters is that thecomponents attach to each other in the same sequence.

0t also means that voltage measurements between sets of "electrically common" points will bethe same. That is, the voltage between points : and > directly across the battery- will be the

same as the voltage between points 6 and < directly across the resistor-. Take a close look atthe following circuit, and try to determine which points are common to each other&


25/40

%ere, we only have 6 components e!cluding the wires& the battery and the resistor. Though the

connecting wires take a convoluted path in forming a complete circuit, there are severalelectrically common points in the electrons3 path. ?oints :, 6, and < are all common to each

other, because they3re directly connected together by wire. The same goes for points >, 7, and5.

The voltage between points : and 5 is :8 volts, coming straight from the battery. %owever,since points 7 and > are common to 5, and points 6 and < common to :, that same :8 volts

also e!ists between these other pairs of points&

Between points 1 and 4 = 10 volts


Between points 3 and 4 = 10 volts (directly across the resistor)




Between points 1 and 6 = 10 volts (directly across the battery)



'ince electrically common points are connected together by (ero resistance- wire, there is no

significant voltage drop between them regardless of the amount of current conducted from one

to the ne!t through that connecting wire. Thus, if we were to read voltages between commonpoints, we should show practically- (ero&

Between points 1 and 2 = 0 volts Points 1, 2, and 3 areBetween points 2 and 3 = 0 volts electrically coon


Between points 4 and 5 = 0 volts Points 4, 5, and 6 are

Between points 5 and 6 = 0 volts electrically coon


This makes sense mathematically, too. ith a :8 volt battery and a 7 + resistor, the circuit

current will be 6 amps. ith wire resistance being (ero, the voltage drop across any continuous

stretch of wire can be determined through 1hm3s aw as such&


26/40

0t should be obvious that the calculated voltage drop across any uninterrupted length of wire ina circuit where wire is assumed to have (ero resistance will always be (ero, no matter what the

magnitude of current, since (ero multiplied by anything equals (ero.

;ecause common points in a circuit will e!hibit the same relative voltage and resistance

measurements, wires connecting common points are often labeled with the same designation.

This is not to say that the terminal connection points are labeled the same, just the connecting

wires. Take this circuit as an e!ample&

?oints :, 6, and < are all common to each other, so the wire connecting point : to 6 is labeled

the same wire 6- as the wire connecting point 6 to < wire 6-. 0n a real circuit, the wirestretching from point : to 6 may not even be the same color or si(e as the wire connecting

point 6 to .

Knowing that electrically common points have (ero voltage drop between them is a valuable

troubleshooting principle. 0f 0 measure for voltage between points in a circuit that are supposed

to be common to each other, 0 should read (ero. 0f, however, 0 read substantial voltagebetween those two points, then 0 know with certainty that they cannot be directly connected

together. 0f those points are supposed to be electrically common but they register otherwise,then 0 know that there is an "open failure" between those points.

1ne final note& for most practical purposes, wire conductors can be assumed to possess (eroresistance from end to end. 0n reality, however, there will always be some small amount of

resistance encountered along the length of a wire, unless its a superconducting wire. Knowingthis, we need to bear in mind that the principles learned here about electrically common points

are all valid to a large degree, but not to an absolute degree. That is, the rule that electricallycommon points are guaranteed to have (ero voltage between them is more accurately stated

as such& electrically common points will have very little voltage dropped between them. That

small, virtually unavoidable trace of resistance found in any piece of connecting wire is boundto create a small voltage across the length of it as current is conducted through. 'o long as you


27/40

understand that these rules are based upon ideal conditions, you won3t be perple!ed when youcome across some condition appearing to be an e!ception to the rule.

• REVIEW:

• $onnecting wires in a circuit are assumed to have (ero resistance unless otherwise

stated.

• ires in a circuit can be shortened or lengthened without impacting the circuit3s

function )) all that matters is that the components are attached to one another in thesame sequence.

• ?oints directly connected together in a circuit by (ero resistance wire- are

considered to be electrically common.

• lectrically common points, with (ero resistance between them, will have (ero

voltage dropped between them, regardless of the magnitude of current ideally-.

• The voltage or resistance readings referenced between sets of electrically common

points will be the same.

• These rules apply to ideal conditions, where connecting wires are assumed to

possess absolutely (ero resistance. 0n real life this will probably not be the case, but

wire resistances should be low enough so that the general principles stated here stillhold.

Polarity of voltage drops

e can trace the direction that electrons will flow in the same circuit by starting

at the negative )- terminal and following through to the positive L- terminal of the battery,

the only source of voltage in the circuit. rom this we can see that the electrons are movingcounter)clockwise, from point 5 to 7 to > to < to 6 to : and back to 5 again.

