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Transformation of Resistances (Star to Delta and Delta to Star)

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Transformation of resistances is a key tool in solving many problems related to equivalent resistance around a given circuit, etc. It reduces the math work and acts as a bonus while problem solving in competitive exams.

Star and Delta formations of Resistances

Delta to star transformation, star to delta transformation, question section (beginner), question section (intermediate), question section (advanced).

In this section we will understand what are star and delta formations of resistances and also try to identify them in simple circuits.

Star and delta formations of resistances is a standard 3-phase circuit or network of resistances connected in the same way as their name suggests.

Star formation of resistances looks like this:

Delta formation of resistances looks like this:

The key to solving problems is to identify them in a simple circuit.

In this section we will convert Delta formation of resistances to Star formation resistances.

Here is the formula for transformation-

\(R_{12} = \frac{R_1.R_2}{R_1+R_2+R_3}\)

Note that the above formula is cyclic in nature hence it works the same for \(R_{23}\) and \(R_{31}\).

Show that \(R_{12} = \frac{R_1.R_2}{R_1+R_2+R_3}\) Lets consider the resistance from \(A\) to \(C\): \(R_{AC,left} = R_2\) in parallel with \(R_1 + R_3\) \(R_{AC,left} = \frac{R_2 (R_1+R_3)}{R_2+R_1+R_3}\) \(R_{AC,right} = R_{12} + R_{23}\) Equating the two: \(\frac{R_2 (R_1+R_3)}{R_2+R_1+R_3} = R_{12} + R_{23}\) Similarly: \(\frac{R_3 (R_2+R_1)}{R_3+R_2+R_1} = R_{23} + R_{31}\) \(\frac{R_1 (R_3+R_2)}{R_1+R_3+R_2} = R_{31} + R_{12}\) Adding the first two equations, and subtracting the middle equation gives: \((R_{31} + R_{12}) + (R_{12} + R_{23}) - (R_{23} + R_{31}) = \frac{R_1 (R_3+R_2)}{R_1+R_3+R_2} + \frac{R_2 (R_1+R_3)}{R_2+R_1+R_3} - \frac{R_3 (R_2+R_1)}{R_3+R_2+R_1}\) \(2R_{12} = \frac{2R_1R_2}{R_1+R_2+R_3}\) \(R_{12} = \frac{R_1R_2}{R_1+R_2+R_3}\) QED

In this section we will convert Star formation of resistances to Delta formation resistances.

We will do this by finding equivalent resitances in place of resistances given in the problem.

For example,

In order to replace \(R_1\) and \(R_2\) (See Star formation) in the given diagram we will be there equivalent, that is \(R_{12}\) (See Delta formation).

\(R_{12}=R_1 + R_2 + \frac{R_1.R_2}{R_3}\)

Prove that \(R_{12}=R_1 + R_2 + \frac{R_1.R_2}{R_3}\) This is the same setup as in the previous setup, with the resistors renamed as follows: \(R_1 \rightarrow R_{23}\) \(R_2 \rightarrow R_{13}\) \(R_3 \rightarrow R_{12}\) \(R_{23} \rightarrow R_1\) \(R_{13} \rightarrow R_2\) \(R_{12} \rightarrow R_3\) So, the general results of the above proof become: \(R_{3} = \frac{R_{23}R_{13}}{R_1+R_{13}+R_{12}}\) And similarly: \(R_{1} = \frac{R_{13}R_{12}}{R_1+R_{12}+R_{23}}\) \(R_{2} = \frac{R_{12}R_{23}}{R_1+R_{23}+R_{13}}\) So, \(R_1 + R_2 + \frac{R_1R_2}{R_{12}} = \frac{R_{12}(R_{23}+R_{13})}{R_{23}+R_{13}+R_{12}} + \frac{R_{13}R_{12}R_{23}R_{12}}{R_{23}R_{13}} \) \(R_1 + R_2 + \frac{R_1R_2}{R_{12}} = \frac{R_{12}(R_{23}+R_{13}+R_{12})}{R_{23}+R_{13}+R_{12}} \) \(R_1 + R_2 + \frac{R_1R_2}{R_{12}} = R_{12} \) QED

Find the equivalent resistance in the given circuit diagram (in terms of \(R\))- In order to solve this question, we will transform the circuit and apply the formula side by side- Notice the highlighted area in the circuit, and then observe the change. Can you identify this transformation (Is it Star-to-Delta or Delta-to-Star)? Next we will do the same to the other side of the circuit. It will look like this- Now observe- Congratulations! We have transformed a complex looking circuit into a simple circuit we can easily solve. Wasn't that easy? Note that the method shown in this example is not the only way to solve the question. Try transforming other points, make your own way. Overall answer to the question is \(\boxed{\frac{2R}{3}}\).

