star delta transformation

Star Delta Transformation

Star Delta Transformations allow us to convert impedances connected together from one type of connection to another. We can now solve simple series, parallel or bridge type resistive networks using Kirchoff´s 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

star resistor network


As we have already seen, we can redraw the T resistor network above to produce an electrically equivalent Star or Υ type network. But we can also convert a Pi or π type resistor network into an electrically equivalent Delta or Δ type network as shown below.

Pi-connected and Equivalent Delta Network.

delta resistor 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 circuit transformations allow us to change the three connected resistances (or impedances) 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.

delta star transformation


Compare the resistances between terminals 1 and 2.

delta star resistance

Resistance between the terminals 2 and 3.

resistance between terminals

Resistance between the terminals 1 and 3.

resistance between terminals 1 and 3

This now gives us three equations and taking equation 3 from equation 2 gives:

resistance equation

Then, re-writing Equation 1 will give us:

second resistance equation

Adding together equation 1 and the result above of equation 3 minus equation 2 gives:

resistor equation

From which gives us the final equation for resistor P as:

star resistor p


Then to summarize a little about 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:

star resistor q

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:

star resistor r

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 equations as:

Delta to Star Transformations Equations

star resistance pstar resistance qstar resistance r

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

Delta – Star Example No1

Convert the following Delta Resistive Network into an equivalent Star Network.

delta to star example delta to star equation

Star Delta Transformation

Star Delta transformation is simply the reverse of above. We have seen 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 below.

Star to Delta Transformation

star delta transformation

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:

resistor a

with respect to terminal 3 and resistor B is given as:

resistor b

with respect to terminal 2 with resistor C given as:

resistor c

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

equivalent resistor aequivalent resistor bequivalent resistor c

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

Star – Delta Example No2

Convert the following Star Resistive Network into an equivalent Delta Network.

star to delta example star to delta equation

Both Star Delta Transformation and Delta Star Transformation allows us to convert one type of circuit connection into another type in order for us to easily analyse the circuit. These transformation techniques can be used to good effect for either star or delta circuits containing resistances or impedances.


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  • t

    i would like to recieve those notes daily

  • l
    luckson siamujompa chikowa

    I would like to be sending me this type of questions on D.C theorems and delta/star…thank u

  • m
    moses mdzagada

    thank you for the notes

  • r
    ramesh g

    hi sir this is ramesh., i have a doubt on conversion from star to delta, in my analysis two inductors on top and capacitor is connected in middle to each inductor, so how can convert it into delta form,,,if any one knows send to my email:rameshg44@gmail.com

    • Wayne Storr

      If you know the supply frequency, find the Inductive Impedances, XL and Capacitive Impedance, Xc and transpose.

  • c
    chauhan digvijaysinh j.

    I like it..

  • N


  • M

    Delta to star is okay. But how do you decide ‘ directly opposite’ in case of Star to delta?

    Please clear my confusion.

  • L

    I really like your tutorial. Easy to understand. Thanks a lot.

  • v

    We know that delta connection is like parallel connection and star connection is like series connection . So the resistance of series connection is always greater than that of parallel connection . So how do we get RDELTA = 3RSTAR i.e. resistance of delta connection is 3 times that of star .

    Please clear my doubt.
    Thanks in advance

    • Wayne Storr

      Hello Vaibhav, No, Delta is like a Pi connection not parallel and star is like a T connection and not series. If P, Q, and R are the same value, for example 1 Ohms then do the maths as:

      A = PQ/R + Q + P = 1×1/1 + 1 + 1 = 1 + 1 + 1 = 3 Ohms or Delta leg A = 3 x Star.

  • n

    I have a got a Delta connected load with each limb having a wattage of 1.6 kw. I jus wanted to know what if i connect this load in star instead of delta..As u said the effective resistance in delta=3 x effective resistance in star… so will the current gets increased in star as compared to that in delta??? i aslo knw that for a star delta starter for motor starting current is 3 times lesser than the running current ..i.e. wen it is in star current is 3 times lesser than that is in delta.
    So i feel it is contradictory…. im jus confused regarding current consumption in star and delta…
    pls helpme out

    • Wayne Storr

      When a load is connected in delta, full line voltage appears across each winding as Vphase = Vline. At starting, the slip of the motor is 100 per cent and for a small moment the motor acts as a low impedance transformer with the stator acting as the primary and the rotor as the secondary. As such the starting current can be over three times the full-load rated current.

      The voltage appearing across each winding of a star connected machine is only 58 per cent (Vph = VL/root(3)) of the full delta connected voltage as there are now two windings in series across the supply so the motor represents a higher impedance to the supply and the starting current is reduced to one-third of what it would be in a delta connection. This is why star-delta starting is used for large motors instead of direct-on-line starting.

      Star-delta or delta-star transformations allow the analysis of complex circuits which may include resistances and impedances to be converted from one form into another into a much simpler equivalent circuit. If the impedances are equal (balanced) then the equivalent delta impedance, Zdelta per leg is three times the star impedance, Zstar. Therefore, Zdelta = 3Zstar, or Zstar = Zdelta/3 the other way.

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