9nodes_corr_v05

1

(a)

Figure 1 9 nodes problem which has 81 coefficients. It can be solved by eliminate

undesired coefficients 63 times

The first step in principle is to eliminate one variable per one equation if the coefficient is not duplicate more than one time with both two equations.

(1)

2

The second step is to substitute (1) into (2) as following and then grouping the same node voltage

(2)

(2.2)

(2.3)

The third step is to group the coefficient of (2.2)

(2.4)

3

The fourth step is to update intermediate coefficient of (a)

(b)

The fifth step is to substitute (1) to (3) as following and then grouping the same node voltage

(3)

(3.2)

(3.3)

4

The sixth step is to group the coefficient of (3.2)

(3.4)

The seventh step is to update intermediate coefficient of (b)

(c)

The eighth step is to substitute (1) to (4) as following and then grouping the same node voltage

(4)

(4.2)

(4.3)

5

The ninth step is to group the coefficient of (4.2)

(4.4)

The tenth step is to update intermediate coefficient of (c)

(d)

The eleventh step is to substitute (1) to (5) as following and then grouping the same node voltage

(5)

(5.2)

(5.3)

6

The twelve step is to group the coefficient of (5.2)

(5.4)

The thirteenth step is to update intermediate coefficient of (d)

(d2)

The fourteenth step is to substitute (1) to (6) as following and then grouping the same node voltage

(6)

(6.2)

(6.3)

7

The 15th step is to group the coefficient of (6.2)

(6.4)

The 16th step is to update intermediate coefficient of (d2)

(d3)

The 17th step is to substitute (1) to (7) as following and then grouping the same node voltage

(7)

(7.2)

(7.3)

8

The 18th step is to group the coefficient of (7.2)

(7.4)

The 19th step is to update intermediate coefficient of (d3)

(d4)

The 20th step is to substitute (1) to (8) as following and then grouping the same node voltage

(8)

(8.2)

(8.3)

9

The 21th step is to group the coefficient of (8.2)

(8.4)

The 22th step is to update intermediate coefficient of (d4)

(d5)

The 23th step is to substitute (1) to (9) as following and then grouping the same node voltage

(9)

(9.2)

10

(9.3)

The 24th step is to group the coefficient of (9.2)

(9.4)

The 25th step is to update intermediate coefficient of (d5)

(d6)

The 26th step Let’s considering third step which has only 8 parameters in (2.4) .

Can we write from (2.4)?

(2.4b)

11

The 27th step Let’s substitute equation (2.4b) into equation (3.4) as following

(3.4b)

The 28th step is to group the coefficient of (3.4b) which have the same node voltage 7 nodes left

(3.4c)

12

Now, I must define intermediate coefficient in (3.4c) before run out of memory

(3.4d)

13

The 29th step is to update intermediate coefficient of (d6)

(d7)

The 30th step Let’s substitute equation (2.4b) into equation (4.4) as following

(4.4b)

14

(2.4b)

The 30.2th step is to group the coefficient of (4.4b) which have the same node voltage 7 nodes left

(4.4c)

Now, I must define intermediate coefficient in (4.4c) before run out of memory

15

(4.4d)

The 30.3th step is to update intermediate coefficient of (d7)

16

(d8)

The 31th step Let’s substitute equation (2.4b) into equation (5.4) as following

(5.4b)

(2.4b)

17

The 31.2th step is to group the coefficient of (5.4b) which have the same node voltage 7 nodes left

(5.4c)

18

Now, I must define intermediate coefficient in (5.4c) before run out of memory

(5.4d)

The 31.3th step is to update intermediate coefficient of (d8)

19

(d9)

The 32th step Let’s substitute equation (2.4b) into equation (6.4) as following

(6.4b)

(2.4b)

20

The 32.2th step is to group the coefficient of (6.4b) which have the same node voltage 7 nodes left

(6.4c)

21

Now, I must define intermediate coefficient in (6.4c) before I run out of memory

(6.4d)

The 32.3th step is to update intermediate coefficient of (d9)

22

(d10)

The 33th step Let’s substitute equation (2.4b) into equation (7.4) as following

(7.4)

(7.4b)

(2.4b)

23

The 33.2th step is to group the coefficient of (7.4b) which have the same node voltage 7 nodes left

(7.4c)

24

Now, I must define intermediate coefficient in (7.4c) before I run out of memory

(7.4d)

The 33.3th step is to update intermediate coefficient of (d10)

25

(d11)

The 34th step Let’s substitute equation (2.4b) into equation (8.4) as following

(8.4)

(2.4b)

(8.4b)

26

The 34.2th step is to group the coefficient of (8.4b) which have the same node voltage 7 nodes left

(8.4c)

Now, I must define intermediate coefficient in (8.4c) before I run out of memory

27

The 34.3th step is to update intermediate coefficient of (d11)

28

(d12)

