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Lecture 04

Equivalent transformation of Y- and ∆-connection of resistance

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Motivation of studying Y- and ∆-connection: not all resistors are connected in parallel or series. How to simplify a complex circuit in that case?
  • Star-shape connection is also called Y-connection and Triangle-connection is also called ∆(delta)-connection.
  • In general, it is hard to obtain the equivalent resistance based on only parallel or serial connection.

​

​

  • Example:

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What if the resistors in the right figure do not have equal resistance?

a

b

c

d

R1

R2

R5

R3

R4

Rab-eq=?

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Y- and ∆-connection: tri-terminal networks

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1

2

3

R12

R13

R23

∆-connection

R1

R2

R3

1

2

3

Y-connection

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Transformation of Y- and ∆-connection

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1

2

3

R12

R13

R23

∆-connection

1

2

3

R12

R13

R23

π-connection

R12

R23

R13

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Transformation of Y- and ∆-connection

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R1

R2

R3

1

2

3

Y-connection

R1

R2

R3

1

2

3

R1

R2

R3

3

1

2

T-connection

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Conditions of equivalence of Y- and ∆-connection

1

2

3

R12

R31

R23

+

-

+

+

-

-

u12∆

u31∆

u23∆

i1∆

i2∆

i3∆

R1

R2

R3

1

2

3

+

-

+

-

+

-

i1Y

i2Y

i3Y

u12Y

u23Y

u31Y

i1∆ = i1Y i2∆ = i2Y i3∆ = i3Y

​

u12∆ = u12Y u23∆ = u23Y u31∆ = u31Y

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Conditions:

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Conditions of equivalence of Y- and ∆-connection

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​

​

​

  • Based on Kirchholf current law and voltage law, as long as two of the three equations are satisfied, the third one can also be satisfied.

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i1∆ = i1Y i2∆ = i2Y i3∆ = i3Y

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u12∆ = u12Y u23∆ = u23Y u31∆ = u31Y

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Conditions:

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Equivalent transformation of Y- and ∆-connection of resistance

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(KCL) i1∆ = i12-i31 = u12∆ /R12 - u31∆ / R13

i12

i31

i23

i12

(KCL) i2∆ = i23-i12 = u23∆ /R23 - u12∆ / R12

i31

i23

(KCL) i3∆ = i31-i23 = u31∆ / R13 - u23∆ /R23

Representing currents with voltages

Representing voltages with currents

(KVL) u12Y = uR1-uR2 = i1Y*R1 - i2Y*R2

uR1

+

-

uR2

+

-

(KVL) u23Y = uR2-uR3 = i2Y*R2 – i3Y*R3

uR3

+

-

(KVL) u31Y = uR3-uR1 = i3Y*R3 – i1Y*R1

(KCL) i1Y + i2Y + i3Y =0

(1)

(2)

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Equivalent transformation of Y- and ∆-connection of resistance

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(3)

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Equivalent transformation of Y- and ∆-connection of resistance

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or

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Equivalent transformation of Y- and ∆-connection of resistance

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or

What if the three resistors

are equal?

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Example

1KΩ

1KΩ

1KΩ

1KΩ

+

-

1KΩ

1KΩ

1KΩ

1KΩ

1KΩ

+

-

1KΩ

1KΩ

+

-

1/3 KΩ

1/3 KΩ

1/3 KΩ

1KΩ

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Example

1KΩ

1KΩ

1KΩ

1KΩ

+

-

1KΩ

1KΩ

1KΩ

1KΩ

1KΩ

+

-

1KΩ

1KΩ

3KΩ

+

-

1KΩ

3KΩ

3KΩ

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Equivalent transformation of Y- and ∆-connection of resistance

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  • Example

1Ω

4Ω

90Ω

9Ω

1Ω

+

-

9Ω

9Ω

9Ω

20V

1Ω

4Ω

90Ω

9Ω

1Ω

+

-

9Ω

9Ω

9Ω

20V

27Ω

27Ω

27Ω

1Ω

4Ω

90Ω

9Ω

1Ω

+

-

20V

27Ω

27Ω

27Ω