Fundamentals of Electric Circuits: Sinusoidal Sources and The Concept of Phasor in Circuit Analysis

EE2010 Fundamentals of Electric
Circuits
Lecture 13
Sinusoidal sources and the concept of
phasor in circuit analysis
Sinusoidal waveform
• The time-varying voltage that is commercially
available in large quantities and is commonly
called the ac voltage (ac are an abbreviation
for alternating current)
• The term alternating indicates only that the
waveform alternates between two prescribed
levels in a set time sequence
Sinusoidal waveform
Definition:
Waveform: The path traced by a quantity, such as the voltage in

Fig. plotted as a function of some variable such as time (as
above), position, degrees, radians, temperature, and so on.
Instantaneous value: The magnitude of a waveform at any
instant of time; denoted by lowercase letters (e1, e2 in Fig.).
Peak value: The maximum instantaneous value of a function as
measured from the zero volt level. For the waveform in Fig. the
peak amplitude and peak value are the same, since the average
value of the function is zero volts.
Sinusoidal waveform
Peak-to-peak value: Denoted by Ep-p or Vp-p the full voltage
between positive and negative peaks of the waveform, that is,
the sum of the magnitude of the positive and negative peaks.
Periodic waveform: A waveform that continually repeats itself
after the same time interval. The waveform in Fig. is a periodic
waveform.
Period (T): The time of a periodic waveform.
Cycle: The portion of a waveform contained in one period of
time. The cycles within T1, T2, and T3 in Fig. 13.3 may appear
different in Fig. but they are all bounded by one period of time
and therefore satisfy the definition of a cycle.
Sinusoidal waveform
Frequency ( f ): The number of cycles that occur in 1 s.
The frequency of the waveform in Fig.(a) is 1 cycle per
second, and for Fig. (b), 2.5 cycles per second. If a
waveform of similar shape had a period of 0.5 s [Fig. (c)],
the frequency would be 2 cycles per second.
The unit of measure for frequency is the hertz (Hz), where

Example
For the sinusoidal waveform in Fig.
a. What is the peak value?
b. What is the instantaneous value at 0.3 s and 0.6 s?
c. What is the peak-to-peak value of the waveform?
d. What is the period of the waveform?
e. How many cycles are shown?
f. What is the frequency of the waveform?
Example
FREQUENCY SPECTRUM
FREQUENCY SPECTRUM
Example
• Find the period of periodic waveform with a

frequency of
a. 60 Hz.
b. 1000 Hz.
Solution:
The Sinusoidal Waveform
The sinusoidal waveform is the only alternating

waveform whose shape is unaffected by the
response characteristics of R, L, and C elements.
If the voltage across (or current through) a resistor,
inductor, or capacitor is sinusoidal in nature, the
resulting current (or voltage, respectively) for each
will also have sinusoidal characteristics, as shown in
Fig.
The Sinusoidal Voltage or Current
The basic mathematical format for the sinusoidal waveform is
where Am is the peak value of

the waveform and
α is the unit of measure
for the horizontal axis
The equation α = ωt states that the angle α through which the
rotating vector
The general format of a sine wave can also be written
Example
Sketch e = 10 sin 314t with
a. angle (α) in degrees.
b. angle (α) in radians.
c. time (t) in seconds.
(a) (b)
(c)
PHASE RELATIONS
The phase shift for a sinusoidal function that crosses the

horizontal axis with a positive slope before 0°.
If the waveform passes

through the horizontal
axis with a positive-going
(increasing with time) slope before 0°
as shown in Fig.
PHASE RELATIONS
The terms leading and lagging are used to indicate the

relationship between two sinusoidal waveforms of the same
frequency plotted on the same set of axes.
In Fig. the cosine curve is said to lead the sine curve by 90°,
and the sine curve is said to lag the cosine curve by 90°.
Example
What is the phase relationship between the
sinusoidal waveforms of each of the following
sets?
Solutions:
Ans:
Ans:
EFFECTIVE (rms) VALUES
The equivalent dc value of a sinusoidal current or voltage

is 1/√2 or 0.707 of its peak value
The equivalent dc value is called the rms (root-mean-
square) or effective value of the sinusoidal quantity.

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EE2010 Fundamentals of Electric

Waveform: The path traced by a quantity, such as the voltage in

The unit of measure for frequency is the hertz (Hz), where

• Find the period of periodic waveform with a

The sinusoidal waveform is the only alternating

where Am is the peak value of

The phase shift for a sinusoidal function that crosses the

If the waveform passes

The terms leading and lagging are used to indicate the

The equivalent dc value of a sinusoidal current or voltage

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