Department of Electronic EngineeringNational Taipei University of Technology

••••••• Y ∆

•

Department of Electronic Engineering, NTUT2/47

(Linear Circuits)

• :

• :y = Kx Kx y

R y = Kx

, where 0dy

ax adt

= ≠

, where 0dx

y b bdt

= ≠

v Ri=( ) ( )Kv K Ri R Ki= =

Department of Electronic Engineering, NTUT3/47

1

• 1: i1i2 vg1 ig2

KVL vg1 = i1 i2

KCL i1 + i2 = ig2

×2

+−

3 Ω

i1

6 Ω

i2

vg1 ig2

2 1 13 6 gi i v− + =

1 1 2

2 1 2

1 19 31 2

9 3

g g

g g

i v i

i v i

= + = − +

Department of Electronic Engineering, NTUT4/47

(Practical Sources) I

•

(a)(a)

(b) :

• Rs Rs

Rs Rs

= 0

sL

s L

vi

R R=

+L s

Ls L

R vv

R R

⋅=+

+−

(a)

Rs

vs

(b)

+−

+

−

Rs

iL

vs vL RL

Department of Electronic Engineering, NTUT5/47

II

•

Rg

+

−

is VRg

Department of Electronic Engineering, NTUT6/47

(Equivalent Circuit) I

•

• (a)

• (b)

+ − +

−

(a)

+−vs

v11

s

s s

v vvi

R R

−= =

222

s gs s

g g

i R vvi i i i

R R

−= − = − =

Rs i1

v1

+

−

(b)

i

i2

v2is Rg

i

–

Department of Electronic Engineering, NTUT7/47

II

• i1=i2 , v1=v2

i1= i2

v1= v2

•

21 s gs

s g

i R vv v

R R

−− =

11 s gs

s g

i R vv v

R R

−− =

+ − +

−

(a)

+−vs

v

Rs i1

v1

+

−

(b)

i

i2

v2is Rg

i

and s s g s gv i R R R= =

Department of Electronic Engineering, NTUT8/47

2

• 2: (a) I

(c)

+− 8 V

I

(a)

+−

3 Ω

4 AI

12 Ω 1 Ω

6 Ω 6 Ω 2 Ω32 V

(b)

+− 8 VA

332 I

12 Ω 1 Ω 2 Ω

6 Ω6 Ω3 Ω

12 Ω 3 Ω

6 Ω2 ΩA332

Department of Electronic Engineering, NTUT9/47

2

+−

16 Ω

16 VI

(g)

(a) (f) (g)

(e)

+−

IV3

64A3

8

12 Ω2 Ω

2 Ω

(d)

+−

IV3

64

12 Ω2 Ω

6 Ω 3 Ω A38

(f)

+−

+−

I

2 Ω 12 Ω 2 Ω

V316

V364

( ) ( )64 162 12 2 1 A

3 3I I+ + = − ⇒ =

Department of Electronic Engineering, NTUT10/47

I

• (Superposition Theorem)

Department of Electronic Engineering, NTUT11/47

II

•

1)

2)

3) (1) (2)

4)

5) ( ) KK

Department of Electronic Engineering, NTUT12/47

3

• 3: (a)I

(b) (c)(b) (c) 4Ia 4Ib

5Ω

(b):

+ −

+−

4 Ω

(a)

I

4I

18 Ω4 A20 V

5 Ω

+ −

+−

(b)

4 Ω 4Ia

18 Ω20 V

Ia

5 Ω

+ −

(c)

4 Ω 4Ib

18 Ω4 ΑIb

5 Ω

4 − Ib

( ) ( )20 4 4 18 5 0.87 Aa a aI I I− = − + + ⇒ = −( ) ( ) ( )4 4 5 4 18 3.13 Ab a b bI I I I= + − − ⋅ ⇒ =

( )2.26 Aa bI I I= + =

Department of Electronic Engineering, NTUT13/47

4

• 4: (a) 5 Ω

+ −

(c) (d)

v+ −

(a)

