CHAPTER
1
Direct Current Circuits 1.1
ELECTRIC CHARGE AND CURRENT
There are two kinds of electric charge, positive and negative. They are so
named because they add and subtract like positive and negative numbers.
All atoms contain charge; the usual picture of an atom is a small (10- 15 m
diameter) positively charged nucleus around which negatively charged
electrons move in orbits of the order of 10- 10 m diameter. Charge is
measured in coulombs in mks units and in statcoulombs in the cgs system.
The basic indivisible unit of charge is the charge on one electron which is
-1.6 x 10- 19 coulombs or -4.8 x 10- 10 statcoulombs. All electric charges
(positive and negative) are, in magnitude, integral multiples of the charge of
the electron. However, in most electronic circuit problems the discrete
nature of electric charge may be neglected, and charge may be considered
to be a continuous variable. One of the most fundamental conservation laws
of physics says "that in any closed system the total net amount of electric
charge is conserved or, in other words, is constant in time. For example, in
a semiconductor if one electron is removed from a neutral atom then the
atom minus the one electron (the ion) has a net electric charge of +1.6
x 10- 19 coulombs.
The flow of electric charge, either positive or negative, is called
current; that is, the current I at a given point in a circuit is defined as the
time rate of change of the amount of electric charge 0 flowing past that
point.
dQ
I=di
(1.1)
The direction of the current is taken by convention to be the direction of
the flow of positive charge. If electrons are flowing from right to left in a
wire, then this electron current is electrically equivalent to positive charge
flowing from left to right; hence, we say the current is to the right.
Current is measured in amperes (A); one ampere of current is the flow
of one coulomb of charge per second, which is 6.25 x 1018 electrons per
second. Other units of current are the milliampere (rnA), which is 10- 3 A,
the microampere (Il-A), which is 10-6 A, the nanoampere (nA), which is
1
r
2
CHAP. 1
Direct C_nt Cln:uib
10- 9 A, and the picoampere (pA) , which is 10- 12 A. Prefixes for various
powers of ten are given in Table 1.1. Typical currents in low-power
transistor electronic circuits, such as small radio receivers and amplifiers,
are of the order of 1 to 10 mA; typical currents in low-power vacuum tube
circuits are about lO to 100 mA. The largest current normally encountered
in vacuum tube circuits is about 500 rnA, but specially designed high­
current transistors carrying currents from 1 to 100 A are available.
TABLE 1.1. Powers-of-Ten Prefixes
Prefix
giga
mega
kilo
milli
micro
nano
pico
Symbol
Meaning
G
M
k
m
9
10
10~
10'
10-'
10-6
10-9
10- 12
IL
n
p
SEC. 1.2
Voltage
3
electrons in the current is similar to ignoring the individual molecules in
water flow.
1.2 VOLTAGE
The voltage, or electric potential, Vat a point in a circuit is defined as the
electrical potential energy W of a positive charge Q at that point divided
by the magnitude of the charge; that is,
Example
Va W
Q
9
1 gigahert2 1 GHz = 10 hertz
1 megohm 1 Mll = 106 ohms
1 kilovolt 1 kV = 10' volts
1 milliampere = 1 rnA = 10-' amperes
1 microvolt = llLV = 10-6 volts
1 nanoampere = 1 nA = 10-9 amperes
1 picofarad = 1 pF = 10- 12 farads
The two general kinds of current are direct current (dc) and alternating
current (ac). In direct current the direction of charge flow is always the
same. If the magnitude of the current varies from one instant of time to
another, but the direction of flow remains the same, this type of current is
called pulsating direct current. If both the direction and the magnitUde of the
charge flow are constant, the current is called pure direct current or simply
direct current. If the charge flow alternates back and forth, the current is
called alternating current (to be discussed further in Chapter 2). These
currents are shown graphically in Fig. 1.1.
I
The voltage is measured relative to some other specified reference point in
the circuit where we say the energy of all charges is zero. This reference
point is usually called ground (earth in British literature). A good ground is
a cold-water pipe or sometimes just the metal chassis or box enclosing the
circuit. The point is that the circuit ground has a constant, unvarying
potential, which we set equal to zero volts by convention. The earth has a
constant potential or voltage simply because it is so large and is a reason­
ably good conductor. Any charge taken from or added to the earth through
a circuit's ground wire will not appreciably change the total charge on the
earth or the earth's voltage. The symbols used for ground in circuit notation
are shown in Fig. 1.2.
-L
7
clw:ssis grounds
Jm
earth ground
FIGURE 1.2 Ground symbols.
01
...
direct current
01
...
pulsating direct clIrrent
altematillg ClIrrent
FIGURE 1.1 Types of current.
It is sometimes useful to compare electric current in a wire to water
flow in a pipe, with the water molecules being analogous to the electrons in
the wire. A flow of water in kilograms per second is analogous to a flow of
electric charge in coulombs per second or amperes. Ignoring the individual
The unit of voltage is the volt (V); one volt is defined as one joule per
coulomb (J/c). Other units of voltage are the kilovolt (kV), which is
lOOOV, the millivolt (mV), which is 10-3 V, the microvolt (J,tV), which is
lO-6V, and the nanovolt (nV), which is 1O-9 V. Notice that the voltage at a
given point has no absolute meaning but only means the potential energy
per unit charge relative to ground. This energy-per-unit-charge definition of
voltage is rather difficult to understand intuitively until one considers the
direction in which the charges tend to move. In all natural processes things
tend to move so as to minimize the potential energy; thus, positive charges
will move from points of higher voltage toward points of lower voltage
(e.g., from a point with a voltage of + 15 V toward a point with a voltage of
+12 V). Similarly, negative charges will tend to move from points of lower
voltage toward points of higher voltage. For example, negative charge
4
CHA~'
DkectCunwntCkcuhs
(electrons) will move from a point of voltage -12 V to a point of voltage
-7 V. In terms of the analogy between current and water flow, the voltage
is analogous to the water pressure, because water tends to flow from points
of higher pressure toward points of lower pressure. It is also sometimes
useful to think of the voltage as causing or forcing the flow of current, just
as one thinks of the water pressure as causing or forcing the flow of water.