As the current encounters the 7 + resistance, voltage is dropped across the resistor3s ends. Thepolarity of this voltage drop is negative )- at point > with respect to positive L- at point


28/40

Between points 3 (!) and 4 (") = 10 volts







hile it might seem a little silly to document polarity of voltage drop in this circuit, it is animportant concept to master. 0t will be critically important in the analysis of more comple!

circuits involving multiple resistors and=or batteries.

0t should be understood that polarity has nothing to do with 1hm3s aw& there will never be

negative voltages, currents, or resistance entered into any 1hm3s aw equationsG There are

other mathematical principles of electricity that do take polarity into account through the use of signs L or )-, but not 1hm3s aw.

• REVIEW:

• The polarity of the voltage drop across any resistive component is determined bythe direction of electron flow through it& negative entering, and positive e!iting.

Computer simulation of electric circuits

$omputers can be powerful tools if used properly, especially in the realms of

science and engineering. 'oftware e!ists for the simulation of electric circuits by computer, and

these programs can be very useful in helping circuit designers test ideas before actually

building real circuits, saving much time and money.

These same programs can be fantastic aids to the beginning student of electronics, allowing thee!ploration of ideas quickly and easily with no assembly of real circuits required. 1f course,

there is no substitute for actually building and testing real circuits, but computer simulationscertainly assist in the learning process by allowing the student to e!periment with changes and

see the effects they have on circuits. Throughout this book, 03ll be incorporating computer

printouts from circuit simulation frequently in order to illustrate important concepts. ;yobserving the results of a computer simulation, a student can gain an intuitive grasp of circuit

behavior without the intimidation of abstract mathematical analysis.

To simulate circuits on computer, 0 make use of a particular program called '?0$, which works

by describing a circuit to the computer by means of a listing of te!t. 0n essence, this listing is akind of computer program in itself, and must adhere to the syntactical rules of the '?0$language. The computer is then used to process, or "run," the '?0$ program, which interprets

the te!t listing describing the circuit and outputs the results of its detailed mathematicalanalysis, also in te!t form. /any details of using '?0$ are described in volume 7 "2eference"-

of this book series for those wanting more information. %ere, 03ll just introduce the basicconcepts and then apply '?0$ to the analysis of these simple circuits we3ve been reading

about.

irst, we need to have '?0$ installed on our computer. As a free program, it is commonly

available on the internet for download, and in formats appropriate for many different operatingsystems. 0n this book, 0 use one of the earlier versions of '?0$& version 6*5, for its simplicity

of use.


29/40

Ie!t, we need a circuit for '?0$ to analy(e. et3s try one of the circuits illustrated earlier inthe chapter. %ere is its schematic diagram&

This simple circuit consists of a battery and a resistor connected directly together. e know thevoltage of the battery :8 volts- and the resistance of the resistor 7 +-, but nothing else about

the circuit. 0f we describe this circuit to '?0$, it should be able to tell us at the very least-,how much current we have in the circuit by using 1hm3s aw 0F=2-.

'?0$ cannot directly understand a schematic diagram or any other form of graphicaldescription. '?0$ is a te!t)based computer program, and demands that a circuit be described

in terms of its constituent components and connection points. ach unique connection point ina circuit is described for '?0$ by a "node" number. ?oints that are electrically common to each

other in the circuit to be simulated are designated as such by sharing the same number. 0tmight be helpful to think of these numbers as "wire" numbers rather than "node" numbers,

following the definition given in the previous section. This is how the computer knows what3sconnected to what& by the sharing of common wire, or node, numbers. 0n our e!ample circuit,

we only have two "nodes," the top wire and the bottom wire. '?0$ demands there be a node 8somewhere in the circuit, so we3ll label our wires 8 and :&

0n the above illustration, 03ve shown multiple ":" and "8" labels around each respective wire to

emphasi(e the concept of common points sharing common node numbers, but still this is agraphic image, not a te!t description. '?0$ needs to have the component values and node

numbers given to it in te!t form before any analysis may proceed.

$reating a te!t file in a computer involves the use of a program called a te't editor . 'imilar to aword processor, a te!t editor allows you to type te!t and record what you3ve typed in the form

of a file stored on the computer3s hard disk. Te!t editors lack the formatting ability of word

processors no italic , bold, or underlined characters-, and this is a good thing, since programssuch as '?0$ wouldn3t know what to do with this e!tra information. 0f we want to create a

plain)te!t file, with absolutely nothing recorded e!cept the keyboard characters we select, ate!t editor is the tool to use.

