Chapter 5: Parallel dc Circuits

1. ENT 114: CIRCUIT THEORY Chapter 5: Parallel dc Circuits

2. RESISTORS IN PARALLEL Resistors that are connected to the same two points/nodes are said to be in parallel.

3. Schematic representations of three parallel resistors.

4. a) Parallel resistors; (b) R1 and R2 are in parallel; (c) R3 is in parallel with the series combination of R1 and R2.

5. PARALLEL RESISTORS • For resistors in parallel, the total resistance is determined from • Note that the equation is for the reciprocal of RT rather than for RT. • Once the right side of the equation has been determined, it is necessary to divide the result into 1 (RT-1) to determine the total resistance (6.1)

6. PARALLEL RESISTORS • Since G=1/R, the equation can also be written in terms of conductance as follows: • As the number of resistors in parallel increases, the input current level will increase for the same applied voltage. • This is the opposite effect of increasing the number of resistors in a series circuit. (Siemens, S) (6.2)

7. PARALLEL RESISTORS • The total resistance of any number of parallel resistors can be determined using • The total resistance of parallel resistors is always less than the value of the smallest resistor. • If the smaller resistor of a parallel combination is much smaller than the other parallel resistors, the total resistance will be very close to the smallest resistor value. (6.3)

8. EXAMPLE 5.1 • Find the total conductance of the parallel network in Figure 6.1. • Find the total resistance of the same network using the result of part (a) and using Eq. (6.3) Applying Eq.6.3

9. PARALLEL RESISTORS • For equal resistors in parallel: Where N = the number of parallel resistors. (6.4)

10. EXAMPLE 5.2 Find the total resistance of the parallel circuit

11. SPECIAL CASE: TWO PARALLEL RESISTORS • A special case: The total resistance of two resistors is the product of the two divided by their sum. • The equation was developed to reduce the effects of the inverse relationship when determining RT (6.5)

12. Summary Summary Special case for resistance of two parallel resistors The resistance of two parallel resistors can be found by either: or Question: What is the total resistance if R1 = 27 kW and R2 = 56 kW? 18.2 kW

13. EXAMPLE 5.3 • Find the total resistance of the same network using Eq. (6.5) Applying Eq.6.3

14. PARALLEL RESISTORS • Parallel resistors can be interchanged without changing the total resistance or input current. • For parallel resistors, the total resistance will always decrease as additional parallel elements are added.

15. Parallel Circuits Parallel circuits A parallel circuit is identified by the fact that it has more than one current path (branch) connected to a common voltage source.

16. For example, the source voltage is 5.0 V. What will a volt- meter read if it is placed across each of the resistors? Parallel Circuits Parallel circuit rule for voltage Because all components are connected across the same voltage source, the voltage across each is the same.

17. Parallel Circuits • Voltage is always the same across parallel elements. V1 = V2 = VS The voltage across resistor 1 equals the voltage across resistor 2, and both equal the voltage supplies by the source.

18. Parallel Circuits • The source current can be determined using Ohm’s Law: • Since the voltage is the same across parallel element, the current through each resistor can also be determined using Ohm’s Law. That is,

19. Parallel Circuits • For single-source parallel networks, the source current (Is) is equal to the sum of the individual branch currents. • For a parallel circuit, source current equals the sum of the branch currents. For a series circuit, the applied voltage equals the sum of the voltage drops.

20. Parallel Circuits • For parallel circuits, the greatest current will exist in the branch with the lowest resistance.

21. Power Distribution in a Parallel Circuit • For any resistive circuit, the power applied by the battery will equal that dissipated by the resistive elements. • The power relationship for parallel resistive circuits is identical to that for series resistive circuits.

22. Continuing with the previous example, complete the parameters listed in the Table. I1= R1= 0.68 kWV1= P1= I2= R2= 1.50 kWV2= P2= I3= R3= 2.20 kWV3= P3= IT= RT= 386 WVS= 5.0 V PT= Summary Summary Parallel circuit Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a parallel circuit. 7.4 mA 5.0 V 37.0 mW 3.3 mA 5.0 V 16.5 mW 2.3 mA 5.0 V 11.5 mW 13.0 mA 65.0 mW

23. The sum of the currents entering a node is equal to the sum of the currents leaving the node.In other words, the algebraic sum of the current at the junction is zero. Notice in the previous example that the current from the source is equal to the sum of the branch currents. I1= R1= 0.68 kWV1= P1= 7.4 mA 5.0 V 37.0 mW I2= R2= 1.50 kWV2= P2= 3.3 mA 5.0 V 16.5 mW I3= R3= 2.20 kWV3= P3= 2.3 mA 5.0 V 11.5 mW IT= RT= 386 WVS= 5.0 V PT= 13.0 mA 65.0 mW Kirchhoff’s Current Law Kirchhoff’s current law is generally stated as:

24. Kirchhoff’s Current Law • The sum of the currents entering a node is equal to the sum of the currents leaving the node.

25. Kirchhoff’s Current Law • The algebraic sum of the current at the junction/node is zero.

26. Example 5.4 • Determine the currents I1, I3, I4 and I5 for the network in figure:

27. Kirchhoff’s Current Law • Most common application of the law will be at the junction of two or more paths of current. • Determining whether a current is entering or leaving a junction is sometimes the most difficult task. • If the current arrow points toward the junction, the current is entering the junction. • If the current arrow points away from the junction, the current is leaving the junction.

28. Current Divider Rule • The current divider rule (CDR) is used to find the current through a resistor in a parallel circuit. • General points: • For two parallel elements of equal value, the current will divide equally. • For parallel elements with different values, the smaller the resistance, the greater the share of input current. • For parallel elements of different values, the current will split with a ratio equal to the inverse of their resistor values.

29. Current Divider Rule

30. Example 5.5 • For the parallel network, determine current I1

31. Current Divider Rule

32. Special case: Two Parallel Resistor Substituting RT into this equation for current I1 results in

33. Summary Summary Current divider When current enters a junction it divides with current values that are inversely proportional to the resistance values. The most widely used formula for the current divider is the two-resistor equation. For resistors R1 and R2, and Notice the subscripts. The resistor in the numerator is not the same as the one for which current is found.

34. Open and Short Circuits • An open circuit can have a potential difference (voltage) across its terminal, but the current is always zero amperes.

35. Open and Short Circuits • A short circuit can carry a current of a level determined by the external circuit, but the potential difference (voltage) across its terminals is always zero volts. Insert Fig 6.44

36. Key Terms One current path in a parallel circuit. Branch Current divider Junction Kirchhoff’s current law Parallel A parallel circuit in which the currents divide inversely proportional to the parallel branch resistances. A point at which two or more components are connected. Also known as a node. A law stating the total current into a junction equals the total current out of the junction. The relationship in electric circuits in which two or more current paths are connected between two separate points (nodes).