Thus, voltage is a "cause," and current is an "effect."
Note that the average drift speed of electrons through a copper wire is
rather slow, of the order of 0.1 mmls (10-4 m/s). But the speed with which
electrical energy (the "Signal") moves is normally O.Sc to 0.9c, where
8
c = 3.0 X 10 mls is the speed of light. The reason is that the electrons
strongly repel each other by their surrounding electric fields; the electrons
act, in some sense, like very hard elastic spheres. As an electron is
accelerated by an applied electric field, it collides with a nearby electron,
which in turn immediately collides with its neighboring electron, and so on.
The situation is somewhat similar to a series of closely packed, steel ball
bearings tightly packed in a pipe. If the ball bearing at one end is struck
(given energy), the energy will propagate through the pipe much faster than
the overall drift speed of the ball bearings themselves.t
SEC. '.3
5
R••inance
The current into the circuit element must exactly equal the current
leaving, from the conservation of charge law. V 2 is the voltage at the
terminal where the current enters the circuit element; VI is the voltage
where the current leaves. If the circuit element is passive, that is, if there is
no energy given to the charge by the element, then the charge loses energy
in the element. Thus, V 2 must be greater than VI. Thus, for all passive
circuit elements, the static or dc resistance is positive. The current-voltage
graph for an ordinary positive resistance is shown in Fig. 1.4(a). The unit of
slope
= IfI
slope =
V2 -
VI = V
Rdynamic
V2
(a) ordinary linear resistance
-
VI
V
(b) nonlinear resistance
(tunnel diode)
1.3 RESISTANCE
RGURE 1.4 Current-voltage curves.
If a current I flows through any two-terminal circuit element, then the static
or dc resistance (usually referred to simply as the resistance) of that circuit
element is defined as the voltage difference, V 2 - VI. between the ter­
minals divided by the current I (see Fig. 1.3). Strictly speaking, this
I
----+
y
~. ---+ I
m=,
ELEMENT
L
R =PA:
R= V2
Definition of resistance R.
definition applies only to a circuit element that converts electrical energy
into heat, but this situation occurs in the overwhelming majority of cases
in electronic circuitry.
R!E V 2
(1.3)
VI
-
I
FIGURE 1.3
resistance is the ohm (0); one ohm is defined as one volt per ampere. Other
units of resistance are the kilohm (kO, sometimes written as just k), which is
103 0, and the megohm (MO or just M), which is 106 O.
A piece of conducting material of length L m and cross-sectional area
A m2 has a resistance of R ohms (0):
-
I
VI
(1.2)
!For a further discussion, see R. T. Weidner and R. L. Sells. Elementary Classical Physics,
2nd ed. (Newton, Mass.: Allyn and Bacon, Inc" 1973), Chapter 27,
where P is the resistivity of the material in O-m. The resistivity expresses the
difficulty an electron has in moving through the material due to the
collisions it experiences with the atoms. Resistivity varies for different
materials; it depends slightly on temperature, typically increasing with
increasing temperature for most metals. For carbon resistors used in most
electronic circuits, resistivity increases by approximately 0.5% to 1% for a
10"C temperature increase. Resistivity does not depend on the object's size
or shape but is a property of the material itself. Resistivity values for
various materials are given in Table 1.2.
A conductor is a substance or material with a "low" resistivity. Most
metals are good conductors; copper or aluminum wire is usually used in
SEC. 1.3
6
CHAP. J
7
Resistance
DIrect Cummt Cln:uifll
always refers to dynamic or ac resistance, defined as the rate of change of
voltage with respect to current:
TABLE 1.2. Resistivity Values of Various
Materials
dV
Resistivity p ([1- m)
at 20 DC
Material
Conductors
Silver
Copper
Aluminum
Tungsten
Nichrome (alloy)
Carbon
1.5 X
1.7 X
2.8 X
5.5 X
100 X
3500 x
RdynamiC
Germanium
0.43
2.6 x 10'
Insulators
12
Glass Mica Quartz (fused)
(1.4)
where dV is the change in voltage across the circuit element, and dI is the
change in current through the element. In Fig. 1.4(b), which is the current­
voltage characteristic for a tunnel diode, the static resistance is always
positive for all values of V, but between A and B the dynamic resistance is
negative because I decreases as V increases. Also note that both static and
dynamic resistances vary with voltage. A circuit element for which the
current-voltage curve is not a straight line has nonlinear resistance.
A resistor is a two-terminal circuit element specifically manufactured to
have a constant resistance. Thus, a 4.7-kO resistor has a resistance of
47000; a 2.2-MO resistor has a resistance of 2,200,000 O.
The resistance of a resistor usually is given by a color code (see Fig.
1.5). Bands of different colors specify the resistance according to the
following rule:
10-8
10- 8
10-"
10- 8
10-8
10-"
Semiconductors
Silicon == di
"" 10
9 X 1013
5 X 10. 6
electronic circuits to carry current. An insulator is a substance or material
with a "high" resistivity that is used to prevent current flow. Most plastics,
rubber, air, mica, and quartz are good insulators; wire is usually covered
with a plastic sheath insulator to confine"the current flow to the wire. Note
that germanium and silicon have resistivities that are much greater than
those of metals but much less than those of insulators. Hence, they are often
called semiconductors.
The diameter and resistance per unit length of selected copper wires
are given in Table 1.3. In a practical electronic circuit, for example, a No.
22 wire 4 in. (10 cm) long would have a resistance of only 0.006 O.
