CHAPTER 2
Theory
This chapter covers the basic concepts of electronics, such as current, voltage, resistance, electrical po'ltrer, capacitance, and inductance. After going through these concepts, this chapter illustrates how to mathematically model currents and voltages
through and across basic electrical elements such as resistors, capacitors, and inductors. By using some fundamental laws and theorems, such as ohm,s law, Kirchoff,s
laws, and rhevenin's theorem, the chapter presents methods for analyzing complex
networks containing resistors, capacitors, and inductors that are clriven by a power
source. The kinds of power sources used to drive these networks, as we will see,
include direct current (dc) sources, alternating current (ac) sources (including sinusoidal and nonsinusoidal periodic sources), and nonsinusoidal, nonperiodic sources.
At the end of the chapter, the approach needed to analyze circuits that contain nonlinear elements (e.g., diodes, transistors, integrated circuits, etc.) is discussed.
As a note, if the math in a particular section of this chapter starts looking scary,
don't worry, As it turns out, most of the nasty math in this chapter is used to prove,
say, a theorem or law or is used to give you an idea of how hard things can get
if you
do not use some mathematical tricks. The actual amount of math
will
need
to
)/ou
know to design most circuits is surprisingly small; in fact, aigebra may be all you
need to know. Therefore, when the math in a particular section in this chapter starts
looking ugly, skim through the section until you locate the useful, nonugly formulas,
rules, etc. that do not have u'eird mathematical expressions in them.
2.a
Current
Current (symbolized with an 1) represents the amount of electrical charge Ae
crossing a cross-sectional area per unit time, which is given by
6r
de)
.AQdQ
Lt dt
Tlre unit of current is called the anpere (abbreviated amp or A) and is equal to
one
. ".:l.,rll.
Lrer Secontl:
PRACTICAL ELECTRONICS FOR INVENTORS
7
A=7 C/s
Electric currents typically are carried by electrons. Each electron carries a charge of
-e, which equals
-e=_ 7.6x10
1eC
Benjomin Fronklin's Positive Charges
Now, there is a tricky, if not crude, subtlety with regard to the direction of current
flow that can cause headaches and confusion later on if you do not realize a historical convention initiated by Benjamin Franklin (often considered the father of electronics). Anytime someone says "current I flows from point A to point 8," you would
undoubtedly assume, from what I just told you about current, that electrons would
flow from point A to point B, since they are the things moving. Tt seems obvious'
Unfortunately, the conventional use of the term current, along with the symbol l used
in the equations, assumes that positive charges are flowing from point A to B/ This
means that the electron flow is, in fact, pointing in the opposite direction as the current flow. What's going on? Why do we do this?
The answer is conaention, or more specifically, Benjamin Franklin's convention of
assigning positive charge signs to the mysterious things (at that time) that were moving and doing work. Sometime later a physicist by the name of joseph Thomson performed an experiment that isolated the mysterious moving charges. However, to
measure and record his experiments, as well as to do his calculations, Thomson had
to stick with using the only laws available to him-those formulated using Franklin's
(which he
positive currents. However, these moving charges that Thomson found
called electrons) were moving in the opposite direction of the conventional current 1
used in the equations, or moving against convention'
(:.