+−4 Ω

3 Ω

5 Ω

5 A

2 A

10 V

+ −

(b)

v1

5 Ω

4 Ω

3 Ω

5 A

v2

5 Ω

4 Ω

3 Ω2 A

+ −−+

v3

5 Ω

4 Ω

3 Ω10 V

( )5

5 41.67 A

4 3 5I Ω

⋅= =+ +

( )1 1.67 5 8.33 Vv = ⋅ =

( )5

2 30.5 A

4 3 5I Ω

⋅= =+ +

( )2 0.5 5 2.5 Vv = ⋅ = ( )3

10 54.17 V

4 3 5v

⋅= =+ +

( )1 2 3 15 Vv v v v= + + = ( )2 215

45 W5

vp

R= = =

Department of Electronic Engineering, NTUT14/47

III

• :

•

• p = i2R p = v2/R

( )2 21

1

8.3313.95 W

5

vp

R= = =

( )2 22

2

2.51.25 W

5

vp

R= = =

( )2 23

3

4.173.48 W

5

vp

R= = =

( )1 2 3 18.63 Wp p p p= + + =

Department of Electronic Engineering, NTUT15/47

IV

•

1)

2)

Department of Electronic Engineering, NTUT16/47

I

• (Thevenin’s Theorem)

• (voc eth) (Rth)

1) voc eth

(a)

2) Rth () (b)

( )

(a)

+

-

voc

a

b(b)

()

Rth

a

b

Department of Electronic Engineering, NTUT17/47

II

•

+

-

(a)

+-

+

-

(b)

voc

I

v

Rth I

v

a

b

a

b

Department of Electronic Engineering, NTUT18/47

5

• 5: (a) v1

Rth

(b)

voc

:

1)(c)

+−

+

−(a)

v1

a

b

6 Ω 4 Ω

3 Ω 2 Ω

2 A

12 V

(b)Rth

a

b

6 Ω 4 Ω

3 Ω

(c)

+− ′Voc

a

b

6 Ω 4 Ω

3 Ω12 V

( )6//3 4 6 thR = + = Ω

( )312 4 V

3 6ocv′ = =+

Department of Electronic Engineering, NTUT19/47

5

a-b (e)

(f)

2 A 4 Ω 3 Ω 6 Ωb c

1)

2) (d)

( )2 4 8 Vocv′′ = ⋅ =

( )2 4 8 Voc ab acv v v′′ = = = ⋅ =

( )12 Voc oc ocv v v′ ′′= + =

( )1

212 =3 V

6 2v =

+

(d)

c

′′voc

a

b

4 Ω6 Ω

3 Ω

2 Α

(e)

+−

6 Ω

12 V +−

(f)

+

−v1

a

b

6 Ω

2 Ω12 V

Department of Electronic Engineering, NTUT20/47

I

• (Norton’s Theorem)

• (Isc) (Rth)

1) Isc:

2) Rth:

Isc

a

b

Department of Electronic Engineering, NTUT21/47

II

•

• (a) a-b(b)

(a) (b)

a

b

Isc Rth

a

b

(a)

Isc

a

b

ocsc

th

vI

R=

oc ab th scv v R I= =

+-

(b)

voc

Rth a

b

Isc

+

-

(c)

a

b

vocRthIsc

Department of Electronic Engineering, NTUT22/47

6

• 6: (a) I

A. (a) Rth

B. a-b (b)

C. a-b (c)

D. (d) I

+− Ω3V24

(a)

I

Rth

a

b

Ω2

Ω6 Ω3

+− V24

(b)

I1

Isc

a

b

Ω6 Ω3

Ω3

A2.8

(c)

a

b

Ω5

(d)

Ia

b

A2.8 Ω5 Ω2

( )6 //3 3 5 thR = + = Ω

( )1

425.6 A

6 3//3I = =

+3

5.6 2.8 A3 3scI = =

+

52.8 2 A

5 2I = =

+

Department of Electronic Engineering, NTUT23/47

III

• () Rth

•Rth Rth

voc Isc

(Driving Point)v( I ) I(

v)

octh

sc

vR

I=

th

vR

I=

Department of Electronic Engineering, NTUT24/47

IV

• (a)(b)

v I+ −

+ −Rth

(a)

+ −+−

S.C.