In a formal sense a battery has a negative resistance because the
battery gives energy to the charge, making Vt greater than V 2 ; but this
occurs because chemical energy is converted into electrical energy in a
battery. Some circuit elements are said to have negative resistance; but this
TABLE 1.3. Resistance of Various Sizes of Copper Wire
A WG wire sizlI (solid)
Diameter (in.)
Resistance
per lOoolt (J1)
24
22
20
18
16
0.0201
0.0254
0.0320
0.0403
0.0508
28.4
18.0
11.3
7.2
4.5
R (first-color number) (second-color number) x 10 (raised to the third­
color number)
The first digit of the resistance is given by the color band closest to the end
of the resistor. The fourth color band gives the tolerance of the resistor.
Silver means ±10% tolerance, gold means ±5% tolerance, and no fourth
~k
brown.
I1
..J,....-",llll~t
I~
====l
R = I HI ±20% silver
yellow - - - - - ,
violet
yellow
R
470 kO ±10%
FIGURE 1.5 Resistor color code.
COLOR
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
NUMlIER
0
I
2
4
5
6
7
8
9
SEC. 1.4
8
CHAP. 1
colored band means :1:20% tolerance. A 2-kfl 10% resistor will have a
resistance somewhere between 1.8 and 2.2 kfl.
If you have difficulty remembering the color code, note that the colors
follow the colors of the visible spectrum, starting with red for the number
two and going through violet for the number seven. And it is extremely
useful to note that if the third color is brown, the resistor is in the hundreds
of ohms; if red, thousands of ohms; if orange, tens of thousands of ohms; if
yellow, hundreds of thousands of ohms; if green, millions of ohms.
On certain precision metal-film resistors, the resistance value is
specified to three significant figures. These resistors have five colored
bands; the first three bands represent the three significant figures, the fourth
band the multiplier, and the fifth band the tolerance. Certain high-reliability
resistors are tested for failure rates under conditions of maximum power and
voltage, and the results are expressed in percentage failure per thousand
hours. The fifth colored band represents the failure rate per thousand hours
according to the following scheme: brown 1%, red 0.1%, orange 0.01%,
and yellow 0.001 %.
The physical size of a resistor determines how much power in watts it
can safely dissipate, not its electrical resistance in ohms. Characteristics of
various types of resistors are given in Appendix A.
The reciprocal of resistance is called conductance and is usually
represented by the symbols G or y:
1
G=­
(1.5)
R
Conductance is measured in ohms-I, One ohm-I is called a siemen or a
mho ("ohm" spelled backwards).
1.4 OHM'S LAW
For many kinds of circuit elements it is empirically true that the resistance
of the element is constant if the temperature and composition of the
element are fixed. This is true over an extremely large range of voltages and
currents. That is, changing the voltage difference between the two ter­
minals by any factor changes the current by exactly the same factor; that is,
doubling the voltage difference V 2 - VI exactly doubles the current I.
Ohm's law is simply the statement that the resistance is constant. It can
be written in three ways:
R=
V 2 - VI
=
IR
9
Ohm's l.IIw
Direl:t CUmlnt C1rculD
I = V2
-
R
VI
(1.6)
Any two-terminal circuit element of constant resistance is shown in circuit
----vvvvv--­
R
(b) actual resistor
(a) schematic s.vmbol
FIGURE 1.6 Resistance.
diagrams as a zig-zag line [see Fig. 1.6(a)] and is called a resistor. The
larger the voltage difference, the larger the current for a fixed resistance;
the larger the resistance for a fixed voltage drop, the smaller the current.
Thus, "resistance" is a very appropriate name; it literally means "opposition
to current flow."
These three forms of Ohm's law can be thought of in the following
terms. R = (V2 VI) I I means that if there is a voltage difference V 1 - VI
across a circuit element through which a current I is flowing, then the
circuit element must have a resistance (V2 - VI)I I. V 2 - VI = IR means
that if a current I is forced through a resistance R, then a voltage
difference IR will be developed between the two ends of the resistance.
I
(V2 - VI) I R means that if there is a voltage difference V 2 VI across
a resistance, then a current (V2 - VI) I R must be flowing through the
resistance. Perhaps the most important thing to remember about Ohm's law
is that it is only the difference (V2 - VI) in voltage across a resistor which
causes current to flow. Thus a 5-kfl resistor with one end at 35 V and the
other end at 25 V will pass a current of 2 mA because
I
V2
R
VI _
~ = 0.002 A = 2 mA
- 5000fl
A 5-kfl resistor with one end at 1078 V and the other end at 1068 V will
also pass a current of 2 mA because the voltage difference is also 10 V. It
is useful to remember the shortcut that the voltage difference in volts
divided by the resistance in kilohms equals current in milliamperes; in the
previous example I = 10 V/5 kfl = 2 mA.t
If two resistors RI and R2 are connected in series (see Fig. 1.7), that is,
if they are connected end-to-end so that the same current flows through
each of them, then the total effective resistance is simply the sum of the two
t A careful analysis of free-electron drift through a conductor in response to an applied
voltage difference shows that O~ law holds over a wide range of applied voltages because
the thermal speed of a free electron (1(16 mls at room temperature) is much larger than the drift
speed (l0- 4 mls). Thus the average time between collisions is essentially independent of the
applied voltage, and the drift speed of the free electrons is proportional to the applied voltage
or electric field, which is basically Ohm's law.
CHAP. 1
10
1-
V2
Va
VI
RI
Direct Cummt Clrcun.
SEC. 1.4
11
Ohm'. Law
resistance is given by
_I
RIR2
R2
R tota!
1
FIGURE 1.7 Two resistors in series.
RI
individual resistances. This result follows from Ohm's law applied separately
to RI and R 2 •
RIO'"!
V3 - V 2
I
I
-VI
+
=
RI + R 2
(1.7)
Thus, a l·kO resistor and a 3·kO resistor in series act like a single 4-kO
resistor. This rule can be extended to N resistors in series, in which case the
total effective resistance is equal to the sum of all the N individual
resistances.
