# Chapter 7. Chapter 7

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1 Chapter 7

2 Combination circuits Most practical circuits have combinations of series and parallel components. You can frequently simplify analysis by combining series and parallel components. An important analysis method is to form an equivalent circuit. An equivalent circuit is one that has characteristics that are electrically the same as another circuit but is generally simpler.

3 Combination circuits For example: R1 1.0 k R k is equivalent to R k There are no electrical measurements that can distinguish the boxes.

4 Combination circuits Another example: is equivalent to R1 R2 1.0 k 1.0 k R 1,2 500 There are no electrical measurements that can distinguish the boxes.

5 is equivalent to R1 1.0 k R k R k R 3.7 k 1,2 3 R 4.7 k is equivalent to R 1,2, k There are no electrical measurements that can distinguish between the three boxes.

6 Combination circuits Kirchhoff s voltage law and Kirchhoff s current law can be applied to any circuit, including combination circuits. For example, applying KVL, the path shown will have a sum of 0 V. 0 VR 2 VR4 VR5 VR V 0.417V V S 5.0 V Start/Finish V S 0.417V 0.417V R V R V R R V R V R V

7 Combination circuits Kirchhoff s voltage law and Kirchhoff s current law can be applied to any circuit, including combination circuits. So will this path! 0 VR 2 VR V V S 1.25V V S 5.0 V 5V + - St a rt/finish R R V 1.25V R V R R R 5 100

8 Kirchoff s current law can also be applied to the same circuit. What are the readings for node A? V S 5.0 V + I ma I I T = I R1 + I R2 R1 A V R S 1 I ma 5V 270 I ma R I R2 R R 3 R 2 5V VS [ R3 ( R4 R5 R6)] 5V 470 [ ] R R 6 8.0mA R V

9 Combination circuit V S + 10 V R 2 R R 3 Tabulating current, resistance, voltage and power is a useful way to summarize parameters. Solve for the unknown quantities in the circuit shown. I 1 = 21.6 ma R 1 = 270 V 1 = 5.82 V P 1 = 126 mw I 2 = 12.7 ma R 2 = 330 V 2 = 4.18 V P 2 = 53.1 mw I 3 = 8.9 ma R 3 = 470 V 3 = 4.18 V P 3 = 37.2 mw I T = 21.6 ma R T = 464 V S = 10 V P T = 216 mw

10 Kirchhoff s laws can be applied as a check on the answer. V S + 10 V R 2 R R 3 Notice that the current in R 1 is equal to the sum of the branch currents in R 2 and R 3. The sum of the voltages around the outside loop is zero. I 1 = 21.6 ma R 1 = 270 V 1 = 5.82 V P 1 = 126 mw I 2 = 12.7 ma R 2 = 330 V 2 = 4.18 V P 2 = 53.1 mw I 3 = 8.9 ma R 3 = 470 V 3 = 4.18 V P 3 = 37.2 mw I T = 21.6 ma R T = 464 V S = 10 V P T = 216 mw

11 Loading effect of a voltmeter V S + 10 V Assume V S = 10 V, but the meter reads only 4.04 V when it is across either R 1 or R 2. Can you explain what is happening? R k R 2 47k All measurements affect the quantity being measured. A voltmeter has internal resistance, which can change the resistance of the circuit under test. In this case, a 1 M internal resistance of the meter accounts for the readings V V

12 Loading effect on Voltage-dividers I Bleeder is the leftover current that flows in the divider after splitting through the load(s) k If R L = 1kΩ, then V OUT = 10V V 3.33 I R1 = I R2 + I RL I T IT I T 5V I T I R2 = I Bleeder I RL

13 Loading effect on Voltage-dividers As R L increases, less load current is drawn from the divider, larger percentage of I T flows through R 2, V OUT approaches maximum value. As R L decrease, more load current is drawn from the divider, smaller percentage of I T flows through R 2, V OUT approaches 0V. V OUT = Maximum V OUT = Approaches 0V V OUT = Approaches Maximum