Now we are capable of solving questions related to transformation of resistances.

Find the resistance between A and B if each resistor measures 1\(\Omega.\)

Determine the resistance in ohms between the points A and B (equivalent resistance) of the circuit shown below. All the values of the resistances are given in ohms.

Above shows an arrangement of resistors. Each resistor has a resistance of 1 ohm. Calculate the equivalent resistance of the arrangement

If the equivalent resistance between points \(A\) and \(B\) of the circuit above is \(R_{eq}\) in ohms, find \(\lfloor 10^3 R_{eq} \rfloor\).

In the figure below, all resistors have resistance \(R=1~\Omega\). Find the equivalent resistance in Ohms between the points A and O.

What is the equivalent resistance (in Ω) between \(1\) and \(3\) in this circuit, to 2 decimal places?

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Star To Delta Conversion Formula (Delta to Wye)

Delta To Star Conversion

Star to delta conversion, video presentation of delta to star transformation, leave a comment cancel reply.

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Star Delta Transformation Solved Problems

Star connection:.

When three resistance is so connected that either the starting point or the finishing point of each resistance is shorted together, then it is called a star connection .

Delta Connection:

When three resistances are so connected that the finishing point of one resistance is connected to the starting point of the other resistance, then is said to be a delta connection .

When it is found to be very difficult to solve the problems with the help of Kirchhoff’s law, then these complicated electrical networks can be solved easily by replacing successively star connections with equivalent delta connections and delta connections with equivalent star connections.

Star to Delta Transformation:

The equivalent delta resistance can be found out is this way. Star to Delta Transformation is the summation of two-star resistances, where the delta resistance meets plus the product of these two-star resistances divided by the third-star resistance.

Delta to Star Transformation:

The equivalent star resistances can be found out in this way. Delta to Star Transformation is the product of the two delta resistance, where the star resistance meets its end divided by the summation of three delta resistances.

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How to solve this star delta problem?

Please help me solve this star delta problem:

My attempt:

What to do next to solve this completely?

• \$\begingroup\$ It looks like homework so you need to show some effort. You seem to know that you'll need to do a star-delta (or delta-star) transformation so where are you stuck? What do your course notes or your web research tell you? Please edit to show your work. \$\endgroup\$ –  Transistor Commented Apr 22, 2021 at 19:52
• 2 \$\begingroup\$ You've done the hard part. Can you see that you have 12.2 in parallel with 7.4? \$\endgroup\$ –  Transistor Commented Apr 22, 2021 at 20:04
• 2 \$\begingroup\$ I didn't check the exact numbers, but it looks like you already did the hard part. Maybe a short break will help? \$\endgroup\$ –  jippie Commented Apr 22, 2021 at 20:05
• \$\begingroup\$ Yes, I solved it Total Resistance=3.69 ohm \$\endgroup\$ –  Shivam Jaiswal Commented Apr 22, 2021 at 20:07
• 2 \$\begingroup\$ Can you post your solution as an answer and then accept it so the system knows your question is solved. Otherwise it keeps popping up hoping for an answer. Thanks. \$\endgroup\$ –  Transistor Commented Apr 22, 2021 at 21:46

Yes, I solved it Total Resistance=3.69 ohm

• \$\begingroup\$ This answer isn't going to be much help to anyone with a similar problem. \$\endgroup\$ –  Transistor Commented Apr 24, 2021 at 8:34
• \$\begingroup\$ The theme is Delta-Star, so instead of taking the easy route, you prudently turned the delta 7, 3.2, 7.4 to a star… \$\endgroup\$ –  greybeard Commented Feb 15, 2023 at 8:51

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Basic Electrical Engineering Questions and Answers – Star Delta Transformation

This set of Basic Electrical Engineering Multiple Choice Questions & Answers (MCQs) focuses on “Star Delta Transformation”.

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Star to Delta Conversion

In this article we will discuss about star to delta conversion. We will first discuss about star and delta connections, current and voltage equations in these circuits. Then we will discuss about the differences between the star and delta connection. We will also discuss in detail the steps followed to derive the equations for conversion. In addition to this we will look into some solved examples that will help us understand the concept better. Later in the article we will discuss about the applications, advantages and disadvantages of this conversion.

Table of Content

• What is Start and Delta connection?
• Characteristics
• Difference between star and delta connection
• Conversion of Star to Delta
• Solved Example on Start to Delta conversion

What is Star and Delta Connection?

In electrical circuit analysis there are certain type of complex circuits that have resistances connected in either series or parallel. These complex arrangements are usually connected in the T, Y, Delta or pi connections. Among these star and delta are some common types of connection.

