Sinusoidal Alternating Waveforms Lecture 01 Course Conducted by – Shuvodip Das, Lecturer, Department of Electronics and Telecommunication Engineering Prime University, Dhaka.
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Sinusoidal Alternating Waveforms
Lecture 01
Course Conducted by –
Shuvodip Das,
Lecturer, Department of Electronics and Telecommunication EngineeringPrime University, Dhaka.
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Introduction
Previously we studied DC circuits. Where Current and Voltage arefixed in magnitude except for transient effects.
• Which was called DC current and DC voltage.
• AC i.e alternating or time varying quantities are used mainly for
commercial supplies.
• The term alternating indicates only that the waveform alternates
between two prescribed levels in a set time sequence.
•
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Waveforms
Fig: Alternating Waveforms
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Sources of AC Power
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Sinusoidal ac voltage: Definitions
Note: Vertical scaling is in volts or amperes and horizontal scaling is in units of time.
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 of voltage and 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. , the peak amplitude and peak value are the same, since the average value of the function is
zero volts.
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Sinusoidal ac voltage: Definitions
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 of Fig. 13.3 is a periodic waveform.
Period (T ): The time interval between successive repetitions of a periodic waveform
(the period T1 T2 T3 in Fig. 13.3), as long as successive similar points of the periodic
waveform are used in determining T.
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Sinusoidal ac voltage: Definitions
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Sinusoidal ac voltage: Definitions
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Sinusoidal ac voltage: Definitions
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Sinusoidal ac voltage: Definitions
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Sinusoidal ac voltage: Definitions
Angular Velocity: The velocity with which the radius vector rotates about the center, called the
angular velocity, can be determined from the following equation:
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Sinusoidal ac voltage: Definitions
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Sinusoidal ac voltage: Definitions
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GENERAL FORMAT FOR THE SINUSOIDAL
VOLTAGE OR CURRENT
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PHASE RELATIONS
The terms lead and lag are used to indicate the relationship
between two sinusoidal waveforms of the same frequency plotted on
the same set of axes. In Fig. 13.25, 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°. The 90° is referred to as the phase angle between the
two waveforms. In language commonly applied, the waveforms are
out of phase by 90°.
H.W: Introductory Circuit Analysis, Example: 13.12
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The Average/Mean Value of an AC Waveform
The average or mean value of a continuous DC voltage will always be equal to its maximum peak
value as a DC voltage is constant. In a pure sine wave if the average value is calculated over the full
cycle, the average value would be equal to zero as the positive and negative halves will cancel each
other out. So the average or mean value of an AC waveform is calculated or measured over a half
cycle only and this is shown below.
Average or Mean ValueThe amplitude of an AC waveform is its height as depicted on a graph over time. An amplitude
measurement can take the form of peak, peak-to-peak, average, or RMS quantity.
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Average/Mean Value: Continued
Average Value of a Non-sinusoidal Waveform
Where: n equals the actual number of mid-ordinates used.
• Average amplitude is the mathematical "mean" of all a waveform's points over the period of one cycle.• For a sine wave, the average value so calculated is approximately 0.637 of its peak value.
Average Voltage, VAV = VPK x 0.637 or
Average Current, IAV = IPK X 0.637
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RMS or Effective Value"RMS" stands for Root Mean Square, and is a way of expressing an AC quantity
of voltage or current in terms functionally equivalent to DC.
The root-mean-square (rms) value or effective value of an ac waveform is a
measure of how effective the waveform is in producing heat in a resistance.
Example: If you connect a 5 Vrms source across a resistor, it will produce the same
amount of heat as you would get if you connected a 5 V dc source across that same
resistor. On the other hand, if you connect a 5 V peak source or a 5 V peak-to-
peak source across that resistor, it will not produce the same amount of heat as a 5
V dc source.That's why rms (or effective) values are useful: they give us a way to compare ac
voltages to dc voltages.
To show that a voltage or current is an rms value, we write rms after the unit: for
example, V rms = 25 V rms.
It is also known as the "equivalent" or "DC equivalent" value of an AC voltage or current. For a sine wave, the RMS value is approximately 0.707 of its peak value.
VRMS = VPK x 0.707 and
IRMS = IPK x 0.707
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RMS or Effective ValueRelationship Between Peak Values & RMS Values
•For a sine wave, to convert from peak values to rms values, use these equations:
V rms
≈ 0.707 × V p
I rms ≈ 0.707 × I p
To convert in the other direction (from rms values to peak values), use these
equations: V p ≈ 1.414 × V rms
I p ≈ 1.414 × I rms
NOTE:
When we use a multimeter to measure ac voltage or current, it gives you effective (or
rms) values, not peak values or peak-to-peak values.So if we measure the same voltage with both the multimeter and the oscilloscope , we've
got to realize that we're getting an effective voltage from the meter and you're getting a
peak (or peak-to-peak) voltage from the oscilloscope. To compare the two values,we need
to convert one of them using the equations given above.
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Crest factor and Form factor
The crest factor of an AC waveform is the ratio of its peak (crest) to its RMS value.
The crest factor or peak-to-average ratio (PAR ) or peak-to-average power ratio (PAPR ) is a
measurement of a waveform, calculated from the peak amplitude of the waveform divided by the RMS
value of the waveform.
The form factor of an AC waveform is the ratio of its peak (crest) value to its average
value.