University Physics: Waves and Electricity

Ch27. Circuit Theory

Lecture 11

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

11/2January–April 2010 University Physics: Wave and Electricity

Homework IX (IT)

(a) Find the equivalent resistance between points a and b in the circuit diagram below.

(b) Calculate the current in each resistor if a potential difference of 34 V is applied between points a and b.

11/3January–April 2010 University Physics: Wave and Electricity

Solution of Homework IX (IT)

(a) eq 4 (7 ||10) 9R 7 10

4 97 10

17.12

(b)

4 4.12 9

≡eq

ViR

34

17.12 1.986 A

4 9 1.986 Ai i

4.12 7 10V V V

4.12 (1.986)(4.12)V 8.182 V

4.12 4.12 4.12V i R

11/4January–April 2010 University Physics: Wave and Electricity

Solution of Homework IX (IT)

≡

4.12 7 10V V V

77

Vi

R

8.182

7

1.169 A

1010 10

Vi

8.182

10

0.8182 A

4.12 4.12 7 7 10 10i R i R i R c d

c d

4.12 7 10 8.182 VcdV V V V

11/5January–April 2010 University Physics: Wave and Electricity

Example: Single-Loop Circuit

1 2R 2 1R 2 6 VE

1 12 VE

ba

For the circuit shown, determine Vab.

i1 2 1 2 0iR iR E E12 1 2 6 0i i

3 6i 2 Ai

• a a, cw

• a b, ccw

2 1a bV iR V +E

2 1a bV V iR E

6 (2)(2)abV

10 V

• Assume a current direction• Perform the calculation• i > 0 means that the current flows in

the assumed direction• i < 0 means that the current flows in

opposite direction

• Try again for i assumed to be ccw

11/6January–April 2010 University Physics: Wave and Electricity

2R 2V

E

1R 1V

i

Voltage Divider and Current Divider

1 2

iR R

E

1 1V iR

Voltage Divider Current Divider

11

1 2

RV

R R

E

2 2V iR 22

1 2

RV

R R

E

2R 2V

E

1R 1V

i

1i 2i

11

iR

E

22

iR

E

eqiRE

eq1

1

Ri i

R 2

11 2

Ri i

R R

12

1 2

Ri i

R R

eq

22

Ri i

R

11/7January–April 2010 University Physics: Wave and Electricity

2R

1R

Req of Two Resistance in Parallel

eq 1 2

1 1 1

R R R 2 1

1 2

R R

R R

1 2eq

1 2

R RR

R R

(3)(1)

3 10.75 eqR

2V 2

182 0.75 4

5.33 V

4V 4

182 0.75 4

10.67 V

3 1V V 18 5.33 10.67 2 V

0.75V 0.75

182 0.75 4

2 V

11/8January–April 2010 University Physics: Wave and Electricity

Example: Finding Voltage and Current

10

40

15V

6

12 1i

1V

2V

2i

For this circuit, determine i1, i2, V1, and V2.

15V

1V

4

2V

≡

1

415

4 8V

5 V

2

815

4 8V

10 V

11 12

Vi 0.417 A5

12

22 40

Vi 0.25 A10

40

6 ?i 10 ?i through the battery?i

11/9January–April 2010 University Physics: Wave and Electricity

Example: Two-Loop Circuit

1E 4 6R

2E

3E

5 4R

6 2R

1 A

2 A

1

2

If E1 = 20 V, determine E2 and E3.

1 4 2 5(1) (1) 0R R E E• Loop 1, cw

i

1 Ai 1 2i

220 (1)(6) (1)(4) 0 E

2 18 VE

5 2 6 3(1) (2) 0R R E E• Loop 2, cw

3(1)(4) 18 (2)(2) 0 E

3 10 VE

3 6 4 1(2) (1) 0R R E E• Outer Loop, ccw

3 (2)(2) (1)(6) 20 0 E

3 10 VE

11/10January–April 2010 University Physics: Wave and Electricity

Multiloop Circuits

Junction RuleThe sum of the currents entering any junction must be equal to the sum of the currents leaving that junction

• Remember again Loop Rule, Resistance Rule, and Emf Rule

Our basic tools for solving complex circuits are the loop rule and the junction rule.

1 3 2i i i • Junction Rule, at d

• Loop Rule, left-hand loop, b b, ccw

1 1 1 3 3 0i R i R E

• Loop Rule, right-hand loop, b b , ccw

3 3 2 2 2 0i R i R E• Afterward, find the unknowns

from these three equations

. . . . . . . . . .(1)

. . . . . .(2)

. . . . . .(3)

11/11January–April 2010 University Physics: Wave and Electricity

Example: Multiloop Circuits

The value of the elements in the following circuits are:

E1 = 3 V, E2 = 6 V,R1 = 2 Ω, R2 = 4 Ω

Find the magnitude and direction of the current in each of the branches.

3 1 2i i i • Junction Rule, at a

• Loop Rule, left-hand loop, a a, ccw

1 1 1 1 1 2 2 2 0i R i R i R E E

• Loop Rule, right-hand loop, a a , cw

. . . . . . . . . .(1)

. . .(2) . . .(3)1 1 22 3 2 6 4 0i i i

1 24 4 3i i

3 1 2 3 1 2 2 2 0i R i R i R E E

3 3 22 6 2 6 4 0i i i

2 34 4 0i i

11/12January–April 2010 University Physics: Wave and Electricity

Example: Multiloop Circuits

3 1 2i i i . . . . . . .(1)

. . .(2)

. . .(3)1 24 4 3i i

2 34 4 0i i

• Insert (1) into (3),

2 1 24 4( ) 0i i i

1 24 8 0i i . . .(4)

• Option A: Elimination Method

1 24 4 3i i

1 24 8 0i i

1

11 24 4 3i i

1 24 8 0i i

212 3i

• Option B: Substitution Method

21

3 4

4

ii

• From (2)

• Insert (5) into (4)

22

3 44 8 0

4

ii

2 23 4 8 0i i

212 3i

2 0.25 Ai

1

3 4( 0.25)

4i

2 0.25 Ai

. . . . . . . .(5)

0.5 A

3 0.5 ( 0.25)i 0.25 A1 2

8

4i i 0.5 A

3 0.5 ( 0.25)i 0.25 A

11/13January–April 2010 University Physics: Wave and Electricity

Homework X (IT)For the circuit depicted below, determine the currents (i1, i2, i3) and the potential difference across the resistors (V1, V2, V3).

5V

2

2V

1i 3i

2i

8

3 V

4

1V

3V