ELECTRIC CIRCUIT ANALYSIS - I

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Chapter 13 – Sinusoidal Alternating Waveforms Lecture 14 by Moeen Ghiyas 08/06/22 1

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ELECTRIC CIRCUIT ANALYSIS - I. Chapter 13 – Sinusoidal Alternating Waveforms Lecture 14 by Moeen Ghiyas. TODAY’S lesson. CHAPTER 13 – Sinusoidal Alternating Waveforms. TODAY’S LESSON CONTENTS. Average Value Effective (RMS) Value Assignment # 3 Application and Safety Concerns. - PowerPoint PPT Presentation

Transcript of ELECTRIC CIRCUIT ANALYSIS - I

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Chapter 13 – Sinusoidal Alternating Waveforms

Lecture 14

by Moeen Ghiyas

21/04/23 1

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CHAPTER 13 – Sinusoidal Alternating Waveforms

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Average Value

Effective (RMS) Value

Assignment # 3

Application and Safety Concerns

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The longer the distance, the lower is the average value

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If the distance parameter includes a

depression, as shown in fig., some of the

sand will be used to fill the depression,

resulting in an even lower average value

for the landscaper.

Similarly for a sinusoidal waveform, the

depression would have the same shape

as the mound of sand (over one full

cycle), resulting in an average value at

ground level (or zero volts for a

sinusoidal voltage over one full period).

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For example, if a person travelled for 5 h, find the average

speed and distance travelled?

= 245 /5 mi/h

= 49 mi/h

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Thus for any variable quantity, such as current or voltage, if

we let G denote the average value

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Thus for any variable quantity, such as current or voltage, if

we let G denote the average value

The average value of any current or voltage is the value

indicated on a dc meter (or dc mode in DMM).

In the analysis of electronic circuits to be considered in a

later course, both dc and ac sources of voltage will be

applied to the same network.

In other words, average value indicates the dc component

value in a sinusoidal alternating waveform.

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EXAMPLE - Determine the average value of the waveform

of fig.

By inspection, the area above the axis equals the area

below over one cycle, resulting in an average value of zero

volts

Mathematically

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EXAMPLE - Determine the average value of the

waveforms of Fig.

Solution

In reality, the waveform of fig.

(b) is simply the square wave

of fig.(a) with a dc shift of 4 V;

that is,

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Using Area = 1/2 b x h for the area of a right-triangle we can

approximate the area of a pulse with a single triangle

However, we can obtain a good approximation of the area of a

half sine wave by attempting to reproduce the original wave

shape using a number of small rectangles or other familiar shapes

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A closer approximation than a

single triangle might be obtained

by a rectangle with two similar

triangles

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But calculus(integration) gives the best solution

Finding the area under the positive pulse of a sine

wave using integration, we have

Here dα indicates that

we are integrating with

respect to α.

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Since now we know the area under the positive (or negative)

pulse,

we can easily determine the average value of the positive (or

negative) region of a sine wave pulse by applying eq.

Note the average is

same for half pulse as

for a full pulse

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Example – Determine the average value of the sinusoidal

waveform of fig.

Solution

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Example – Determine the average value of the waveform of fig.

Solution - peak-to-peak value = 16 mV + 2 mV = 18 mV

. peak amplitude = 18 mV/2 = 9 mV

Counting down 9 mV from 2 mV (or 9 mV up from -16 mV)

results in an average or dc level of -7 mV, as noted by the

dashed line

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Example – Determine the average value of the waveform

of fig.

Solution

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Measuring Average Value or dc component using oscilloscope by

choosing DC – GND – AC option for vertical channel

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Measuring Average Value or dc component using oscilloscope by

choosing DC – GND – AC option for vertical channel

Fig (a) AC option for vertical channel while fig (b) with DC option

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Question arises, how is it possible for a sinusoidal ac quantity to

deliver a net power (P=VI as in dc) if, over a full cycle, the net

current in any one direction is zero (average value 0)?

However, understand that irrespective of direction (positive or

negative), current of any magnitude through a resistor will deliver

power to that resistor.

The power delivered at each instant will, of course, vary with the

magnitude of the sinusoidal ac current.

Effective value (or rms value) co-relates dc and ac quantities

with respect to the power delivered to a load.21/04/23 20

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A fixed relationship between ac and dc voltages and currents can

be derived from the experimental setup shown in Fig

If switches are closed alternately, a dc current I or i will flow as

per ohm’s law.

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First the temperature reached by the water is determined by the dc

power dissipated in the form of heat by the resistor.

Later the ac input ‘i’ is varied until the temperature is the same as that

reached with the dc input.

When this is accomplished, the average electrical power delivered to

the resistor R by the ac source is the same as that delivered by the dc

source.

