1. Electrical Network Analysis & Traffic Flow 1

2. Electrical Network Analysis Systems of linear equations are used to determine the currents through various branches of electrical networks. The following two laws, which are based on experimental verification in the laboratory, lead to the equations. 2

3. Kirchhoff’s Laws Gustav Robert Kirchhoff German Physicist 1824 - 1887 1. Junctions: All the current following into a junction must flow out of it. 2. Paths: The sum of the IR terms (I denotes current, R resistance) in any direction around a closed path is equal to the total voltage in the path in that direction. 3 Remark: A closed path is a sequence of branches such that the beginning point of the first branch coincides with the end point of the last branch.

4. Example Consider the electrical network in the below figure. Determine the currents through each branch of this network. 4 Batteries Resistances

5. Solution Two Batteries are 8 and 16 volts. Three resistances one 1-ohm, one 4-ohms, and two 2-ohms. Let the currents in the various branches of the circuit be I 1, I 2, and I 3. Kirchhoff’s laws refer to junctions and closed paths. There are two junctions in the circuit, namely the points B and D. There are three closed paths, namely ABDA, CBDC, and ABCDA. 5

6. Apply the laws to the junctions. Junctions: Junction B, I 1 + I 2 = I 3 Junction D, I 3 = I 1 + I 2 These two equations result in a single linear equation I 1 + I 2 - I 3 = 0 6

7. Apply the laws to the paths. Paths: Path ABDA, 2I 1 + 1I 3 + 2I 1 = 8 Path CBDC, 4I 2 + 1I 3 = 16 It is not necessary to look further at path ABCDA. We now have a system of three linear equations with three unknowns, I 1, I 2, and I 3. Path ABCDA in fact leads to an equation that is a combination of the last two equations; there is no new information. 7

8. I 1 + I 2 - I 3 = 0 4I 1 + I 3 = 8 4I 2 + I 3 = 16 8

9. The currents are I 1 = 1, I 2 = 3, and I 3 = 4 amps. The solution is unique, as to be expected in this physical situation. 9

10. Another Example Consider the electrical network in the below figure. Determine the currents through each branch of this network. Note: This example illustrates how one has to be conscious of direction in applying law 2 for closed paths. 10

11. Solution Two Batteries are 12 and 16 volts. Three resistances one 1-ohm and two 2-ohms. Let the currents in the various branches of the circuit be I 1, I 2, and I 3. Kirchhoff’s laws refer to junctions and closed paths. There are two junctions in the circuit, namely the points B and D. There are two closed paths, namely ABCDA and ABDA. 11

12. Apply the laws to the junctions. Junctions: Junction B, I 1 + I 2 = I 3 Junction D, I 3 = I 1 + I 2 These two equations result in a single linear equation I 1 + I 2 - I 3 = 0 12

13. Apply the laws to the paths. Paths: Path ABCDA, 1I 1 + 2I 3 = 12 Path ABDA, 1I 1 + 2(-I 2 ) = 12 + (-16) Observe we have selected the direction ABDA around this last path. The current along the branch BD in thie direction is -I 2, and the voltage is -16. We now have a system of three linear equations with three unknowns, I 1, I 2, and I 3. 13

14. I 1 + I 2 - I 3 = 0 I 1 + 2I 3 = 12 I 1 - 2I 2 = -4 Solving these equations, we get: I 1 = 2, I 2 = 3, and I 3 = 5 amps 14

15. Traffic Flow Network analysis, as we saw in the previous discussion, plays an important role in electrical engineering. In recent years, the concepts and tools of network analysis have been found to be useful in many other fields, such as information theory and the study of transportation systems. 15

16. Example Consider the typical road network shown below. 16

17. The streets are all one-way with arrows indicating the direction of traffic flow. The traffic is measured in vehicles per hour (vph). The figures in and out of the network given here are based on midweek peak traffic hours, 7 A.M. to 9 A.M. and 4 P.M. to 6 P.M. Let us construct a mathematical model that can be used to analyze the flow x 1,…,x 4 within the network. 17

18. Assume the following traffic law applies. All traffic entering an intersection must leave that same intersection. 18

19. This conservation of flow constraint (compare it to the first Kirchhoff’s laws for electrical networks) leads to a system of linear equations. These are, by intersection: A: Traffic in = x 1 + x 2. Traffic out = 400 + 225. Thus x 1 + x 2 = 625. B: Traffic in = 350 + 125. Traffic out = x 1 + x 4. Thus x 1 + x 4 = 475. C: Traffic in = x 3 + x 4. Traffic out = 600 + 300. Thus x 3 + x 4 = 900. D: Traffic in = 800 + 250. Traffic out = x 2 + x 3. Thus x 2 + x 3 = 1,050. 19

20. System of Linear Equations The constraints on the traffic are described by the following system of linear equations. x 1 + x 2 = 625 x 1 + x 4 = 475 x 3 + x 4 = 900 x 2 + x 3 = 1,050 20

21. Augmented Matrix 21

22. X 1 + x 4 = 475 x 2 - x 4 = 150 x 3 + x 4 = 900 Expressing each leading variable in terms of the remaining variable, we get x 1 = -x 4 + 475 x 2 = x 4 + 150 x 3 = -x 4 + 900 22

23. As was perhaps to be expected the system of equations has many solutions – there are many traffic flows possible. One does have a certain amount of choice at intersections. Let us now use this mathematical model to arrive at information. Suppose it becomes necessary to perform road work on the stretch DC of Monroe Street. It is desirable to have as small a flow x 3 as possible along this stretch of road. The flows can be controlled along various branches by means of traffic lights. 23

24. What is the minimum value of x 3 along DC that would not lead to traffic congestion? We use the preceding system of linear equations to answer this question. All traffic flows must be nonnegative (a negative flow would be interpreted as traffic moving in the wrong direction on a one-way street). The third equation, x 3 = -x 4 + 900, tells us x 3 will be a minimum when x 4 is as large as possible, as long as it does not go above 900. 24

25. The largest value x 4 can be without causing negative values of x 1 or x 2 is 475. Thus the smallest value of x 3 is -475 + 900, or 425. Any road work on Monroe Street should allow for at least 425 vph. 25