Chapter 13 – Sinusoidal Alternating Waveforms Lecture 12 by Moeen Ghiyas 23/11/2015 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 (e 1, e 2 ) Peak amplitude: The maximum value of a waveform as measured from its average, or mean, value, denoted by uppercase letters (such as E m for sources V m 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 E p-p or V p-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 T 1, T 2, and T 3 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.

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Example – Determine the frequency of the waveform of fig Solution 23/11/

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 ?

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):

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

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