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Schedule Date Day Class Title Chapters HW Lab Exam No. Due - - PowerPoint PPT Presentation

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Schedule Date Day Class Title Chapters HW Lab Exam No. Due date Due date 1 Oct Wed 9 Equivalent Circuits 3.6 2 Oct Thu 3 Oct Fri Recitation HW 4 4 Oct Sat 5 Oct Sun 6 Oct Mon 10 Energy Storage 3.7, 4.1 NO LAB 7

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SLIDE 1

ECEN 301 Discussion #9 – Equivalent Circuits 1

Date Day Class No. Title Chapters HW Due date Lab Due date Exam 1 Oct Wed 9 Equivalent Circuits 3.6 2 Oct Thu 3 Oct Fri Recitation HW 4 4 Oct Sat 5 Oct Sun 6 Oct Mon 10 Energy Storage 3.7, 4.1 NO LAB 7 Oct Tue NO LAB 8 Oct Wed 11 Dynamic Circuits 4.2 – 4.4

Schedule…

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SLIDE 2

ECEN 301 Discussion #9 – Equivalent Circuits 2

Equivalence - Equality

Mosiah 29: 38 38 Therefore they relinquished their desires for a king, and became exceedingly anxious that every man should have an equal chance throughout all the land; yea, and every man expressed a willingness to answer for his own sins.

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SLIDE 3

ECEN 301 Discussion #9 – Equivalent Circuits 3

Current Sources

All current sources can be modeled as voltage sources (and vice-versa)

 Many sources are best modeled as voltage sources (batteries, electric

  • utlets etc.)

 There are some things that are best modeled as current sources:

  • Van de Graaff generator

Behaves as a current source because of its very high output voltage coupled with its very high output resistance and so it supplies the same few microamps at any output voltage up to hundreds of thousands of volts

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SLIDE 4

ECEN 301 Discussion #9 – Equivalent Circuits 4

Lecture 9 – Equivalent Circuits

Thévenin Equivalent Norton Equivalent

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SLIDE 5

ECEN 301 Discussion #9 – Equivalent Circuits 5

Network Analysis

Network Analysis Methods:

Node voltage method Mesh current method Superposition Equivalent circuits

Source transformation Thévenin equivalent Norton equivalent

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SLIDE 6

ECEN 301 Discussion #9 – Equivalent Circuits 6

Equivalent Circuits

It is always possible to view a complicated circuit in terms of a much simpler equivalent source and equivalent load circuit.

va

R1 R3 + – R2 R5 R4 i + v – Load + v – i Source

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SLIDE 7

ECEN 301 Discussion #9 – Equivalent Circuits 7

Equivalent Circuits

It is always possible to view a complicated circuit in terms of a much simpler equivalent source and equivalent load circuit.

va

R1 R3 + – R2 R5 R4 i + v – Load + v – i Source

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SLIDE 8

ECEN 301 Discussion #9 – Equivalent Circuits 8

Equivalent Circuits

Equivalent circuits fall into one of two classes:

Thévenin: voltage source vT and series resistor RT Norton: current source iN and parallel resistor RN

vT

+ – RT Load + v – i

iN

RN Load + v – i Thévenin Equivalent Norton Equivalent NB: RT = RN

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SLIDE 9

ECEN 301 Discussion #9 – Equivalent Circuits 9

Equivalent Circuits

Thévenin Theorem: when viewed from the load, any network comprised of independent sources and linear elements (resistors), may be represented by an equivalent circuit.

  • Equivalent circuit consists of an ideal voltage source vT in series with an

equivalent resistance RT vT

+ – RT Load + v – i “View from load” A fancy way of saying: “The circuit that includes everything except for the load”

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SLIDE 10

ECEN 301 Discussion #9 – Equivalent Circuits 10

Equivalent Circuits

Norton Theorem: when viewed from the load, any network comprised of independent sources and linear elements (resistors), may be represented by an equivalent circuit.

