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Introduction to DC Theory

The study of electronics is a very exciting challenge for anyone, with something new to be learned each day.  Since it is a field that is constantly expanding with new ideas, systems, and products, even those who have been involved with the field for years still have to study to keep up with this ever expanding field.  A thorough study of this field, for entry into the workplace as an electronics technician, generally takes about two years.  To make this easier, the material is generally broken down into modules or courses with students progressing from the more basic to the more advanced.  Most instructors would agree that all need to start with certain core or foundational material before trying the more advanced.  What is core and what is not is the subject of some disagreement, but most would put the following four courses in the core:  DC Theory, AC Theory, Semiconductor Devices and Circuit Theory, and Digital Electronics.  After completing these, the remaining course requirements and sequence often depend on the exact branch of electronics one is preparing for, with specialties like communications, computers, industrial controls, robotics, consumer products, and biomedical electronics.   This introduction is written to be an overview of the concepts of DC theory, which most would agree is the preferred starting point to begin a study of electronics.  This first course introduces the foundation material which all the other courses build upon.  This page is prepared as a brief introduction with the intentions of helping you decide if this is the career field for you.  Please consult the Terms and Definitions Page and the Electronics Symbols page as needed .

The study of electronics begins with a review of atomic theory, specifically the electron theory of matter.  This is material that most will have already covered in a general science class.  Remember that matter is anything that has mass and occupies space, and exists as a solid, liquid, or gas. All matter is made of the basic elements, substances that cannot be reduced to simpler substances by normal chemical means. A chemical combination of two or more elements forms a compound of which the smallest part is a molecule. The smallest unit of an element, having all the characteristics of that element is the atom. The model of the atom was developed by the physicist Niels Bohr. In this model, an atom consist of a dense, central, positively charged nucleus surrounded by a system of electrons.  Most of the mass of an atom is in the nucleus which consist of neutrons and positive (+) charged particles called protons.  The negatively (-) charged electrons rotate around the nucleus and are arranged in shells or energy levels.  Normally, an atom has an equal number of electrons and protons making it electrically neutral,  the attraction between the opposite charges is what causes the electrons to orbit the nucleus. Note that the electron is very light compared to the proton.  This will be very important later.

A periodic table of the elements list the atomic number, which is the total number of protons, for each element.  With the number of electrons being equal to the number of protons, Bohr's model would show the distribution of electrons for each shell.   The orbit that is the farthest from the nucleus is called the valence shell and can never have more than eight electrons.  It is this shell that is of main interest to the field of electronics.  Since the atoms of this shell are at the greatest distance from the nucleus, they can be set free easier than the others and become free electrons. Usually some extra energy is required to do this but heat is always present in our environment as an energy source.  The flow of free electrons in a conductor is one definition for electricity.

There are several theories used to explain how the number of electrons in this shell affect the properties of the atom and how it reacts with other atoms.  One model defines the shells as energy levels.  All of the electrons in the valence shell share the energy for that shell. If a material has eight electrons, each electron only has one eighth of that shell's energy, thus it would takes a great amount of outside energy to give any electron enough energy to become free.  These materials normally do not have free electrons and are called insulators.  Any material with five to seven have similar characteristics.  On the other end of the extreme, there are materials with only one valence electron.  This electron has all of the energy of the shell and consequently can be set free with little extra energy.  Materials with two or three valence electrons have similar characteristics to a lesser degree. These materials generally have many free electrons which can be made to flow as current.  We call these materials conductors. 

Another simple model says that materials strive for a complete outer shell.  If a material has a complete valence shell of eight electrons it is very stable and would not have free electrons. Likewise materials that have five to seven valence electrons would have few free electrons because they would tend to want to capture any free electrons that came along to complete their outer shell.  These materials form insulators which resist the flow of electrons.  Materials with one to three would find it easier to get rid of the few than find enough to be complete.  These materials would always have a great number of free electrons.  We refer to these as conductors because electrons can easily flow through them. Most metals are good conductors with silver being the best followed by copper, gold and aluminum. 