14 Loaded Voltagedividers Current paths Find all voltages and currents! Without R L2 and R L1, V A 12V and V B 6V

15 Loading effect on Voltage-divider SOLUTION 10.7k 22.7k Find V A = V RL1 : 24V 11.31V

16 Loading effect on Voltage-divider SOLUTION Find V R1 : VS VA 24V 11.31V 12. 7V

17 Loading effect on Voltage-divider SOLUTION 5.84k 12.04k Find V B = V RB = V RL2 = V R3 : V A 11.31V 5. 49V

18 Loading effect on Voltage-divider SOLUTION Find V R2 : VA VB 11.31V 5.49V 5. 82V

19 Loading effect on Voltage-divider SOLUTION All Voltages Found! 11.31V 5.49V 12.7V 5.82V 5.49V 5.49V 11.31V

20 Loading effect on Voltage-divider SOLUTION Find I T : Check: VR1 R V 12k 1.06mA R 1 VS 24V RA 22.7k 1.06mA 11.31V 12.7V 5.82V 5.49V 5.49V 5.49V 11.31V

21 Loading effect on Voltage-divider Find I RL1 : ( R2 R ( R2 RB) 113.9uA B Disregard rounding errors ) 12.04k ( I RL k T 1.06mA) 11.31V SOLUTION Check: V R RL1 L1 12.7V 5.82V 11.31V 100k 113.1uA 5.49V 5.49V 5.49V 11.31V

22 Loading effect on Voltage-divider Find I R2 : I T I RL1 Disregard rounding errors (1.06mA) (113.9uA) 0.946mA 11.31V SOLUTION Check: VRL2 R V 5.82V 5.82V 6.2k 0.939mA 5.49V 5.49V 5.49V 11.31V

23 Loading effect on Voltage-divider Find I R3 : RL2 100k R3 RL k 891uA Disregard rounding errors I R mA 11.31V SOLUTION Check: VR3 R V 5.82V 5.49V 6.2k 885.5uA 5.49V 5.49V 5.49V 11.31V

24 Loading effect on Voltage-divider Find I RL2 : I R2 I R3 55uA Disregard rounding errors (0.946mA) (891uA) 11.31V SOLUTION Check: V R RL2 L2 12.7V 5.82V 5.49V 100k 54.9uA 5.49V 5.49V 5.49V 11.31V

25 Loading effect on Voltage-divider SOLUTION All Currents Found! 1.06mA 0.946mA 55uA 113.9uA 891uA

26 Voltage-divider with Positive and Negative Voltage Taps

27 Wheatstone bridge The Wheatstone bridge consists of four resistive arms forming two voltage dividers and a dc voltage source. The output is taken between the dividers. Frequently, one of the bridge resistors is adjustable. When the bridge is balanced, the output voltage is zero, and the products of resistances in the opposite diagonal arms are equal. R R4 R2 R3 V 1 1V 4 V 2 V 3 R R R 1 R 2, 1 3 R R 2 4 R 3 V V 1 3 R 4 V V 2 4

28 Wheatstone bridge Example: What is the value of R 1 if the bridge 12 V is balanced when R 3 is adjusted to 384 Ω? R R 1 1 R R3 R

29 Wheatstone Bridge with Strain-gauges

30 Balanced bridge Bleeder current Load Key Terms A bridge circuit that is in the balanced state is indicated by 0 V across the output. The current left after the load current is subtracted from the total current into the circuit. An element (resistor or other component) connected across the output terminals of a circuit that draws current from the circuit.

31 Key Terms Unbalanced bridge Wheatstone bridge A bridge circuit that is in the unbalanced state is indicated by a voltage across the output that is proportional to the amount of deviation from the balanced state. A 4-legged type of bridge circuit with which an unknown resistance can be accurately measured using the balanced state. Deviations in resistance can be measured using the unbalanced state.

