Problems

Section 17-3: T-to-T1 Transformations P17.2-1

P17.2-2

1

P17.2-3

21 1 2 212 i

22 L 1 22 L

(forward current gain)

z I I zI Az R I z R− −

= ⇒ = =+ +

( )12 i 11 11 1 12 2 12 21

in 11 111 1 1 22 L

(input resistance)z A IV z I z I z zR z z

I I I z R+

= = = − = −+

i L2

2 2 L i L 1 1 in 1 v1 in

and (forward voltage gain)A RVV I R A R I V R I A

V R= − = = ⇒ = =

L2p i v i

in

R

A A A AR

∴ = =

P17.2-4 First, simplify the circuit using a Δ-Y transformation:

1eq || 5 || 20 4

3R

R R= = = Ω

Mesh equations: 2

2

30 18 1050 10 20

I II I

= −= −

Solving for the required current: 30 1050 20 100 0.385 A

18( 20) ( 10)10 260I

−− −= = =

− − − −

2

P17.2-5

3

Section 17-3: Equations of Two-Port Networks P17.3-1

18 6

6 9⎡ ⎤

= Ω⎢ ⎥⎣ ⎦

Z

12

11 12 11

22 21 22

6

12 18

3 9

Z

Z Z Z

Z Z Z

= Ω

− = Ω ⇒ = Ω

− = Ω ⇒ = Ω

2

1

1

111

1 0

1 212 21

2 20

2 222

2 20

1 S14

6 1 S (6 12) 21

/ 7 1 S7

V

V

V

IYV

I IYV V

I VYV V

=

=

=

= =

−= = = − = Υ

+

= = =

1 1 14 21= S1 1 21 7

⎡ ⎤−⎢ ⎥⎢ ⎥−⎣ ⎦

Υ

P17.3-2

12 21

11 12 11

22 21 22

4

2 2 4

2 2 4

z z j

z z z j

z z j z j j j

= = − Ω

− = Ω ⇒ = − Ω

2 − = Ω ⇒ = − = − Ω

2 4 4

4 2 j j

j j− −⎡ ⎤ = Ω⎢ ⎥− −⎣ ⎦

Z

1

P17.3-3

22

1 211 21

1 1 00

and VV

I IY YV V

==

= =

1 2 1 2 2

1 11 1 2

and I I I I IV bG G G

V+ += + =

1

so 1 1 2 1 1 2 2 1 ( ( 1) ) 1 and ( 1) 3 I G b G V V I b G V V= − − = − = − =

Finally 11 211 S and 3 SY Y= − =

Next

1 2 2

2 23 2

and I I I

V VG G+ −

= =

1

1

112 2

2 0

222 2 3

2 0

1 S

4 S

V

V

IY G

V

IY G G

V

=

=

= = − = −

= = + =

P17.3-4

Using Fig. 17.3-2 as shown:

12 21 12 21

11 12

22 21

0.1 S or 0.1 S 0.2 0.3 S 0.05 0.15 S

Y Y Y YY YY Y

− = − = = = −= − == − =

P17.3-5

12 21

11 12

11

10 S

13.33 S

23.33 S

Y Y

Y Y

Y

μ

μ

μ

= − =

+ =

=

22 21 2220 S 30 SY Y Yμ μ+ = ⇒ =

2

P17.3-6

2

2

111

1 0

2 121

1 1 0

3 3 2 (3 )

2 2

I

I

VZ j jI

V j IZ jI I

=

=

j= = + − = + Ω

−= = = − Ω

1

1

112

2 0

222

2 0

2

2

I

I

VZ jI

VZ jI

=

=

= = −

= = −

Ω

Ω

P17.3-7

11 21

1121 12

2

22 21 22

Z 41 4 1 41Z

1 2 1 2 2

ZsZssZ

ssZ Z s Z sss

− = ⎫+⎪ ⇒ = + =⎬

− = ⎪⎭+

− = ⇒ = + =

P17.3-8 Given:

1 1 =

1 1

ss

s

+⎡ ⎤−⎢ ⎥⎢ ⎥− +⎣ ⎦

Y

Try a π circuit as shown at the right.

