lecture 01 sinusoidal alternating waveforms

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Sinusoidal Alternating Waveforms

Lecture 01

Course Conducted by –

Shuvodip Das,

Lecturer, Department of Electronics and Telecommunication EngineeringPrime University, Dhaka.



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.

•



Waveforms

Fig: Alternating Waveforms



Sources of AC Power



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.




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.




Angular Velocity: The velocity with which the radius vector rotates about the center, called the

angular velocity, can be determined from the following equation:



GENERAL FORMAT FOR THE SINUSOIDAL

VOLTAGE OR CURRENT



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



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.



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



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



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.



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.

lecture 01 sinusoidal alternating waveforms

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