R'ota!
= RI + R2 + R3 + ... .f RN
(1.8)
Notice that the total resistance for a series connection always is greater than
any of the individual resistances. Also notice that a straight line connecting
the two resistances in a circuit diagram represents a wire or electrical
connection of zero resistance. Thus all points of a straight line in a circuit
diagram must be at exactly the same voltage; there is no voltage drop along
a wire of zero resistance. In an actual circuit the resistance of the wire used
to connect various elements is usually negligibly small; for example, a 2-in.
length of No. 18 copper wire (0.04 in. diam) has a resistance of only
0.00110.
Ohms law in terms of conductance is simply
1= G(V2
VI)
where (V2 - VI) is the voltage drop across a circuit element with conduc­
tance G.
If two resistors are connected in parallel or side by side (see Fig. 1.8) so
that the same voltage appears across each one, then the total effective
RI
'vWV\
1,­
1 - ----'--iA
/1­
-'\.NINI
R1
FIGURE 1.8 Two resistors in parallel.
VI
-I
1
+
(1.9)
RI + R2
R2
This result follows from Ohm's law and the conservation of current.
point A, the current [ entering splits up into two parts, [ = [I + [2.
Rtotol
V2
-
[
VI
V2
[I
V2
VI
+
[2
-
--"--- +
VI
V2
-
1
1
-+
RI R2
VI
~--"
R2
At
RIR2
RI + R2
For example, a 6-kO resistor and a 4-kO resistor in parallel act like a single 2A-kO resistor. If we have N resistors R I, R 2 , R 3 , · · ., RN all connected in parallel, the total effective resistance is given R to"I-
11 +
-+
RI
R2
1
R3
+ ...
1
(1.10)
+~
RN
Notice that the total resistance for a parallel connection is always less than
any of the individual resistances and that the voltage drop is the same
across all of the resistors in parallel. Ohm's law applied to each resistor
shows that the current divides among the various resistors in such a way
that the most current flows through the smallest resistance, and vice versa.
For example, in the circuit of Fig. 1.8:
If R, = 10 kfl and R2 2 kfl, then R ,ota' = (10 kn)(2 kfl)/l2 kfl = 1.67 kil.
If V 2 - V, = 8 V, then [
8 V/1.67 kfl = 4.8 rnA.
The current I, flowing through R, = 10 kfl is I, = 8 V/IO kfl = 0.8 rnA.
The current flowing through R2 2 kil is 12 = 8 V12 kH 4 rnA.
In general, [t! [2 R2! R" which follows from the fact that RI and R2 have
[2 R 2.
the same voltage drop across them: [I RI
In terms of conductance, for N resistances in series the total conduc­
tance is
1
(1.11)
G tota!
1
1
1
GI
+
G2
+ ... + ­
GN
For N resistances in parallel the total conductance is
Gto!.I"" G I + G 2 + .,. + G N
(1.12)
SEC. 1.fi
12 CHAP. f
Direct Current Cin:uks
Variable resistors, often called potentiometers or "pots," are also avail­
able. They come in many sizes and styles, and are usually adjusted by
manually turning a shaft, as shown in Fig. 1.9. They have three terminals:
one at each end of the resistor, and one for the variable position of the tap.
The total resistance RT between the two end terminals A and B is always
constant and equals the resistance value of the pot. The resistance Rl
between A and the tap and the resistance R2 between B and the tap vary as
the shaft is turned. Notice that if the shaft is turned fully clockwise, the tap
B\W.
13
currents. Thus,.Ohm's law does not apply to a zener diode: in other words,
the resistance of a zener diode varies with the current through the diode.
Consider a resistor and a zener diode connected in series with a battery as
shown in Fig. 1.10. The current through R and the zener diode must be
I
+.....\
+
---
\ V R voltage drop
across R
J
I
A
battery
RT{::~~'"
,
+,
I
I
t
\
I Vz
,I
voltage drop across
zener diode
I
B
FIGURE 1.10 Zener diode circuit.
(bl actual device
(a) schematic symbol equal because they are connected in series, and the voltage of the battery,
V blt , must equal the voltage across R, VR , plus the voltage across the zener,
V z : Vb"
VR + V z . Ohm's law applied to the resistor alone implies VR =
IR. Thus V bo = IR + V z . If Vob decreases, the constant zener voltage V z
implies that I must decrease, so the IR drop across R always equals the
difference between V bo and V z .
Ra
Ra Rr
IR
shaft rotation Vob - V z (1.13)
shaft rotation
This circuit is often used to produce a constant output voltage across the
zener diode. If Voo = 12 V and Vz 6.8 V, then IR = 12 V 6.8 V =
FIGURE 1.9 Variable reSistor.
5.2 V. If I = 10 rnA, then R = 5.2 V /10 rnA = 52011 by Ohm's law. For
R 52011, if Vbb falls to 10 V, then I must decrease from 10 rnA to
V 6.8 V)f520 = 3.2 V 1520 6.16 rnA.
is electrically connected to terminal A and R2 = R T , Rl = O. The variation
may be linear taper, with shaft rotation as shown in Fig. 1.9(c), or logarith­
mic taper as in Fig. 1.9(d). For example, a 100-k11 pot has RT = RJ + R2 =
100 kG regardless of the shaft rotation, but Rl and R2 depend on the shaft
1.5 BATTERIES
position-always subject to the condition RI + R2 = 100 k11. A logarithmic
taper is usually used in volume controls for audio equipment; a linear taper
A battery is a two-terminal device in which chemical energy is converted
is more commonly used in scientific apparatus.