In the middle are materials with four valence electrons having characteristics of both conductors and insulators.  They are referred to as semiconductors but generally are very poor conductors at low temperatures but can become fairly good conductors at high temperatures.  What happens with semiconductors is that they align themselves in a way to share their electrons such that each one thinks it has eight.  This relationship is referred to as covalent bonding.  Examples are silicon and germanium, which are then modified to produce the material (P and N) that in turn can be fashioned into diodes and transistors.  These are the primary components of integrated circuits (ICs) and the electronic brains of today's high-tech devices. 

It is also very common for material to lose or acquire electrons through the process called ionization.  One or more atoms of a material gives up its electron/s and becomes positively charged while the one receiving the electron/s becomes negatively charged. These charges atoms, called ions, remain bound together because of the  attraction of opposite charges. These arrangements produced through ionic bonding may have very different characteristics than the materials that produced them. An example is two gases bonding to produce water.  Remember that these chemical combinations of two or more elements form compounds of new material with a molecule of that material being the smallest part.  Ionic bonds are very stable arrangements with few free electrons, consequently many insulators are produced this way.

After having reviewed basic atomic theory,  the next step is to realize that electricity is the flow of electrons.  For electrons to flow there has to be a source of  electrons.  Most people have experienced what is referred to as static electricity, defined as charge at rest.  Static electricity is produced by friction such as walking across a carpet while wearing rubber-soled shoes.  This separation of charge produced by friction between two objects is referred to as triboelectricity.  Dynamic electricity or current is charge in motion.  It is the flow of negative electron charges from atom to atom toward a lesser negative point or more positive point. This is what you actually experience after walking across the carpet and then touching a metal object. One of the fundamental laws of electricity is that like charges repel and unlike charges attract.  So a source is something that causes  a separates of charges, creating an accumulation of negative charges at one terminal and positive charges at the other.  Then when a path between the two is made current flow occurs with the the very light negative electron charges being attracted  toward the positive point which has an electron deficiency.  The unit for measuring electrical charge (Q) is the coulomb (C)  with one coulomb equal to the charge of 6.25 X 10¹⁸ electrons.  A object that is negatively charged has gained electrons while an object that has lost electrons is positively charged.  Current is more accurately defined as the rate at which electrons flow and its symbol is I  which stands for intensity.  The unit of current flow is the ampere (A) and one ampere is that current flow when one coulomb of charge is flowing past a point in one second. Current can be explained by the equation:   I = Q/t.