O.C.

v

(b) Rth

I

a

b

a

b

th

vR

I=

Department of Electronic Engineering, NTUT25/47

7

• 7 (a) a-b

A. isc a-b (b)

+−

6 Ω4 ΩA10

2i1V

i1

a

b(a)

3 Ω

2 110 sci i i= − −

( )1 1 14 10 2 6 0sci i i i− − − − + =

16 3 0sci i− + =

5 Asci =

A10

+−V

(b)

4 Ω 6 Ω

3 Ω2i1

i2 i1 isc

Department of Electronic Engineering, NTUT26/47

7

B. Rth

Rth

(c)

(d)

C. a-b(e)

+−A10

+

−

a

b(c)

(e)

16ocv i=( )1 1 16 2 4 10i i i= + −

30 Vocv⇒ =

( )306

5oc

thsc

vR

i= = = Ω

13 6v i i= +( )1 1 16 2 4i i i i= + −

( )1 0.5 , 6 , 6 th

vi i v i R

i= = = = Ω

3 ΩV2i1i1

4 Ω 6 Ωi110−i1 voc

(d)

+−

+−

a

b

v

Rth

i4 Ω 6 Ω

V2i1

i−i1 i1

3 Ω

a

b

5 A6 Ω

Department of Electronic Engineering, NTUT27/47

8

• 8 (a)

A. (b)

+−a

b(a)

6 Ω

V2i1

i14 Ω

0, 0oc scv i= =

( ) 1th

v vR v

i= = = Ω

121

4 6

v i v− + =

16v i= ( ) ( )1 0.5 A , 3 Vi v= = ( )3 thR = Ω

(b) Rth

+−

b

a

1A

+

−v

Rth

vV2i1

v−2i1

6 Ω4 Ωi1

Department of Electronic Engineering, NTUT28/47

8

(c) Rth

(d)

C. (d)

B. (c)

16v i=( )1 1 16 2 4i i i i= + −

1 0.5 , 3i i v i= =

( )3 th

vR

i= = Ω

v

+−

b

a

V

+

−v

Rth

vV2i1

i−i1

6 Ω4 Ωi1 +

−

i

a

b

3 Ω

Department of Electronic Engineering, NTUT29/47

I

•

•

• ,

•

sL

s L

vi

R R=

+

( )

2 22

2s s L

L L Ls L s L

v v Rp i R R

R R R R

= = ⋅ = + +

0L

dp

dR=

( ) ( )( )

2 2 2

4

2s L s s L s L

L s L

R R v v R R Rdp

dR R R

+ − + =+

( ) ( )( )

2 2

4

2s L s L s s L

s L

R R R R v v R

R R

+ + − =+

( )( )( )

22 2

3 3s L ss s L s

s L s L

R R vR v R v

R R R R

−−= =+ +

s LR R= 0L

dp

dR=

s LR R=( )

2 2

2 4s L s

Ls L

v R vp

RR R= =

+

+−

Rs

+

−vs

iL

RLvL

Department of Electronic Engineering, NTUT30/47

II

• Rs RL = Rs

RL

•

•

RL=Rth RL

+−

Rth

+

−eth RL = Rth

2

max 4s

L

vp

R=

2

max 4g gR i

p =

2

max 4th

th

ep

R=

Department of Electronic Engineering, NTUT31/47

9

• 9 (a) RL

+ −30V

RL

6 Ω

3A8 Ω

20V

1A

4 Ω

2 Ω

+ −20V

(a)

30V

6 Ω

3A8 Ω

20V

1A

4 Ω

2 Ω

20V + −1A

3A

2A

+ −eth+ −

(b)

(b)

