Method helps determine branch current in parallel-series network in relation to total input current by inspection. Method is particularly useful for circuits with many elements when branch elements are described as admittances. If element values are variables, then these values may be expressed as admittances to find currents readily in desired branches

Tapia, M. A.

English

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• August 1974	 B74-10093 
NASA TECH BRIEF 
•	 Langley Research Center 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
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be directed to the Technology Utilization Office, NASA, Code 1(1, Washington, D.C. 20546. 
Generalized Current Distribution Rule 
A method has been developed which helps to de-
termine the branch current in a parallel-series network 
in relation to the total input current by inspection. 
The method is particularly useful for circuits with many 
elements when the branch elements are described as 
admittances. If the element values are variables, then 
these values may be expressed as admittances to find 
currents readily in desired branches. 
A parallel-series network, N, with k branches b, 
lik, in parallel, with branch b comprising ad - 
mittances g1 1' gj 2'	 in series, has current I 
entering its ' •••termihal node 1 and leaving its terminal 
node 1' as shown in Figure 1. The objective is to obtain 
an expression for the current ratio Is/I, where Ii is the 
current in the branch b1. 
By letting P1 be the product of all admittances in 
i-th branch, P1 becomes
ni 
P =
	
(1)
Further, Qi
 is the sum of products of admittances in the 
i-th branch taken (ni—I) at a time, if (ni-1)>O and 
defined by Qi = 1, if ni = i. Next define Q as
(2) 
Finally, Ti is defined as the sum of admittance products 
of all trees that intersect in the branch b 1 , which is the 
i-th tree admittance products sum given by 
Ti = PQQ 1	 (3) 
To facilitate evaluation of T, a nontopological definition 
of Ti is given later after equation 5. The sum T of tree 
admittance products of N becomes 
k 
T = T1	 (4) 
i 1
S	 Figure 1. Parallel-Series Network (continued overleaf) 
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Now the current ratio I a/I is equal to the ratio of 
the admittance of branch b 1 to the admittance of the 
total network. The admittance ratio. can be shown to 
be:
sum of tree admittance products of branch b 
sum 'Tof 2-tree admittance products of b 
sum of 2-tree adriiittance products of N 
sum of tree admittance products of N 
Hence, using the previously defined terms, the current 
ratio is described by 
III = (P1/Q1)(Q/T) = (PQQ 1)/T 
= T/T	 (5) 
Equation 5 gives the generalized current distribution 
rule in terms of topological concepts, which can be easily 
obtained by inspection. Familiarity with topological con-
cepts, however, is not necessary in order to find the right-
hand side of equation 5 in terms of g1 's. The sum Tican 
be looked upon as a product of k terms, each correspond-
ing to a branch of N, with the i-th term consisting of the 
product of all admittances in the i-th branch and the
As an example, the developed method is applied to 
the network shown in Figure 2. Using equation 5, 
current ratio 1 2 /I is evaluated as follows: 
12/I = (9 1 +9 2) (939495) (9697+9698+9798) 
[(9 1 92 ) (9394+9395 +9495) (9697+9698+9798) 
+(91+92)(939495)(9697+9698+9798) 
+(91+92)(9495+9493+9593)(969798)1-1 
Observe that:
T2 
12/I=T2/T=
T1+T2+T3 
where 
T 1 = (9 1 92 ) (9394+9395 +9495) (9697+9698+9798) 
T2 = (9 1 +92 ) (939495) (96979698 +9798) and 
T3 =(91+92)(9394+9395+9495)(969798) 
Thus, in a parallel-series network, the current that flows 
in a network branch is proportional to the sum of 
admittance products of trees of the network that include 
the branch. 
Note: 
No further documentation is available. Specific ques-
tions, however, may be directed to: 
Technology Utilization Officer 
Langley Research Center 
97	 Mail Stop 139-A 
Hampton, Virginia 23665 
Reference: B74-10093 
Patent status: 
NASA has decided not to apply for a patent.
S 
Figure 2. Parallel-Series Network 
With Eight Admittance 
Elements 
j-th term, for all j i, consisting of the sum of the 
products of all possible (nj-1) admittances in the j-th 
- - branch, the j-th term being unity if and only if nj = i -
Source: Moiez A. Tapia of
Georgia Institute of Technology 
under contract to
Langley Research Center 
(LAR-1 15 65) 
B74-10093
	
Category 02,09 •


Generalized current distribution rule

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