  • Equivalent circuit consists of an ideal current source iN in parallel with

an equivalent resistance RN

“View from load” A fancy way of saying: “The circuit that includes everything except for the load”

iN

RN Load + v – i

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SLIDE 11

ECEN 301 Discussion #9 – Equivalent Circuits 11

Thévenin and Norton Resistances

Computation of Thévenin and Norton Resistances:

1. Remove the load (open circuit at load terminal) 2. Zero all independent sources

 Voltage sources short circuit (v = 0)  Current sources

  • pen circuit (i = 0)

3. Compute equivalent resistance (with load removed)

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SLIDE 12

ECEN 301 Discussion #9 – Equivalent Circuits 12

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

is

R1 R3 R2 R6 R5

vs

+ – R4 R7 RL

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SLIDE 13

ECEN 301 Discussion #9 – Equivalent Circuits 13

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

is

R1 R3 R2 R6 R5

vs

+ – R4 R7 1. Remove the load

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SLIDE 14

ECEN 301 Discussion #9 – Equivalent Circuits 14

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

R1 R3 R2 R6 R5 R4 R7 2. Zero all independent sources

  • short circuit voltage sources
  • Open circuit current sources
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SLIDE 15

ECEN 301 Discussion #9 – Equivalent Circuits 15

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

R1 R3 R2 R6 R5 R4 R7 3. Compute equivalent resistance

10 6 4

2 1 1

R R REQ 1 4 4 ) 2 ( ) 2 ( ) 2 )( 2 (

5 4 5 4 2

R R R R REQ

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SLIDE 16

ECEN 301 Discussion #9 – Equivalent Circuits 16

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

REQ1 R3 R6 REQ2 R7 3. Compute equivalent resistance

5 20 100 ) 10 ( ) 10 ( ) 10 )( 10 (

3 1 3 1 3

R R R R R

EQ EQ EQ

1 10

2 1 EQ EQ

R R

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SLIDE 17

ECEN 301 Discussion #9 – Equivalent Circuits 17

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

REQ3 R6 REQ2 R7 3. Compute equivalent resistance 5 1

3 2 EQ EQ

R R

6 1 5

2 3 4 EQ EQ EQ

R R R

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SLIDE 18

ECEN 301 Discussion #9 – Equivalent Circuits 18

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

REQ4 R6 R7 3. Compute equivalent resistance

2 9 18 ) 3 ( ) 6 ( ) 3 )( 6 (

6 4 6 4 5

R R R R R

EQ EQ EQ

6

4 EQ

R

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SLIDE 19

ECEN 301 Discussion #9 – Equivalent Circuits 19

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

REQ5 R7 3. Compute equivalent resistance

7 5 2

7 5

R R R

EQ EQ

2

5 EQ

R

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SLIDE 20

ECEN 301 Discussion #9 – Equivalent Circuits 20

Thévenin and Norton Resistances

 Example1: find the equivalent resistance as seen by the load RL

 is =0.5A, vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω , R4 = 2Ω , R5 = 2Ω , R6 = 3Ω, R7 = 5Ω

REQ 3. Compute equivalent resistance

7

EQ

R

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SLIDE 21

ECEN 301 Discussion #9 – Equivalent Circuits 21

Thévenin Voltage

Thévenin equivalent voltage: equal to the open-circuit voltage (voc) present at the load terminals (load removed)

vT

+ – RT i = 0 + – voc = vT

vT

+ – RT Load + v – i Remove load

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SLIDE 22

ECEN 301 Discussion #9 – Equivalent Circuits 22

Thévenin Voltage

Computing Thévenin voltage:

1. Remove the load (open circuit at load terminals) 2. Define the open-circuit voltage (voc) across the load terminals 3. Chose a network analysis method to find voc

 node, mesh, superposition, etc.