into electrical energy, and a voltage difference is generated between the
two battery terminals. A battery tends to spew positive charge out of the
It is worthwhile to note that Ohm's law must apply to a resistance even
positive terminal and draw it into the negative terminal. A battery also
though it is connected with another circuit element that does not obey
tends to spew negative charge out of the negative terminal and draw it into
Ohm's law. For example, a zener diode is a solid-state device with the
the positive terminaL In an actual battery negatively charged electrons are
nonlinear property that the voltage drop across its two terminals is essen­
tially constant, regardless of the current through it, over a wide range of
(c) linear taper (d) logarithmic taper
SEC. 1.5
14
CHAP. 1
spewed out the negative terminal and are drawn in the positive terminal,
even though we may, for convenience, speak of positive charge flowing. A
dry battery has chemicals that are essentially dry or in a paste form (e.g., a
flashlight battery). A wet battery contains liquids (e.g., an automobile
battery that contains sulfuric acid solution). The circuit symbol for a battery
is a series of long and short paralleilines, as shown in Fig. 1.11: The longer
1
voj
v~{2]~
(b) current flow
(a) schematic symbol
OV
+6V
+
Vbb "" 6 V -;;;;; +
R
15
Blltterie.
Direct CUrrent Cfn:uits
v~ 6v~1l
-6V
(c) voltages with different ({rounds
FIGURE 1.11 Batteries.
line represents the positive terminal; the shorter line, the negative terminaL
Thus, if a resistor R is connected between the two battery terminals as
shown in Fig. l.11(b), then positive current will flow out of the battery "+"
terminal, through the resistor, and back into the battery "-" terminal.
Notice that the (positive) current flows out of the battery + terminal and
into the + end of the resistor; this means the battery is discharging. if
(positive) current flows into the battery + terminal, that battery is being
charged by some other battery with a higher voltage.
Notice again that the straight lines drawn in the circuit diagram
represent wires with zero resistance. Thus, there is no change in voltage
along the wires represented by straight lines. The voltage at the positive
battery terminal is exactly the same as the voltage at the top of the
resistance in Fig. 1.11(b). To have a voltage difference between two points
in a circuit, there must be some resistance between these two points (from
Ohm's law, Vz VI IR); that is, if R 0, V 2 VI 0 even if If o.
Notice also that the + and - signs represent the relative polarity of the
voltages; that is, the top of the resistor is positive with respect to the
bottom. Also notice that we may ground anyone point in a circuit. Two
such cases are shown in Fig. 1.11(c). In either case the voltage difference
between the battery terminals is 6 V, and the current is the same. Notice
that no current flows into the ground connection; the current flowing out of
the positive terminal of the battery flows back into the negative ter­
minal. The ground connection merely sets the zero voltage level.
A battery is rated in volts; its voltage rating Vb" indicates the
difference in voltage that the battery will maintain between its two ter­
minals. A perfect or ideal battery will always maintain the same voltage
difference between its terminals regardless of how much current I it
supplies to the rest of the circuit. However, the voltage of any real battery
will decrease as more and more current is drawn from it. Thus, for a real
battery, Vbb represents the terminal voltage when no current is drawn from
the battery. Vbb is often called the open-circuit voltage. In general, the
larger the battery's physical size (for the same voltage rating) the less the
voltage will decrease as more current is drawn; in other words, a large
battery can supply more current than a small battery at the same voltage.
This behavior can be explained by the fact that a real battery has an internal
resistance r, as shown in Fig. 1.12(a). The actual current must flow through
both rand R, which are in series, so I = Vbbl(r + R). Notice that the
battery terminal voltage VA - VB must equal Vbb Ir because of the
polarity difference between the Ir voltage drop across the internal resis­
tance rand Vbl>'
The larger the battery for a given voltage the smaller r is (r = 0 for an
ideal battery). The internal resistance of a battery can be determined by
measuring the terminal voltage for various measured currents I drawn from
the battery. The internal resistance r is then the negative slope of the graph
of the terminal voltage plotted versus I, as shown in Fig. 1.12(b). Or, if the
terminal voltage is measured for two currents, then r is given by [see Fig.
1.12(b)] r = (Vi - V 2)/(I2 - II)'
VA - VB
V~T.\
.r
+I
Vbb
slope = --r
+
battery
terminals
~)
R
v,
V.
VB
I,
~
FIGURE 1.12
Internal resistance r of battery_
~
~=~
h
16
CHAP. 1
Dil1H:f CilmJnf C1n:uIts
When a battery goes "bad," its internal resistance r increases sharply.
The typical good 12-V automobile battery has an internal resistance of
about 0.03 n. A size D 1.5-V carbon-zinc dry cell, sucll as is used in
flashlights and in some portable transistor circuits, has r = O.S n. A I"
x !" x l" 9-V battery, often used to power portable transistor radios, has
r = 13 n. Usually the internal resistance r is omitted from circuit diagrams,
but this omission is valid only when r is much less than any other series
resistance in the circuit.
In scientific circuitry a battery that very gradualIy goes bad (that is,
whose voltage slowly decreases) is a real disadvantage, because the circuit
behavior may become erratic and difficult to diagnose as the battery slowly
wears out. However, with a mercury battery, which goes bad very abruptly
(after a long life), the battery voltage is either all right or extremely low, in
which case the circuit will usually not function at all. Thus, in most scientific
instruments mercury batteries are used, particularly if proper circuit
behavior depends strongly upon a certain minimum battery voltage. The
nickel-cadmium battery also goes bad abruptly after a long life, and it has
the additional advantage of being rechargeable. It is, however, more
expensive than the mercury battery. For a brief summary of the six different
types of batteries, see Appendix B.
1.6 POWER
Power is defined as the time rate of doing work or the time rate of
expending energy; that is, P == d W I dt, where W is work or energy and t is
time. The units of power are thus joules per second (J/s); 1 watt (W) ==
1 J/s. We will now show that a de current I flowing through a resistor R
develops a power of 12 R or VI or y2 I R, where V is the voltage drop
across the resistor.