32 Quiz 1. Two circuits that are equivalent have the same a. number of components b. response to an electrical stimulus c. internal power dissipation d. all of the above

33 Quiz 1. Two circuits that are equivalent have the same a. number of components b. response to an electrical stimulus c. internal power dissipation d. all of the above

34 Quiz 2. If a series equivalent circuit is drawn for a complex circuit, the equivalent circuit can be analyzed with a. the voltage divider theorem b. Kirchhoff s voltage law c. both of the above d. none of the above

35 Quiz 2. If a series equivalent circuit is drawn for a complex circuit, the equivalent circuit can be analyzed with a. the voltage divider theorem b. Kirchhoff s voltage law c. both of the above d. none of the above

36 Quiz 3. For the circuit shown, a. R 1 is in series with R 2 R 1 R 2 b. R 1 is in parallel with R 2 c. R 2 is in series with R 3 V S + - R 3 d. R 2 is in parallel with R 3

37 Quiz 3. For the circuit shown, a. R 1 is in series with R 2 R 1 R 2 b. R 1 is in parallel with R 2 c. R 2 is in series with R 3 V S + - R 3 d. R 2 is in parallel with R 3

38 Quiz 4. For the circuit shown, a. R 1 is in series with R 2 b. R 4 is in parallel with R 1 R 1 R 4 R 2 c. R 2 is in parallel with R 3 V S + R 3 d. none of the above -

39 Quiz 4. For the circuit shown, a. R 1 is in series with R 2 b. R 4 is in parallel with R 1 R 1 R 4 R 2 c. R 2 is in parallel with R 3 V S + R 3 d. none of the above -

40 Quiz 5. The total resistance, R T, of the group of resistors is a. 1.0 k b. 2.0 k c. 3.0 k d. 4.0 k R T R R 5 R 3 R k 1.0 k R4 1.0 k

41 Quiz 5. The total resistance, R T, of the group of resistors is a. 1.0 k b. 2.0 k c. 3.0 k d. 4.0 k R T R R 5 R k R k R4 1.0 k RT R1 R5 [ R3 ( R2 R4)]

42 Quiz 6. For the circuit shown, Kirchhoff's voltage law a. applies only to the outside loop b. applies only to the A junction. c. can be applied to any closed path. d. does not apply. V S + 10 V A R R 2 R 3

43 Quiz 6. For the circuit shown, Kirchhoff's voltage law a. applies only to the outside loop b. applies only to the A junction. c. can be applied to any closed path. d. does not apply. V S + 10 V A R R 2 R 3

44 Quiz 7. The effect of changing a measured quantity due to connecting an instrument to a circuit is called a. loading b. clipping c. distortion d. loss of precision

45 Quiz 7. The effect of changing a measured quantity due to connecting an instrument to a circuit is called a. loading b. clipping c. distortion d. loss of precision

46 Quiz 8. Assume R 2 is adjusted until the Wheatstone bridge is balanced. At this point, the voltage across R 4 is measured and found to be 5.0 V. The voltage across R 1 will be a. 4.0 V b. 5.0 V c. 6.0 V d. 7.0 V V S 12 V + - R 1 R 3 R 2 R L + - R V

47 Quiz 8. Assume R 2 is adjusted until the Wheatstone bridge is balanced. At this point, the voltage across R 4 is measured and found to be 5.0 V. The voltage across R 1 will be a. 4.0 V b. 5.0 V c. 6.0 V d. 7.0 V V S 12 V + - R 1 R 3 R 2 R L + - 0V R V VR1 VR3 VS VR4 12V 5V 7V

48 Quiz 9. An unbalanced Wheatstone bridge has the voltages shown. The voltage across R 4 is a. 4.0 V b. 5.0 V c. 6.0 V V S 12 V + - R 1 R V + R L V d. 7.0 V R 2 R 4

49 Quiz 9. An unbalanced Wheatstone bridge has the voltages shown. The voltage across R 4 is a. 4.0 V b. 5.0 V c. 6.0 V d. 7.0 V V S 12 V + - R 1 R V 8V + R L V 5V R 2 R 4 4V

50 Quiz 10. For the circuit shown, if R 3 opens, the voltage at point A will a. decrease b. stay the same. c. increase. V S + 10 V A R 2 R R 3

51 Quiz 10. For the circuit shown, if R 3 opens, the voltage at point A will a. decrease b. stay the same. c. increase. V S + 10 V A R 2 R R 3

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