EE 101 – Introduction To Electrical Fundamentals, Fall 2004
Lecture 1
Electronic Materials and Basic Electrical Circuit Concepts
Introduction
From our high school chemistry and physics courses, we know that matter is made up of
atoms. Each atom contains protons and neutrons that reside in the atom’s nucleus and
electrons that orbit the nucleus. Electrons, which are of particular interest to us here, are
negatively charged particles and are attracted to the atom’s nucleus as the nucleus
contains the positively charged protons.
In a very general sense, an electric circuit may be thought of as a connection of elements
assembled to move charges along specified paths. How these charges (electrons) move
in a circuit, thus creating an electric current, depends on the components and connections
that comprise the circuit. Some elements allow electrons to flow easily, others hinder
electron flow to relative degrees, and yet others allow electrons to flow under only
specific conditions if at all. The extent to which a component will allow current flow is,
in large measure, related to the materials from which the component is constructed. Later
on in your studies (e.g. EE 409: Material Science) you will learn a great deal about
various classes of materials, for now we will take a very simplified look at three basic
types of materials: conductors, insulators and semiconductors. As examples, we will take
aluminum (conductor), sodium chloride (insulator) and silicon (semiconductor).
Electronic Materials
Conductors
Consider the metal aluminum (Al). You might be aware that Al is a good electrical
conductor, that is, it allows electrons to flow freely or to “conduct”. Why is this so? An
aluminum block (or wire) is made up of Al atoms that are held together by so-called
“metallic” bonding. Metallic bonds are characterized by the existence of a “sea of
electrons” that are able to flow from one atom to the next. Since these electrons are able
to move quite easily, Al may be used to efficiently transport electrons from one part of a
circuit to another. Examples of other materials that act in a similar fashion include
copper (Cu), gold (Au) and silver (Ag). Each of these materials are held together by
metallic bonding.
Insulators
Insulators act quite differently than conductors. Instead of allowing the free flow of
electrons, insulators tend to prohibit their flow. Simple table salt, sodium chloride,
(NaCl) is an interesting example of an insulator.
NaCl is held together by what is known as “ionic” bonding. Ions are charged atoms. An
isolated atom has the same number of protons (positive charge) and electrons (negative
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
charge) and thus has zero net charge. Thus, alone, sodium (Na) and chlorine (Cl) are
uncharged. When we bring Na and Cl together to create NaCl, each Na atom tend to give
up one of its eleven electrons (recall the periodic table) which the chlorine atoms kindly
accept. This creates positively charged sodium ions (Na+) as each loses an electron and
negatively charged chlorine atoms (Cl-) as each gains an electron. Due to attractive
forces between the positive and negative ions, NaCl forms through ionic bonding. In this
case, all of the electrons are tightly bound, there is no “sea of electrons” as in the case of
a conductor. Thus, NaCl does not conduct current effectively. Indeed, it takes a
significant amount of energy to free the electrons in NaCl before any appreciable amount
of current will flow. It is interesting to note that when salt is dissolved in water, the
sodium and chlorine ions are set free. Since ions are charged, they can create an electric
current!
Semiconductors
As the name suggests, semiconductors such a silicon, are not quite conductors. Instead,
they are insulating, but under the right conditions become reasonably good electric
conductors. Like the case of insulators, semiconductors are not characterized by a sea of
electrons. However, the energy required to free electrons from their bonds in a
semiconductor is much less than that required in an insulator. The energy to free
electrons may come in the form of heat or light for example, and thus semiconductors
may be used to detect changes in temperature and light intensity. The conductivity of
semiconductors may also be dramatically changed through a process known as “doping”.
Doping consists of adding foreign atoms to a previously pure material. For example,
phosphorus may be added to silicon to create free electrons without the need for an
increase in temperature or light intensity. Later in the course we will begin to learn about
“diodes” and “transistors”, both critically important electronic devices which may be
built from doped semiconductors. Diodes allow current to flow in only one direction and
transistors may be used to create efficient switches to control current flow and may be
used to increase the size (amplify) of a signal.
So there we have it, conductors, insulators and semiconductors. In electric circuits we
will use conductors to transport charge efficiently from one component to the next,
insulators to restrict current flow and to create components which store charge (more on
that later) and semiconductors to create devices which allow us to do seemingly magical
things with charge. Integrated circuits (ICs) including those used in your PC employ all
three types of materials.
Charge, Current and Voltage
Charge (denoted q or Q) is measured in units of Coulombs [C] in honor of the French
scientist, Charles Augustin de Coulomb. Coulomb was a pioneer in the field of
electricity and magnetism. Compared to a value of 1 C, the charge of an individual
electron is miniscule. Specifically, one electron has a charge of -1.602 x 10-19 C. That is,
it takes 6.242 x 1018 electrons to make 1 C of charge.