The external force, created by the separation of charges, that causes electrons to flow is called electromotive force (emf), and is also referred to as potential difference, difference of potential, or just voltage.  The symbol E or V may be used, and the unit of measurement is the volt (V).  One volt is defined as that electromotive force that will cause one coulomb of  charge to flow through one ohm of resistance in one second.  The ohm and resistance will be discussed shortly.  To use electricity effectively, a constant supply of charges is required.  Other than the  triboelectric effect, electricity can be produced by several other methods.  Through chemical reactions we have basic energy cells which can be used individually or combined to form batteries.  These can be designed as primary cells that only last until all of the internal materials have reacted as much as possible, or as secondary cells which can be recharged.   Cells and batteries represent our largest category of portable sources.  By mechanically moving conductors through a magnetic field, the magnetoelectric effect, we have our greatest source of electrical energy.  The devices using this effect are referred to as generators and alternators.  Some source of mechanical energy is required to rotate the armature winding in the magnetic field.  This could be an internal combustion engine as our car engine or it could be through a turbine being powered by wind, falling water, or steam. The power company that supplies our homes generates energy primarily this way.  A hydroelectric dam allows water to flow through a large pipe to turn the turbine. A steam plant would burn a fossil fuel such as coal or natural gas to make steam which in turn would rotate the turbine. A nuclear point just uses a nuclear reaction to make steam.  Light may also be used to produce electricity through the photovoltaic effect.  It was discovered that certain semiconductor materials would cause a separation of charge in the presence of light.  These devices are referred to as photovoltaic cells and, if use with sunlight, solar cells. Each cell only produces about 400 millivolts, and even that depends on the light intensity. The more intense the light the greater the voltage and the larger the cell the greater the current output. They can be combined in different arrangement to produce greater voltage and current requirements.   These devices use available sunlight and have a very long life expectancy. Their potential for future applications is tremendous and one day may serve as our primary energy source.  A common everyday application is to provide power for a pocket calculator.  Two other sources of electricity are worth mentioning.  The thermoelectric effect produces electricity directly from heat.  Two dissimilar material joined at one end forms a thermocouple.  When the junction is heated a very small potential is formed across the free ends.  This potential is very dependent on the temperature of the junction, is predictable and repeatable for a very large range of temperatures.  This makes thermocouples excellent sensors for very high temperatures.  Thermocouples can also be arranged to get greater voltage and/or current.  These arrangements are referred to as thermopiles. The final source is created when certain crystals are put under pressure.  Named for the Greek term for pressure, the piezoelectric effect will generate a small charge when the pressure is changing ,or dynamic, but produces nothing when the pressure is static.  A common application would be a transducer to change sound pressure into an electrical signal, the crystal microphone.

When current flows, there is an opposition to that flow.  This opposition is called resistance (R) and the unit of measurement is the ohm ().  Resistance is a natural occurrence, as electrons move through a material they collide with countless other electrons in their natural orbits. Also, atoms that have lost electrons are now positively charged and attract the passing electrons.  The resistance causes heat in the conductor when current flows.  This occurs much less in conductors than other materials but it is this resistance that establishes the maximum safe current (ampacity) for conductors.  Insulators have so much resistance that they allow virtually no electron flow.  Resistance is often used to change electric energy to heat, examples being  water heaters,  stoves, coffee makers, toaster, hair dryers, clothes dryers, and electric heating.  It is also used in incandescent lamps to produce light.  In electronics, resistance is often added to a circuit to limit current or to produce a voltage division.  Components made for this purpose are called resistors and come in many ohmic values and physical sizes which determine how much heat they can safely dissipate. Many are made to be variable to control some circuit parameter like volume.  Two common variable resistor configurations are the potentiometer and the rheostat.

There are two more very important terms that need to be introduced, energy and power.  Energy can be defined as the capacity to do work. When electrons are forced to move they have kinetic energy which can be released in some device to produce work.  The unit of electrical work is the joule (J) and one joule is the amount of energy carried by one coulomb of charge being propelled by an electromotive force of one volt. Power is defined as the rate of doing work or converting energy to a different form.  Electric power (P) is measured in watts (W) with one watt being energy converted at the rate of one joule per second.  Since energy in joules is coulombs times voltage (Q x V) and power is joules/second, we can equate that power is  (Q x V)/t. And since I = Q/t, we can simplify the power equation to be P = I V.  Another  interesting  point about energy and power comes through the way that energy consumption is expressed.  Energy consumption is reported in units of watthours (Wh), which is the rating in watts multiplied by time in hours.  A 100 W light bulb operated for 4 hours would consume 400 watthours.  Now realize that our utility company does not really charge for power, but for energy used.  If you consider that one watt is 1 joule/second and 1 hour is 3600 seconds then 1 watthour is 1 joule/sec x 3600 sec. for 3600 joules.  The 400 watthours from the above example would have to be expressed as 1,440,000 joules.  So the watthour expression is just a more practical expression.  Just remember that it is an expression of energy and not power.  Typically, we use the larger expression of  kilowatthours (kWh) for power consumption over longer periods. One kWh equal 1000 Wh. Utility bills typically show the usage for a month. To figure the cost of operating a device for a month, first figure the total consumption in kilowatthours by multiply the power rating times the number of hours per day times the number of days per month, then dividing by 1000.  Then find the cost by multiplying the number of kilowatthours by the cost per kWh. If you do not know the cost per kWh then use a figure of 7 to 8 cents/kWh. If the 100W bulb already mentioned was used 4 hours each night for the entire month, we would find the cost as  100W/1000W/KW x 4 hours x 30 days x 0.07/kWh = $ 0.84 or 84 cents.  Obviously that is not a large amount but realize that a typical home might have many lights on at a time.  The real energy users are the larger devices such as heating devices.  A portable electric heater rated at 1500W on for half the time would cost 1500/1000 x 12 x 30 x 0.07 = $37.80.  Considering that we use electricity so much for heating devices like water heaters, clothes dryers, and just heating, it is understandable why they are the real culprits in making energy bills high.