The 35th step Let’s substitute equation (2.4b) into equation (9.4) as following

(9.4)

(2.4b)

(9.4b)

29

The 35.2th step is to group the coefficient of (9.4b) which have the same node voltage 7 nodes left

(9.4c)

Now, I must define intermediate coefficient in (9.4c) before I run out of memory

30

(9.4d)

The 35.3th step is to update intermediate coefficient of (d12)

31

(d13)

The 36th step Let’s see equation d13, we saw that first two row is in good position, so we can eliminate coefficients further by writing V3 as a function of other node voltage, then substitute into other row starting from third row until nine row to eliminate the total of 7 coefficients at third column

(3.4e)

From formula (3.4e), you can write as following

(3.4f)

Substitute (3.4f) into all remaining 6 row equations except the first row equation as following

32

(3.4f)

The 36.2th Grouping of the same node voltage in equation (3.4f)

(3.4g)

The 36.3th step is to update intermediate coefficient of (d13)

33

(d14)

The 37th step Let’s see equation d14, we saw that first fourth row is in good position, so we can eliminate coefficients further by writing V4 as a function of other node voltage, then substitute into other row starting from fourth row until ninth row to eliminate the total of 5 coefficients at fourth column.

(4.4e)

From formula (4.4e), you can write as following

34

(4.4f)

Substitute (4.4f) into all remaining 5 row equations except the first row equation as following

(4.4g)

The 37.2th Grouping of the same node voltage in equation (4.4g)

(4.4h)

The 37.3th step is to update intermediate coefficient of (d14)

35

(d15)

The 38th step Let’s see equation d15, we saw that first fifth row is in good position, so we can eliminate coefficients further by writing V5 as a function of other node voltage, then substitute into other row starting from sixth row until ninth row to eliminate the total of 4 coefficients at fifth column.

(5.4e)

From formula (5.4e), you can write as following

36

(5.5f)

Substitute (5.5f) into all remaining 4 row equations except the first row equation as following

(5.5g)

The 38.2th Grouping of the same node voltage in equation (5.5g)

(5.5h)

The 38.3th step is to update intermediate coefficient of (d15)

37

(d16)

The 39th step Let’s see equation d16, we saw that first sixth row is in good position, so we can eliminate coefficients further by writing V6 as a function of other node voltage, then substitute into other row starting from sixth row until ninth row to eliminate the total of 3 coefficients at sixth column.

(6.4e)

From formula (6.4e), you can write as following

38

(6.4f)

Substitute (6.4g) into all remaining 3 row equations in (6.4e) except the first row equation as following

(6.4g)

The 39.2th Grouping of the same node voltage in equation (6.4g)

(6.4h)

The 39.3th step is to update intermediate coefficient of (d16)

39

(d17)

The 40th step Let’s see equation d17, we saw that first seventh row is in good position, so we can eliminate coefficients further by writing V7 as a function of other node voltage, then substitute into other row starting from eighth row until ninth row to eliminate the total of 2 coefficients at seventh column.

(7.4e)

From formula (7.4e), you can write as following

40

(7.4f)

Substitute (7.4f) into all remaining 2 row equations in (7.4e) except the first row equation as following

(7.4g)

The 40.2th Grouping of the same node voltage in equation (7.4g)

(7.4h)

(7.4k)

The 40.3th step is to update intermediate coefficient of (d17)

(d18)

41

The 40.4th step we found that the first ratio of voltage was derived. After that you can determined other ratio from row 7 to row 1 which should have no duplication of both two parameters in the ratio. Substitute V8 from (7.4h) into seventh row equation as following

(7.4k)

The 40.5th step is to update intermediate coefficient of (d18)

(d19)

The 40.6th step we found that the first ratio of voltage was derived. After that you can determined other ratio from row 6 to row 1 which should have no duplication of both two parameters in the ratio. Substitute V7 from (7.4k) and V8 from (7.4h) into sixth row equation (6.4e) as following

42

(6.4f)

(6.4g)

(6.4h)

The 40.7th step is to update intermediate coefficient of (d19)

43

(d20)

The 41th step we found that the first ratio of voltage was derived. After that you can determined other ratio from row 5 to row 1 which should have no duplication of both two parameters in the ratio. Substitute V6 , V7 , V8 into fifth row equation (5.4e) as following

(5.4e)

(5.4f)

Determine the ratio of equation (5.4f) by grouping the same node voltage

(5.4g)

The 41.2th step is to update intermediate coefficient of (d20)

44

(d21)

The 42th step we found that the first ratio of voltage was derived. After that you can determined other ratio from row 4 to row 1 which should have no duplication of both two parameters in the ratio. Substitute V5 , V6 , V7 , V8 into fourth row equation (4.4e) as following

(4.4e)

(4.4f)

Determine the ratio of equation (4.4f) by grouping the same node voltage

(4.4g)

(4.4h)

The 42.2th step is to update intermediate coefficient of (d21)

45

(d22)