0f using a /icrosoft operating system such as 41' or indows, a couple of te!t editors arereadily available with the system. 0n 41', there is the old %dit te!t editing program, which may


30/40

be invoked by typing edit at the command prompt. 0n indows


31/40

This line of te!t tells '?0$ that we have a voltage source connected between nodes : and 8,

direct current 4$-, :8 volts. That3s all the computer needs to know regarding the battery. Iowwe turn to the resistor& '?0$ requires that resistors be described with a letter "r," the

numbers of the two nodes connection points-, and the resistance in ohms. 'ince this is acomputer simulation, there is no need to specify a power rating for the resistor. That3s one nice

thing about "virtual" components& they can3t be harmed by e!cessive voltages or currentsG


32/40

Iow, '?0$ will know there is a resistor connected between nodes : and 8 with a value of 7 +.This very brief line of te!t tells the computer we have a resistor "r"- connected between the

same two nodes as the battery : and 8-, with a resistance value of 7 +.

0f we add an #end statement to this collection of '?0$ commands to indicate the end of the

circuit description, we will have all the information '?0$ needs, collected in one file and readyfor processing. This circuit description, comprised of lines of te!t in a computer file, is

technically known as a netlist , or dec$ &

1nce we have finished typing all the necessary '?0$ commands, we need to "save" them to afile on the computer3s hard disk so that '?0$ has something to reference to when invoked.

'ince this is my first '?0$ netlist, 03ll save it under the filename "circ$it1#cir" the actual

name being arbitrary-. Jou may elect to name your first '?0$ netlist something completely

different, just as long as you don3t violate any filename rules for your operating system, suchas using no more than BL< characters eight characters in the name, and three characters in

the e!tension& 123456%{- in 41'.

To invoke '?0$ tell it to process the contents of the circ$it1#cir netlist file-, we have toe!it from the te!t editor and access a command prompt the "41' prompt" for /icrosoft users-

where we can enter te!t commands for the computer3s operating system to obey. This

"primitive" way of invoking a program may seem archaic to computer users accustomed to a"point)and)click" graphical environment, but it is a very powerful and fle!ible way of doing

things. 2emember, what you3re doing here by using '?0$ is a simple form of computerprogramming, and the more comfortable you become in giving the computer te!t)form

commands to follow )) as opposed to simply clicking on icon images using a mouse )) the moremastery you will have over your computer.

1nce at a command prompt, type in this command, followed by an NnterO keystroke thise!ample uses the filename circ$it1#cirH if you have chosen a different filename for your

netlist file, substitute it-&


33/40

spice ' circ$it1#cir

%ere is how this looks on my computer running the inu! operating system-, just before 0

press the NnterO key&

As soon as you press the NnterO key to issue this command, te!t from '?0$3s output shouldscroll by on the computer screen. %ere is a screenshot showing what '?0$ outputs on my

computer 03ve lengthened the "terminal" window to show you the full te!t. ith a normal)si(e

terminal, the te!t easily e!ceeds one page length-&


34/40

'?0$ begins with a reiteration of the netlist, complete with title line and #end statement.

About halfway through the simulation it displays the voltage at all nodes with reference to node

8. 0n this e!ample, we only have one node other than node 8, so it displays the voltage there&:8.8888 volts. Then it displays the current through each voltage source. 'ince we only have

one voltage source in the entire circuit, it only displays the current through that one. 0n thiscase, the source current is 6 amps. 4ue to a quirk in the way '?0$ analy(es current, the value

of 6 amps is output as a negative )- 6 amps.

The last line of te!t in the computer3s analysis report is "total power dissipation," which in this

case is given as "6.88L8:" watts& 6.88 ! :8:, or 68 watts. '?0$ outputs most figures inscientific notation rather than normal fi!ed)point- notation. hile this may seem to be more

confusing at first, it is actually less confusing when very large or very small numbers areinvolved. The details of scientific notation will be covered in the ne!t chapter of this book.


35/40

1ne of the benefits of using a "primitive" te!t)based program such as '?0$ is that the te!tfiles dealt with are e!tremely small compared to other file formats, especially graphical formats

used in other circuit simulation software. Also, the fact that '?0$3s output is plain te!t meansyou can direct '?0$3s output to another te!t file where it may be further manipulated. To do

this, we re)issue a command to the computer3s operating system to invoke '?0$, this timeredirecting the output to a file 03ll call "o$tp$t#tt"&

'?0$ will run "silently" this time, without the stream of te!t output to the computer screen as

before. A new file, o$tp$t1#tt, will be created, which you may open and change using a te!t

editor or word processor. or this illustration, 03ll use the same te!t editor (im- to open thisfile&


36/40

Iow, 0 may freely edit this file, deleting any e!traneous te!t such as the "banners" showing

date and time-, leaving only the te!t that 0 feel to be pertinent to my circuit3s analysis&

1nce suitably edited and re)saved under the same filename o$tp$t#tt in this e!ample-, the

te!t may be pasted into any kind of document, "plain te!t" being a universal file format foralmost all computer systems. 0 can even include it directly in the te!t of this book )) ratherthan as a "screenshot" graphic image )) like this&


37/40

y irst circ$it

v 1 0 dc 10

r 1 0 5

#end

node volta*e

( 1) 10#0000

volta*e so$rce c$rrents

nae c$rrent

v "2#000+!00

total power dissipation 2#00+!01 watts

0ncidentally, this is the preferred format for te!t output from '?0$ simulations in this bookseries& as real te!t, not as graphic screenshot images.

