# Ch. 27 - Electrical Circuits and their analysis

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7 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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Electrical Circuits and their analysis

Schematic Diagrams

Circuits are a collection of devices connected together in such a way that the flow of
electrical charges is controlled in such a way to achieve a specific result. This might be a
simple as a com
mon flashlight or as complicated as a modern computer.

Circuits are usually designed with the aide of diagrams called schematic diagrams, in this
case electrical schematic diagrams. They are diagrams of how individual components are
connected together, t
he components being represented by symbols agreed on by
convention. Schematic diagrams are used for many different kinds of systems and are not
unique to electrical systems. Many none electrical, even none engineering, applications are
described efficien
tly by symbolic diagrams. They might include heating and air conditioning
systems to sports diagrams of “plays” or tactics.
The central feature is the symbolic
representations of elements and a graphical presentation that is relatively easy to
understand
.
It transcends language barriers, and usually facilitates analysis.

In electrical schematics, each element is represented by a drawn symbol that in some way is
almost intuitively obvious as to the element it represents. We examine a few of these
elemen
ts in a moment.

In electrical schematics, which we will hence forth shorten to a schematic, all of the “action” i
s
considered to take place with
in the element represented. Connections are shown as lines
between elements and are assumed to be perfect cond
uctors. As perfect conductors, it is
therefore assumed that anything connected to on of these conductors is at the same
potential.

The schematic is designed to represent how the circuit is connected together with the
arrangement of circuit elements shown

in a way that facilitates an understanding how the
circuit operates. It is not meant to be a representation of a physical arrangement of the
actual devices. Certainly a mechanical engineering schematic diagram of a fluid processing
plant would usually b
e much closer to the physical reality.

In electrical designs, schematics are usually only one representation of the electrical system.
Other diagrams would show how the actual parts are positioned to satisfy other mechanical
and electrical constrains. D
issipation of heat and packaging considerations in modern
electronic devices are among the major considerations of their design.

Conductors and the
F
low of
C
urrent

(a small diversion)

When we think of electrical current we usually think of charge carrier
s moving at incredible
speeds along or within conductors. Common sense seems to demonstrate this to us all the
time. Turn on a light switch and the light across the room goes on immediately. Telephones
or telegraphs connect
ed

by very long lengths of wir
es seem to also demonstrate this very
high speed for the carriers. But let’s take a closer look at what we call
current
in a normal
conductor.

If we examine a metal conductor, it like most materials, is composed of an unimaginable
number of atoms in any
substantial amount of the material, say a copper wire. The atoms
themselves are relatively large, massive, and more or less fixed in position in the material.
However in metals, particularly something like copper, the outer most electrons of these
atoms
are free to move about fairly freely in what is commonly called a conduction band.
This conduction band is a broad range of available energies for the electrons resulting in the
electrons acting without any association with their parent atoms.

These “con
duction” electrons travel at very high speeds
6

m/s
. The problem is
that they do not go very far before they hit something, either another electron or more
probably a copper atom. In the process, they loose all memory of what they were doi
ng
before the collision and take off in a new, totally random direction. The freedom of an
electron to free itself from its parent atom and become part of a larger group of conduction
electrons is really a function of the particular material and in large
measure determines the
ease of conduction of current through different metal conductors. Silver is the best
conductor, then copper, gold and aluminum.

Silver is too reactive to use on PC boards and too expensive to use in wires. Aluminum wires
were us
e for home wiring when copper became too expensive
.

Unfortunately,

habit of starting fires in homes as the aluminum wires and brass screws of electrical connectors
caused a reaction that got
very
hot when current passed across the metal jun
ction
.

As a result,
aluminum wire is absolutely forbidden for use in homes or buildings. Copper wire is now the
only choice and it is a good one. Gold is used on PC boards almost universally since it is quite
inert and an excellent conductor.

Let’s get

back to those conduction electrons. If no net electric field exists, then this extremely
large number of conduction electrons will be moving very fast between collisions with atoms
or other electrons but on average, the net movement of electrons is zero.

If an electric field
is present, perhaps caused by connecting the wire between two points of different potential,
then there will be a net drift of carriers in a direction from lower electrical potential to higher
electrical potential. (Recall that elec
trons are negatively charged and therefore the force they
experience is opposite to the field direction that is defined by positive charge)

Now consider
current.
Current is defined as the rate of charge passing through a cross
-
section. This is defined a
s the amount of charge per unit time that passes. Thus one
AMPERE,
the unit of current
,

is defined as one coulomb/sec. Recalling that the electronic
charge is approximately 1.6 x 10
-
19

C,
and then

one ampere represents a flow of about 6.28 x
10
18

electro
ns/sec. This may seem impressive but considering that there are about 10
24

conduction electrons per mole of copper, the average drift speed of the sea of conduction
electrons is quite small, about a fraction of a mm/s.