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
A few questions may arise as to motion of charges in an electric circuit.
• What is the rate at which charge moves in a circuit?
• How much energy must be spent to move the charge?
What is the rate at which charge moves in a circuit?
The rate at which electric charge moves through a given portion of a circuit is the current
through that portion. Current is denoted “I” for the French word intensité (intensity) and
has the units of Amperes [A], or Amps for short.
Therefore,
Current ≡ I ≡
∂q
C
→ →A
s
∂t
where ∂q denotes change in charge, ∂t denotes change in time, C indicates Coulombs and
s, seconds
Current has polarity. In circuit theory, current is thought of as motion of positive charge.
Thus, the direction of current flow is opposite that of electron flow. Why?
How much energy must be spent to move the charge?
It takes work to move a charge. From physics, we know that work is defined as the
product of the force and the distance. The relevant force in electric circuits is the
electromotive force. You will learn more about these concepts in subsequent courses
such as EE 206 and EE 207. For now, let’s simply remember that the energy required to
move a unit charge [+1 C] between two points in a circuit is the potential difference
(voltage) between these two points.
The unit of energy is the Joule [J] (after the British physicist James Joule) and the unit of
potential difference is the Volt (after the Italian physicist Alessandro Giuseppe Antonio
Anastasio Volta).
Potential difference ≡ Voltage ≡
Energy
J
→ → Volt [V]
Unit charge
C
Current can be produced in a circuit when a potential difference exists somewhere in the
circuit. While electrons are in constant motion even without a driving voltage, without
such a potential difference, the electron motion is random and has no identifiable
direction. Thus, current is the concerted motion of electrons and thus has a definite
direction.
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
Ohm’s Law and Resistance
The German physicist Georg Simon Ohm (1787-1854) is credited with formulating the
current-voltage relationship for a resistor. A resistor is a circuit element that can be used
to control current flow. He found that the voltage across the resistor is directly
proportional to the current through the resistor. The constant of proportionality is termed
the “resistance” with units of ohms [Ω]. That is,
Voltage across a resistor = (Current through the resistor) x (Resistance)
V = IR
The above equation can be rearranged to solve for the resistance of the resistor:
R=
V
I
A resistor “resists” the flow of current. Thus for a given voltage, less current will flow
through a resistor with a large resistance as opposed to a resistor with a small resistance.
How can we understand a resistor in light of our knowledge of conductors, insulators and
semiconductors?
Recall that good conductors allow current to flow quite freely. Since such materials
present little hindrance to current flow, they are characterized by very low values of
resistance (less than one ohm for small lengths of wire). On the other hand, insulators
allow almost no current to flow and thus are characterized by very high resistance values
(billions of ohms). For standard resistance values, we seek something in between these
two extremes.
Common resistors are made with materials that are rather poor conductors, but not so
poor as to be considered insulators. Carbon is a typical material used in the construction
of resistors. Below is a sketch of the cross section of a carbon resistor. In such a resistor,
a film of carbon is spiraled around a rod of insulating material (typically a ceramic). Low
resistance metal leads are made to the two ends of the carbon film and are used to
connect the resistor to other circuit elements. An insulting material is used to coat the
carbon film to isolate it from the environment. Various markings are placed on the
outside of the insulating coating to indicate the value of resistor.
Carbon film
Metal lead
Insulating
coating
Insulating
rod
Figure 1: Schematic diagram of a typical carbon resistor
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
The phenomenon of resistance (the restraining of charge flow) is due to collisions
between electrons and atoms in the resistive material. To increase resistance one may
either seek materials that exhibit higher resistivity (more collisions on average) or require
electrons to travel a longer distance through the resisitive material.
In integrated circuits, semiconductor materials are often used as resistors. The value of
such resistors can be controlled by both the size of the resistor and the amount of doping
in the semiconductor.
Power
Power (P) is defined as the rate at which energy changes. That is:
Power ≡
Change in Energy
J
→ → Watt [W]
Unit time
s
As indicated in the above formula, power is given in units of Watts (W). James Watt, in
whose honor we name the unit of power, was a Scottish engineer who designed an engine
to make steam power practical.
Let’s multiply voltage and current together and see what we get.
Voltage × Current: Volts×Amps →
J C
J
× = =W
C s
s
Indeed, power in an electrical circuit may be computed by multiplying the appropriate
current by the appropriate voltage. That is,
P = VI
Let’s proceed with an example.
Consider the following circuit and assume that the voltage source has a value of 5 volts
and the resistor, a value of 100 Ω.