So far we have introduced a number of terms.  Of these, four stand out as to their importance in solving dc circuits, these are electromotive force or voltage (E or V), current (I), resistance (R), and power (P) . Georg Simon Ohm discovered that electricity always acted in a predictable manner.  His research led to what we now call Ohm's Law which expresses the relationship between voltage, current, and resistance. In his research, Ohm discovered that if he maintained the voltage constant, that current would vary inversely to changes in resistance, and if he maintained resistance constant, that current would vary directly with voltage.  Simply stated Ohm's Law says "current is directly proportional to the applied voltage and inversely proportional to circuit resistance".  As an equation, I = V/R.  This equation allows us to solve for any unknown as long as we know the other two variables.   The other forms then are V = IR and R = V/I.  Already introduced, the power equation, P=IV, may also be expressed in terms of the other variables yielding I=P/V and V=P/I. These six formulas are listed below.

With the above six equations, most dc circuits can be solved for any unknown if at least two other variables are given .  In some cases it may be required to find a third variable. For example, one might want to know the power in a resistor when voltage and resistance are known.  Since the power equation only has I and V and we are only given V and R, I could be solved with I = V/R and then P could be found.  There is away to get there directly but it requires doing a little mathematical manipulation of the two base formulas, I = V/R and P =IV. 

Since

and

, substituting V/R for I in the first equation yields

and

Since

and

, substituting IxR for V in the P equation yields

and

Solving the two new power equations in terms of their other variables gives us four additional equations. This gives us twelve altogether, and allows us to solve for any variable as long as two are known.  Actually each term can be solved by one of three equations depending on what is known about a circuit.

Using the equations from above, one can now  begin to apply them to the solution of basic DC circuits. But what is meant by the term DC?  This is a term to classify the manner in which current flows in a circuit.  Direct current produces a flow that  flows in one direction only in a circuit.  This means that the source terminal always maintain the same polarity, as in a battery.  This is in contrast to alternating current (AC) which periodically reverses direction. An ac source periodically reverses the polarity at its terminal causing the current reversal.  An example of this is the type of electricity provided to us by our utility companies.  Another point is which way does current flow. There are two different theories on this issue, electron flow and conventional flow.  The original theory, conventional flow, states that current flows from positive to negative. This is based on  the flow of positive ions, and is also referred to as the fluid theory of flow.  Since this is the oldest, many symbols reflect this in their design. Electron theory is based on the electron theory of matter and states that electrons flow from negative to positive, which they actually do.  The idea hear is that the free electrons are much lighter than the positive charges attracting them and thus they are what current flow consist of.  Both camps still argue their point relentlessly, but either can be used if one consistently applies the same assignment.  So if you see different books showing current flowing in different directions do not be to concerned.

Now that we have some theory and some circuit formulas we are ready to look at some circuit arrangements. Note that in the more complex circuits, the formulas can be applied to the source or to a circuit component.  A subscript can be used to distinguish the terms. From the source prospective V_S , V_T, or  V_A might indicate the source, total, or applied voltage,  while V₁ would indicate the voltage at R_1.  In terms like R_x or V_part,  the subscripts would be used to generically refer to any of the load parameters without specifically identifying them.  Just be sure to plug in the appropriate values. 