4. Thévenin voltage vT = voc

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SLIDE 23

ECEN 301 Discussion #9 – Equivalent Circuits 23

Thévenin Voltage

Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω

R2 R1 RL R3

vs

+ – iL

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SLIDE 24

ECEN 301 Discussion #9 – Equivalent Circuits 24

Thévenin Voltage

Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω

+ R2 – +R1– + R3 –

vs

+ – 1. Remove the load 2. Define voc + voc –

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SLIDE 25

ECEN 301 Discussion #9 – Equivalent Circuits 25

Thévenin Voltage

Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω

+ R2 – +R1– + R3 –

vs

+ – 3. Choose a network analysis method

  • Voltage divider

+ voc – i

s

  • c

v R R R v

2 1 2

i3 = 0

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SLIDE 26

ECEN 301 Discussion #9 – Equivalent Circuits 26

Thévenin Voltage

Example2: find the Thévenin voltage vs = 10V, R1 = 4Ω, R2 = 6Ω , R3 = 10Ω

+ R2 – +R1– + R3 –

vs

+ – 4. vT = voc + voc – i

s T

v R R R v

2 1 2

i3 = 0

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SLIDE 27

ECEN 301 Discussion #9 – Equivalent Circuits 27

Thévenin Equivalent Circuit

Computing Thévenin Equivalent Circuit:

  • 1. Compute the Thévenin resistance RT
  • 2. Compute the Thévenin voltage vT

vT

+ – RT Load + v – i

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SLIDE 28

ECEN 301 Discussion #9 – Equivalent Circuits 28

Thévenin Equivalent Circuit

Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω

R2 R1 RL R3

vs

+ – iL

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SLIDE 29

ECEN 301 Discussion #9 – Equivalent Circuits 29

Thévenin Equivalent Circuit

Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω

1. Compute RT

  • Remove RL

+ R2 – +R1– + R3 –

vs

+ –

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SLIDE 30

ECEN 301 Discussion #9 – Equivalent Circuits 30

Thévenin Equivalent Circuit

Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω

1. Compute RT

  • Remove RL
  • Zero sources

+ R2 – +R1– + R3 –

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SLIDE 31

ECEN 301 Discussion #9 – Equivalent Circuits 31

Thévenin Equivalent Circuit

Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω

1. Compute RT

  • Remove RL
  • Zero sources
  • Compute RT = REQ

REQ

2 1 3

|| R R R RT

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SLIDE 32

ECEN 301 Discussion #9 – Equivalent Circuits 32

Thévenin Equivalent Circuit

Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω

1. Compute RT 2. Compute vT

  • (previously computed)

s T

v R R R v

2 1 2

R2 R1 RL R3

vs

+ – iL

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SLIDE 33

ECEN 301 Discussion #9 – Equivalent Circuits 33

Thévenin Equivalent Circuit

Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω

s T

v R R R v

2 1 2

R2 R1 RL R3

vs

+ – iL

vT

+ – RT RL iL

2 1 3

|| R R R RT

NB: iL is the same in both circuits

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SLIDE 34

ECEN 301 Discussion #9 – Equivalent Circuits 34

Thévenin Equivalent Circuit

Example3: find iL by finding the Thévenin equivalent circuit vs = 10V, R1 = 4Ω, R2 = 6Ω, R3 = 10Ω, RL = 10Ω

s T

v R R R v

2 1 2

vT

+ – RT RL iL

2 1 3

|| R R R RT

A R R R R v R R R R R v i

L s L T T L

27 . 4 . 22 6 4 . 2 20 1 10 60 ) 10 ( ) 6 ( || ) 4 ( ) 10 ( 1 ) 6 ( ) 4 ( ) 10 )( 6 ( || 1

2 1 3 2 1 2

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SLIDE 35

ECEN 301 Discussion #9 – Equivalent Circuits 35

Norton Current

Current equivalent current: equal to the short-circuit current (isc) present at the load terminals (load replaced with short circuit)

Short circuit load

iN

RN Load + v – i

iN

RN iN = isc

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SLIDE 36

ECEN 301 Discussion #9 – Equivalent Circuits 36

Norton Current

Computing Norton current:

1. Replace the load with a short circuit 2. Define the short-circuit current (isc) across the load terminals 3. Chose a network analysis method to find isc

 node, mesh, superposition, etc.