Maqnified
view of wire
Convenlional aurrent'flow
(l)
rr(@->
v
,ttt@)),,r(/ist,,@ ,r{@-) "t(@'->
rtt6->
(
I
Eleotronflow
{
o
i
I
a6
<_z
FIGURE
2.1
ol
/Druvmrnn/:\ <-"
I
+t
9atterv
What does this mean to us, to those of us not so interested in the detailed physics
and such? Well, not too much. I mean that we could pretend that there were positive
charges moving in the wires, and electrical devices and things would work out fine' In
fact, all the formulas used in electronics, such as Ohm's Iaw (V = 1R), "pretend" that the
current 1 is made up of positive charge carriers. We will always be stuck with this convention. In a nutshell, whenever you see the term current or the symbol I, pretend that
positive charges are moving. However, when you see the tetrn electron fo-tu, make sure
you realize that the conventional current flow l is moving in the opposite direction'
Theory
2.2
Voltage
When two charge distributions are separated by a distance, there exists an electrical
force between the two. If the distributions are similar in charge (both positive or both
negative), the force is opposing. If the charge distributions are of opposite charge
(one positive and the other negative), the force is attractive. If the two charge distributions are fixed in place and a small positive unit of charge is placed within the system, the positive unit of charge will be influenced by both charge distributions. The
unit of charge will move toward the negatively charged distribution ("pulled" by the
negatively charged object and "pushed" by the positively charged object). An electrical t'ield is used to describe the magnitude and direction of the force placed on the positive unit of charge due to the charge distributions. When the positive unit of charge
moves from one point to another within this configuration, it will change in potential
energy. This change in potential energy is equivalent to the work done by the positive
unit of charge over a distance. NoW if we divide the potential energy by the positive
unit of charge, we get what is called a uoltage (or electricnl potential-not to be confused with electrical potential energy). Often the terms potential and electromotiae t'orce
(emf) are used instead of ooltage.
Voltage (symbolized V) is defined as the amount of energy required to move a unit
of electrical charge from one place to another (potential energy/unit of charge). The
unit for voltage is the zolf (abbreviated with a Y which is the same as the symbol, so
watch out). One volt is equal to one joule per coulomb:
1V =11/C
In terms of electronics, it is often helpful to treat voltage as a kind of "electrical
pressure" similar to that of water pressure. An analogy for this (shown in Fig. 2.2) can
be made between a tank filled with water and two sets of charged parallel plates.
Elealrical 1yetem
Waler System
::?,
1
-=
=?
FIGURE 2.2
system, water pressure is greatest toward the bottom of the tank
because of the weight from the water above. If a number of holes are drilled in the
side of the tank, water will shoot out to escape the higher pressure inside. The further
In the tank
PRACTICAL ELECTRONICS FOR INVENTORS
it. The
down the hole is drilled in the tank, the further out the water will shoot from
exiting beam of water will bend toward the ground due to gravity'
Now, if we take the water to be analogous to a supply of positively charged particles
the elecand take the water pressure to be analogous to the voltage across the plates in
plate
trical system, the positively charged particles will be drawn away from the positive
"escaping
(a) and move toward the negatively charged plate (b). The charges will be
from the
from the higher voltage to the lower voltage (analogous to the water escaping
tank). As the charges move toward plate (b), the voltage acloss
plates
(c) and (d) will bend the beam of positive charges toward plate (d)-positive charges
bending as
again are moving to a lower voltage. (This is analogous to the water beam
it escapes the tank.) The higher the voltage between
(d).
plates (a) and (b), the less the beam of charge will be bent toward plate
in
Understanding voltages becomes a relativity game. For example, to say a point
another point in the cira circuit has a voltage of 10 v is meaningless unless you have
charge-absorbing
infinite
its
cuit with which to compare it. Typically, the earth, with
the
ability and net zero charge, acts as a good point for comparison. It is considered
here:
shown
is
ground
the
0-V reference point or grounil point. The symbol used for
a
result of the force of gravity
as
FIGURE 2.3
to
There are times when voltages are specified in circuits without reference
specleft
simply
the
to
ground. For example, in Fig. 2.4, the first two battery systems
to the
ify one battery terminal voltage with respect to another, while the third system
right uses ground as a reference point'
6V
3V
FIGURE 2.4
2.3
Resistance
is the term used to describe a reduction in current flow' All conductors
can be
intrinsically have some resistance built in. (The actual cause for the resistance
heating,
a number of things: electron-conducting nature of the material, external
are
reslslors
impurities in the conducting med.ium, etc.). In electronics, devices called
electronics is
specifically designed to resist current. The symbol of a resistor used in
Resistance
shown next:
Theory
---^/v\-FIGURE 2.5
If a voltage is placed between the two ends of a resistor, a current will flow
through the resistor that is proportional to the magnitude of the voltage applied
(called
across it. A man by the name of Ohm came up with the following relation
Ohm's law) to describe this behavior:
V =IR
(abbreviR is called lhe resistance and is given in units of volts per ampele, or ohms
ated Q):
7
A=7V /A
Electricol Power
within the electron current that runs through a resistor is
converted into thermal energy (vibration of lattice atoms/ions in the resistor). The
Some of the kinetic energy
power lost to these collisions is equal to the current times the voltage. By substituting
Ohm's law into the power expression, the power lost to heating can be expressed in
two additional forms. All three forms are expressed in the following way:
P
2.4
=IV =I2R=V2/R
DC Power Sources
Power sources provide the voltage and current needed to run circuits. Theoretically,
power sources can be classified as ideal uoltage sources or ideql current s7urces.
