1. DC CIRCUITS: CHAPTER 2 DET 101/3 Basic Electrical Circuit 1

2. RESISTIVE CIRCUITS –Series/Parallel Equivalent Circuits –Voltage Divider Rule (VDR) –Current Divider Rule (CDR) –Voltage and Current Measurements –Wheatstone Bridge –Delta (or Pi) and Wye (or Tee) Equivalent Circuit

3. Series/Parallel Equivalent Circuits Most common connection found in circuit analysis.Most common connection found in circuit analysis. Circuit simplifying technique.Circuit simplifying technique. Several resistors are combined to represent a single equivalent resistance.Several resistors are combined to represent a single equivalent resistance. Equivalent resistance depends on two (2) factors:Equivalent resistance depends on two (2) factors: –Type of connection –Point of terminals

4. Series Equivalent Circuit The equivalent resistance for any number of resistors in series connection is the sum of each individual resistor.The equivalent resistance for any number of resistors in series connection is the sum of each individual resistor. (2.1)

5. Series Equivalent Circuit (Continued…) Apparently the single equivalent resistor is always larger than the largest resistor in the series connection.Apparently the single equivalent resistor is always larger than the largest resistor in the series connection. Series resistors carry the same current thru them.Series resistors carry the same current thru them. Voltage across each of the resistors obtained using voltage divider rule principle or Ohm’s law.Voltage across each of the resistors obtained using voltage divider rule principle or Ohm’s law.

6. Parallel Equivalent Circuit The equivalent resistance for any number of resistors in parallel connection is obtained by taking the reciprocal of the sum of the reciprocal of each single resistor in the circuit.The equivalent resistance for any number of resistors in parallel connection is obtained by taking the reciprocal of the sum of the reciprocal of each single resistor in the circuit. (2.2)

7. Parallel Equivalent Circuit (Continued…) Apparently, the single equivalent resistor is always smaller than the smallest resistor in the parallel connection.Apparently, the single equivalent resistor is always smaller than the smallest resistor in the parallel connection. Voltage across each resistor must be the same.Voltage across each resistor must be the same. Currents thru each of them are divided according to the current divider rule principle.Currents thru each of them are divided according to the current divider rule principle.

8. Parallel Equivalent Circuit (Continued…) Special simplified formula if the number of resistors connected in series is limited to two elements i.e. N=2.Special simplified formula if the number of resistors connected in series is limited to two elements i.e. N=2. **When just two resistors connected in parallel the equivalent resistance is simply the product of resistances divided by its sum. (2.3)

9. Special Cases of Connections: Open Circuit (O.C) An opening exists somewhere in the circuit.An opening exists somewhere in the circuit. The elements are not connected in a closed path.The elements are not connected in a closed path. O.C: i = 0 A KVL: V oc = V s Ohm’s Law: R ab = V/I = ∞

10. Special Cases of Connections: Short Circuit (S.C) Both of its terminal are joint at one single node.Both of its terminal are joint at one single node. The element is bypassed. The element is bypassed. S.C: R ab = 0  Ohm’s Law: i = V s /(R 1 + R 3 ) : V sc = 0 V

11. Practice Problem 2.9 Q: By combining the resistors in Figure below, find R eq.

12. Practice Problem 2.10 Q: Find R ab for the circuit in Figure 2.39.

13. Practice Problem 2.11 Q: Calculate G eq in the circuit of Figure 2.41.

14. Voltage Divider Rule (VDR) Whenever voltage has to be divided among resistors in series use voltage divider rule principle.Whenever voltage has to be divided among resistors in series use voltage divider rule principle.

15. VDR (Continued…) In general, to find the voltage drop across the nth resistor in the voltage divider circuit configuration we use this formula:In general, to find the voltage drop across the nth resistor in the voltage divider circuit configuration we use this formula: Where n = 1, 2, 3,.....N (2.4)

16. Practice Problem 2.12 Find V 1 and V 2 in the circuit shown in Figure 2.43. Also calculate i 1 and i 2 and the power dissipated in the 12  and 40  resistors.Find V 1 and V 2 in the circuit shown in Figure 2.43. Also calculate i 1 and i 2 and the power dissipated in the 12  and 40  resistors.

17. Current Divider Rule (CDR) Whenever current has to be divided among resistors in parallel, use current divider rule principle.Whenever current has to be divided among resistors in parallel, use current divider rule principle.

18. CDR (Continued…) Circuit with more than two branches…Circuit with more than two branches… n = 1, 2, 3…..N In general, for N-conductors the formula represents: (2.5)

19. Practice Problem 2.13 Find (a)V 1 and V 2 (b) the power dissipated in the 3 k  and 20 k  resistors and (c) power supplied by the current source.Find (a)V 1 and V 2 (b) the power dissipated in the 3 k  and 20 k  resistors and (c) power supplied by the current source.