Recalling that voltage is electrical potential energy per unit charge, we
see that a charge has less electrical potential energy when it leaves a resistor
than it has when it enters because of the decrease in voltage, or voltage
"drop" across the resistor. The time rate at which the flowing charge gives
up electrical potential energy is thus the amount of charge flowing per
second multiplied by the energy lost per unit charge, which is exactly equal
to the current I multiplied by the voltage drop V. Thus, P =' IV. But, from
Ohm's law we know that V = IR; thus the power can also be expressed as
P'" 12R (see Fig. 1.13). And, again from Ohm's law, I = VIR; thus
another way of expressing the power is P = V2 I R. These three expressions
for the power are equivalent and apply only to direct current flowing
through a resistor. For alternating current the phase angle between the
current and the voltage must be taken into account (more about this in
Chapter 2).
The power developed in a resist()r shows up as heating of the resistor.
SEC. 1.6
Power
17
Vs V2 - VI
...
I
V2
'V\IV\J\
+
R
VI
-
...
P
IV = fUR) = f2R
P
!V= R V="If
f
(V)
V2
FIGURE 1.13 Power dissipated in a resistor.
In other words, the loss of electrical potential energy (due to the IR voltage
drop) of the charge flowing through the resistor is converted into random
thermal motion of the molecules in the resistor. The kinetic energy of the
flowing charges remains approximately constant everywhere in the circuit.
Electrical potential energy is converted into heat energy in any dc circuit
element across which there is a voltage drop and through which current
flows.
If too many watts of power are converted into heat in a resistor or in
any circuit element, the resistor may "burn up," in which case the resistor
turns brown or black and may actually fragment, thus breaking the elec­
trical circuit. In other words, the resistor is open or has an infinite resis­
tance. Or, if the resistor is heated too much, its resistance value may
increase tremendously, thus changing the operation of the circuit drastic­
ally. For these reasons resistors are rated by the manufacturer according to
how much power they can safely dissipate without being damaged. Resistors
are commonly available with wattage ratings of 1W, 1W (for low-power
transistor circuits), !W, 1 W, 2W, SW, lOW, 20W, 50W, WOW, and
200 W. The larger the resistor is physically, the more power it can safely
dissipate as heat. The actual sizes of several commonly used resistors rated
for various power dissipations are shown in Fig. 1.14.
----c::J--
1/8 watt
---=e=:J-=--
1/4 watt
1/2 watt
I watt
2 watt
actual size
AGURE 1.14 Resistor sizes for different power ratings.
18
CHAP. f
Direct Cilmmt Circuits
We emphasize that the resistance does not depend upon the actual
physical size. The physical size determines the power rating; for example, a
!-W, 2.2-kO resistor is the same size as a !-W, 470-kO resistor. Circuit
designers usually choose a power rating of at least three or four times the
expected power. For example, if a 1.5-kO resistor is to carry 20 rnA of
direct current, then the power dissipated as heat in the resistor will be
P
= 12 R = (20 X 10-3 A)2(1.5
X 103 0)
= 0.6 W
In the actual circuit a 1.5-kO, 2-W resistor would be used, or perhaps even
a L5-kO, 5-W resistor if the circuit were very sensitive to heat. If a large
wattage is developed in a certain part of a circuit, care should be taken to
provide an adequate vertical flow path for air around the hot element so
that the heat can be carried away by the resulting convection air currents.
A resistor dissipating a large amount of power should never be placed in a
closed chassis. Heat is an enemy of transistors as well as other circuit
elements.
Notice also that two !-W, I-kO resistors in parallel are equivalent to
one loW, 500-0 resistor, and two !-W, l-kO resistors in series are
equivalent to one loW, 2-kO resistor.
We will now derive an important theorem about power. If a battery has
a certain fixed internal resistance r and we connect a load resistor RL across
its terminals, as shown in Fig. 1.15, how do we maximize the power
SEC. f.B
19
Pow.r
of RL the power dissipated in RL is maximized. Let us set up an expression
for PL as a function of RL and maximize it.
= 1VL
PL
But
I
Vbb
(r
-(~)RL
r + RL
VL
+ Rd and
1RL -
Therefore,
RL
L (r +VbbRd . (r +VbbRL) . RL = (~)2
r + RL
P =
aPL
aR L
=
2
(r
+ RL)2
Vbb
(r
- 2Rdr
+ Rd 4
(
battery
terminals
r
+,,
+
RL <;
+
IVI•
I
A'
FIGURE 1.15 Circuit illustrating power transfer from source to load Re.
(1.15)
+ Rd
Set aPdaR L = 0 to find the value of RL for which PL is an extremum.
_aPL
aR
0-
V 2bb
(r
+ Rd 4
(r
+ Rd 2
L
(
r
+ Rd
2Rdr
RL = r
/'-
(1.14)
2
- 2Rdr
+ Rd]
+ R L)
(1.16)
It can be shown that PL is a maximum when RL = r by showing that
ilPdaRi is negative at RL = r. In words, the maximum power is dissipated
in the load resistance RL when it equals the internal resistance of the
battery. Under this condition the voltage V L across the load equals one half
of V bb , which is the open-circuit battery voltage. One half of the total power
is also dissipated in r and one half in R L •
However, suppose we have a fixed load resistance RL , Vb/>, and variable
r, and we ask the question: "How can we maximize the power dissipated in
RL 1" The answer is not r = R L , but r = O! This answer is really obvious
when we realize that any power dissipated in r is wasted as far as the load is
concerned. It also follows mathematically from maximizing P L =
VEbRL/(r + Rd 2 with respect to r.
PL = 1VL dissipated in RL 1 If RL is very small, I is large; but
VL
1RL
is small, so the power in RL is small. If RL is very large, the voltage V L
across RL is nearly equal to Vb/>; but then the current is small, so again the
power in RL is small. It seems reasonable that for some intermediate value
, To sum up, if the load RL is fixed, the smaller the internal resistance r
the better. If we are presented with a fixed internal resistance r, then the
load RL gets maximum power when RL = r. Also note that when r is fixed,
the load voltage is very large when RL is very large, and the load current is
very large when RL is very small.