Resistor
Voltage
Source
Assumed
direction of
current flow
+
-
Figure 2: A simple electric circuit
Notice that the assumed direction of current flow follows convention and is thus directed
from the voltage source’s positive terminal to its negative terminal. (Which direction do
the electrons flow?) Let’s make a couple of simple calculations.
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
The value of the current flowing in the resistor may be determined using Ohm’s law as
follows.
I=
V
5V
=
= 0.05 A = 50 mA
R 100 Ω
Notice that the current is given in both amps (A) and milliamps (mA), where 1 milliamp
equals 1/1000 of an amp. At the end of these lecture notes is a list of common prefixes
used to concisely write the vast array of numerical magnitudes that are commonly
encountered in engineering.
In our circuit, a current of 50 mA flows through the resistor (and through the wires and
voltage source; current flow requires a closed loop). Due to the polarity of the voltage
source, current flows in a clockwise fashion around the loop. If the polarity of the
voltage source is switched (negative on top, positive on bottom), 50 mA of current would
flow in a counterclockwise direction around the loop.
Let’s compute the power.
P = VI = ( 5 V )( 50 mA ) = 0.25 W
From our calculation we see that there is 0.25 W of “energy change” in our circuit. But
where does this power go? Recall that the phenomenon of resistance can be related to
collisions between moving electrons and (approximately) stationary atoms. Such
collisions produce heat. Since we assume that the wires which connect the resistor to the
battery are excellent conductors (and thus have low resistance), relatively few heat
producing collisions occur in the wire. On the other hand, the resistor is characterized by
significantly more collisions when current flows. Thus, most of the power is “dissipated”
in the resistor and becomes heat. If we are not careful in our circuit design, the resistors
(or other components) in the circuit may be required to dissipate too much power and will
overheat to the point of failure.
For both safety and reliability, clearly we must consider power dissipation in our circuit
and either seek to reduce power dissipation or choose components that are safely able to
dissipate the anticipated amount of power. For example, resistors are not only
characterized by a resistance value but also by their power handling capability. Common
resistor power ratings are ¼ W, ½ W, 1 W, 2 W, etc. Larger diameter resistors tend to
have higher power ratings. While from example above, we might think that a ¼ W
resistor would be the appropriate choice, a good rule of thumb is to add a safety factor of
about two thus making a ½ W resistor a more prudent choice. In choosing a higher
power rated resistor, one ensures that the resistor can do the job while maintaining a
reasonable temperature and allows for the event in which there might be some unintended
increase in voltage and current in the circuit. The disadvantages in choosing a higher
power rated resistor might very well be cost and size. As engineers, we are often called
upon to reduce the size and cost of the circuits we develop and thus we are constantly
faced with cost versus performance decisions.
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
Resistors in Series and Parallel Combination
Often we use combinations of resistors in a circuit. Such combinations may be used to
attain values of resistances that are not commonly available in a single component or to
control voltage and current levels through simple circuits called “voltage dividers” and
“current dividers” respectively. The combination of resistors can come in the form of a
series connection or a parallel connection.
Series connection of resistors
Below is a circuit in which two resistors are connected to a voltage source using a series
connection between the resistors.
Resistor 1
Voltage
Source
+
Resistor 2
-
Assumed
direction
of current flow
Figure 3: A circuit with a series connection of two resistors.
Let’s try to use Ohm’s law to understand how a series combination of resistors behaves.
Take the voltage of the voltage source to be “V” and the resistance of resistor 1 to be
“R1” and that of resistor 2 “R2”. Recall Ohm’s law:
V = IR
In words this equation states: “The voltage drop across a resistor is equal to the product
of the current flowing through the resistance and the resistance of the resistor.”
We have the two resistances R1 and R2, but what about the voltage and current values?
As in our previous example, and in accord with convention, the current is assumed to
flow from the positive terminal of the voltage source, through the circuit, and then return
to the negative terminal. Is the current through R1 the same as that through R2 even if
they have different resistance values? The answer is yes, regardless of the resistance
values, the amount of current flowing through resistors connected in series is in fact the
same. Can we make an argument as to why this is the case?
Recall that current is the measure of the rate at which charge flows in a circuit. If the rate
of charge flow were different in the two resistors connected in series, there would be a
net buildup of charge in some area of the circuit (i.e. more electrons would flow per unit
time through the lower valued resistor and thus build up at one end of the larger valued
resistor). The build up of charge would tend to counteract the current flow as like
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
charges repel one another. Since under steady state conditions there can be no such
charge build up, the current flow through series connected resistors must be the same.
Now what about voltage drops? Is the voltage drop across R1 the same as that across R2?
Again, let’s consult Ohm’s law.
Let V1 and V2 be the voltage drops across R1 and R2 respectively with a similar
designation for the currents.