20. Chapter 2, Problem 34 Determine i 1, i 2, v 1, and v 2 in the ladder network in Fig. 2.98. Calculate the power dissipated in the 2-  resistor.Determine i 1, i 2, v 1, and v 2 in the ladder network in Fig. 2.98. Calculate the power dissipated in the 2-  resistor.

21. Chapter 2, Problem 36 Calculate V o and I o in the circuit of Fig. 2.100.Calculate V o and I o in the circuit of Fig. 2.100.

22. Voltage and Current Measurements To determine the actual and quantitative behavior of the physical system.To determine the actual and quantitative behavior of the physical system. Two most frequently used measuring devices in the laboratories:Two most frequently used measuring devices in the laboratories: – Ammeter –Voltmeter

23. Ammeter Must be placed in series connection with the element whose current is to be measured.Must be placed in series connection with the element whose current is to be measured. An ideal ammeter should have an equivalent resistance of 0  and considered as short circuit equivalent to the circuit where it is being inserted.An ideal ammeter should have an equivalent resistance of 0  and considered as short circuit equivalent to the circuit where it is being inserted.

24. Voltmeter Must be placed in parallel connection with the elements whose voltage is to be measured.Must be placed in parallel connection with the elements whose voltage is to be measured. An ideal voltmeter should have an equivalent resistance of ∞  and considered as open circuit equivalent to the circuit where it is being inserted.An ideal voltmeter should have an equivalent resistance of ∞  and considered as open circuit equivalent to the circuit where it is being inserted.

25. Meter Types Analog metersAnalog meters –Based on the d’Arsonval meter movements. Digital metersDigital meters –More popular than analog meters. –More precision in measurement, less resistance and can avoid severe reading errors. –Measure the continuous voltage or current at discrete instants of time called sampling times.

26. Configuration of Voltmeter and Ammeter In A Circuit

27. Wheatstone Bridge : Practical Application of Resistance Measurement Invented by a British professor, Charles Wheatstone in 1847.Invented by a British professor, Charles Wheatstone in 1847. More accurate device to measure resistance in the mid-range (1  to 1 M  )More accurate device to measure resistance in the mid-range (1  to 1 M  ) In commercial models of the Wheatstone bridge, accuracies about ± 0.1% are achievableIn commercial models of the Wheatstone bridge, accuracies about ± 0.1% are achievable

28. Wheatstone Bridge (Continued…) The bridge circuit consists ofThe bridge circuit consists of –Four resistors –A dc voltage source –A detector known as galvanometer (microampere range) Figure A

29. Balanced Bridge If R 3 is adjusted until the current I g in the galvanometer is zero the bridge its balance state.If R 3 is adjusted until the current I g in the galvanometer is zero the bridge its balance state. No voltage drop across the detector which means point a and b are at the same potential.No voltage drop across the detector which means point a and b are at the same potential. Implies that V 3 = V x when I g = 0 A.Implies that V 3 = V x when I g = 0 A.

30. Balanced Bridge (Continued…) Applying the voltage divider rule (VDR):Applying the voltage divider rule (VDR):

31. Balanced Bridge (Continued…) Since no current flows through the galvanometer,Since no current flows through the galvanometer, hence (2.6)

32. Example 1 The galvanometer shows a zero current through it when R x measured as 5 k . What do you expect to be the value of the adjustable resistor, R 3 ? Show your derivation in getting the formula.The galvanometer shows a zero current through it when R x measured as 5 k . What do you expect to be the value of the adjustable resistor, R 3 ? Show your derivation in getting the formula.

33. Exercise 1 The bridge in Figure A is energized by 6V dc source and balanced when R 1 = 200 , R 2 = 500  and R 3 = 800 .The bridge in Figure A is energized by 6V dc source and balanced when R 1 = 200 , R 2 = 500  and R 3 = 800 . (a) What is the value of R x ?(a) What is the value of R x ? (b) How much current (in miliamperes) does the dc source supply?(b) How much current (in miliamperes) does the dc source supply? (c) Which resistor absorbs the least power and which absorbs the most? How much?(c) Which resistor absorbs the least power and which absorbs the most? How much?

