1© Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current Feature of alternating current and voltage Mean current and mean voltage I r.m.s. and V r.m.s. Relationship between I r.m.s. and I o  V r.m.s. and V o

2 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 298) Alternating current Power cables carry alternating current (a.c.) - Principles for d.c. (Ohm’s Law, Kirchhoff’s Laws, principles used for resistors in series and parallel) are also applicable to a.c. - 3 types of a.c. Go to More to Know 1 More to Know 1

3 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 299) Feature of alternating current and voltage - direction of current changes from one direction to another periodically - voltage varies in the same manner

4 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 300) Feature of alternating current and voltage Physical quantities of a.c. 1. cycle – complete change of I or V from particular value and back to the same value in same direction 2. period (T) – time taken for a complete cycle 3. frequency (f) – no. of complete cycles in 1 s angular frequency (  ) = 2  f 4. T = 1/f 5. Peak current (I 0 ) and peak voltage (V 0 ) – maximum values of I and V For sinusoidal current and voltage I = I 0 sin  t (or I 0 sin2  ft) V = V 0 sin  t (or V 0 sin2  ft) Go to More to Know 2 More to Know 2

5 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 300) Mean current and mean voltage 1. One cycle 2. Half-cycle Mean current = area under I = I 0 sin2  ft = area under straight line from t = 0 to T/2

6 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 301) Mean current and mean voltage 2. Half-cycle - Mean current over +ve half-cycle is - Mean current over –ve half-cycle is

7 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 301) I r.m.s. and V r.m.s. Power dissipated in pure resistor = I 2 R Mean power = Mean value of (I 2 R) = Mean value of I 2 r.m.s. x R

8 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 301) I r.m.s. and V r.m.s. The root-mean-square value of an alternating current (I r.m.s. ) is the value of the steady current which when flowing through the same resistor, produces heat at the same rate as the mean rate of heat produced by the alternating current. The root-mean-square value of an alternating voltage (V r.m.s. ) is the value of the direct voltage which when applied across the same resistor, produces heat at the same rate as the mean rate of heat produced by the alternating voltage. Go to More to Know 3 More to Know 3

9 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 302) Relationship between I r.m.s. and I 0 / V r.m.s. and V 0 Go to More to Know 4 More to Know 4

10 © Manhattan Press (H.K.) Ltd Root-mean-square (r.m.s.) current (SB p. 303) Relationship between I r.m.s. and I 0 / V r.m.s. and V 0 Go to More to Know 5 More to Know 5 Go to Example 1 Example 1 Go to Example 2 Example 2 Go to Example 3 Example 3

11 © Manhattan Press (H.K.) Ltd. End

12 © Manhattan Press (H.K.) Ltd. Waveform of a.c. The waveforms of alternating currents as shown in Fig can be displayed by a CRO. Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 299)

13 © Manhattan Press (H.K.) Ltd. Frequency of a.c. In most countries, the frequency of the alternating current is 50 Hz. Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 300)

14 © Manhattan Press (H.K.) Ltd. V r.m.s. in Hong Kong In Hong Kong, the mains supplies 200 V r.m.s. to our homes. Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 302)

15 © Manhattan Press (H.K.) Ltd. Measurement of I r.m.s. and V r.m.s. An a.c. ammeter and a.c. voltmeter show the r.m.s. values of alternating current ( I r.m.s. ) and alternating voltage ( V r.m.s. ) respectively. Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 302)

16 © Manhattan Press (H.K.) Ltd. I r.m.s. and V r.m.s. The equations are valid to sinusoidal a.c. only. Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 303)

17 © Manhattan Press (H.K.) Ltd. Q: Q: Find the r.m.s. value of the alternating current (I) from the figure Root-mean-square (r.m.s.) current (SB p. 303) Solution

18 © Manhattan Press (H.K.) Ltd. Solution: Consider the current variation over one complete cycle. Time /sCurrent / A 0 – – – r.m.s. value of current Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 303)

19 © Manhattan Press (H.K.) Ltd. Q: Q: Alternating current of r.m.s. value 4.0 A and frequency 50 Hz flows in a circuit containing a resistor. Find (a) the peak current, and (b) the value of the current 6 × 10 –4 s after it changes direction Root-mean-square (r.m.s.) current (SB p. 304) Solution

20 © Manhattan Press (H.K.) Ltd. Solution: (a) Peak current (I 0 ) = I r.m.s. = ×4.0 = 5.66 A (b) From the equation I = I 0 sin 2πft = 5.66 ×sin (2 π ×50 ×6 ×10 –4 ) = 5.66 ×sin (0.188) = 1.06 A Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 304)

21 © Manhattan Press (H.K.) Ltd. Q: Q: In the circuit, an a.c. supply of 240 V r.m.s. is connected to the terminals P and Q. The resistance of the variable resistor R is slowly decreased. The fuse F breaks when the root-mean-square current just exceeds 13 A. When the a.c. supply is substituted with a 120 V d.c. supply and the resistance of R decreased in the same manner, for what value of the current would the fuse F break? 18.1 Root-mean-square (r.m.s.) current (SB p. 304) Solution

22 © Manhattan Press (H.K.) Ltd. Solution: The fuse F is heated when a current flows through it. It breaks when the root-mean-square current exceeds 13 A irrespective of the voltage or type of current. Hence for a 120 V d.c. supply, the fuse breaks when the current exceeds 13 A. Return to Text 18.1 Root-mean-square (r.m.s.) current (SB p. 304)