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Lecture 03

Equivalent transformation of resistance circuit

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Equivalent transformation of resistance circuit

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  • Equivalent transformation of circuit
  • Serial and parallel connection of resistors
  • Equivalent transformation of Y- and ∆-connection of resistance
  • Serial and parallel connection of voltage source and current source
  • Two models of actual power supply and their equivalent transformation
  • Input resistance

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Equivalent transformation of circuit

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  • Resistance circuit: A circuit consisting only of power supply and linear resistors

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  • Methodology: Ohm’s law and Kirchhoff‘s law

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  • Two terminal circuit: Any complex circuit that leads out two terminals and the current flowing from one terminal is equal to the current flowing from the other terminal is called a two-terminal network.
  • Equivalent two-terminal circuits: If the terminal voltage and current of the two-terminal circuits are the same respectively, these two-terminal circuits are equivalent.

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​

​

  • Note: VCR is the same only at terminals, but the inner of circuit might be different.

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B

A

C

A

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Serial and parallel connection of resistors

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  • Serial connection of resistors

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    • Each resistor is connected in sequence, and the current flowing through each resistor is the same
    • The total voltage is equal to the sum of voltages of the serial resistors.
    • The equivalent resistor value is equal to the sum of values of the serial resistors.

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​

​

u

u6

u5

u4

u3

u2

u1

+

-

R1

R2

R3

R4

R5

R6

+

+

+

+

+

+

-

-

-

-

-

-

u

Req

+

-

i

i

u= ⅀ui Req =⅀Ri

ui=Ri*u/Req

The total power

P = i2*(R1+R2+…+R6)=⅀PRi=i2Req

The power distributed to each resistor

PRi = i2*Ri

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Serial and parallel connection of resistors

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  • Serial connection of resistors

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    • When the resistors are connected in series, the power consumed by each resistor is proportional to the resistance value.

​

    • The power consumed by the equivalent resistor is equal to the sum of the power consumed by all resistors.

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Serial and parallel connection of resistors

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  • Parallel connection of resistors
    • The voltage of each resistor is the same.
    • The total current is equal to the sum of the currents flowing through all the resistors.

i6

i5

i4

i3

i2

+

-

R1

R2

R3

R4

R5

R6

i

u

i1

i= ⅀ik= ⅀u/Rk = u ⅀1/Rk

= u/Req = uGeq

+

-

Req

i

u

1/Req = ⅀1/Rk 🡪 Geq = ⅀Gk 🡪

Req<Rk

ik /i= (u/Rk)/ (u/R) =Gk/Geq 🡪 ik =i*(Gk/Geq)

The power distributed to each resistor

PRi = u2*Gi

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Serial and parallel connection of resistors

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  • Parallel connection of resistors
    • When the resistors are connected in parallel, the power consumed by each resistor is proportional to the conductance value (Gi), or inverse proportional to the resistance value.

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    • The power consumed by the equivalent resistance is equal to the sum of the power consumed by all parallel resistors.

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Serial and parallel connection of resistors

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  • Serial and parallel connection of resistors: There are both serial resistors and parallel resistors in the circuit.

+

-

165V

i1

5Ω

6Ω

i2

i3

i4

i5

4Ω

12Ω

18Ω

  • Example: calculate the voltage and current of each branch.

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R1

R2

R3

R4

R5

R45-eq=R4*R5/(R4+R5) = 4*12/16 = 3Ω

R345-eq= R3 + R45-eq = 6+3 = 9Ω

R2345-eq= R2*R345-eq/(R2+R345-eq) = 18*9/27 = 6Ω

R12345-eq= R1 + R2345-eq = 5+6 = 11Ω

i1 = u/R12345-eq = 165/11 = 15A

uR1 = i1*R1 = 15*5 = 75V

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uR2=(u-uR1)=165-75=90V; iR2= uR2/R2=90/18=5A

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i3 = i1-i2 = 15-5 = 10A; uR3 = i3*R3 = 60V

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uR4 = uR5 = uR2-uR3 = 90-60 = 30V;

iR4 = uR4/R4 = 7.5A; iR5 = uR5/R5 = 2.5A

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u

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Serial and parallel connection of resistors

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  • Example: calculate the current i1,i4 and voltage u4

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R34-eq=R3*R4/(R3+R4) = 1Ω

R346-eq=R6+ R34-eq = 2Ω

R2346-eq= R2*R346-eq/(R2+R346-eq) = 1Ω

R23456-eq=R5+ R2346-eq = 2Ω

R123456-eq= R1*R23456-eq/(R1+R23456-eq) = 1Ω

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i1 = u/R123456-eq = 12A

iR1 = u/R1 = 6A

i2 = i1 - iR1 = 6A, uR5 = 6V

uR2 = u-uR5 = 6V, iR2 = uR2/R2 = 3A

i3 = i2 - iR2 = 3A, uR6 = 3V

uR3 = uR2-uR6 = 6-3 = 3V, iR3=uR3/R3 = 1.5A

i3+i4 = iR3 🡪 i4 = iR3 - i3 = -1.5A

u4 = uR2 – uR6 = 3V

+

-

R1

i1

R2

i3

R3

i4

R4

u=12V

i2

+

-

u4

2Ω

2Ω

2Ω

2Ω

1Ω

1Ω

R6

R5

  • Calculate the equivalent resistance or equivalent conductance.
  • Apply Ohm's law to calculate the total current or total voltage.
  • The voltage and current of each branch are calculated by using Ohm's law or the formula of partial voltage principle (partial current principle).

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Serial and parallel connection of resistors

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  • Assignment:

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a

b

c

d

6Ω

5Ω

15Ω

5Ω

Rab-eq=?

Rcd-eq=?

20Ω

40Ω

100Ω

10Ω

60Ω

50Ω

80Ω

a

b

Rab-eq=?

c

d

e

c

a

b

d

e

120Ω

60Ω

60Ω

20Ω

100Ω

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Serial and parallel connection of resistors

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  • Assignment :

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Rab-eq=?

a

b

5Ω

20Ω

6Ω

6Ω

15Ω

7Ω

a

b

5Ω

20Ω

6Ω

6Ω

15Ω

7Ω

a

b

4Ω

3Ω

7Ω

15Ω

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Serial and parallel connection of resistors

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  • Assignment :

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a

b

c

d

R

R

R

R

R

Rab-eq=?

symmetric circuit

a

b

c

d

R

R

R

R

R

equal potential