Please consult the Terms and Definitions Page and the Electronics Symbols page as needed.

Simple Circuit - The most simplest of circuits, requires that there be at least a source, a load, and some conductors to connect everything together into a single complete path for electron flow. The load is the device that is to convert the energy from the source into a practical application. There may be a control device (switch) to turn on or off the circuit and possible an overload  protection device (fuse).  A good example of this would be a flashlight.  The circuit at the right would be a schematic diagram for a simple circuit.  The lamp is rated at 3 watts when 12 volts is connected across it.  Since we know P and V, the lamps resistance (R) and current (I) that will flow when the switch is closed can be found using the appropriate formulas above.	See this circuit solved
SERIES CIRCUITS - A series circuit is one that has multiple loads, multiple sources, or multiple control devices or all three connected such that there is only one path for current to flow.  Typically this means that all parts are connected end-to-end and the same current flows through every part.  This means that the electrons encounter all the resistances additively in flowing around the loop. So total resistance is the sum of all the individual resistances.  From the equations above we can see that current flow in a resistance results in a voltage.  This represents the work being done by the electrons flowing through the resistance. These voltages across the resistors are called voltage drops and since all have the same current, each voltage drop is the current times that resistor's value. Each gets only a part of the total voltage based on its relationship to the total resistance. For this reason, series circuits are referred to as voltage dividers.  This also means that the sum of the voltage drops has to be equal to the source voltage or net source voltage for multiple sources. This is referred to as Kirchhoff's Voltage Law. The rate of consuming energy (P) can be figured for each load. The total power is always the sum of all of the individual load powers. The circuit at right has series connected loads. In solving this circuit, first find RT, then use that to find I.  Next find each voltage drop.  You can check yourself by making sure they all add up to equal VS. Last find the power in each resistor.  Add these to find PT. Again you can check yourself comparing your answer with PT found from any of the three power equations from above solved for the source. Sources may also be wired in series. When connected this way they can be connected as series-aiding or series-opposing. Series-aiding is normally done to get a higher voltage than a single source acting alone.   Just remember that the current in series is the same through all parts, so the sources need to have similar current ratings.  Series-aiding sources are connected end-to-end, positive-to-negative, and the total voltage is the sum of all the individual values.  Series-opposing means that sources are connected such that some cancel out part of the net value of the others.  This is rarely done on purpose but it does often happen by accident, as when someone installs one of several series batteries backward. To see example circuits with series connected sources and to read about Kirchhoff's Voltage Law and voltage drop polarity go to the Circuit Solutions Page. One more interesting point about series circuits is that an open (break in circuit path) anywhere causes the entire circuit to stop working.  If one component opens or if a wire comes disconnected then the circuit is no longer functional.  Keep reading to find out how to circumvent this.	See this circuit solved
	SERIES CIRCUIT FORMULAS I is the same in all parts.     RT  = R1+R2+ . . . . +RN       VT  =  V1+V2+ . . . . +VN   Vx = I Rx or Vx = Rx/RT x VS PT = P1+P2+ . . . . +PN Px = IVx = I2Rx = Vx2/Rx
PARALLEL CIRCUITS - A parallel circuit is one that has multiple loads, multiple sources, or both connected such that one end of each device has a common connection.  The same is true of the other ends.  Note in the diagram to the right that the top of each resistor and the positive end of the source are all connected together when the switch is closed.  The same is true of the bottom of each resistor and the negative end of the source.  This means that each load device has connections to the source terminals. This helps us to understand the main point about parallel circuits and that is that the voltage is the same across all parts and is equal to the source voltage. Since the loads came have different resistances, each load can have different current values. These paths are often referred to as branches.  Also, each load can be operated independently from the others.  