An ideal voltage source is a two-terminal device that maintains a fixed voltage
drop across its terminals. If a variable resistive load is connected to an ideal voltage
source, the source will maintain its voltage even if the resistance of the load changes.
This means that the current will change according to the change in resistance, but the
voltage will stay the same (in I = V f R,I changes with R, but I/ is fixed).
Now a fishy thing with an ideal voltage source is that if the resistance goes lo zero,
the current must go to infinity. Well, in the real world, there is no device that can supply an infinite amount of current. Instead, we define a real aoltage source (e'9., abattery) that can only supply a maximum finite amount of current. It resembles a perfect
voltage source with a small resistor in serres.
-I
ldeal voltage
ReaJ voltage
source
SOUTCC
FIGURE 2.6
Anideal current source is a two-terminal idealization of a device that maintains a constant current through an external circuit regardless of the load resistance or applied volt-
t0
PRACTICAL ELECTRONICS FOR INVENTORS
It must be able to supply any necessary voltage across its terminals. Real current
constant outsources have a limit to the voltage they can Provide, and they do not provide
source'
put current. There is no simple device that can be associated with an ideal current
age.
2.5
Two Sirnple Battery Sources
load conThe two battery networks shown in Fig. 21 wlllprovide the same Power to a
the volttimes
three
nected to its terminals. I{owever, the network to the left will provide
provide only
age of a single battery across the load, whereas the network to the right will
the curthree
times
one times the voltage of a single battery but is capable of providing
rent to the load.
vB
,|
<-
F-€
I
vu
vu
-
i",
vB
I
L
vB
2.6
FIGURE 2.7
Electric Circuits
An electric circuit is any arrangement of resistors, wires, or other electrical compovoltage
nents that permits an electric current to flow. Typically, a circuit consists of a
or other
source and a number of components connected together by means of wires
ciruits
parallel
conductive means. Electric circuits can be categorized as series ciruLits,
'
or series and parallel combination circuits
Bosic Circuit
Vn=3V
A simple light bulb acts as a load (the part of
the circuit on which work must be done to
move current through it). Attaching the bulb to
the battery's terminals as shown to the right
will initiate current flow from the positive terminal to the negative terminal' In the process,
the current will power the filament of the bulb,
and light will be emitted. (Note that the term
cwrenthere refers to conventional positive current-electrons are actually flowing in the
opposite direction.)
Series Circuit
Connecting load elements (lightbulbs) one after
Vs=3Y
FIGURE 2.8
the other forms a series circuit. The current
through all loads in a series circuit will be the
same.In this series circui! the voltage drops by a
third each time current passes through one of
the bulbs. With the same battery used in the
basic circui! each lightwillbe one-third as bright
as the bulb in the basic circuit.The effective resistance of this combination will be three times that
of a single resisdve element (one bulb).
Theory
il
Porollel Circuit
A parallel circuit contains load elements that
+
l,/
their leads attached in such a way that the
voltage across each element is the same. If all
three bulbs have the same resistance values,
have
{,
- lv
V
current from the battery
will be
divided
equally into each of the three branches. In this
arrangement, light bulbs will not have the
dimming effect as was seen in the series circuit, but three times the amount of current will
flow from the battery hence draining it three
times as fast. The effective resistance of this
combination will be one-third that of a singie
resistive element (one bulb).