Star Connection Circuit

Star connection circuit is the circuit where the three resistor in the circuit have a common point. The shape of the circuit can be in the shape of ‘Y’ and ‘T’ alphabet or we say star shape. The common point is grounded in most cases. This type of connections are required when there is need for a neutral point. It is mostly used in low and medium voltage distribution systems.

Star Connection

Delta Connection Circuit

Delta connection is the type of connection where the three resistors are connected in a such way that they form a loop and they every two resistors have a common node. It is sometimes also in the shape of pi and mostly in shape of a triangle or can be referred to as delta. there is no common or neutral point available in the system. This connection type is used in high voltage transmission system.

Delta Connection

Characteristics of Star and Delta Connection

Some of the important characteristics of star and delta connection are as follows:

Current and Voltage in Star Connection

Let there be three phase voltages [Tex]V_R,V_Y,V_B [/Tex] and the line voltages are represented by [Tex]V_{RY},V_{YB},V_{BR} [/Tex]

[Tex]V_R=V_Y=V_B=V_{ph} \newline V_{RY}=V_R+(-V_Y)=V_R-V_Y \newline using \: vector \: parallelogram \: law \newline V_{RY}=\sqrt{V_R^2+V_Y^2+2V_RV_Y\cos60} \newline V_{RY}=\sqrt{V_{ph}^2+V_{ph}^2+2V_{ph}^2\cos60} \newline \therefore V_L=\sqrt3{V{ph}} [/Tex]

So we can say that line voltage = [Tex]\sqrt3 [/Tex] X phase voltage in a star connection.

Similarly for current from the figure we can see that the load is balanced.

[Tex]\therefore I_R=I_Y=I_B=I_{ph} [/Tex]

We know that line current is equal to phase current from the above equation. [Tex]I_L=I_{ph} [/Tex]

Current and Voltage in Delta Connection

Circuit and Phasor Diagram

From the phasor diagram

[Tex]I_R=I_{BR}-I_{RY} \newline I_Y=I_{RY}-I_{YB} \newline I_B=I_{YB}-I_{BR} \newline using \: vector \: parallelogram \: law \newline I_R=\sqrt{I_{BR}^2+I_{RY}^2+2I_{BR}I_{RY}\cos60} [/Tex]

for Balanced load. [Tex]I_{RY}=I_{BR}=I_{YB}=I_{ph} [/Tex]

[Tex]\therefore I_R=\sqrt{I_{ph}^2+I_{ph}^2+2I_{ph}^2\cos60} \newline I_L=\sqrt3I_{ph} [/Tex]

As, [Tex]V_{RY}=V_{YB}=V_{BR}=V_L \newline \therefore V_L=V_{ph} [/Tex]

Difference Between Star and Delta Connection

The resistances of star connection from delta connection are,

[Tex]R_A=\frac{R_1R_2}{R_1+R_2+R_3} \newline R_B=\frac{R_2R_3}{R_1+R_2+R_3} \newline R_C=\frac{R_3R_1}{R_1+R_2+R_3} [/Tex]

Now, to express the resistance of delta network in terms of star network we use the above equations. first we multiply set of two resistances and then add the three sets. After simplifying we will get the following equation.

[Tex]R_AR_B+R_BR_C+R_AR_C= \frac{R_1R_2R_3}{R_1+R_2+R_3} [/Tex] -(1)

Delta connection

The above equation(1) is now divided by the equation of R B ,

[Tex]\frac{R_AR_B+R_BR_C+R_AR_C}{R_B}=R_1 \newline \Longrightarrow R_1 =R_C+R_A+\frac{R_CR_A}{R_B} [/Tex]

The equation(1) is now divided by the equation of R C ,

[Tex]R_2 =R_A+R_B+\frac{R_AR_B}{R_C} [/Tex]

The equation(1) is now divide by the equation of R A ,

[Tex]R_3 =R_B+R_C+\frac{R_BR_C}{R_A} [/Tex]

The above three equation can be used to convert the star connection into delta connection.

Step By Step Approach of Star to Delta Conversion

• First the star connection needs to be identified
• After identifying the circuit, label all the resistors and also label the nodes. Find the central node and accordingly label other nodes.
• Using the above derived equations find the equivalent delta resistances.
• Now replace the star connection with the delta resistances and shape
• Verify the connection and check the resistance values and their positions between the nodes.

