Delta/Star* Transformation

In solving networks (having considerable number of branches) by the application of Kirchhoff s
Laws, one sometimes experiences great difficulty due to a large number of simultaneous equations
that have to be solved. However, such complicated network can be simplified by successively
replacing delta meshes by equivalent star system and vice versa.
Suppose we are given three resistance RI2' R23and R31connected in delta fashion between
terminals 1,2 and 3 as in Fig. 2.185 (a). So far as the respective terminals are concerned, these three
given resistances can be replaced by the three resistances RJ' R2and R3connected in star as shown in
Fig. 2.185 (b).
These two arrangements will be electrically equivalent if the resistance as measured between
any pair of terminals is the same in both the arrangements. Let us find this condition.
First, take delta connection: Between terminals 1 and 2, there are two parallel paths; one having
a resistance of RI2 and the other having a resistance of (R12+R31).
. R x (R + R )
.. ResIstance between terminals 1 and 2 is = n1l22+ (~33+ ;131
Now, take star connection: The resistance between the same terminals 1 and 2 is (R1+ R2).
As terminal resistances have to be the same
.. R1+ R2- _ R12~ ~~~!!"_R31)
RI2 + R23 + R31
Similarly, for terminals 2 and 3 and terminals 3 and I, we get
~3 x (R31+ R12)
R2 + R3 = -.-.
RI2 + R23 + R31
R +R - -R-31 x (R12 + R-2-3)
3 1 - RI2 + R23 + R31
Now, subtracting (ii) from (i) and adding the result to (;;i), we get
R R R R R R
RI=--1L JI_ ; R2= 1Ll.L - and R3= 31 23
RI2+ R23+ R31 RI2+ ~3 + R31 RI2+ R23+ R31How to Remember?
It is seen from above that each numerator is the product of the two sides of the delta which meet
at the point in star. Hence, it should be remembered that: resistance of each arm of the star is given
by the product of the resistances of the two delta sides that meet at its end divided by the sum of the
three delta resistances.
Compensation Theorem

~ortoo's Th1:9FeJr..
This theorem is an alternative to the Thevenin's theorem. In fact, it is the dual of Thevenin's
theorem. Whereas Thevenin's theorem reduces a two-terminal active network of linear resistances
and generators to an equivalent constant-voltage source and series resistance, Norton's theorem
replaces the network by an equivalent constant-current source and a parallel resistance.
This theorem is particularly usful for the following two purposes :
(a) For analysing those networks where the values of the branch elements are varied and for
studying the effect of tolerance on such values.
(b) For calculating the sensitivity of bridge network.
As applied to d.c. circuits, it may be stated in the following to ways :
(i) In its simplest form. this theorem asserts that any resistance R in a branch of a network in
which a current I isflowing can be replaced,for the purposes of calculations, by a voltage
equal to - IR.
OR
(ii) If the resistance of any branch of network is changedfrom R to (R + M) where the current This theorem may be stated as follows :
(i) Any two-terminal active network containinll voltalle sources and resistance when viewed
from its oUtpUIle.,IfUtUlI,:iI,)equIvalent to a consrant-.c-urrentsource and a Darallei resistance. The --~...
constant current is equaf to the current which would flow m a short-circuit placed across the
terminals and parallel resistance is the resistance of the network when viewed from these open-
, circuited terminals after all voltage and current sources have been removed and replaced by their
internal resistances.flowing originally is I, the change of current at any other place in the network may be
calculated by assuming that an e.m.f. - I. M has been injectedinto the modifiedbranch
while all other sources have their e.m.f.s. suppressed and are represented by their internal
resistances only As seen from Fig. 2.202 (a), a short is placed across the terminals A and B of the network with
all its energy sources present. The short-circuit current Isc gives the value of constant-current
source.
For finding Rj' all sources have been removed as shown in Fig. 2.202 (b). The resistance of the
network when looked into from terminals A and B gives Rj'
The Norton's equivalent circuit is shown in Fig. 2.202 (c). It consists of an ideal constantcurrent
source of infinite internal resistance (Art. 2.16) having a resistance of Rj connected in parallel
with it. Solved Examples 2.96, 2.97 and 2.98 etc. illustrate this procedure.
(ii) Another useful generalized form of this theorem is as follows:
The voltage between any two points in a network is equal to ISc. Rj where Iscis the short-circuit
current between the two points and Rj is the resistance of the network as vil;!wedfrom these points
with all voltage sources being replaced by their internal resistances (if any) and current sources
replaced by open-circuits.
Suppose, it is required to find the voltage across resistance R3and hence current through if [Fig.
2.202 (d)]. If short-circuit is placed between A and B, then current in it due to battery of e.m.f. EI is
E/RI and due to the o~er battery is E.jR2'