V1 = I1R1 and V2 = I 2 R 2
Recall that the current through R1 and R2 is the same as the resistors are connected in
series and thus I1 = I2 ≡ I. Therefore,
V1 = IR1 and V2 = IR 2
Clearly, the only way in which the voltage drops across the two resistors will be the same
is if R1 = R2! In the next set of lecture notes we will take a look at two laws which
explain current flow in circuits. These laws will allow us to easily derive the net
resistance in series and parallel connections. For now, let’s simply state the result and
leave the proof for latter.
The net resistance that is seen in the circuit of Figure 3 is:
R net,series = R1 + R 2
Since the current is the same through each of the resistors in a series connection (the
argument can be extended to any number of series connected resistors), the net effect of
the resistors is cumulative. So, if we have N resistors in series, they present a resistance
of:
N
R net,series = R1 + R 2 + R 3 + " + R N = ∑ R i
i=1
where the capital Greek letter sigma (Σ) indicates a summation.
Parallel connection of resistors
Below is a circuit in which two resistors are connected to a voltage source using a parallel
connection between the resistors.
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
I
Voltage
Source
A
I1
+
I2
R2
R1
-
B
I
Figure 4: A circuit with a parallel combination of resistors
In the figure, a current ‘I’ emanates from the positive terminal of the voltage source and
enters the negative terminal of the voltage source. The current through R1 is labeled I1
and that through R2 labeled I2. A distinction to be made between the series and parallel
cases is that in the case of a series combination, the resistors are connected at only one
end; in the parallel case they are connected at both ends (that is, at points A and B in
Figure 4).
Can we make the same claim as that in the case of the series connected resistors that the
current through the parallel resistors is the same? As mentioned above, a current ‘I’
emanates from the voltage source. Unlike in the case of the series connection, the current
‘I’ can head either through R1 or R2 – it has two paths. There is nothing that seems to
suggest that these must be equal and so for the time being we can say with generality that
the current ‘I’ splits into ‘I1’ and ‘I2’. Let’s redraw the figure as follows.
A
I
I1
+
Voltage
Source
R1
I
I2
R2
B
Figure 5: A redrawn version of Figure 4; the electrical behavior is unchanged
While the circuits of Figures 4 and 5 may seem different, their electrical performance is
identical. Resistors R1 and R2 are still connected at both ends (points A and B) and the
voltage source is connected in the same fashion as in Figure 4. However, it is perhaps
somewhat more clear from Figure 5 that the current ‘I’ which emanates from the voltage
source splits into I1 and I2. The currents that flow through R1 and R2 (I1 and I2
respectively) enter point B, recombine and then continue to the negative terminal of the
voltage source. It stands to reason that if we agree that there can be no charge build up at
any point in the circuit, that
I = I1 + I 2
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
We see that we cannot in general state that I1 = I2, only that the sum of these two currents
equals I. Resistors R1 and R2 do share something in common however; since they share
the same nodes (A and B), they share the same voltage drop.
The voltage drop across R1 is the voltage level at point A minus the voltage level at point
B. The same is true for the voltage drop across R2. That is,
Voltage Drop Across R1 = Voltage Drop Across R 2 = VA - VB = V
where ‘V’ is the voltage drop across the voltage source as it is connected between points
A and B as well. Keeping these facts in mind, under what condition in Figure 4 (and thus
Figure 5) would I1 = I2?
Saving the proof for later, it can be shown that the circuit of Figure 4 presents an
equivalent resistance to the voltage source of:
R net,parallel
 1
1 
=
+

 R1 R 2 
−1
If we have N resistors in parallel, the net resistance is given by:
−1
R net,parallel
 1
1
1
1 
 N 1 
=
+
+
+"+
=

∑ R 
RN 
 i=1 i 
 R1 R 2 R 3
We will seek to prove the series and parallel formulas next lecture.
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EE 101 – Introduction To Electrical Fundamentals, Fall 2004
Commonly Used Unit Prefixes
In our work as engineers, we will often encounter both very large numbers and very small
numbers. For example, we might be asked to use a resistor with a value of 2500000 Ω or
we might calculate a current in a circuit and find it to be 0.0000000067 A. It is tedious to
write out such numbers (and quite easy to miss a zero or two). So instead, we use
prefixes to indicate powers of ten. For example, the above resistance could be given as
2.5 MΩ and the current as 6.7 nA. Below is a list of commonly used prefixes. Study
them, commit them to memory and practice using them.
1012
109
106
103
10-3
10-6
10-9
10-12
→ tera (T)
→ giga (G)
→ mega (M)
→ kilo (k)
→ milli (m)
→ micro (µ)
→ nano (n)
→ pico (p)
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