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5.3 Thévenin and Norton Equivalent Circuits 5.4 Maximum Power Transfer

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회로이론-І 2013년2학기 이문석 1

Handy Circuit Analysis Techniques

5.1 Linearity and Superposition 5.2 Source Transformation

5.3 Thévenin and Norton Equivalent Circuits 5.4 Maximum Power Transfer

5.5 Delta-Wye Conversion

5.6 Selecting an Approach: A Summary of Various Techniques

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• Linear Elements and Linear Circuits

linear element: a passive element that has a linear voltage-current relationship linear dependent source: a dependent source whose output is proportional only to the first power of a specified variable in the circuit (or to the sum of such quantities)

linear circuit: a circuit composed entirely of independent sources, linear dependent sources, and linear elements.

=0.5 +3 vs. =0.5

2

In any linear resistive network, the voltage across or the current through any resistor or source may be calculated by adding algebraically all the individual voltages or currents caused by the separate independent sources acting alone, with

Superposition Theorem

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회로이론-І 2013년2학기 이문석 3

=1.5 +0.25 , =0.25 +0.875

Node 1: 0.7 0.2

Node 2: 0.2 1.2

Node 1: 0.7 0.2

Node 2

:

0.2 1.2 0

=1.5 , =0.25

=0.25 , =0.875

Node 2: 0.2 1.2

Node 1

:

0.7 0.2 0

∶ = + , =

(4)

Example 5.1 Use superposition to determine the branch current ix

3

6 9

3 15

6

6 9

6

152 12 15

(by current division)

3

15

12 15 1

set 0 set 0

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회로이론-І 2013년2학기 이문석 5

6

100 64

6 164

64

100 64 , 100

100 64

6

164

64 164 , 6

164

100 164

⇒ 1/4

100 50

⇒ 1/4

64 62.5

6 164

64

164 50 10

⇒ 86.59 10 164

64 222

6 164

100

164 62.5 10

⇒ 25.91 10 164

100 42.5

positive Ix

_

42.5

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Example 5.3 Use the superposition principle to determine the current ix

10 2 2 0

→ 2

0

2 3 2

1 0

→ 3 2 2 0

→ 0.6 2

∴ 2 0.6 1.4

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회로이론-І 2013년2학기 이문석 7

Practical Voltage Sources

ideal 12 V voltage source

independent of the current flowing through them 1 Ω

12

1 12

1 Ω

12

10 12,000

0 Ω

∞ ?

What happens?

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practical 12 V voltage source

12 0.01 0

→ 0.01 12

→ 0.01 . 12

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회로이론-І 2013년2학기 이문석 9

0

∞ open circuit 0

open circuit voltage

0 short circuit 0

short circuit current internal resistance

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Practical Current Sources

∞ open circuit 0

open circuit voltage

0 short circuit 0

short circuit current

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회로이론-І 2013년2학기 이문석 11

Equivalent Practical Sources

equivalent if

6 3 2Ω

=

(12)

Equivalent Practical Sources

4 Ω

4 Ω 2 Ω

6 V

3

4

2 46 4 6

2 4 1 4

6 1 6

2 1 2

2

2 4 3 4 4 2

2 43 1 4

Equivalent at load

2 Ω

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회로이론-І 2013년2학기 이문석 13

45 5 4.7 3 3 0

→ 42

5 4.7 3 3.307 Example 5.5 Calculate the current through the 2  resistor

7.5 3.5 51 26

2 9 0

2

⇒ 31.5 51 2 1.5

→ 1.5

102 31.5 21.28

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source transformation: OK!

: current source and resistor are in parallel

source transformation: invalid!

: voltage source and resistor are NOT in series

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회로이론-І 2013년2학기 이문석 15

Regardless of value, may omit

Voltage source and resistor are in parallel

does not alter the voltage across, the current through, or the power dissipated by

Current source and resistor are in series

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Thévenin equivalent network Norton equivalent network

≡ ≡

TH N

TH N N

R R

V I R

Voltage, current, and

power at the load is

maintained !!!

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회로이론-І 2013년2학기 이문석 17

Thévenin’s Theorem

1. Given any linear circuit, rearrange it in the form of two networks, A and B, connected by two wires. A is the network to be simplified; B will be left untouched.

2. Disconnect network B. Define a voltage V

oc

as the voltage now appearing across the terminals of network A.

3. Turn off or “zero out” every independent source in the network A to form an inactive network. Leave dependent sources unchanged.

4. Connect an independent voltage source with value V

oc

in series with the inactive network. Do now complete the circuit; leave the two terminals disconnected (open).

5. Connect network B to the terminals of the new network A. All currents and voltages in B will remain unchanged.

6. The only restriction that we must impose on A or B is that all dependent sources in A have their control variables in A, and similarly for B

7. The inactive network A can be represented by a single equivalent

resistance R

TH

, which we will call the Thévenin equivalent resistance .

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Norton’s Theorem

1. Given any linear circuit, rearrange it in the form of two networks, A and B, connected by two wires. Network A is the network to be simplified; B will be left untouched. As before, if either network contains a dependent

source, its controlling variable must be in the same network

2. Disconnect network B and short the terminals of A. Define a current i

sc

as the current now flowing through the shorted terminals of network A.

3. Turn off or “zero out” every independent source in the network A to form an inactive network. Leave dependent sources unchanged.

4. Connect an independent voltage source with value i

sc

in parallel with the

inactive network. Do now complete the circuit; leave the two terminals

disconnected.

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회로이론-І 2013년2학기 이문석 19

Zero-out the independent sources in network A.

Network A Network B

3 2 5 Ω

| |

| 4

| 2 10 2 10 4

| | 4 4 8

Thévenin equivalent network Norton equivalent network Source

transform

| |

= 2=

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When Dependent sources are present

- Controlling variable and its associated element(s) should be in the same network.

Example 5.9 Determine the Thévenin equivalent circuits

4 2 10

4000 0

→ 8

Useless form of Thévenin circuit

1 1 2

0.1 - Finding RTH

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회로이론-І 2013년2학기 이문석 21

0

0

0

0 ∶

1 0.6Ω

1 2

1.5

→ 6 3 23 1.5 1 2

→ 3

5

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Digital Multimeter: Voltage

1||1 500 Ω 0.5k

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회로이론-І 2013년2학기 이문석 23

9 1000 1000 0

→ 9

2000

0.1 Ω → 4.4998 Less than 4.5 mA (ideal value)

Digital Multimeter: Resistance

|

10 Ω 10Ω

→ || 9.9999Ω

10 Ω 10 Ω

→ || 5 Ω

DMM actually measure ||

Ideal DMM

0 for current measurment

∞ for voltage measurmetn

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- An independent voltage source in series with a resistance Rs, or an independent current source in parallel with a resistance Rs, delivers maximum power to a load resistance RL such that RL = Rs.

2 0

⇒ 2

0 → 0,

∞ → 0

Minimum power transfer

Condition for maximum power transfer

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회로이론-І 2013년2학기 이문석 25

1 0.03 30

||

. || 2.5 440

⇒ 69.6 sin 440

4

1.211 sin 440 1 Ω

(26)

∆ network Y network

= =

0

v

ac

, v

bc

, i

1

,

and i

3

are

same in both

networks

(27)

회로이론-І 2013년2학기 이문석 27

1 3

1 3 4

3 3 4 8

1 3 4

12 4 1 8

1 3 4

4 8

- Due day : 5장 수업 끝나고 일주일 후 수업시작 전까지 제출.

Homework : 5장 Exercises 5의 배수 문제 (70번 문제까지)

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용하였음.

All figures at this slide file are provided from The McGraw-hill company.

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