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The power delivered by the dc supply at any instant of time is

P = VI = V2/R = I2R

Thus

From Trigonometric identity

Therefore,

and

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The average power delivered by the ac source is just the first

term, since the average value of a cosine wave is zero even

though the wave may have twice the frequency of the original

input current waveform.

Equating the average power delivered by the ac generator to

that delivered by the dc source

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The equivalent dc value is called the effective value of the

sinusoidal quantity.

In summary

In other words, it would require an ac current with a peak value of

√2(10) = 14.14 A to deliver the same power to the resistor in fig. as

a dc current of 10 A.

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From above experiment, the effective value of any quantity plotted as a

function of time can be found by

Thus, to find the effective value, the function i(t) must first be squared.

Next the area under the curve is found by integration.

It is then divided by time period T, to obtain the average or mean value of

the squared waveform.

The final step is to take the square root of the mean value.

This procedure gives us another designation for the effective value, the

root-mean-square (rms) value. 21/04/23 26

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EXAMPLE - Find the rms values of the sinusoidal

waveform in each part of Fig

Note that frequency did not change the effective value in

(b) above compared to (a).

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Example – A 120-V dc source of fig. (a)

delivers 3.6 W to the load. Determine peak

value of the applied voltage (Em) and

current (Im) if ac source fig. (b) is to deliver

same power to load.

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Example – Calculate

the rms value of the

voltage of Fig.

Solution

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The rms values of sinusoidal quantities such as voltage or

current will be represented by E and I.

These symbols are the same as those used for dc voltages

and currents.

To avoid confusion, the peak value of a waveform will

always have a subscript m associated with it as in Im

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When finding the rms value of the positive pulse of a sine

wave, note that the squared area is not simply (2Am)2 = 4A2m;

it must be found by a completely new integration. This will

always be the case for any waveform that is not rectangular.

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A unique situation arises if a waveform has both a dc and an ac

component. What is the rms value of the voltage vT?

. Vac (rms) = 3/2 * 0.707 = 1.06V

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Ch 13 – Q.1, Q.8, Q.16, Q.24, Q.31 fig (b), Q.39 fig (a),

Q. 39 and Q. 49

Submission date & time: 09:00 next week

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Frequency selection?

220V vs 120V?

DC vs AC?

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Frequency selection? Small difference so general agreement

exists

Frequency selection originally focused on freq that would not

exhibit flicker in the incandescent lamps available in old days.

Another important factor was the effect of frequency on the size

of transformers; size of a transformer is inversely proportional

to frequency.

The result is that transformers operating at 50 Hz must be

larger (on mathematical basis about 17% larger) than those

operating at 60 Hz.

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However, higher frequencies result in increased arcing, increased

losses in transformer core due to eddy current and hysteresis losses,

and skin effect phenomena.

Since accurate timing is such a critical part of our technological

design, significant motive could have been the fact that 60 Hz is an

exact multiple of 60 seconds in a minute and 60 minutes in an hour.

Whereas 50 Hz has close affinity to the metric system. Keep in mind

that powers of 10 are all powerful in the metric system, with 100 cm in

a meter, 100°C the boiling point of water, and so on. Note that 50 Hz is

exactly half of this special number.

All in all, difference may have been simply political in nature.

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The difference in voltage (Almost 100% difference)

Larger voltages such as 220 V raise safety issues beyond those raised

by voltages of 120 V.

However, when higher voltages are supplied, there is less current in

the wire for the same power demand, permitting the use of smaller

conductors—a real money saver. In addition, motors, compressors,

and so on, found in common home appliances and throughout the

industrial community can be smaller in size.

Higher voltages, however, also bring back the concern about arcing

effects, insulation requirements, and, due to real safety concerns,

higher installation costs.

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For instance, under dry conditions, most human beings can survive a

120-V ac shock such as obtained when changing a light bulb, turning

on a switch, and so on. However, if hit by 220 V, the response (though

still possible to survive) will be totally different.

For voltages beyond 220 V rms, the chances of survival go down

exponentially with increase in voltage. It takes only about 10 mA of

steady current through the heart to put it in defibrillation.

In general, therefore, always be sure that the power is disconnected

when working on the repair of electrical equipment.

Never use extension cords or plug in TVs or radios in the bathroom or

wet places.

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AC vs DC

You have probably seen in

movies or comic strips that

people are often unable to let

go of a hot wire

If you happen to touch a “hot”

120-V ac line, you will get a

good sting, but you can let go.

But in case of a dc line, you

will probably not be able to let

go, and a fatality could occur.

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Average Value

Effective (RMS) Value

Application and Safety Concerns

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