SEC. f. 7
20 CHAP. f
21
Tc""pereture Variation.
Direct CUrrent Citl:uIts
I
0
1.7 TEMPERATURE VARIATIONS OF
RESISTIVITY AND RESISTANCE
For most metallic conductors, the resistivity increases slowly with increasing
temperature. At higher temperatures the thermal motion of the atoms
increases; hence the average distance moved by a free electron between
collisions decreases, producing a slightly lower drift speed. The resistance
of a metallic conductor depends upon temperature according to the relation
t
constant (
current
source
RT
v
v = IRT
V = IR"fI + a (T
T,,)J
FIGURE 1.16 Resistance thermometer.
RT = Ro(1 + aliT) (1.17)
where Ro is the resistance in ohms at some reference temperature To,
Ii T = (T To) is the temperature rise in °C, and a is the linear tem­
perature coefficient of resistance in eC)-I. a is typically O.OOSeC)-1 for
most metals or about O.S%/"C. For example, if a piece of copper has a
resistance of 1000 at 10°C, at 300°C, it will have a resistance of
R = RT = (100 0)[1 + (0.0039fOC)(300°C - 10°C)J = 2130
Values of a for various materials are given in Table 1.4.
TABLE 1.4. linear Temperature Coefficients of
Various Materials
Material arC)~l
Manganin alloy Carbon Iron Copper Silver Aluminum Platinum Tungsten Iron Nickel +0.00001
-0.0005
+0.005
+0.0039
+0.0038
+0.0039
+0.0039
+0.0045
+0.0050
+0.0067
The resistance of pure platinum is often used as a standard of tem­
perature over a wide temperature range, from -190°C to over 600°C; the
device is called a resistance thermometer. A constant known current is
forced through a pure piece of platinum wire, and the voltage across the
platinum wire is measured, as shown in Fig. 1.16. This voltage drop is
linearly proportional to the resistance of the platinum wire, which in turn
depends upon the temperature, according to equation (1.17). Notice that
the voltmeter must draw zero current. Usually a potentiometer is used (this
will be described later).
A thermistor is a special two-terminal device designed to have a
resistance that is a strong function of temperature. The resistance, R T , of a
thermistor is given by
RT = R oe A (IIT-llTo) (1.18)
where A is a constant in kelvins (K) whose value depends upon the
particular thermistor, Ro is the resistance at temperature To (K), T = the
temperature of the thermistor (K), and e = 2.718 (the basis of the natural
logarithms). The constant A depends slightly on temperature. Using (1.18)
over a SO°C range with A constant would typically produce a ±3°C error.
For greater accuracy the thermistor should be calibrated experimentally
over the temperature range expected.
The resistance change for a thermistor is typically ten times the
resistance change for copper for the same temperature change. The ther­
mistor resistance decreases with increasing temperature, which is opposite
to the temperature dependence for most metals.
Commercial thermistors are usually made from sintered mixtures of
Mn2 0 3 and NiO or platinum alloys and are often encapsulated in a thin
glass bead with two wire leads, as shown in Fig. 1.17. They are available
in a wide range of resistance values. The thermistor resistance at 2SoC (R 2S )
RTI
I~
(a) resistance
VI.
FIGURE 1.17 Thermistor.
temperature
. r
glass bead
T
(b) actual thermiSlOr
SEC. 1.B
'I"
22 CHAP. 1
can range (rom 30n to 20Mn (or various types, and the ratio of resistance
at 2S"C to resistance at 125°C may range from 10:1 to 100:1.
The thermistor takes a certain time to come to equilibrium if its
surrounding temperature is changed. The thermistor time constant is
defined as the time required for the thermistor resistance to change by 63%,
witb 100% being the total change in resistance for an infinite time. For
example, consider a thermistor with a 10-kfl resistance at 100°C, and a
110-kO resistance at 30'C, and a time constant of 100 ms. If the thermistor
is initially in equilibrium at 100°C and is suddenly immersed in a 30°C
environment, then its resistance will increase to
1l0kn+0.63(1l0kn
KirChhoff's Laws and Networlc Analysis 23
Direct Current Circuits
lOkf!) 10kfl+0.63(100kH)
10 kH + 63 kH = 73 kH
in the lirst 100 ms. Bare thermistors with no glass covering are the fastest;
they are available with time constants as short as 4 ms in water and 100 ms
in air. Larger thermistors encapsulated in glass would, of course, have
longer time constants, perhaps 200 ms in air and 5 s in water.
Finally, it should be pointed out that the self-heating of the thermistor
due to the current I flowing through it should be as small as possible. The
manufacturer will specify a self-heating or dissipation constant, which gives
the maximum self-heating ([2 R T ) power per degree Celsius error produced.
This might be 0.5 mWrC in air and 2.5 mWrC in water for a typical
thermistor. For such a thermistor in water, the internal self-heating should
be kept much less than 2.5 mW for the temperature error to be much less
than 1°C.
1.8 KIRCHHOFF'S LAWS AND NETWORK ANALYSIS The two basic laws of electricity that are most useful in analyzing circuits
are Kirchhoff's current law and VOltage law:
!
I
!
I
I
destroyed at the junction in question; that is, the total current entering
equals the total current leaving. The voltage law says that there is no net
gain or loss in electrical potential energy for any charge making a trip
around any closed loop; that is, the energy the charge gains (in passing
through a battery) must be all lost (as heat, radiation, etc.) in the rest of the
loop. (If a changing magnetic field is present, then an induced "emf" must
be placed in series with the loop just as if a battery were actually present.)
To solve for the currents and voltages in a circuit or network, using
Kirchhoff's laws, we first must assume a current direction in each bram:h of
the circuit and define a current symbol such as 11 for that current.
