Have you
ever imagined a situation where the resistors in a circuit are neither in
series nor in parallel? Often, circuits are drawn complex and therefore it
becomes difficult to identify whether the resistors are in series or in
parallel.
Consider the
following example and try figuring out the series and parallel resistors!
What’s your
conclusion? Try matching it with mine.
There are no
resistances which are present in series. Resistors R2 and R6
are present in parallel. This is what is observable from this circuit. The
resistors R5, R6, and R7 are neither in series
nor in parallel with any other resistance. The resistors R5, R6,
and R7 make what is called a Delta connection.
Generally
there are two types of such networks that represent a situation where the
resistances are neither in series nor in parallel. They are called Delta and
Wye (Star) networks. Solving these networks is not a big problem. Our main aim
here is in how to identify them when they occur as a part of a network and how
to apply wye – delta transformation in the analysis of that network.
STAR (WYE)
AND DELTA CONNECTIONS –
WYE NETWORK
DELTA NETWORK
WYE-DELTA SUPERIMPOSED
DELTA TO WYE
TRANSFORMATION –
If the two
arrangements shown in the above figure are to be equivalent, the resistance
between any pair of terminals (AB, BC, or CA) of the two circuits has to be the
same, when the third line is left open. Suppose, terminal A is left open, and
we equate the resistance between B and C, to get
${{R}_{B}}+{{R}_{C}}=\frac{{{R}_{1}}({{R}_{2}}+{{R}_{3}})}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$
…Eq.1
Similarly, we can write
${{R}_{C}}+{{R}_{A}}=\frac{{{R}_{2}}({{R}_{1}}+{{R}_{3}})}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$ …Eq.2
and
${{R}_{A}}+{{R}_{B}}=\frac{{{R}_{3}}({{R}_{1}}+{{R}_{2}})}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$ …Eq.3
Adding the
above three equations and dividing by 2, we get
${{R}_{A}}+{{R}_{B}}+{{R}_{C}}=\frac{{{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$ …Eq.4
Now,
subtracting Eq.1 from Eq.4, we get
${{R}_{A}}=\frac{{{R}_{2}}{{R}_{3}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$
…Eq.5
Similarly,
we get
${{R}_{B}}=\frac{{{R}_{3}}{{R}_{1}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$ …Eq.6
and
${{R}_{C}}=\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$ …Eq.7
Eq.5, 6 and
7 are the set of equations which transform a delta connection to its equivalent
wye (star) connection. These relationships can therefore be remembered as –
The equivalent
star resistance connected to a given terminal is equal to the product of the
two delta resistances connected to the same terminal divided by the sum of
delta resistances.
NOTE – If in
Delta – connection, R1=R2=R3=R, then
\[{{R}_{Y}}(\text{Resistances
of Y connection})=\frac{R.R}{R+R+R}=\frac{R}{3}\]
WYE TO DELTA
TRANSFORMATION –
If the
situation demands that the presence of a wye connected system would make the
circuit easier to solve, then we need to convert the delta connection to wye
connection.
To obtain
the formulae for wye to delta transformation, we multiply Eq.5 and 6 to get
${{R}_{A}}{{R}_{B}}=\frac{{{R}_{1}}{{R}_{2}}R_{3}^{2}}{{{\left(
{{R}_{1}}+{{R}_{2}}+{{R}_{3}} \right)}^{2}}}$
…Eq.8
Similarly,
we get
${{R}_{B}}{{R}_{C}}=\frac{R_{1}^{2}{{R}_{2}}{{R}_{3}}}{{{\left(
{{R}_{1}}+{{R}_{2}}+{{R}_{3}} \right)}^{2}}}$ …Eq.9
and
${{R}_{C}}{{R}_{A}}=\frac{{{R}_{1}}R_{2}^{2}{{R}_{3}}}{{{\left(
{{R}_{1}}+{{R}_{2}}+{{R}_{3}} \right)}^{2}}}$ …Eq.10
Adding the
above three equations, we get
\[{{R}_{A}}{{R}_{B}}+{{R}_{B}}{{R}_{C}}+{{R}_{C}}{{R}_{A}}=\frac{{{R}_{1}}{{R}_{2}}R_{3}^{2}+R_{1}^{2}{{R}_{2}}{{R}_{3}}+{{R}_{1}}R_{2}^{2}{{R}_{3}}}{{{\left(
{{R}_{1}}+{{R}_{2}}+{{R}_{3}} \right)}^{2}}}\]
$=\frac{{{R}_{1}}{{R}_{2}}{{R}_{3}}}{{{R}_{1}}+{{R}_{2}}+{{R}_{3}}}$
…Eq.11
Dividing
Eq.11 by each of Eq.5, 6 and 7, we get
\[{{R}_{1}}=\frac{{{R}_{A}}{{R}_{B}}+{{R}_{B}}{{R}_{C}}+{{R}_{C}}{{R}_{A}}}{{{R}_{A}}}\]
Similarly,
\[{{R}_{2}}=\frac{{{R}_{A}}{{R}_{B}}+{{R}_{B}}{{R}_{C}}+{{R}_{C}}{{R}_{A}}}{{{R}_{B}}}\]
And
\[{{R}_{3}}=\frac{{{R}_{A}}{{R}_{B}}+{{R}_{B}}{{R}_{C}}+{{R}_{C}}{{R}_{A}}}{{{R}_{C}}}\]
To remember
these equations:
The
equivalent delta resistances is the sum of possible products of wye resistors
taken two at a time, divided by the opposite wye resistor.
NOTE: If in
wye connection, RA = RB = RC = R then
\[{{R}_{\Delta
}}=\frac{{{R}^{2}}+{{R}^{2}}+{{R}^{2}}}{R}=3{{R}_{Y}}\]
Note that in
making the transformations, we do not take anything out or put anything into
the circuit. We merely substitute an equivalent three terminal network in place
of other so as to find REQ of the circuit easily.
