Delta to Star Conversion

In this article, we will discuss about delta to star conversion. Firstly, we will try to understand the two types of connections i.e. star and delta connections. The understanding will be developed by learning about their characteristics through current and voltage equations in these circuits. We will proceed to see the differences between the star and delta connections based on different parameters. Results will be derived from the beginning to ensure clarity of formulas.

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 the applications, advantages, and disadvantages of this conversion. The article will conclude with some frequently asked questions that readers can refer to.

What is Star and Delta Connection?

In electronics and communication, we often need to analyze the circuit to simplify certain tasks. For this purpose, certain types of complex circuits have been introduced which simplify the task of circuit solving. These complex arrangements are usually connected in the zig-zag, series, and parallel, and sometimes they can include both types in a single circuit. Among these connections, star, and delta are some common types of connection. Let us study the characteristics of these circuits individually to develop a better understanding.

Formally,

Star and Delta Connection are two types of connections used in the 3-phase circuit to simplify the task of solving resistance in the circuit.

Delta Connection Circuit

Delta connection is the type of connection where ending terminal of one winding joins to the starting terminal of other winding resulting in a closed circuit. Therefore, this type of network is called a mesh network. It is sometimes also in the shape of pi and mostly in the shape of a triangle or can be referred to as a delta.

The circuit diagram has been provided for better understanding about delta circuits. Note that in this type of connection, three lines run from the 3 common points formed by any two terminals. These lines are called Line Conductors.

Formally, A Delta Connection Circuit is a circuit that has no common or neutral point in the system due to which line voltage is equal to the phase voltage

👁 Delta Connection

Star Connection Circuit

A star connection circuit is the circuit where the three resistors in the circuit have a common point this point is called star or the neutral point. These circuits usually have a shape like the alphabet 'Y' which resembles a star and therefore these circuits get the name Star Connection Circuit .

The common point is grounded in most cases. Note that in this type of connection, all the windings terminal have one common junction and other terminal of each winding is joined to a separate junction.

Formally, Star Connection Circuit are a types of connection that have a neutral wire and the voltage developed across line is root three times the phase voltage.

👁 Star Connection

Properties of Star and Delta Connection

Some of the important electrical properties of star and delta connection are current and voltage so let us study them individually.

Current and Voltage in Delta Connection

Let us take a look at the circuit and the phasor diagram.

👁 Circuit Diagram of Delta Connection

The figure clearly shows us the arrangement in a delta connection. We have assumed that the current flowing in each of the three windings in I_R, I_Y, I_B. We will use the phasor diagram to find the relationship between the quantities.

Mathematical Equations

From the phasor diagram

👁 Current and Voltage in Delta Connection

I_R = I_BR - I_RY

and I_Y = I_RY - I_YB

and I_B = I_YB - I_BR

Now we will use vector parallelogram law to calculate the equivalent

∴ I_R= √(I_BR² +I_RY²+ 2I_BRI_RYcos(60)

We know that for a balanced load, I_RY =I_BR =I_YB =I_ph. (I_ph is the phase current)

I_R= √(I_ph²+I_ph²+2I_ph² cos(60))

∴ I_L= √3 I_ph

Also as V_RY= V_YB= V_BR= V_L

∴ V_L=V_ph

Current and Voltage in Star Connection

Let us take a look at the circuit diagram of star connection

👁 Circuit Diagram of Star Connection

The figure shows the arrangement in a delta circuit. We have assumed that there are three phase voltages V_R, V_Y, V_B and the corresponding line voltages are V_RY, V_YB, V_BR. We will use the phasor diagram to find the relationship between the quantities.

Mathematical Equations

From the phasor diagram

👁 Phasor of Star Connection

V_R=V _Y=V_B=V_ph

Also V_RY =V_R +(−V _Y)=V_R−V_Y

​Now we will use vector parallelogram law to calculate the equivalent

∴ V_RY = √(V_R²+V_Y²+2V_R​V_Y cos60)

∴ V_RY=√(V_ph²+ V_ph²+2V_ph²cos60)

Therefore V_L=√3 V_ph

From the above calculation, we can say that line voltage is √3 times phase voltage in a star connection. Also from the figure, we can say that load is balanced hence current must be equal

I_R=I_Y=I_B=I_ph

Difference Between Star and Delta Connection

Let us look at the difference between delta and star connection

Parameter	Star Connection	Delta Connection
Presence of Node	In this type of circuit the all the three branches have a common node and branches initiate form this node.	In this type of circuit the three are connected in way that they form a loop and no common node is present.
Common Terminal	One terminal is common for all the branches in the circuit.	For every two branches one terminal remains common.
Relation between line and phase current	IL=Iph	IL=√3 Iph
Relation between line and phase voltage	VL=√3 Vph	VL=Vph
Presence of neutral point	There is a neutral point present in the circuit	No neutral point is present in this type of circuit
Power Supply	It receives less power than the supply.	It receives the full power.
Used in	Used in power transmission networks	Used in power distribution networks
Load Type	It can have both balanced and unbalanced load	It can only have balanced load.

Delta to Star Conversion

Delta to Star Conversion allows to convert Delta circuit connection into Star circuit connection by formation of resistances in both the side. 

Let us consider three resistances R_AB, R_BC, R_AC are connected in delta to three terminals A, B and C

If this circuit is converted to star connection by three resistances R_A, R_B, R_C as shown here.