4. Norton current iN = isc

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SLIDE 37

ECEN 301 Discussion #9 – Equivalent Circuits 37

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

R1 R2 RL R3 – +

vs

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SLIDE 38

ECEN 301 Discussion #9 – Equivalent Circuits 38

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

R1 R2 R3 – +

vs

1. Short-circuit the load 2. Define isc isc

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SLIDE 39

ECEN 301 Discussion #9 – Equivalent Circuits 39

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

+ R1 – + R2 – + R3 – – +

vs

isc 3. Choose a network analysis method

  • Node voltage

Node a Node b Node d i1 i2 i3 1. va is independent 2. vb is dependent (actually va and vb are dependent on each other but choose vb) vb = va + vs iv

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SLIDE 40

ECEN 301 Discussion #9 – Equivalent Circuits 40

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

s v a v s

i i R v i i i

1 1

: a node at KCL

is

+ R1 – + R2 – + R3 – – +

vs

isc 3. Choose a network analysis method

  • Node voltage

Node a Node b Node d i1 i2 i3 iv

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SLIDE 41

ECEN 301 Discussion #9 – Equivalent Circuits 41

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

+ R1 – + R2 – + R3 – – +

vs

isc 3. Choose a network analysis method

  • Node voltage

Node a Node b Node d i1 i2 i3

3 2 3 2 3 2 3 2 3 2

1 1 1 1 ) ( ) ( : b node at KCL R R v R R v i R v v R v v i R v R v i i i i

s a v s a s a v b b v v

iv

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SLIDE 42

ECEN 301 Discussion #9 – Equivalent Circuits 42

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

12 6 30 5 6

a v a v

v i v i

is

+ R1 – + R2 – + R3 – – +

vs

isc 3. Choose a network analysis method

  • Node voltage

Node a Node b Node d i1 i2 i3 iv

V v A i

a v

3 5 . 2

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SLIDE 43

ECEN 301 Discussion #9 – Equivalent Circuits 43

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

+ R1 – + R2 – + R3 – – +

vs

isc 3. Choose a network analysis method

  • Node voltage

Node a Node b Node d i1 i2 i3 iv

A v v R v i

b s b sc

2 3 2 ) (

3

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SLIDE 44

ECEN 301 Discussion #9 – Equivalent Circuits 44

Norton Current

Example4: find the Norton equivalent current iN

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

+ R1 – + R2 – + R3 – – +

vs

isc 4. iN = isc i1 i2 i3 iv

A i i

sc N

5 . 1

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SLIDE 45

ECEN 301 Discussion #9 – Equivalent Circuits 45

Norton Equivalent Circuit

Computing Norton Equivalent Circuit:

  • 1. Compute the Norton resistance RN
  • 2. Compute the Norton current iN

iN

RN Load + v – i

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SLIDE 46

ECEN 301 Discussion #9 – Equivalent Circuits 46

Norton Equivalent Circuit

Example5: find the Norton equivalent circuit

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

R1 R2 RL R3 – +

vs

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SLIDE 47

ECEN 301 Discussion #9 – Equivalent Circuits 47

Norton Equivalent Circuit

Example5: find the Norton equivalent circuit

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

is

R1 R2 R3 – +

vs

1. Compute RN

  • Remove RL
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SLIDE 48

ECEN 301 Discussion #9 – Equivalent Circuits 48

Norton Equivalent Circuit

Example5: find the Norton equivalent circuit

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

R1 R2 R3 1. Compute RN

  • Remove RL
  • Zero sources
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SLIDE 49

ECEN 301 Discussion #9 – Equivalent Circuits 49

Norton Equivalent Circuit

1. Compute RN

  • Remove RL
  • Zero sources
  • Compute RN = REQ

REQ

2 1 3

|| R R R RN

Example5: find the Norton equivalent circuit

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

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SLIDE 50

ECEN 301 Discussion #9 – Equivalent Circuits 50

Norton Equivalent Circuit

Example5: find the Norton equivalent circuit

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

4 ||

2 1 3

R R R RN

1. Compute RN 2. Compute iN

  • (previously computed)

A i i

sc N

5 . 1

is

R1 R2 RL R3 – +

vs

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SLIDE 51

ECEN 301 Discussion #9 – Equivalent Circuits 51

Norton Equivalent Circuit

Example5: find the Norton equivalent circuit

vs = 6V, is = 2A, R1 = 6Ω, R2 = 3Ω, R3 = 2Ω, RL = 10Ω

4

N

R A iN 5 . 1

is

R1 R2 RL R3 – +

vs iN

RN RL