34. Unbalanced Bridge To find I g when the Wheatstone bridge is unbalanced, use Thevenin equivalent circuit concept to the galvanometer terminals.To find I g when the Wheatstone bridge is unbalanced, use Thevenin equivalent circuit concept to the galvanometer terminals. Assuming R m is the resistance of the galvanometer yields,Assuming R m is the resistance of the galvanometer yields, (2.7)

35. Delta (or Pi) and Wye (or Tee) Equivalent Circuit Stuck with neither series nor parallel connection of the resistors in part of a circuit.Stuck with neither series nor parallel connection of the resistors in part of a circuit. Simplify the resistive circuit to a single equivalent resistor by means of three-terminal equivalent circuit.Simplify the resistive circuit to a single equivalent resistor by means of three-terminal equivalent circuit.

36. Wye/Tee Circuit Same type of connectionsSame type of connections

37. Delta/Pi Circuit Same type of ConnectionsSame type of Connections

38. Delta-to-Wye and Wye-to-Delta Transformation Remember that before and after transformation using either Wye-to-Delta or Delta-to-Wye, the terminal behavior of the two configurations must retain.Remember that before and after transformation using either Wye-to-Delta or Delta-to-Wye, the terminal behavior of the two configurations must retain. Then only we can say that they are equivalent to each other.Then only we can say that they are equivalent to each other.

39. Special Case of  -Y Transformation A special case occur when R 1 = R 2 = R 3 = R Y or R a = R b = R c =R  under which the both networks are said to be balanced. Hence the transformation formulas will become:A special case occur when R 1 = R 2 = R 3 = R Y or R a = R b = R c =R  under which the both networks are said to be balanced. Hence the transformation formulas will become: R Y = R  /3orR  = 3R Y By applying Delta/Wye transformations, we may find that this final process leads to series/parallel connections in some parts of the circuit.By applying Delta/Wye transformations, we may find that this final process leads to series/parallel connections in some parts of the circuit.

40. Delta to Wye Transform To obtain the equivalent resistances in the Wye- connected circuit, we compare the equivalent resistance for each pair of terminals for both circuit configurations.To obtain the equivalent resistances in the Wye- connected circuit, we compare the equivalent resistance for each pair of terminals for both circuit configurations.

41. Delta to Wye Transform(Continued…) To retain the terminal behavior of both configurations i.e. R  = R YTo retain the terminal behavior of both configurations i.e. R  = R Y So that,So that, (2.8) (2.9) (2.10)

42. Delta to Wye Transform(Continued…) To obtain the resistance values for Y-connected elements, by straightforward algebraic manipulation and comparisons of the previous three equations gives,To obtain the resistance values for Y-connected elements, by straightforward algebraic manipulation and comparisons of the previous three equations gives, (2.11) (2.12) (2.13)

43. Wye to Delta Transform By algebraic manipulation, obtain the sum of all possible products of the three Y-connected elements; R 1, R 2 and R 3 in terms of  -connected elements; R a, R b and R c.(From Eq. (2.11 – 2.13)By algebraic manipulation, obtain the sum of all possible products of the three Y-connected elements; R 1, R 2 and R 3 in terms of  -connected elements; R a, R b and R c.(From Eq. (2.11 – 2.13) (2.14)

44. Wye to Delta Transform(Continued…) Then we divide Eq. (2.14) by each of Eq. (2.11) to (2.13) to obtain each of the  -connected elements as to be found variable in your left-side and its equivalent in Y-connected elements.Then we divide Eq. (2.14) by each of Eq. (2.11) to (2.13) to obtain each of the  -connected elements as to be found variable in your left-side and its equivalent in Y-connected elements. (2.14) / (2.11):

45. Wye to Delta Transform(Continued…) Using the same manner,Using the same manner, (2.14) / (2.12): (2.14) / (2.13): (2.14) / (2.11): (2.15) (2.16) (2.17)

46. Superposition of Delta and Wye Resistors “Each resistor in the Y-connected circuit is the product of the two resistors in two adjacent  branches divided by the sum of the three  resistors”“Each resistor in the Y-connected circuit is the product of the two resistors in two adjacent  branches divided by the sum of the three  resistors” “Each resistor in the  -connected circuit is the sum of all possible products of Y resistors taken two at a time divided by the opposite Y resistors”“Each resistor in the  -connected circuit is the sum of all possible products of Y resistors taken two at a time divided by the opposite Y resistors”

47. Practice Problem 2.15 Q: For the bridge circuit in Fig. 2.54, find R ab and i.

48. Exercise 2 Use  -to-Y transformation to find the voltages v 1 and v 2.Use  -to-Y transformation to find the voltages v 1 and v 2.

49. Exercise 3 Find the equivalent resistance R ab in the circuit below.Find the equivalent resistance R ab in the circuit below.

50. Exercise 4 Find R ab in the circuit below.Find R ab in the circuit below.