2. 1© Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors Resistor and capacitor in series (RC circuit) Resistor and inductor in series (LR circuit) Resistor, capacitor and inductor in series (LCR circuit)

3. 2 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 331) Resistor and capacitor in series (RC circuit) I = I 0 sin  t 1. Phasor diagram V R = IR (in phase with I)

4. 3 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 332) Resistor and capacitor in series (RC circuit) 2. Impedance (Z) Impedance – measure of opposition to I Unit - ohm

5. 4 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 332) Resistor and capacitor in series (RC circuit) 2. Impedance (Z) Z is large at low input frequency,and is small at high input frequency (high-pass filter) Go to More to Know 10 More to Know 10

6. 5 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 333) Resistor and capacitor in series (RC circuit) 3. Phase difference (  ) I leads V RC by . I = I 0 sin  t, V RC = V o sin(  t -  ) Go to Example 11 Example 11 Go to Example 12 Example 12

7. 6 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 335) Resistor and inductor in series (LC circuit) I = I 0 sin  t 1. Phasor diagram V R = IR (in phase with I)

8. 7 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 336) Resistor and inductor in series (LC circuit) 2. Impedance (Z) Z is low at low input frequency,and is high at high input frequency (low-pass filter)

9. 8 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 337) Resistor and inductor in series (LC circuit) 3. Phase difference (  ) V LR leads I by . I = I 0 sin  t, V LR = V o sin(  t +  ) Go to Example 13 Example 13 V LR

10. 9 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 338) Resistor, capacitor and inductor in series (LCR circuit) I = I 0 sin  t 1. Phasor diagram V R = IR (in phase with I)

11. 10 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 339) Resistor, capacitor and inductor in series (LCR circuit) 2. Impedance (Z)

12. 11 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 339) Resistor, capacitor and inductor in series (LCR circuit) 2. Impedance (Z)

13. 12 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 339) Resistor, capacitor and inductor in series (LCR circuit) 3. Phase difference (  ) Notes: 1. For  > 0, V LCR leads I 2. For  = 0, V LCR and I are in phase when 3. For  < 0, I leads V LCR

14. 13 © Manhattan Press (H.K.) Ltd. 18.5 Series combination of resistors, capacitors and inductors (SB p. 340) Resistor, capacitor and inductor in series (LCR circuit) 3. Phase difference (  ) I = I 0 sin  t, V LCR = V o sin(  t +  ) For  > 0 Go to Example 14 Example 14 Go to Example 15 Example 15

15. 14 © Manhattan Press (H.K.) Ltd. End

16. 15 © Manhattan Press (H.K.) Ltd. Impedance and reactance In a resistive circuit, the ratio (V/I) is called impedance. In a non-resistive circuit, the ratio (V/I) is called reactance. Their units are the same, ohm. Return to Text 18.5 Series combination of resistors, capacitors and inductors (SB p. 332)

17. 16 © Manhattan Press (H.K.) Ltd. Q: Q: An alternating current of r.m.s. value 1.5 mA and angular frequency  = 100 rad s –1 flows through a resistor of 10 kΩ and a 0.5 μF capacitor joined in series. Calculate (a) the r.m.s. voltage across the resistor, (b) the r.m.s. voltage across the capacitor, (c) the total r.m.s. voltage across the resistor and capacitor, (d) the r.m.s. voltage across the resistor when the current is maximum, and (e) the impedance of the circuit. 18.5 Series combination of resistors, capacitors and inductors (SB p. 334) Solution

18. 17 © Manhattan Press (H.K.) Ltd. Solution: (a) Voltage across the resistor (V R ) = IR = (1.5 ×10 –3 ) × (10 ×10 3 ) = 15 V (b) Voltage across the capacitor (V C ) = IX C (c) From the phasor diagram, the total voltage (V) across the resistor and capacitor is given by: (d) The voltage (V R ) across the resistor is in phase with the current whereas the voltage (V C ) across the capacitor lags behind the current by  /2. Therefore, when the current is maximum, V C is zero and V R is maximum. Hence, V R when the current is maximum = I 0 R =  2 I r.m.s. R =  2 (1.5 ×10 –3 ) × (10 ×10 3 ) = 21.2 V (e) Impedance of the circuit (Z) Return to Text 18.5 Series combination of resistors, capacitors and inductors (SB p. 334)