8 4 2 6 20 thR = + + + = Ω30 20 8 1 4 2 2 2 6 3 72 Vth the e+ + = ⋅ − ⋅ − ⋅ − ⋅ ⇒ = −

( )22 7264.8 W

4 4 20th

mth

ep

R

−= = =

⋅Department of Electronic Engineering, NTUT32/47

• (Millman’s Theorem)

• N a

• N

+−v1

+−

+−

b

a+

−vv2 vN

R1 R2 RN

1 2

1 2

0N

N

v vv v v v

R R R

−− −+ + + =⋯

( ) ( ) ( )( ) ( ) ( )1 1 2 2

1 21 1 1N N

ocN

v R v R v Rv v

R R R

+ + += =

+ + +⋯

⋯ ( ) ( ) ( )1 2

1

1 1 1thN

RR R R

=+ + +⋯

Department of Electronic Engineering, NTUT33/47

10

• 10

+−

+− +

−

4 Ω 4 Ω

3 Ω

2 Ω

24V32V40V

+

−

v

( )40 32 0 24

184 4 3 2 4.5 V1 1 1 1 44 4 3 2

v+ + −

= = =+ + +

Department of Electronic Engineering, NTUT34/47

Y ∆ I

• Y ∆

∆

RY

R R y

∆∆

=

′a ′b

′c

ba

c

R1 R2

R3RA RC

RB( ) 1 2//ab a b B A CR R R R R R R′ ′= ⇒ + = +

( ) 2 3//bc b c C B AR R R R R R R′ ′= ⇒ + = +

( ) 3 1//ca c a A B CR R R R R R R′ ′= ⇒ + = +

1 2 2 3 3 1

2A

R R R R R RR

R

+ +=

1 2 2 3 3 1

3B

R R R R R RR

R

+ +=

1 2 2 3 3 1

1C

R R R R R RR

R

+ +=

Department of Electronic Engineering, NTUT35/47

Y ∆ II

• Y ∆

• Y

Ry

R

R= ∆

∆

′a ′b

′c

ba

c

R1 R2

R3RA RC

RB( ) 1 2//ab a b B A CR R R R R R R′ ′= ⇒ + = +

( ) 2 3//bc b c C B AR R R R R R R′ ′= ⇒ + = +

( ) 3 1//ca c a A B CR R R R R R R′ ′= ⇒ + = +

1A B

A B C

R RR

R R R=

+ +

2B C

A B C

R RR

R R R=

+ +

3C A

A B C

R RR

R R R=

+ +

Department of Electronic Engineering, NTUT36/47

11

• 11 (a) I vab

2 Ω+

−vab

4 Ω

4 Ω

2 Ω

3 Ω

5 Ω

I

2.5 Ωd b

+ −200V

c a3.4 Ω

(a)

∆ (b) Y

+

−

d b

+ −200V

c a

I1

IT

2.5 Ω

3.4 Ω

1 Ω0.6 Ω

1.5 ΩI2

vab

0.8 Ω

16 Ω0.8 Ω

(b)

( ) ( ) ( )1 0.6 3.4 0.8 // 1.5 2.5 0.8 1.6 5 TR = + + + + + + = Ω ( )20040 AT

T

IR

= =

( )1 2 2 20 ATI I I= = = ( )1 20.6 1.5 18 Vcdv I I= − + =

( )6 A3cdv

I = = ( )1 20.8 0.8 0 Vabv I I= − =

Department of Electronic Engineering, NTUT37/47

I

• Y - ∆ Y ( ) → ∆( )

•

1 2 2 3 3 1 1 2 2 3 3 1 1 212 1 2

3 3 1 2 1 2 3

1 1 1R R R R R R R R R R R R R RR R R

R R R R R R R

+ + + += = = + +

1

1,2, ,1

,where 1,2, ,N

ij i jp p

i N

R R R j NR

i j=

== =

≠∑

⋯

⋯

∆ ( )

23

1

23

1

Y( )

R1

R3 R2

R31 R12

R23

Department of Electronic Engineering, NTUT38/47

II

• N = 4

(1) (Star to Mesh Conversion)

