Download File - Electric Circuit Analysis

Chapter 13 – Sinusoidal Alternating Waveforms
Lecture 12
by Moeen Ghiyas
29/04/2017
1
CHAPTER 13 – Sinusoidal Alternating Waveforms
 Introduction – Sinusoidal Alternating Waveforms
 Definitions
 Defined Polarities & Direction
 The Sine Wave
 AC networks are those in which the magnitude of the source is
the time-varying voltage or current
 Letters 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
 The pattern of particular interest is the sinusoidal ac waveform
for voltage
 Referance the diagram, the vertical scaling is in volts or amperes
and the horizontal scaling is always in units of time
 Waveform: The path traced by a quantity, such as the voltage
 Instantaneous value: The magnitude of a waveform at any
instant of time; denoted by lowercase letters (e1, e2)
 Peak amplitude: The maximum value of a waveform as
measured from its average, or mean, value, denoted by
uppercase letters (such as Em for sources Vm for the voltage
drop across a load)
 Peak value: The maximum instantaneous value of a function as
measured from the zero-volt level. For the waveform of Fig shown,
the peak amplitude and peak value are the same, since the average
value of the function is zero volts.
 Peak-to-peak value: Denoted by Ep-p or Vp-p, the full voltage
between positive and negative peaks of the waveform
 Periodic waveform: A waveform that continually repeats itself after
the same time interval.
 Period (T ): The time interval between successive repetitions
 Cycle: The portion of a waveform contained in one period of time.
The cycles within T1, T2, and T3 of fig (top) may appear different in
fig (below), but they are all bounded by one period of time and
therefore satisfy the definition of a cycle
 Frequency ( f ): The number of cycles that occur in 1 s. The frequency of
the waveform of fig below(a) is 1 cycle per second, and for fig below (b),
2 ½ cycles per second. If a waveform of similar shape had a time period
of 0.5 s [fig below (c)], the frequency would be 2 cycles per second.
29/04/2017
10
29/04/2017
11
 Example – Determine the frequency of the waveform of fig
 Solution
29/04/2017
12
 The polarity and current direction will be for an instant of time
 A lowercase letter is employed for each to indicate that the
quantity is time dependent
 Why use sinusoidal waveform in ac and why not square or
triangular?
 Ans. The sinusoidal waveform is the only alternating waveform
whose shape is unaffected by the response characteristics of
resistor R, inductor L, and capacitor C elements.
 The unit of measurement on vertical
axis can be voltage or current.
 The unit of measurement for the
horizontal axis can be time (seconds)
or phase angle θ in degrees (o) or
radians (rads).
How we arrive at 1 cycle of waveform
being equivalent to 2π or 3600 ?
 We define radian as shown in diagram
 We know that circumference of the circle is C = 2 π r
 Thus the quantity π is the ratio of the circumference of a circle to its
diameter i.e, π = C / 2r
or
π=C/d
 A sinusoidal waveform can be derived from the length of the
vertical projection of a radius vector rotating in a uniform circular
motion about a fixed point
 The velocity with which the radius vector rotates about the centre,
called the angular velocity, can be determined from the following
equation (S = vt):
 The velocity with which the radius vector rotates about the
centre, called the angular velocity, can be determined from the
following equation (S = vt):
 Substituting into equation (above) and assigning the Greek letter
omega (ω) to the angular velocity, we have
 Now we know that the angular velocity (ω) is
 The time required to complete one revolution is equal to the
period (T) of the sinusoidal waveform .
 The radians subtended in this time interval are 2π.
Substituting, we have
 Now that the angular velocity (ω) for time period (T = 2π) is
 In other words, this equation states that the smaller the period
of the sinusoidal waveform, the greater must be the angular
velocity (ω) of the rotating radius vector and vice versa.
 But we also know that f = 1/ T.
Thus,
 EXAMPLE - Determine the angular velocity of a sine wave
having a frequency of 60 Hz.
 EXAMPLE - Given ω = 200 rad/s, determine how long it will
take the sinusoidal waveform to pass through an angle of 90°.
 EXAMPLE - Find the angle through which a sinusoidal
waveform of 60 Hz will pass in a period of 5 ms.
 EXAMPLE - Determine the angular velocity of a sine wave
having a frequency of 60 Hz.
 Solution
 EXAMPLE - Given ω = 200 rad/s, determine how long it will
take the sinusoidal waveform to pass through an angle of 90°.
 Solution
 EXAMPLE - Find the angle through which a sinusoidal waveform
of 60 Hz will pass in a period of 5 ms.
 Solution
 Note:
Whenever, the relationship involving ω is to be used α will
be used or interpreted in radians, because of π factor in ω = 2πf.
 Introduction – Sinusoidal Alternating Waveforms
 Definitions
 Defined Polarities & Direction
 The Sine Wave
29/04/2017
27