When an external field is applied,
it is the electric field disturbance that travels down (or more
properly on) the wire at near light speeds that gets all the electrons drifting

more or less
together
.

Historically, when people realized that charge came in two flavors, positive and negativ
e,
they assumed that the carriers were positive and therefore moved form a higher potential to a
lower potential. This so called
conventional current

became deeply embedded in the science
of “electricity” and only when it was discovered that electrons wer
e the charge carriers was it
realized what the true nature of the conduction process was. As it turns out of course it really
doesn’t much matter what the flavor the actual carriers are since positive charge moving high
potential to low is effectively the

same as negative charge going low to high as far as circuit
analysis goes.

Thus conventional current remains with us, i.e. positive carriers moving from
high to low potential. With semiconductors, that can have both kinds of carriers, it really
becomes
a moot point.

Now that we have allowed conductors to be less than perfect, we need to introduce the
concepts of resistance or conductivity, which ever way you want to look at it.

Since copper wires or gold films deposited on a PC board or the surface of
a semiconductor,
are so much better conductors that almost any other material we will assume them perfect
conductors and ascribe conductivity related issues entirely to other components like
RESISTORS.

Some Components Used On Schematic Diagrams

(The symbo
ls will be drawn on the board. You might want to annotate these notes
with drawn images of the component symbols.

1.

Resistor

A resistor is a device that has a substantially lower conductivity then the interconnecting
conductors and therefore becomes a
device that will have an easily discernible effect on the
current flowing. For a given potential difference across a resistor, a certain current will flow.
Thus we can define the resistance of this object as the ratio of the potential difference divided
by the current flowing. Departing from the scientific notation of potential difference as “V”, we
DEFINE the resistance “R” as

Where E is the potential difference across the device. (Not to be confused with the more
common scienti
fic notation of E being the electric field)

The current passing through it is
denoted by “
I”

and I = E/R.

This equation for
I

is merely a statement of the fact that the current through a devise is
dependent upon both R and E. In a plot of I vs. E,
I/
R is

merely the slope of the curve

at
some particular point
. The above relationship of R, E, and I
IS NOT OHM’S LAW!!!!!
The
functional dependence of the current on the voltage is not necessarily a constant,
called R, the resistance
.
R is merely the ratio o
f the voltage to the current and is not
necessarily a constant
.

OHM’S LAW

is an assertion that the current through a device is always directly
proportional to the potential difference applied to the device. This assertion is
correct only in certain situa
tions, e.g.
a metal conductor at
constant temperature.
Devices that in fact do obey Ohm’s law, over some definable set of physical
conditions, are said to be “ohmic”. Non linear devices, like diodes whose
resistance is dramatically different depending up
on the polarity of the applied
vo
ltage, are none ohmic.

2.

Capacitor

A capacitor is physical configuration of electrodes and insulating material between the
conductors that stores electrical energy by allowing a charge separation to exist. The
symbol is su
ggestive of a parallel plate capacitor. Like other elements, it is assumed
that the metal conducting electrodes have no significant resistance.

3.

Switches

Switches are used to control and direct the flow of current and are schematically
represented by a m
oveable conductor(s) and portions of fixed conductors and contact
points.

4.

Battery

A battery, or the symbol used for it, is used for a source of charge separated by a
constant potential difference. The symbol includes an indication of the polarity of

the
electrodes. By convention the longer, thinner line is positive and the short fat line
negative. The battery is often considered “perfect”, that is it has a negligible internal
resistance. If, however, the internal resistance of the battery is signi
ficant, then it is
shown as a resistor, usually labeled R
i
, in series with the battery. In fact the symbol
for the battery in series with the internal resistor is often referred to as its schematic
diagram.

When a battery, or any source of electrical pote
ntial, is used it is important to
understand the effect of a non
-
negligible internal resistance. We will shortly,
use a schematic diagram of a battery, with its internal resistance, connected to a
“load resistance” that represents a use for the electrical

current, as a way to
understand how a source of potential difference delivers useful energy.

Batteries, or whatever the source, supply energy by supplying charges at some
potential difference determined by the chemistry or some mechanical means of the
sou
rce. Since the electrical energy of each charge carrier is q V, then the rate at
which energy is delivered, the power, is flow rate of the charge, x the potential at
which they are delivered.