To alter a component value in the simulation, we need to open up the netlist file

circ$it1#cir- and make the required modifications in the te!t description of the circuit, then

save those changes to the same filename, and re)invoke '?0$ at the command prompt. This

process of editing and processing a te!t file is one familiar to every computer programmer. 1neof the reasons 0 like to teach '?0$ is that it prepares the learner to think and work like a

computer programmer, which is good because computer programming is a significant area ofadvanced electronics work.

arlier we e!plored the consequences of changing one of the three variables in an electriccircuit voltage, current, or resistance- using 1hm3s aw to mathematically predict what would

happen. Iow let3s try the same thing using '?0$ to do the math for us.

0f we were to triple the voltage in our last e!ample circuit from :8 to


38/40

#ust as we e!pected, the current tripled with the voltage increase. $urrent used to be 6 amps,but now it has increased to 5 amps )5.888 ! :88-. Iote also how the total power dissipation in

the circuit has increased. 0t was 68 watts before, but now is :B8 watts :.B ! :86-. 2ecallingthat power is related to the square of the voltage #oule3s aw& ?F6 =2-, this makes sense. 0f

we triple the circuit voltage, the power should increase by a factor of nine


39/40

/e*end ! = vbranch

""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

sweep vbranch"2#00e!01 "1#00e!01 0#00e!00

"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

0#000e!00 0#000e!00 # # !

5#000e!00 "1#000e!00 # # ! #

1#000e!01 "2#000e!00 # # ! #1#500e!01 "3#000e!00 # # ! #

2#000e!01 "4#000e!00 # # ! #

2#500e!01 "5#000e!00 # # ! #

3#000e!01 "6#000e!00 # # ! #

3#500e!01 "%#000e!00 # # ! #

4#000e!01 "e!00 # # ! #

4#500e!01 ".#000e!00 # # ! #

5#000e!01 "1#000e!01 # ! #

5#500e!01 "1#100e!01 # ! # #

6#000e!01 "1#200e!01 # ! # #

6#500e!01 "1#300e!01 # ! # #

%#000e!01 "1#400e!01 # ! # #

%#500e!01 "1#500e!01 # ! # #e!01 "1#600e!01 # ! # #

Ǵe!01 "1#%00e!01 # ! # #

.#000e!01 "1#&00e!01 # ! # #

.#500e!01 "1#.00e!01 # ! # #

1#000e!02 "2#000e!01 ! # #

"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

sweep vbranch"2#00e!01 "1#00e!01 0#00e!00

0n both output formats, the left)hand column of numbers represents the battery voltage at

each interval, as it increases from 8 volts to :88 volts, 7 volts at a time. The numbers in the

right)hand column indicate the circuit current for each of those voltages. ook closely at thosenumbers and you3ll see the proportional relationship between each pair& 1hm3s aw 0F=2-

holds true in each and every case, each current value being :=7 the respective voltage value,

because the circuit resistance is e!actly 7 +. Again, the negative numbers for current in this'?0$ analysis is more of a quirk than anything else. #ust pay attention to the absolute value of

each number unless otherwise specified.

There are even some computer programs able to interpret and convert the non)graphical data

output by '?0$ into a graphical plot. 1ne of these programs is called &utmeg, and its outputlooks something like this&


40/40

Iote how Iutmeg plots the resistor voltage v(1) voltage between node : and the implied

reference point of node 8- as a line with a positive slope from lower)left to upper)right-.

hether or not you ever become proficient at using '?0$ is not relevant to its application in

this book. All that matters is that you develop an understanding for what the numbers mean ina '?0$)generated report. 0n the e!amples to come, 03ll do my best to annotate the numerical

results of '?0$ to eliminate any confusion, and unlock the power of this ama(ing tool to helpyou understand the behavior of electric circuits.

Contriutors

$ontributors to this chapter are listed in chronological order of their

contributions, from most recent to first. 'ee Appendi! 6 $ontributor ist- for dates and contact

information.

Larry Cramblett 'eptember 68, 688>-& identified serious typographical error in "Ionlinear

conduction" section.

James Boorn #anuary :B, 688:-& identified sentence structure error and offered correction.Also, identified discrepancy in netlist synta! requirements between '?0$ version 6g5 andversion

basic ohms law

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