Alternating voltages and currents

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1 Alternating voltages and currents

2 Introduction - Electricity is produced by generators at power stations and then distributed by a vast network of transmission lines (called the National Grid system) to industry and for domestic use. - It is easier and cheaper to generate alternating current (a.c.) than direct current (d.c.) and a.c. is more conveniently distributed than d.c. since its voltage can be readily altered using transformers. - Whenever d.c. is needed in preference to a.c., devices called rectifiers are used for conversion

3 Waveforms - If values of quantities which vary with time t are plotted to a base of time, the resulting graph is called a waveform. - Waveforms (a) and (b) are unidirectional waveforms, - although they vary considerably with time, they flow in one direction only (i.e. they do not cross the time axis and become negative).

4 Waveforms (continued) - Waveforms (c) to (g) are called alternating waveforms since their quantities are continually changing in direction (i.e. alternately positive and negative). - A waveform of the type shown in Figure(g) is called a sine wave. - It is the shape of the waveform of e.m.f. produced by an alternator - One complete series of values is called a cycle (i.e. from O to P in Figure(g)). - The time taken for an alternating quantity to complete one cycle is called the period or the periodic time, T, of the waveform. - The number of cycles completed in one second is called the frequency, f, of the supply and is measured in hertz, Hz.

5 A.c. values - Instantaneous values are the values of the alternating quantities at any instant of time. They are represented by small letters, i, υ, e, etc. - The largest value reached in a half cycle is called the peak value or the maximum value or the amplitude of the waveform. Such values are represented by Vm, Im, etc. - A peak-to-peak value of e.m.f. is shown in Figure (g) and is the difference between the maximum and minimum values in a cycle. - The average or mean value of a symmetrical alternating quantity, (such as a sine wave), is the average value measured over a half cycle, (since over a complete cycle the average value is zero). Average values are represented byvav, IAV, etc. - The effective value of an alternating current is that current which will produce the same heating effect as an equivalent direct current. The effective value is called the root mean square (rms) value.

6 A.c. values (Continued)

7 The equation of a sinusoidal waveform - In Figure below, OA represents a vector that is free to rotate anticlockwise about 0 at an angular velocity of ω rad/s. - A rotating vector is known as a phasor. - A sine curve may not always start at 0. A periodic function is represented by y=sin(ωt ±φ),where φ is the phase (or angle) difference compared with y= sin ωt. a), - y2 =sin(ωt +φ) starts φ radians earlier than y1 = sin ωt and is thus said to lead y1 byφ radians.

8 The equation of a sinusoidal waveform (continued) - y4 =sin(ωt φ) starts φ radians later than y3 = sinωt and is thus said to lag y3 by φ radians. - Given the general sinusoidal voltage, v =Vmsin(ωt ±φ), Where : (i) Amplitude or maximum value=vm (ii) Peak-to-peak value=2 Vm (iii) Angular velocity=ω rad/s (iv) Periodic time, T =2π/ω seconds (v) Frequency, f =ω/2π Hz (since ω=2π f) (vi) φ =angle of lag or lead (compared with v= Vm sinωt)

9 Example. An alternating voltage is given by v =282.8 sin314 t volts. Find (a) the rms voltage, (b) the frequency and (c) the instantaneous value of voltage when t =4 ms. Answer:

10 Example. An alternating voltage is given by v = 75 sin(200πt 0.25) volts. Find (a) the amplitude, (b) the peak-to-peak value, (c) the rms value, (d) the periodic time, (e) the frequency, and (f ) the phase angle (in degrees and minutes) relative to 75 sin 200πt. Answer:

11 Combination of waveforms The resultant of the addition (or subtraction) of two sinusoidal quantities may be determined either: (a) by plotting the periodic functions graphically or, (b) by resolution of phasors by drawing or calculation. Example. The instantaneous values of two alternating currents are given by i1 =20 sin ωt amperes and i2 =10 sin(ωt +π/3) amperes. By plotting i1 and i2 on the same axes, using the same scale, over one cycle, and adding ordinates at intervals, obtain a sinusoidal expression for i1 +i2.

12 Example 2: Two alternating voltages are represented by v1 =50 sin ωt volts and v2 =100 sin (ωt π/6) V. Draw the phasor diagram and find, by calculation, a sinusoidal expression to represent v1 +v2. Answer: Phasors are usually drawn at the instant when time t =0. Thus v1 is drawn horizontally 50 units long and v2 is drawn 100 units long lagging v1 by π/6 rads, i.e. 30 degree. Procedure to draw phasor diagram to represent v1 +v2: (i) Draw v1 horizontal 50 units long, i.e. Oa of Figure (b) (ii) Join v2 to the end of v1 at the appropriate angle, i.e. ab of Figure (b) (iii) The resultant vr =v1 +v2 is given by the length Ob and its phase angle φ may be measured with respect to v1

13 Example( continued) A more accurate solution is obtained by calculation, using the cosine and sine rules. Using the cosine rule on triangle Oab of Figure(b) gives:

CHAPTER 14 ALTERNATING VOLTAGES AND CURRENTS

CHAPTER 14 ALTERNATING VOLTAGES AND CURRENTS CHAPTER 4 ALTERNATING VOLTAGES AND CURRENTS Exercise 77, Page 28. Determine the periodic time for the following frequencies: (a) 2.5 Hz (b) 00 Hz (c) 40 khz (a) Periodic time, T = = 0.4 s f 2.5 (b) Periodic