By using “Ohm’s Law”, and “Engineering
notation for voltage

Looking at these relations for how power is used up, we note that a “user” of power is
a resistor. A resistor receives charge at one potential, and allows it to pass through
and leave at a lower potential. Thu
s charges passing through are losing potential
energy. If that is the case, where does the lost energy go? It goes into heating the
resistor. That heat may be useful (your electric stove) or just wasted. Following the
argument above for power, we now s
ee that the same expressions apply for power
USED or DISSAPATED by a resistor. By looking at the relation P = I
2
R, you should
see why log distance electrical power lines are operated at very voltages.

5.

A look at the effect of “Internal Resistance” in a ba
ttery or electrical power
source. Insert you schematic diagram for a battery connected to an external
load resistor. (A perfect source, sometimes called an emf, for “electromotive
force, is in series with the internal resistance R
i
, the two ends connecte
d to the
terminals of the battery. If we were to determine the potential difference across
the terminals, it would be just the emf. The reason is that no current is flowing
and therefore there is no voltage drop across the internal resistance. This
“ope
n
-
circuit” voltage is that of the perfect emf source. If we now attach a load
resistor to the terminals, current will now begin to flow around the closed loop of
the three elements in series. With the current now flowing through the internal
resistor, R
L
, a voltage drop occurs equal to I
R
i
.

If we now measure the voltage
across the terminals it would be emf

I
R
i
.

This is the so called “terminal
voltage”. Notice that the power delivered to the load resistor R
L
is now the
“terminal voltage” x the curre
nt. Since the terminal voltage changes with R
L
, it
is instructive to see what the value of R
L

should be to maximize the power
delivered to the load, after all that’s where we usually want it. It should be
obvious that for R
L

equal to either zero or a ver
y large value, the power
delivered to the load will be either zero or something diminishing small. This
leads us to surmise that there must be a value somewhere between zero and a
very large value where the power delivered is maximized. A little calculus

will
show you that the maximum power delivered to the load is when load resistor is
equal to the internal resistance. This of course means that the total energy
supplied is dissipated equally in the internal resistance and the load resistance.
That may
or may not be good for other reasons.

Determining the currents and or v
oltages for each element of a circuit

For the moment, we
will deal

with “direct current”, DC for short. In DC circuits, the voltage
sources are time independent, i.e. they provide a

constant, unidirectional potential difference
across their terminals
.

There are three basic equations that are used to analysis circuits. One is the relationship
between R, E, and I, and two conservation principles. The two conservation principles are
conservation of charge and the conservation of energy. The two conservation principles are
known as
“Kirchhoff’s rul
es

.

1.

The Junction Rule:

The Net flow of current into a junction is zero. That is to say that the current flowing into
any junction must b
e equal in magnitude to the current flowing out. This is nothing more
that the fact that charge cannot be created or destroyed.

2.

The Loop Rule

The sum of the potential changes around ANY closed path is zero. This of course is merely
a restatement of the c
onservation of energy. Equivalent to moving up and down a mountain
where your total change in gravitational potential energy is zero as long as you return to the
same height you started at.

Other tools we use to analyze circuits are the expressions for ce
rtain
like

elements
connected in a series or parallel fashion.

The series and parallel equivalent values for
capacitors and resistors I will assume you have all in some detail in PS 16.

The following general procedure is outlined as a guide to systematic
ally applying
Kirchhoff’s Rules.

1.

Sketch a circuit diagram and label all known voltages, currents, and
resistances. Show + and

signs on potentials.

2.

Indicate a current direction in each branch of the circuit. If the direction is not
known, choose a direc
tion. A negative current solution will indicate that the
current is flowing in the opposite direction to the direction assumed.

3.

Assign symbols to all unknown currents, voltages, and resistances.

4.

Apply Kirchhoff’s voltage law to circuit loops and Kirchhoff
’s current law at
junctions to obtain as many independent equations as there are unknowns in
the problem.

5.

When applying the loop law, i.e. writing the equation for the voltage rises and
falls around the loop, the potential across a resistor drops from the
+ current
entry end through the resistor to the

current exit end. Conventional current
flows from the high potential end to the low potential end. Label the polarity of
the ends of the resistors according to your assumed current flow. Therefore if
you

are going around a loop through the resistor from + end to

end, then you
will have a voltage drop and if you go the other way will have a voltage rise
from the

end to the + end. You must be consistent. The sum of all the
voltage changes around the l