= ( ) × ( )

4 3

1 2

R1 R2

R4 R3

4 3

21

R23

R34

R41

R13

R24

R12

(a) N = 4 ( )

(b) (1/2)N(N-1) = 6 ( )

23 2 31 2 3 4

1 1 1 1R R R

R R R R

= + + +

34 3 4

1 2 3 4

1 1 1 1R R R

R R R R

= + + +

41 4 1

1 2 3 4

1 1 1 1R R R

R R R R

= + + +

13 1 31 2 3 4

1 1 1 1R R R

R R R R

= + + +

24 2 4

1 2 3 4

1 1 1 1R R R

R R R R

= + + +

12 1 2

1 2 3 4

1 1 1 1R R R

R R R R

= + + +

Department of Electronic Engineering, NTUT39/47

III

(2) (Mesh to Star Conversion)

N

R ( ) = R ( ) / N R ( ) = N × R ( )

12 1 21 2 3 4

23 2 31 2 3 4

24 2 41 2 3 4

1 1 1 1

1 1 1 1

1 1 1 1

R R R RR R R R

R R R RR R R R

R R R RR R R R

= + + + =

= + + + =

= + + + =

⋮

12 23 24R R R R= = = =⋯

2312 241 2 3 4 4 4 4 4

RR R RR R R R= = = = = = = =⋯

Department of Electronic Engineering, NTUT40/47

•

()

D

A B

C

C

D

10 ΩA B

D

A B

C

(C, D )

(C, D )

Department of Electronic Engineering, NTUT41/47

12

• 12 (a) A-B Req

(a)

C

A4 Ω 4 Ω

6 Ω

2 Ω2 Ω

B

D

10

Ω

C

A4Ω 4Ω

6Ω

2 Ω2 Ω

B

D

C

A4Ω 4Ω

6Ω

2 Ω2 Ω

B

D

∆

C

A

RA RB

B

D

10 ΩRC

(b)

C

A4 Ω 4 Ω

6 Ω

2 Ω2 Ω

B

D

10

Ω

(C, D ) (C, D )

( )24

13eqR = Ω

Department of Electronic Engineering, NTUT42/47

13

• 13 AC Req

E

CD

A B10 Ω

10 Ω12 Ω

12 Ω

E

CD

A B10 Ω

10 Ω12 Ω

12 Ω

E

CD

A

B

10 Ω

10 Ω12 Ω

12 Ω

(D-E, E-B )

(D-E, E-B )

( ) ( ) ( ) ( )4012 12 // 6 6 // 10 10

7eqR = + + + = Ω

( ) ( ) ( )4012 //6 //10 12 //6 //10

7eqR = + = Ω

Department of Electronic Engineering, NTUT43/47

•

()

C

5 Ω

A B

D

(a)

C D,

5 ΩA B

12 Ω

(b)

(C, D )

Department of Electronic Engineering, NTUT44/47

14

• 14 AD Req

1.

2. BC 10 ΩAD

Req ( )

B

A5 Ω 2 Ω

2 Ω

12 Ω

10 Ω12 Ω

C

D

B C,

A D5 Ω

12 Ω

2 Ω

B

A

5 Ω

2 Ω12 Ω

10 Ω

12 Ω

C

D

(a)

( ) ( ) ( )3512 12 2 // 2 //5

12eqR = + + = Ω

(B, C )

Department of Electronic Engineering, NTUT45/47

15

• 15 R = 10 ΩAB Req

( )

10 Ω

G

H

BA

D

CF

E

5 Ω

5 Ω

5 ΩF

G E× D

B H× C

A

5 Ω

5 Ω

10 Ω

10 Ω

(B-H, G-E )

10 Ω

GF

E D

H C

BA

( ) ( )355 10 5 //5 5 5 //10

6eq eqR R′= = + + + + = Ω

Department of Electronic Engineering, NTUT46/47

•

•

•

• Y ∆

Department of Electronic Engineering, NTUT47/47