ECE 3183
Electrical Engineering Systems
Course Objectives
Students who complete ECE 3183 Electrical Engineering Systems
will
(1) possess a working knowledge of the fundamental
concepts of electrical engineering including circuit
components, basic DC and AC circuit analysis techniques,
and electric motors.
(2) be prepared to answer questions on the Fundamentals of
Engineering (FE) Examination related to these electrical
engineering concepts.
Disciplines Within Electrical Engineering
Communications
Power Systems
Control Systems
Signal Processing
Electromagnetics
Digital Systems
Solid State Electronics
The topics covered in ECE 3183 (and ECE 3283 Electronics) are some of
the foundational principles used in the various disciplines of electrical
engineering.
Circuit Elements
The basic elements (or components) of an electrical circuit are shown
in the example circuit diagram shown below. The following elements are
shown in the circuit diagram:
Source elements
(1)
(2)
Voltage source - delivers energy to the circuit in the form of
voltage, units of voltage = volt (V).
Current source - delivers energy to the circuit in the form of
current, units of current = amp (A).
Passive elements
(1)
(2)
(3)
Resistor [energy dissipation device] - dissipates energy in the
form of heat, units of resistance = ohms (S).
Capacitor [energy storage device] - stores energy in an electric
field, units of capacitance = farads (F).
Inductor [energy storage device] - stores energy in a magnetic
field, units of inductance = henries (H).
Circuit Parameters - Current and Voltage
The operation of a circuit element is most often defined in terms of
the circuit parameters of current (i) and voltage (v). Current and voltage
are defined according to the behavior of the electric charge (q) in the
element. The SI unit of electric charge is the coulomb (C). The charge on
an electron (designated as q e) is
For a given circuit element, the current and voltage are defined as:
Current (i) [through the element] - the time rate of flow of (positive)
charge through the element.
Voltage (v) [across the element] - the change in energy per (positive)
charge as the charge moves through the element.
Note the importance of defining the direction of the current and the polarity
of the voltage. The charges that constitute the current are known as the
carriers. Carriers can be charged particles of either sign (positive or
negative) .
If the rate of charge flow is constant, a steady current or direct
current (DC) is the result. If the rate of charge flow varies with time, a
time-varying current is the result.
A special case of the time-varying current is the alternating current (AC).
An alternating current contains periodic oscillations and is characterized by
three parameters: the current amplitude (io), current phase angle (2i), and
the radian frequency (T). The general equation for an AC current is
The radian frequency (T) in rad/s is related to the frequency in Hz (f ) by
while the period (T) of the current is simply the inverse of the frequency in
Hz.
Example (Alternating current)
Plot the following current as a function of time: a 60 Hz alternating
current characterized by an amplitude of 4A and a phase angle of 60 o.
This current oscillates between a maximum of +4A and a minimum of !4A.
The period of the current is T = 1/60 = 16.7 ms.
The position of the cosine waveform is dictated by the phase angle 2i. The
position in time of the peak associated with a zero-valued cosine argument
(located at t = tmax) is given by
Defining the phase angle in radians (changing 60 o to B/3 radians) gives
Note that a positive-valued phase angle shifts the cosine function to an
earlier time location (tmax < 0) while a negative-valued phase angle will shift
the cosine function to a later time (tmax > 0).
To determine the charge q(t) based on a known current i(t), we may
integrate both sides of the current equation over a time interval of (to, t)
yielding
where to is an initial time at which the charge is known. Solving this
equation for the charge q(t) gives
The voltage across a circuit element may be a DC voltage (constant)
or a time-varying voltage, such as an AC voltage. An AC voltage is
defined using the same form as the AC current. An AC voltage is
characterized by the voltage amplitude (vo), voltage phase angle (2v), and
the radian frequency (T). The general equation for an AC voltage
The unit of voltage (energy/charge) is the volt and is defined by
As shown previously, the polarity of the voltage in relation to the current
is critical in differentiating between a source element and a passive
element. Anytime we solve a problem and find a negative-valued voltage,
the polarity of the actual voltage is in the opposite direction.
Circuit Parameters - Power and Energy
Other quantities of interest in the operation of a circuit element are
power (p) and energy (w). The power and energy associated with a circuit
element can be determined directly from the element current i and voltage
v. From the definitions of voltage and current in terms of charge, we have
Taking the product of voltage and current yields energy/time or power.
Thus, the power supplied by a source element or absorbed by a passive
element is defined as
where the corresponding units on power are
The total amount of energy w supplied by a source element or absorbed by
a passive element can be found by integrating the power function over a
specified time interval. Integrating p(t) over the time interval of (t1, t2)
yields
The polarity of the element voltage relative to the direction of the element
current for source elements and passive elements are drawn below using the
same voltage polarity for both elements.
If the source element and the passive element are connected together as
shown below, the source element will supply energy to the passive element,
where the energy will be absorbed. We adopt what is called a passive sign
convention to differentiate between supplied energy and absorbed energy.
In the passive sign convention, positive power indicates that the element is
absorbing energy while negative power indicates that the element is
supplying energy. Using the passive sign convention, the element current
is defined as
positive if the current enters the “+” terminal of the element
negative if the current exits the “+” terminal of the element
Example (Power and energy)
If the voltage and current for the given circuit
element are
v(t) = (20 !2t) V
i(t) = 10 mA
(a.) determine the element power as a function
of time, (b.) the energy transferred between t1 =
0 and t2 = 10 s, and (c.) whether this net energy
is supplied or absorbed by the element.
(a.) current exits the “+” terminal of the element (current is
negative)
(b.)
( c.) Since w < 0, energy is being supplied by the element.
Kirchoff’s Current and Voltage Laws
When elements are connected to form an electric circuit, the resulting
circuit voltages and currents are governed by two fundamental circuit
analysis laws known as Kirchoff’s voltage law (KVL) and Kirchoff’s
current law (KCL). These laws are based on the conservation of charge
and energy. In order to apply KVL and KCL, we must define two
quantities associated with the circuit topology. These quantities are the
circuit node and the circuit loop.
Node - a point at which two or more circuit elements are connected.
Loop - any closed path where no node is encountered more than once.
For example, the circuit shown below consists of 6 elements with a total of
4 nodes (labeled a, b, c, and d) and 6 possible loops.
Kirchoff’s Voltage Law (KVL) - the algebraic sum of voltages around a
closed path in a circuit is zero.
Kirchoff’s Current Law (KCL) - the sum of currents entering a node is
equal to the sum of currents leaving the node.
To apply KVL to the six loops defined for the given circuit, we
assume a loop current flowing in the direction specified for the loop, and
sum the voltage rises and drops in the direction of the loop current.
Note that the voltage drops (the negative terms) in the preceding equations
can be moved to the opposite side of the equals sign (making them
positive). In this way, we may interpret KVL as “the sum of the voltage
rises must equal the sum of the voltage drops around a closed loop”.
To apply KCL to the four nodes defined in the given circuit, we
equate the sum of the incoming currents to the sum of the outgoing currents
at each node.
Series and Parallel Connections
The number of unknowns (voltages or currents) in a given circuit can
be reduced by recognizing when elements are connected in series or
parallel.
Elements connected in series carry the same current.
Elements connected in parallel have the same voltage across them.
Examples of series and parallel connected elements are contained in the
previously considered circuit.
Elements å and æ are connected in series (they carry the same
current) such that i4 = i5.
Elements ã and ä are connected in parallel (they have the same
voltage across them) such that v2 = v3.
The series combination of elements å and æ is connected in parallel
with element ç such that v4 + v5 = v6.