Ac theory

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1. Alternating Current Prepared by Md. Amirul Islam Lecturer Department of Applied Physics & Electronics Bangabandhu Sheikh Mujibur Rahman Science & Technology University, Gopalganj – 8100

2. Average Value of AC Voltage & Current

3. How to calculate the average value? Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.6, Page – 539 Example I – Average Speed of a Car: Q Ans Fig: Plotting speed versus t ime graph for a car From the graph, we can see that the car travelled at different speed in different time. If we want to calculate the average speed, we should first calculate the total distance travelled. Thus, we need to calculate the area A 1 and A 2 then divide the area by travelled time (in this case 5 hours).

4. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.6, Page – 539 Example II – Average Height of Sand: Fig: (a) a pile of sand (b) leveled off keeping the distance unchanged (c) leveled off with an increased distance The average height of the sand may be required to determine the volume of sand available. The average height of the sand is that height obtained if the distance from one end to the other is maintained while the sand is leveled off.

5. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.6, Page – 539 Fig: (a) a pile of sand (b) leveled off keeping the distance unchanged (c) leveled off with an increased distance The average height can be calculated by determining the area ( b×h ) and then dividing the area by distance (d). If the sand is spread over an extended distance, the average height decreases.

6. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.6, Page – 539 From the two examples, we can say that, The algebraic sum of the areas must be determined, since some area contributions will be from below the horizontal axis. Areas above the axis will be assigned a positive sign, and those below, a negative sign. A positive average value will then be above the axis, and a negative value , below .

7. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Example – 13.13, Page – 541 Math. Problem: Determine the average value of the waveforms as shown in the following figures. (a) By inspection, the area above the axis equals the area below over one cycle, resulting in an average value of zero volts . By calculation:

8. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Example – 13.13, Page – 541 (b) By using the equation of average value, we get,

9. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Example – 13.14, Page – 542 Math. Problem: Find the average value of the following waveform over one full cycle.

10. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Example – 13.14, Page – 543 Math. Problem: Find the average value of a sinusoidal wave for (a) one half cycle (b) full cycle (a) We Know, Average Value = To determine the area under the half cycle, we should integrate the equation of the wave within the limit 0 to π .  

11. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Example – 13.14, Page – 543 (b) The average value for full cycle is, Thus, the average value is,

12. Average Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Example – 13.14, Page – 544 Math. Problem: Determine the average value for a full cycle of the waveform as shown in the figure.

13. Effective or RMS Value of AC Voltage & Current

14. Effective (RMS) Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.7, Page – 544 How a sinusoidal AC quantity can deliver power to a load where the net current over a full cycle in any one direction is zero? The equation of power at any instant of time on a load resistance R connected with an AC source is, P = v i = i 2 R = v 2 /R From the above equation, we can see that power is never negative. When v is positive then i is also positive, thus power is also positive. When v is negative then i is also negative, thus power is positive . Thus, regardless of the direction of current and voltage, over a full cycle the power is always delivered to the resistive load and the power is never zero for a full cycle. Here, v = V m sin ω t and i = ( V m /R) sin ω t

15. Effective (RMS) Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.7, Page – 544 You have a dc source which can deliver power P to a resistive load R. Which sinusoidal signal can deliver the same power to that load? A resistor in a water bath is connected by switches to a dc and an ac supply. If switch 1 is closed, a dc current I, determined by the resistance R and battery voltage E, will be established through the resistor R. The temperature reached by the water is determined by the dc power dissipated in the form of heat by the resistor. Fig: Experimental setup to derive the relation between dc and ac quantities

16. Effective (RMS) Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.7, Page – 545 If switch 2 is closed and switch 1 left open, the ac current through the resistor will have a peak value of I m . The temperature reached by the water is now determined by the ac power dissipated in the form of heat by the resistor. The ac input 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. Fig: Experimental setup to derive the relation between dc and ac quantities

17. Effective (RMS) Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.7, Page – 546 The power delivered by the ac supply at any instant of time is - 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 .

18. Effective (RMS) Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.7, Page – 546 Equating the average power delivered by the ac generator to that delivered by the dc source, Thus, the equivalent dc value of a sinusoidal current or voltage is 0.707 of its maximum value.

19. Effective (RMS) Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad , Topic – 13.7, Page – 546 In summary, This experimental result can be also derived by integrating the following equation:

20. Effective (RMS) Value of AC Voltage & Current Reference: Circuit Analysis by Robert Boylestad Math. Problem: 13.19, 13.20, 13.21, 13.22 Page – 548, 549

21. The End