CHAP. 4]

39

ANALYSIS METHODS

ðRA þ RB ÞI1

 R B I2

¼ Va

RB I1 þ ðRB þ RC þ RD ÞI2  R D I3 ¼ 0 RD I2 þ ðRD þ RE ÞI3 ¼ Vb Placing the equations in matrix form, 2 RA þ RB 4 RB 0

RB RB þ RC þ RD RD

32 3 2 3 0 Va I1 RD 54 I2 5 ¼ 4 0 5 Vb RD þ RE I3

Fig. 4-3 The elements of the matrices can be indicated 2 R11 4 R21 R31

in general form as follows: 32 3 2 3 V1 R12 R13 I1 R22 R23 54 I2 5 ¼ 4 V2 5 R32 R33 I3 V3

ð6Þ

Now element R11 (row 1, column 1) is the sum of all resistances through which mesh current I1 passes. In Fig. 4-3, this is RA þ RB . Similarly, elements R22 and R33 are the sums of all resistances through which I2 and I3 , respectively, pass. Element R12 (row 1, column 2) is the sum of all resistances through which mesh currents I1 and I2 pass. The sign of R12 is þ if the two currents are in the same direction through each resistance, and  if they are in opposite directions. In Fig. 4-3, RB is the only resistance common to I1 and I2 ; and the current directions are opposite in RB , so that the sign is negative. Similarly, elements R21 , R23 , R13 , and R31 are the sums of the resistances common to the two mesh currents indicated by the subscripts, with the signs determined as described previously for R12 . It should be noted that for all i and j, Rij ¼ Rji . As a result, the resistance matrix is symmetric about the principal diagonal. The current matrix requires no explanation, since the elements are in a single column with subscripts 1, 2, 3, . . . to identify the current with the corresponding mesh. These are the unknowns in the mesh current method of network analysis. Element V1 in the voltage matrix is the sum of all source voltages driving mesh current I1 . A voltage is counted positive in the sum if I1 passes from the  to the þ terminal of the source; otherwise, it is counted negative. In other words, a voltage is positive if the source drives in the direction of the mesh current. In Fig. 4.3, mesh 1 has a source Va driving in the direction of I1 ; mesh 2 has no source; and mesh 3 has a source Vb driving opposite to the direction of I3 , making V3 negative.

The matrix equation arising from the mesh current method may be solved by various techniques. One of these, the method of determinants (Cramer’s rule), will be presented here. It should be stated, however, that other techniques are far more eﬃcient for large networks. EXAMPLE 4.4 Solve matrix equation (6) of Example 4.3 by the method of determinants. The unknown current I1 is obtained as the ratio of two determinants. The denominator determinant has the elements of resistance matrix. This may be referred to as the determinant of the coeﬃcients and given the symbol R . The numerator determinant has the same elements as R except in the ﬁrst column, where the elements of the voltage matrix replace those of the determinant of the coeﬃcients. Thus,        V1 R12 R13 , R11 R12 R13   V R12 R13       1  1     I1 ¼  V2 R22 R23   R21 R22 R23   V2 R22 R23    V3 R32 R33   R31 R32 R33  R  V3 R32 R33 

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An