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) Voltage source - delivers energy to the circuit in the form of

voltage, units of voltage = volt (V).

(2) Current source - delivers energy to the circuit in the form of

current, units of current = amp (A).

Passive elements

(1) Resistor [energy dissipation device] - dissipates energy in the

form of heat, units of resistance = ohms (S).(2) Capacitor [energy storage device] - stores energy in an electric

field, units of capacitance = farads (F).

(3) 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

ean electron (designated as q ) 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) .

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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

iothree parameters: the current amplitude (i ), current phase angle (2 ), andthe 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.

iThe position of the cosine waveform is dictated by the phase angle 2 . Theposition in time of the peak associated with a zero-valued cosine argument

max(located at t = t ) is given by

Defining the phase angle in radians (changing 60 to B/3 radians) giveso

Note that a positive-valued phase angle shifts the cosine function to an

maxearlier time location (t < 0) while a negative-valued phase angle will shift

maxthe cosine function to a later time (t > 0).

To determine the charge q(t) based on a known current i(t), we may

ointegrate both sides of the current equation over a time interval of (t , t)

yielding

owhere t 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

vocharacterized by the voltage amplitude (v ), voltage phase angle (2 ), andthe 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

1 2specified time interval. Integrating p(t) over the time interval of (t , t )

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) Vi(t) = 10 mA

(a.) determine the element power as a function

1of time, (b.) the energy transferred between t =

20 and t = 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.

Kirchoffs 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 Kirchoffs voltage law (KVL) and Kirchoffs

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.

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Kirchoffs Voltage Law (KVL) - the algebraic sum of voltages around a

closed path in a circuit is zero.

Kirchoffs 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 and are connected in series (they carry the same

4 5current) such that i = i .

Elements and are connected in parallel (they have the same

2 3voltage across them) such that v = v .

The series combination of elements and is connected in parallel

4 5 6with element such that v + v = v .

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