Solved Example on Star to Delta Conversion

1. Given a network of 9 resistors, find the equivalent resistance between point E and F. The connection between A, B, C is in delta connection which is converted to its equivalent star connection with the common node O

[Tex]R_{AO}=\frac{4X6}{2+4+6}=2 \Omega \newline R_{BO}=\frac{2 X 6}{2+4+6}=1\Omega \newline R_{CO}=\frac{2 X 4}{2+4+6}=\frac{2}{3}\Omega [/Tex]

The network can be further simplified by adding the resistances in series combination i.e. [Tex]R_{DO}=2+6=8 \Omega \newline R_{EO}= \frac{2}{3}+\frac{10}{3}=\frac{12}{3}=4\Omega \newline R_{FO}=7+1=8\Omega [/Tex]

Now again the star connection in the inner part of the triangle is again converted into delta connection.

[Tex]R_1=4+8+\frac{8X4}{8}=16\Omega \newline R_2=8+8+\frac{8X8}{4}=32\Omega \newline R_3=8+4+\frac{8X4}{8}=16\Omega [/Tex]

Now, we will apply parallel combination between the nodes DG and EI, EI and FH, FH and DG.

After that we will get the resistances as, [Tex]R_{DE}=8\Omega \newline R_{EF}=8\Omega \newline R_{FD}=\frac{32}{3}\Omega [/Tex]

Star to Delta conversion

The equivalent resistance between E and F will be

[Tex]R_{EF}=\frac{8 X(8+32/3)}{8+(8+32/3)}=\frac{8X56}{80}=5.6 \Omega [/Tex]

2. Find the current drawn from the 5v battery in the network given below.

Let us consider some points in the given network as A, B, C, D and G. If we consider the points A, B, C, D we can see that it is in star connection with common point as B.

Converting it into delta connection,

[Tex]R_1=2+2+\frac{2X2}{3}=\frac{16}{3} \Omega \newline R_2=2+3+\frac{2X3}{2}=8 \Omega \newline R_3=2+3+\frac{2X3}{2}=8 \Omega [/Tex]

Now applying the parallel combination we get resistances as [Tex]R_{AD}=\frac{8}{9}\Omega \newline R_{AC}=\frac{16}{3}\Omega \newline R_{CD}=\frac{8}{3}\Omega [/Tex]

Now we will use parallel combination and series combination to find an equivalent resistance,

[Tex]\frac{\frac{16}{3}X\frac{32}{9}}{\frac{16}{3}+\frac{32}{9}}=\frac{32}{15}\Omega \newline \frac{32}{15}+3=\frac{77}{15}\Omega [/Tex]

From the above circuit, we can calculate the value of I as [Tex]I=\frac{5}{77/15}=0.974A [/Tex]

Applications of Star to Delta Conversion

• Its common application is to analyze and simplify complex circuits. It thus helps in easier calculation and less time consuming methods.
• It is used in power distribution systems, where loads are connected in either star or delta connection. Depending on the parameters the connection type is fixed. The conversion method gives the flexibility in changing the connection type as per requirement and analysis.
• It is used in impedance matching of a load with source which helps in optimizing the efficiency of power transfer.
• It is used in three-phase motors where the conversion method gives the flexibility of changing the connection type depending on the scenarios.

Advantages and Disadvantages of Star to Delta Conversion

There are some list of Advantages and Disadvantages of Star to Delta Conversion given below :

Advantages of Star to Delta Conversion

• Its use gives a easy approach for simplification of complex circuits.
• It gives flexibility in choosing the right connection type as per the requirements.
• It is used in step-up and step-down of voltages and currents depending on the connection type.
• It is highly compatible with three phase system.

Disadvantages of Star to Delta Conversion

• It is not applicable everywhere and mostly limited to three phaser system.
• Sometimes conversion can lead to voltage imbalance.
• It is not suitable for all type of loads and work only for specific types of load.
• It increases the complexity of circuits when applied to simple circuits.

Star and Delta are two types of connections that are used to simplify and analyze complex electrical circuits. They are mainly implemented to the three phase networks and are widely used for power distribution and circuit design. The process of expressing the resistances of delta connection in terms of star provides three equations to convert a star network into a delta network. This technique is quite useful as it gives the flexibility of changing the connection type as the required parameters. Although it is very useful but there certain limitation as well which includes it being mostly limited to three phase networks.

FAQs on Star to Delta Conversion

Why conversion from star to delta is required.

The conversion is needed when we have to change the circuit design for simplification process. It is also done for optimizing the electrical circuit for various applications.

What is the main difference between star and delta?

The main difference between in star and delta is the presence of common node in star connection and its absence in the delta connection. This will also make the shape of both the circuits different.

When is star connection and delta connection preferred?

Star connection is preferred when there is a neutral point required or when the load is unbalanced. Delta connection is preferred in case of high voltage transmission and when common point is not required.

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Wednesday 2 April 2014

Solved Examples Problems On Star-Delta Transformation Or Conversion

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8 comments:

thank you very much for your approach ,my name is Ernest Magesa, a student of university of Dar es salaam

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