19. 18 © Manhattan Press (H.K.) Ltd. Q: Q: A box containing one or more electrical components is joined to an a.c. supply of constant voltage but variable frequency. The current that passes through the box varies with the supply frequency as shown in the graph. State and explain what components are likely to be in the box. 18.5 Series combination of resistors, capacitors and inductors (SB p. 335) Solution

20. 19 © Manhattan Press (H.K.) Ltd. Solution: The components consist of a resistor of resistance R in series with a capacitor of capacitance C. The impedance of the circuit (Z) Return to Text When f  , I  V/R which is a constant. Alternative Method: The variation of impedance Z for a resistor of resistance R connected in series with a capacitor of resistance C in an a.c. circuit is shown in the figure. Since current I = V/Z, the graph for I against frequency will be of the shape given in the question. 18.5 Series combination of resistors, capacitors and inductors (SB p. 335)

21. 20 © Manhattan Press (H.K.) Ltd. Q: Q: A solenoid, which is connected with an a.c. supply, has inductance 2.5 H and resistance 6.0 Ω. (a) Sketch a graph to show how the resistance R and the reactance X of the solenoid vary with the frequency of the a.c. supply. (b) At what frequency is the resistance of the solenoid equal to 1% of its reactance? 18.5 Series combination of resistors, capacitors and inductors (SB p. 337) Solution

22. 21 © Manhattan Press (H.K.) Ltd. Solution: (a) Return to Text (b) When R = 1% of reactance 18.5 Series combination of resistors, capacitors and inductors (SB p. 337)

23. 22 © Manhattan Press (H.K.) Ltd. Q: Q: A 40 Ω resistor, a 0.4 H inductor, and a 10 μF capacitor are connected in series with an a.c. source that generates 120 V, 60 Hz a.c. (a) Find the impedance of the circuit. (b) What is the phase angle? (c) Determine the effective current in the circuit. 18.5 Series combination of resistors, capacitors and inductors (SB p. 340) Solution

24. 23 © Manhattan Press (H.K.) Ltd. Solution: (a) Return to Text (b) The negative phase angle indicates that the voltage lags behind the effective current. (c) 18.5 Series combination of resistors, capacitors and inductors (SB p. 340)

25. 24 © Manhattan Press (H.K.) Ltd. Q: Q: A coil of inductance L and resistance R is connected in series with a capacitor of capacitance C and a variable frequency sinusoidal supply of constant r.m.s. values as shown in the figure. (a) (i) Explain why it is possible for the current in the circuit either to lead or lag behind the applied voltage. (ii) What is the condition for the current to be in phase with the applied voltage? (b) The sinusoidal supply has a frequency of 50 Hz. When the coil is replaced by a temperature- dependent resistor of resistance 5 000 Ω at 0°C, the phase angle between the applied voltage and the current in the resistor is 42°. What is the capacitance of the capacitor? (c) When the temperature of the resistor is raised to 25°C, the phase angle changes to 65°. What is the resistance of the resistor at this temperature? 18.5 Series combination of resistors, capacitors and inductors (SB p. 341) Solution

26. 25 © Manhattan Press (H.K.) Ltd. Solution: (a) (i) Since the inductor has resistance, the circuit actually consists of a resistor of resistance R, an inductor of inductance L and a capacitor of capacitance C in series (Fig. (a)). The voltage: V R = IR, V L = IX L and V C = IX C If V L V C, then from the phasor diagram in Fig. (c), the current I lags behind the applied voltage V. (ii) If V L = V C, then the applied voltage V = V R which is in phase with the current I. Fig. (a) Fig. (b) Fig. (c) 18.5 Series combination of resistors, capacitors and inductors (SB p. 341)

27. 26 © Manhattan Press (H.K.) Ltd. Solution (cont’d): (b) When the temperature-dependent resistor of resistance R is connected in series with the capacitor of capacitance C (Fig. (d)), the phasor diagram is shown in Fig. (e). Return to Text Fig. (d) Fig. (e) 18.5 Series combination of resistors, capacitors and inductors (SB p. 342)