Combinotion of Series ond Porollel
A circuit with load elements placed both in
series and parallel will have the effects of
I'n=3v
both lowering the voltage and dividing the
current.The effective resistance of this combination will be three-halves that of a single
resistive element (one bulb).
OV
FIGURE
2.8
(Continued)
Circuit Anolysis
Following are some important laws, theorems, and techniques used to help predict
what the voltages and currents will be within a purely resistive circuit Powered by a
direct current (dc) source such as a battery.
2.7
Ohm's Law
Ohm's law says that a voltage difference V across a resistor will cause a current 1=
VIR to flow through it. For example, if you know R and V, you plug these into
Ohm's law to find 1. Likewise, if you know R and I, you can rearrange the Ohm's
law equation to find V. If you know V and I, you can again rearrange the equation
to find R.
V=]R
I
V
I =V/R
R =V/I
FIGURE 2"9
2.8
Circuit Reduction
Circuits with a number of resistors usually can be broken down into a number of
series and parallel combinations. By recognizing which portions of the circuit
have resistors in series and which portions have resistors in parallel, these portions can be reduced to a single equivalent resistor. Here's how the reduction
works.
l2
PRACTICAL ELECTRONICS FOR INVENTORS
Resistors in Series
I=v"
Req
R'
' =Rr +Rr
y,
=R. +R,
v,"
R'
v.=
' Rr *Rz v,^
FIGURE 2.10
WhentworesistorsRlandR2areconnectedinseries,thesumofthevoltagedroPs
(Vi")'
the applied voltage across the combination
across each one (Vr and VJ witl equal
Vin=Vt*
Vz
v1 and
both resistors, we can substitute IR, for
since the same current I flows through
result is
IR2 for V, (using Ohm's law)' The
V6 = IR1 +
IR2:
I(R1 + R2) = IR",,
ThesumR'+R,iscalledtheequiaalentresistancefortworesistorsinseries'Thismeans
thatseriesresistorscanbesimplifiedorleducedtoasingleresistorwithanequivalent resistance R"q equal to Rt + Rz'
the preceding equation or, in other
To find the current 1, we simply rearrange
be R"o:
voltage to be Vi' and the resistance to
wotds, apply Ohm's law, taking the
I
=Y^'
Req
Tofigureouttheindividualvoltagedropsacrosseachresistorinseries,ohm,slawis
applied:
O'=
R' U,u,n
&*R,
or=
v,
Lr\2 =!+
v 2=fR,
-
V,"
R' + R'
=\
Vr=/Rr=
V^
Oo
R..r
useful formuaoltage diaider relations-incredibly
These two equations are called the
frequently'
las to know. You'll encounter thern
Foranumberofresistorsinseries,th:equivalentresistanceisthesumoftheindividual resistances:
R=Rr+R2+"'*Rn
Resistors
in Parollel
Rz
R,Rz
1'
= R, + R, /'"
P -,'req
R, +R,
Rr
12=
FIGURE 2.1].
--Rt
+
R2Ii"
Theory I i
When two resistors R1 and R, are connected in parallel, the current d,. dir.ide.
between the two resistors in such a way that
I*=
Ir
I
Iz
Using Ohm's law, and realizing that voltage across each resistor is the same (both V,,,),
we can substitute Vr,/ R, for l, and Vr^/ R for 12 into the preceding equation to get
_ v,n v,n
I=:*r=l/-l-+-l
Rr R2
1\
lt
\ R' Rrl
The equivalent resistance for these two resistors in series becomes
1
1
Re,l Rr
1
R2
or
RR,
R^.
crr= --------'
R, + R,
To figure out the current through each resistor in parallel, we apply Ohm's law again:
,
_ /','R"q _ R, ,
''--V^
R, Rr -R,+Rr''n
,-v*-l1$t-
''- R
R2
R'
Rr+R2
,
I'n
These two equations represent what are called current diaider relations. Like the voltage divider relations, they are incredibly useful formulas to know.
To find the equivalent resistance for a larger number of resistors in parallel, the
following expression is used:
1
1 1
_l(ee Rl R2
1
R,
Practical Electronics
for lnventors
Paul Scherz
McGraw-Hill
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