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Electrical and Computer Engineering

ECE 140

Michael Ibrahim

Fall

Description

ECE 140 - Linear Circuits
Kevin Carruthers
Fall 2012
Basic Components and Electric Circuits
Units and Scales
The international standard is SI units. Under this system, electric current is measured in
Amperes (A), work and energy is measured in Joules (J), and power, or the rate in which
work is done, is measured in Watts (W).
▯18
In increasing order, from 10 and ascending in factors of three, the pre▯xes we use are
atto, femto, pico, nano, micro, milli, kilo, Mega, Giga, and tera.
Charge, Current, Voltage, and Power
Charge
The basic unit of positive charge is the proton. The unit of negative charge is the electron.
We can neither create nor destroy charges.
Electric current is de▯ned as charge in motion, and follows the direction of the
ow of
positive charge (opposite the direction of electron
ow). The fundamental unit of charge is
the Coulomb (C).
▯19
A proton or electron has a charge of ▯1:602 ▯ 10
Current
Moving charges create an electrical current. In moving charges from one place to another,
we may also transfer energy. By changing the current (with respect to time), we can transfer
information. Current, then, is the rate at which the charges are moving past a given reference
point in a speci▯c direction.
1 The charge transfered between times t an0 t can1be expressed as
Z Z
1 t1
dq(▯) = i(▯) d▯
0 t0
and the total charge transfered is given by
Z t1
q(t1) = q(t0) + i(▯) d▯
t0
Voltage
Let a general 2-terminal circuit element have two terminals (ie. resistors, inductors, batter-
ies...). There are thus two paths by which the current may enter or leave the element. If
you want to push charges through a circuit element, you have to expend some energy. The
energy used to push charge through an element is de▯ned as the voltage between the two
terminals, or the potential di▯erence.
In other words, the voltage across a terminal pair is a measure of the work required to move
charge through that element.
Voltage is measured in Volts (V), where
1J
1V =
1C
A voltage can exist between a pair of terminals whether or not a current is
owing. According
to the Conservation of Energy, the energy expended in forcing charge through an element
must appear elsewhere (ie. transformed into heat energy).
Energy could be supplied to an element or by an element.
Power
Power is de▯ned as the rate at which work is done or energy is expended. It is measured in
Watts (W), where
1J
1W =
1s
If one Joule of energy is expended in transfering one Coulomb of energy through a device in
1J 1J 1C
one second, then we have 1W = 1s= 1C ▯ 1s
As such, we have
P = V I
If we have current entering the positive terminal of an element (ie. the terminal with a
larger voltage), we follow the passive sign convention. As such, the power absorbed by
the element is vi and the power generated or supplied by the element is ▯vi.
2 Voltage and Current Sources
The mathematical models used for circuit analysis are only approximations. They depend
on the relation between voltage across their terminals and the current going through them.
Relationship Between v and i Element
v / di Resistor
vZ/ dt Inductor
v / i(t) dt Capacitor
v 6/ i Independant Voltage Source
i 6/ v Independant Current Source
v or i / v or i Dependant Source
0 0
Independant Voltage Sources
The voltage of an independant voltage source is completely independant of the current.
It is ideal because it does not exactly represent any real physical device, but it is a reasonable
approximation of some. For example, household electrical outlets can be approximated as
independant voltage sources providing 115 2cos(120▯▯t) volts. This approximation is valid
for currents less than 20 A.
If the terminal voltage is constant, we have a DC voltage source. Otherwise (ie. it provides
a sinusoidal output) our source is AC.
It is denoted by a circular shape with plus and minus symbols denoting the di▯erence in
voltage. An optional tilde denotes AC voltage.
Independant Current Sources
The current of an independant current source is completely independant of the voltage
across the source. If it provides constant current, we have a DC source, otherwise it is an
AC source.
Voltage across a current source is not known, it depends on the circuit connected to it.
It is denoted by a circular shape with an arrow in the direction of current
ow. An optional
tilde denotes AC current.
Dependant Sources, aka Controlled Sources
A dependant source is one where the source quantity of either voltage or current is de-
termined by a voltage or current elsewhere in the circuit. They usually appear in equivalent
electrical models for devices such as operational ampli▯ers or transistors.
3 It is denoted by a diamond shape.
Networks and Circuits
An electrical network is the interconnection of two or more simple circuit elements. An
electrical circuit is a network which contains at least one closed path. As such, every
circuit is a network, but not all networks are circuits.
An active network is one which contains at least one active element (ie. independant
source). Alternatively, we have passive networks.
Resistance
Ohm’s Law
Ohm’s Law describes the relationship between th voltage across a resistor and the current
passing through it as
v = iR
1V
where resistance is measured in Ohms (
) and 1
= 1A
Note that a linear resistor is an idealized circuit element. Actual resistors only act like linear
resistors within certain ranges of current, voltage, or power, and also depend on temperature
and other environmental factors.
Power Absorbtion
Assuming we are following the passive sign convention, we have
v 2
P = vi = i R =
R
is the power absorbed by a resistor. Note that resistors dissipate energy in the form of
heat and/or light as they cannot store or generate it.
For a complete circuit, the absorbed power is equal to the generated power, as per conser-
vation of energy.
Resistance of Wires
Each material has a property called resistivity (▯) which is the measure of how "easily"
electrons can travel through that material. The units of resistivity are
▯ m, thus we have
l
R = ▯ A
Usually, the resistance of wires can be approximated as 0
4 Conductance
The conductance is the inverse of the resistance, or
G = 1
R
Short or Open Circuits
Short circuits are ones where R = 0
thus V = 0V for any i. Open circuits are ones
where R = 1
thus i = 0A for any v. We will assume wires to be a perfect short circuit.
Voltage and Current Laws
Nodes, Paths, Loops, and Branches
A node is a point at which two or more elements in a circuit have a common connection.
Since we assume all wires are perfectly conducting, any wire attached to a node is considered
part of a node.
Note that each element has a node at each end.
A path is a route through a circuit where no node is encountered more than once. If we end
at the same node as we started at, we have a path. A branch is a single path in a network
composed of one simple element and the nodes at each end of that element.
Kircho▯’s Current Law (KCL)
Theorem: The algebraic sum of all currents entering any node is zero.
KCL is based on the principle of conservation of charge. Basically, charge can not accumulate
at a node, so any charge entering a node must leave it. Consider an intersection: the number
of cars entering an intersection is exactly equal to the number of cars leaving that intersection.
KCL can also be written as "the sum of the currents entering a node is equal to the sum of
currents leaving a node".
Kircho▯’s Voltage Law (KVL)
Theorem: The algebraic sum of the voltage around any closed path is zero.
This is based on the fact that the enrgy required to move a charge from point A to point B
must have the same value independant of the path between points.
5 Series and Parallel Connections
Components in Series
Circuit components that carry the same current are said to be in series. Note that they
must have the same cur

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