Star Delta Transformation
Star Delta Transformation
We can now solve simple series, parallel or bridge type resistive networks using
Kirchoffīs Circuit Laws,
mesh current analysis or nodal voltage analysis techniques but in a balanced 3-phase circuit we can use
different mathematical techniques to simplify the analysis of the circuit and thereby reduce the amount
of math's involved which in itself is a good thing.
Standard 3-phase circuits or networks take on two major forms with names that represent
the way in which the resistances are connected, a Star connected network which has the symbol
of the letter, Υ (wye) and a Delta connected network which has
the symbol of a triangle, Δ (delta). If a 3-phase, 3-wire supply or even a 3-phase
load is connected in one type of configuration, it can be easily transformed or changed it into an equivalent
configuration of the other type by using either the Star Delta Transformation or
Delta Star Transformation process.
A resistive network consisting of three impedances can be connected together to form a
T or "Tee" configuration but the network can also be redrawn to form a
Star or Υ type network as shown below.
T-connected and Equivalent Star Network
As we have already seen, we can redraw the T resistor network
to produce an equivalent Star or Υ type network. But we can also
convert a Pi or π type resistor network into an
equivalent Delta or Δ type network as shown below.
Pi-connected and Equivalent Delta Network.
Having now defined exactly what is a Star and Delta connected network
it is possible to transform the Υ into an equivalent Δ
circuit and also to convert a Δ into an equivalent Υ
circuit using a the transformation process. This process allows us to produce a mathematical relationship
between the various resistors giving us a Star Delta Transformation as well as a
Delta Star Transformation.
These transformations allow us to change the three connected resistances by their equivalents
measured between the terminals 1-2, 1-3 or 2-3 for either a star
or delta connected circuit. However, the resulting networks are only equivalent for voltages and currents external to
the star or delta networks, as internally the voltages and currents are different but each network will consume the
same amount of power and have the same power factor to each other.
Delta Star Transformation
To convert a delta network to an equivalent star network we need to derive a transformation
formula for equating the various resistors to each other between the various terminals. Consider the circuit below.
Delta to Star Network.
Compare the resistances between terminals 1 and 2.
Resistance between the terminals 2 and 3.
Resistance between the terminals 1 and 3.
This now gives us three equations and taking equation 3 from equation 2 gives:
Then, re-writing Equation 1 will give us:
Adding together equation 1 and the result above of equation 3 minus equation 2 gives:
From which gives us the final equation for resistor P as:
Then to summarize a little the above maths, we can now say that resistor P
in a Star network can be found as Equation 1 plus (Equation 3 minus Equation 2) or
Eq1 + (Eq3 - Eq2).
Similarly, to find resistor Q in a star network, is equation 2 plus
the result of equation 1 minus equation 3 or Eq2 + (Eq1 - Eq3) and this
gives us the transformation of Q as:
and again, to find resistor R in a Star network, is equation 3 plus
the result of equation 2 minus equation 1 or Eq3 + (Eq2 - Eq1) and this
gives us the transformation of R as:
When converting a delta network into a star network the denominators of all of the transformation
formulas are the same: A + B + C, and which is the sum of ALL the delta resistances. Then
to convert any delta connected network to an equivalent star network we can summarized the above transformation
Delta to Star Transformations Equations
If the three resistors in the delta network are all equal in value then the resultant resistors
in the equivalent star network will be equal to one third the value of the delta resistors, giving each branch in
the star network as: RSTAR = 1/3RDELTA
Convert the following Delta Resistive Network into an equivalent Star Network.
Star Delta Transformation
We have seen above that when converting from a delta network to an equivalent star network that
the resistor connected to one terminal is the product of the two delta resistances connected to the same terminal,
for example resistor P is the product of resistors A and
B connected to terminal 1.
By rewriting the previous formulas a little we can also find the transformation formulas for converting
a resistive star network to an equivalent delta network giving us a way of producing a star delta transformation as shown
Star to Delta Network.
The value of the resistor on any one side of the delta, Δ
network is the sum of all the two-product combinations of resistors in the star network divide by the star
resistor located "directly opposite" the delta resistor being found. For example, resistor A
is given as:
with respect to terminal 3 and resistor B is given as:
with respect to terminal 2 with resistor C given as:
with respect to terminal 1.
By dividing out each equation by the value of the denominator we end up with three separate
transformation formulas that can be used to convert any Delta resistive network into an equivalent star network
as given below.
Star Delta Transformation Equations
Star Delta Transformations allow us to convert one circuit type of circuit connection
to another in order for us to easily analyise a circuit and one final point about converting a star resistive network to
an equivalent delta network. If all the resistors in the star network are all equal in value then the resultant resistors
in the equivalent delta network will be three times the value of the star resistors and equal, giving:
RDELTA = 3RSTAR