Network Theory - Star to Delta Conversion
Network Theory - Star to Delta Conversion
In the previous chapter, we discussed about the conversion of delta network into an equivalent star network. Now, let us discuss about the conversion of star network into an equivalent delta network. This conversion is called as Star to Delta Conversion.
In the previous chapter, we got the resistances of star network from delta network as
RA=R1R2R1+R2+R3??=?1?2?1+?2+?3 Equation 1
RB=R2R3R1+R2+R3??=?2?3?1+?2+?3 Equation 2
RC=R3R1R1+R2+R3??=?3?1?1+?2+?3 Equation 3
Delta Network Resistances in terms of Star Network Resistances
Let us manipulate the above equations in order to get the resistances of delta network in terms of resistances of star network.
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Multiply each set of two equations and then add.
RARB+RBRC+RCRA=R1R22R3+R2R23R1+R3R21R2(R1+R2+R3)2????+????+????=?1?22?3+?2?32?1+?3?12?2(?1+?2+?3)2
⇒RARB+RBRC+RCRA=R1R2R3(R1+R2+R3)(R1+R2+R3)2⇒????+????+????=?1?2?3(?1+?2+?3)(?1+?2+?3)2
⇒RARB+RBRC+RCRA=R1R2R3R1+R2+R3⇒????+????+????=?1?2?3?1+?2+?3 Equation 4
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By dividing Equation 4 with Equation 2, we will get
RARB+RBRC+RCRARB=R1????+????+??????=?1
⇒R1=RC+RA+RCRARB⇒?1=??+??+??????
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By dividing Equation 4 with Equation 3, we will get
R2=RA+RB+RARBRC?2=??+??+??????
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By dividing Equation 4 with Equation 1, we will get
R3=RB+RC+RBRCRA?3=??+??+??????
By using the above relations, we can find the resistances of delta network from the resistances of star network. In this way, we can convert star network into delta network.
Example
Let us calculate the resistances of delta network, which are equivalent to that of star network as shown in the following figure.
Given the resistances of star network as RA = 6 Ω, RB = 18 Ω and RC = 3 Ω.
We know the following relations of the resistances of delta network in terms of resistances of star network.
R1=RC+RA+RCRARB?1=??+??+??????
R2=RA+RB+RARBRC?2=??+??+??????
R3=RB+RC+RBRCRA?3=??+??+??????
Substitute the values of RA, RB and RC in the above equations.
R1=3+6+3×618=9+1=10Ω?1=3+6+3×618=9+1=10Ω
R2=6+18+6×183=24+36=60Ω?2=6+18+6×183=24+36=60Ω
R3=18+3+18×36=21+9=30Ω?3=18+3+18×36=21+9=30Ω
So, we got the resistances of delta network as R1 = 10 Ω, R2 = 60 Ω and R3 = 30 Ω, which are equivalent to the resistances of the given star network.