Delta Star & - GATE EE Network Theory Free MCQ Test with solutions

 Following the delta to star conversion:
R1=10*5/(10+5+3)
R2=10*3/(10+5+3)
R3=5*3/(10+5+3).

After converting to star, each star connected resistance is equal to the product of the resistances it is connected to and the total sum of the resistances. Hence R1=Ra*Rb/(Ra+Rb+Rc), R2=Rb*Rc/(Ra+Rb+Rc), R3=Rc*Ra/(Ra+Rb+Rc).

The 6 ohm and 9 ohm resistances are connected in parallel. Their equivalent resistances are: 6*9/(9+6)=3.6 ohm.
The 3 3.6 ohm resistors are connected in delta. Converting to star:
R1=R2=R3= 3.6*3.6/(3.6+3.6+3.6)=1.2 ohm.

The star connection is also known as the Y-connection because its formation is like the letter Y.

The 3 5 ohm resistors are connected in delta. Changing it to star:
R1=R2=R3= 1.67 ohm.
One of the 1.67 ohm resistors are connected in series with the 2 ohm resistor and another 1.67 ohm resistor is connected in series to the 3 ohm resistor.
The resulting network has a 1.67 ohm resistor connected in series with the parallel connection of the 3.67 and 4.67 resistors.
The equivalent resistance is: 3.725A.
I=2/3.725= 0.54A.

Using the delta to star conversion formula:
R1=2*6/(2+6+4)
R2=2*4/(2+6+4)
R3=4*6/(2+6+4).

Using the delta-star conversion formula:
R1=4*3/(2+3+4)
R2=2*3/(2+3+4)
R3=2*4/(2+3+4).

Using the star to delta conversion:
R1=4.53+6.66+4.53*6.66/1.23
R2=4.53+1.23+4.53*1.23/6.66
R3=1.23+6.66+1.23*6.66/4.56.

 After converting to delta, each delta connected resistance is equal to the sum of the two resistance it is connected to+product of the two resistances divided by the remaining resistance. Hence R1=Ra+Rb+Ra*Rb/Rc, R2=Rc+Rb+Rc*Rb/Ra, R3=Ra+Rc+Ra*Rc/Rb.

The two 2ohm and one 5ohm resistors are connected in delta, changing them to star, we have R1=R3= 5*2/(5+2+2) = 10/9ohm,  R2 = 2*2/(9)ohm
The one 10/9ohm and one 4/9ohm resistors are connected in series to the 10 ohm and 2 ohm respectively.
This network can be further reduced to a network consisting of a 11.11ohm and 2.44ohm resistor connected in parallel and then series with 10/9 whose resultant is intern connected in parallel to the 10 ohm resistor.

Delta connection is also known as mean connection because its structure is like a mesh, that is, a closed loop.

 In a Y-to-Δ transformation the resistor opposite a given node equals the sum of the products of all pairs of star resistances divided by the star resistance at that node. Thus the delta resistor between B and C (opposite A) is (Rb*Rc + Rc*Ra + Ra*Rb)/Ra which simplifies to Rb + Rc + (Rb*Rc)/Ra.

Using the formula for star to delta conversion:
R1=8/9+4/3+(8/9)*(4/3)/(2/3)
R2=8/9+2/3+(8/9)*(2/3)/(4/3)
R3=2/3+4/3+(2/3)*(4/3)/(8/9).

Concept:

• In a balanced three-phase system, the relationship between the impedances of delta (Δ) and star (Y) connections is given by Z_Y = Z_Δ / 3.
• This relationship applies to both magnitude and phase angle.

Given:

• Delta-connected impedance (Z_Δ) = 3∠30° Ω/phase

To find the equivalent star-connected impedance (Z_Y):

Z_Y = Z_Δ / 3

Substitute the given delta impedance:

Z_Y = 3∠30° Ω / 3
Z_Y = 1∠30° Ω

Therefore, the equivalent star-connected load impedance is 1∠30° Ω/phase.

Key Points

• Star Connection:
  • Also known as 'Y' or 'Wye' connection.
  • In this configuration, one end of each of the three coils is connected to a common point, called the neutral.
  • The other ends of the coils are connected to the three-phase lines.
  • Because it resembles the shape of the letter 'Y', it is sometimes referred to as 'T' connection. Hence, statement I is correct.
• Delta Connection:
  • Also known as 'Δ' connection.
  • In this configuration, the end of each coil is connected to the start of the next coil, forming a closed loop that resembles the shape of the Greek letter 'Δ'.
  • This configuration is sometimes referred to as 'π' (pie) connection because of its looped structure. Hence, statement II is correct.

Additional Information

• Star (Y) Connection Advantages:
  • Allows for the use of a neutral wire, which can help in balancing the load.
  • Provides two different voltage levels: line-to-line voltage and line-to-neutral voltage, making it versatile for different applications.
  • Commonly used in distribution systems.
• Delta (Δ) Connection Advantages:
  • Does not require a neutral wire, simplifying the wiring in certain applications.
  • Can handle higher power loads compared to star connection, making it suitable for heavy machinery.
  • Commonly used in transmission systems and for industrial applications.
• Applications:
  • Star Connection: Ideal for long-distance power transmission and distribution where voltage levels need to be stepped down.
  • Delta Connection: Suitable for short-distance power transmission and industrial applications where high power loads are common.

Key Points

• In electrical circuits, the transformation between Delta (Δ) and Star (Y) configurations is a common technique to simplify complex resistor networks.
• The given problem involves converting a Delta network with resistances R₁, R₂, and R₃ into an equivalent Star network with resistances R_x, R_y, and R_z.
• For the Delta to Star transformation, the resistances in the Star network can be calculated using the following formulas:
  • R_x = (R₁ * R₃) / (R₁ + R₂ + R₃)
  • R_y = (R₁ * R₂) / (R₁ + R₂ + R₃)
  • R_z = (R₂ * R₃) / (R₁ + R₂ + R₃)
• Therefore, the correct formula for R_x is (R₁ * R₃) / (R₁ + R₂ + R₃).
• This transformation is useful in circuit analysis, especially when dealing with complex networks where simplification can make calculations more manageable.

Additional Information

• Incorrect Options Explained
  • Option R_x = (R₁ * R₂) / (R₁ + R₂ + R₃):
    • This formula is incorrect for R_x but is actually the formula for R_y in the Star network.
  • Option R_x = R_Y = R_Z:
    • This option is incorrect because the resistances R_x, R_y, and R_z in a Star network are generally not equal unless R₁, R₂, and R₃ are all equal in the Delta network.
  • Option R_x = (R₂ * R₃) / (R₁ + R₂ + R₃):
    • This formula is incorrect for Rx but is actually the formula for Rz in the Star network.
• Delta and Star Transformation:
  • The Delta (Δ) configuration is also known as a Pi (π) network, and it is commonly used in three-phase power systems.
  • The Star (Y) configuration is also known as a Tee (T) network and is often used in electrical engineering for simplifying circuit analysis.
  • Understanding these transformations is crucial for analyzing complex circuits in both AC and DC systems.
  • These transformations are not limited to resistors but can also be applied to impedance in AC circuits.

Concept:

Star-to-delta conversion


Calculation:

Given


= 6 +2 + 4 = 12

= 18 

Delta to Star conversion:

The star equivalent of delta connected resistor is:

If all resistances connected in the delta are 'R', then the star equivalent resistance is:

Concept:

1) Delta to Star conversion:

2) Star to Delta conversion:

Application:

Let, R_AB = 20 Ω, R_AC = 30 Ω, R_BC = 50 Ω
According to the question, by converting the given Delta to star, we get