Our homes are wired for this type of operation, in that 120 V is wired to each device and each device is designed to operate from 120 volts.  Each device can be operated as needed without effecting the other devices. This is not true in a series circuit. Since the source has to supply all of the devices, in a parallel circuit the individual branch currents, are added to find the source current. Another interesting point is that the total resistance in a parallel is less than the smallest branch resistance.  This may be hard to understand but remember that R is inversely proportional to I when V is constant.   If you think about the circuit at right, if only the 500 ohm resistor was operating, then I1 = 12V/500 W = 0.024 amps. If the second resistor was now added, it would have its own current, I2 = 12/1.5K =  0.008 amps.  The source would be supplying both of these currents so its current would be 0.024 + 0.008 = 0.032 amps.  Then from Ohm's Law,    RT = V/IT = 12/0.032 = 375 W which is less than the 500 W.  Obviously, as other loads are added, more current is required of the source and for IT to increase, RT has to decrease. The formulas for finding RT are shown at right.  Also, note that the power each load uses can always be figured using one of the power equations above for that part.  Then the total power for the circuit is the sum of all the individual load powers. The parallel circuit shown has parallel-connected loads.  To solve this circuit, first find each branch current.  Then add all branch currents to get IT.  Next, Find RT using Ohm's Law and as a check find RT using the general formula for any number of resistors. Then find the individual powers and then PT by either adding all of the part powers or as a check by any of the three power equations solved for the source.   Sources may also be wired in parallel and this is done for two different reasons.  One reason is to deliver more current to a load than what a single source acting alone could deliver. Remember that voltage is the same across all parts of a parallel circuit so parallel sources have to have the same voltage rating.  The other reason is related to when the source voltage is being supplied by batteries.  Two or more batteries in parallel can operate a given load for a longer time interval before having to recharge or replace the batteries. To read more about  parallel connected sources and see an example circuit    go to the Circuit Solutions Page.	See this circuit solved.   PARALLEL CIRCUIT FORMULAS V is the same across all parts.   IT = I1 + I2 + . . . . + IN Each branch current is Ix = V/Rx RT can be found several ways RT = V/IT  Px = IxV = Ix2Rx = V2/Rx PT = P1+P2+ . . . . +PN
SERIES-PARALLEL CIRCUITS - Often circuit components need to be connected in more complex arrangements.  A series-parallel circuit is one that has some of the properties of both series and parallel circuits.  In other words, there has to be parts with different voltages and parts with different currents.  The rules for solving these circuits are the same as for the previous circuits.  Apply the series rules for the series components and parallel rules for parallel components.   The two circuits at right are the simplest series-parallel circuits. In the first one, R1 and R2 are in parallel, and as a combination they are then in series with R3.  Being in parallel, R1 and R2 have the same voltage but different currents.  Their combined current will flow in R3 which has its own unique voltage drop.  To solve this circuit, first find the parallel equivalent for R1 and R2, then add the result to R3 to get RT.  Next find IT and realize that it also is the value of I3.  Using this current find the voltage across R3 and the voltage across the equivalent value for R1 & R2.  Now using  the voltage across R1 and R2, find I1 and I2. As a check, add these two to see that they equal IT.  In the next circuit note there are two paths for current to flow.  One is through R1 & R2 connected in series, and the other is through R3.  Since R3 is connected directly to the source it sees the full source voltage, thus I3 = VS/R3.  Since R1 and R2 are in series they will have the same current but will divide the voltage.  To solve this part, solve for the total branch resistance (R1+R2) and divide VS by this value to get the branch current. Next multiply each resistor by this current value to get V1 and V2.  As a check, these two should add to equal VS.  RT could be found several ways.  The R1-R2 branch current can be added to I3 to get IT  and RT can be found using Ohm's Law.  RT can also be found by finding the parallel equivalent of the sum of R1&R2 with R3. In both circuits power can be found for the circuit or for each part. Just remember that total power is always the sum of all part powers.	See this circuit solved
	See this circuit solved

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