We use arrows to draw the currents, and we label the polarities of
the voltage drops (+ and -) across the resistances, remembering that
positive (conventional) current always flows into the "plus end" and out of
the "minus end" of a resistance. It is useful to draw the current arrows in
the direction the current actually flows in the circuit; if this is done, the
numerical value obtained for the current at the end of the calculation will
turn out to be positive. However, if we guessed wrong as to the current
direction, the numerical value obtained for the current will be negative but
of the same magnitude as if we had guessed the current direction correctly.
In other words, a negative current at the end of a calculation is merely a
"flag" that we guessed wrong when we drew the current arrow on the
circuit diagram. It should also be emphasized that once the current direc­
tions are chosen (or guessed), the polarities of the voltage drops are fixed.
There are basically three methods for calculating the currents and
voltages in a circuit, that is, network analysis. Only experience will enable
you to choose the easiest method. In the following paragraphs, a branch is
simply one path or wire through which one current can flow, a loop is a
closed path in the circuit (an electron can flow around any loop and return
to its starting point without leaving the actual circuit), and a junction or node
is a point in a circuit where three or more wires come together (e.g., joined
solder). In all three methods we must have a clear circuit diagram with
all the voltages and currents clearly defined. It is also useful in all three
methods to look for simple series and parallel combinations of resistances
and to replace them immediately by their equivalent resistances. We must
also remember that no current flows in the ground connection.
Kirchhoff's Current Law (KCL)
At any junction of wires in a circuit, the sum of all the currents entering the
junction exactly equals the sum of all the currents leaving the junction. In
other words, electric charge is conserved.
Kirchhoff's Voltage Law (KVL)
Around any closed loop or path in a circuit, the algebraic sum of all the
voltage drops must equal zero. In other words, energy is conserved.
The current law merely says that no electric charge is being created or
The Branch Method
In this method we draw the current in each branch of the circuit, and we
label the voltages at all batteries. Then we can write a KCL equation at
each junction and a KVL equation for each closed loop in the circuit.
However, a little thought will show that such a procedure applied to each
junction and each closed loop will produce a number of nonindependent
equations; that is, we might obtain five equations in three unknowns.
Obviously, we wish to obtain n independent equations to solve for the n
CHA~1
24
~ctC~entC;reuft5
unknown currents. It can be shown that if there are k junctions in the
circuit, there are only k 1 independent KCL equations. For example, if
there are four junctions, there are three independent KCL equations.
It can also be shown that there are only n - (k - 1) independentKVL
equations. Thus, the total number of independent equations (KCL and
KVL) is
(k
1) + n
1) = n
(k
We can solve these n equations for the n unknown currents. In writing the
KVL loop equations, we must cover each branch in the circuit at least
once.
Let us solve for the currents and voltages in the circuit of Fig. 1.18.
+ R,
The KVL says that starting at G, which is ground (0 V), and going
around the circuit clockwise, we obtain
+ Vb!>
+ Vbb
or
+
Vbb~ 24 V
R2
51 k!1
I2Rz
IR4
0
IR,
I3R3
lR4
0
(1.20)
(1.21)
and 13 , or
1.1
-
IR,
where (1.20) has been obtained by going through R2 in our loop, and (1.21) has been obtained by going through R 3 • Substituting (1.19) for I in (1.20) and in (1.21) immediately gives us two equations in the two unknowns 12 + I3)R 1
Vbb -
(12
Vbb
(12 + I 3)R 1
B
3kn
25
Kin;hhofl's Lews end Networlc Analysis
SEC. 1.B
I2R2
(12 + 13)R4 == 0 I3 R3
(12 + 13)R4
0
Vbb
=
(R1
+ R2 + R 4)I2 + (R, + R4)13
(1.22)
Vbb
= (R l
+ R 4)I2 + (R l + R3 + R4)13
(1.23)
Ra ~ 2 k!1
With the resistances in kD and the currents in rnA, we have
R.
c
5 k!1 +
13
24
912 + 8 13
(1.24)
24
812 + 1013 (1.25)
AGURE 1.18 Circuit problem.
Solving equations (1.24) and (1.25) simultaneously for 12 and 13 yields This problem can, of course, be solved by simply combining resistances in
series and in parallel. R2 and R3 are in parallel, and their combined parallel
resistance is 0.667 kD. Rt. (R2 11 R3), and R4 are all in series, so
I == -----;,.:..- - ­
R, +
R 3 ) + R4
24V
8.667 kD
2.77 mA
12 + 13
(1.19)
1.85 mA
and
I == 12 + 13
From (1.19)
We can then find 12 • 13 , and the voltages by Ohm's law.
Let us illustrate the branch method by solving for the currents and
voltages. First, we use arrows to draw the currents in all parts of the circuit;
then we label the polarities of the voltage drops. For this circuit there are
two junctions, B and
so k 2. Thus, there is k - 1 = 1 independent
KCL equation. There are three unknown currents (n = 3) I, 12 , and 13 , so
there are n (k 1) '" 3 - 1 2 independent KVL equations.
The KCL says that the total current entering a junction equals the total
current leaving the junction. So I
12 + 13 at junction Band 12 + 13 = I at
junction C, which is, of course, the same equation. So we have one KCL
equation:
I
12
13
0.92 mA
2.77 mA
which is the same answer we obtained from the simple series and parallel
resistance analysis. However, we emphasize that the branch method and
Kirchhoff's laws will always yield a solution, while the series and
analysis will work only for relatively simple circuits.
We can calculate the voltages at points Band C hv usinl!. Ohm's law
and the values for 12 and
Vc == +IR4 == VB =
kil)
13,85 V
_ IRl = 24 V -' (2.77 mA)(3.0 kD)
15.7 V Or,
VB = +IR4 + I1. R 2 == (2.77 mA)(5 kD) + (1.85 mA)(l kD)
15.7 V