👁 Delta to Star conversion

We are calculating equivalent circuit therefore, the resistance measured between any two of the terminals A, B, and C must be the same in two cases.

Resistance between A and B for star = Resistance between A and B for delta connection

Mathematically, we can write

R_A+R_B=(R_BC+R_AC)R_AB/(R_AB+R_BC+R_AC) →1

R_B+R_C=(R_AB+R_AC)R_BC/(R_AB+R_BC+R_AC) →2

R_C+R_A=(R_AB+R_BC)R_AC/(R_AB+R_BC+R_AC) →3

Adding the above three equations

2(R_A+R_B+R_C)=2(R_ABR_BC+R_ACR_BC+R_ABR_AC)/(R_AB+R_BC+R_AC)

∴ R_A+R_B+R_C= (R_ABR_BC+R_ACR_BC+R_ABR_AC)/(R_AB+R_BC+R_AC) →4

Subtract Equation 2 from Equation 4

R_A+R_B+R_C−(R_B+R_C)=

equals (R_ABR_BC+R_ACR_BC+R_ABR_AC)/(R_AB+R_BC+R_AC) -(R_AB+R_AC)R_BC/(R_AB+R_BC+R_AC)

∴ R_A=R_ACR_AB/( R_AB+R_BC+R_AC)

Similarly, Subtract Equation 3 from Equation 4

R_B=R_BCR_AB/( R_AB+R_BC+R_AC)

Similarly, Subtract Equation 1 from Equation 4

R_c=R_ACR_BC/(R_AB+R_BC+R_AC)

We can use these relations to calculate the resistances of star network from the resistances of delta network

Steps to Solve Delta to Star Conversion

Let us summarize the steps to be followed for delta to star conversion :

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

Solved Example on Delta to Star Conversion

Find the equivalent star connection of the circuit in star with resistances R_ac=4 ohms ,R_bc=8 ,R_ab=12 ohms respectively.

👁 Figure 1

From our previous derivations, we can say that

R_c=R_acR_bc/ (R_ac+R_ab+R_bc)

R_a=R_abR_ac/ (R_ac+R_ab+R_bc)

R_b=R_bcR_ab/ (R_ac+R_ab+R_bc)

we have R_ac=4 ,R_bc=8 and R_ab=12

On putting these values in above equation

R_c= 4*8/24 = 4/3 ohms

R_a= 4*12/24= 2 ohms

R_b= 8*12/24 = 4 ohms

The resistor value in a Y network that is equivalent to a Δ containing 3 resistors of R ohm.

A Y network means a star network. We are given that the three resistances in star connection are R ohms and we have to find the equivalent resistance in star connection. Consider the diagram

👁 Question

Given R_A=R_B=R_C=R ohms

R₁=R_AR_B/R_A+R_B+R_C

R₂=R_AR_C/R_A+R_B+R_C

R₃=R_CR_B/R_A+R_B+R_C

From symmetry of the question we can say

R₁=R₂=R₃

On putting the value R ohm ,we get

∴ R₁=R₂=R₃=R/3 ohms

Applications of Delta to Star Conversion

Let us see some applications of delta to star conversion

• The most common application is simplification of complex circuits which makes circuit analysis easier. It helps in making the calculations easier and less time-consumes even for circuits with complex parameters.
• It is used in connections involving transformers. Whenever there is a requirement of impedance matching in transformer, the windings of transformer can be arranged in star or delta configuration to match the impedance or the voltage.
• It is used in motors where it helps in optimizing the efficiency of connections. This can be done by arranging winding in different configurations to form a great power system.
• Delta and Star connections are used in high school education to simplify the understanding for students and help them understand complex circuits.
• The transformation circuits can be used for calculating power factor and even helps in correcting it using fault analysis. This helps in optimizing the performance by mitigating the errors.

Advantages of Delta to Star Conversion

Let us see some advantages of delta to star conversion

• It simplifies the task of solving of complex circuits. It is crucial for calculating currents, voltages, and power in the complex systems.
• It helps us in circuit designing as per our need due to various features provided by these circuits. Delta and star connection can serve the diverse needs and can meet specific requirements .
• Due to the ease of designing, they are a common choice for circuit design in everyday appliances like they are used in step-up and step-down motors.
• The transformation from one circuit to another circuit type allows us to identify any potential cause of error therefore helping in fault analysis.
• It allows impedance matching between the components of the system. This is particularly useful in transformers and motors.

Disadvantages of Delta to Star Conversion

Let us see some disadvantages of delta to star conversion

• There are some restrictions associated with using these connections. They are mostly used in three phaser system and there scope can't expand beyond that.
• The manual application of these circuits can be a tedious task. This can be a major problem in circuits that require huge mathematical calculations.
• Sometimes, the transformation of circuits from one form to another can lead to inaccuracy between the original and transformed circuit. Therefore, it can introduce some errors in circuit.
• It increases the complexity of circuits when applied to simple circuits making it challenging to calculate the corresponding circuit values.
• Sometimes, implementing the transformed circuit can involve more components resulting in a complicated overall design of the circuit in addition to the increased overall cost of system.

Conclusion

We have studied two important types of circuit connections in this article. We have also seen the relationship between the different parameters of the Star and Delta circuits. In addition to this, we have seen how the two types of connection are different from each other and therefore, we devised the necessary equations that can be used to convert the delta circuit to a Star circuit. Although, this method is helpful but there are some limitations to it. Some solved examples have been provided for better understanding of the concept. Readers are requested to refer to FAQs in case any doubt still exists.