( )

12 21

11 12 11

22 21 22

1 S

1 1 1

1 1

Y Y

sY Y Ys s

Y Y s Y s s

= = −

++ = ⇒ = − =

+ = ⇒ = − − = +

1s

3

P17.3-9 Given:

2

2 2

2

2 2

2 2 1 1 1 =

1 1 1 1

s ss s s s

ss s s s

⎡ ⎤+ +⎢ ⎥+ + + +⎢ ⎥

+⎢ ⎥⎢ ⎥+ + + +⎣ ⎦

Z

Try :

From the circuit, we calculate:

( ) ( )221 1 2 12

11 1 1 2 22 2

2

1

1 1 1

R L s LC R s R R C L s R RR L sC sz R RR C s LC s LC s R C sR L s

C s

+ + + + ++= + = + =

+ + + ++ +

2

Comparing to the given yields: 11z

12

21

1 2

1 2

11

11

11 H

21 F

2

LCR

R CR

LC RL

R R C LC

R R

= ⎫= Ω⎧⎪= ⎪⎪ = Ω⎪ ⎪= ⇒⎬ ⎨ =⎪ ⎪+ =

⎪ ⎪ =⎩+ = ⎪⎭

Then check . The are all okay. If they were not, we would have to try a different circuit structure..

12 21 22, and z z z

P17.3-10 It is sufficient to require that the input resistance of each section of the circuit is equal to Ro, that is

Then

2 2(2 ) 2 3 ( 33

oo o

o

R R R 1)R R R R R R RR R

+= ⇒ = − ± + = − ± =

+R−

4

5

Section 17-4: Z and Y Parameters P17.4-1

1 12 1

1 1

1 21 2

1 2

( ) and

V bi I

R R

b R R2

1

RV

R

I I i VR R

+= − = −

⎛ ⎞+ += − − = ⎜ ⎟⎜ ⎟

⎝ ⎠

2 2

1 2 11 211 21

1 11 2 0

( ) and

V V

b R R b RI IY YV VR R

= =

+ + += = = = −

1 20 R R

2 1

22 2 2 2

2

I IVV R I IR

= −

= ⇒ =

1 1

2 122 12

2 2 2 0

1 1 and V V

I IY YV R V

= =

= = = = −20 R

Finally 1 2

1 2 2

1

1 2 2

1

1

b R RR R R

b RR R R

+ +⎡ ⎤−⎢ ⎥

⎢ ⎥= ⎢ ⎥+⎢ ⎥−⎢ ⎥⎣ ⎦

Y

P17.4-2

( )1 12

2 1

1 3 40

3v i

iv i

⎧ = + =⎪= ⇒ ⎨ =⎪⎩

1i

therefore 11 214 and 3 Z Z= Ω = Ω

1

( )

( )1 2 2

1

1 2 2

30

3 2

v i ii

v i i

α

α

⎧ = +⎪= ⇒ ⎨= + +⎪⎩ 2i

therefore ( )12 213 1 and 5 3Z Zα α= + = +

Finally, 4 3(1 )

= 3 5 3

αα+⎡ ⎤

⎢ ⎥+⎣ ⎦Z

P17.4-3 Treat the circuit as the parallel connection of two 2-port networks:

The admittance matrix of the entire network can be obtained as the sum of the admittance matrices of these two 2-port networks

1 0 2 1 2 2 1 2 2 1 2

s s s ss s s s

− + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Y

When ( ) ( )1 :i t u t=

1 1 12

2 2

2 1 1 1( ) ( ) 2 2 1

( ) ( ) 3 6 10 0

s sV s V s s s

s sV s V s s s

−

+⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤ − +⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⇒ = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Y Y1

0s

so

2 2( 2) 1 6 1.25 7.25( )

(3 6 1) 3 1.82 0.184S

S S S S SV s

s⎡ ⎤− − −

= = + +⎢ ⎥+ + + +⎣ ⎦

Taking the inverse Laplace transform

1.82 0.1842

1(t) = 6 1.25 7.25 t 0 3

t tv e e− −⎡ ⎤− − + ≥⎣ ⎦

2

P17.4-4

KVL: ( )1 1 2 2 11 2 02

i v v v v− + + − =

KCL: 1 1 14 2i v v v− = + 2

1 1 11 1 11

11 1 2

1 1 2 1 2 21 2 21

1

5 23 6 2 93 6

5 2 6 15 23 1

i v vi v z

ii v vi v v i v v

i v zi

−⎧

8

= − ⇒ = =⎪= − ⎫ ⎪⇒⎬ ⎨= + +⎭ ⎪

Ω

= + ⇒ = = −⎪⎩Ω

KVL: 2 1 1 11 11 52 2

v v v v= + + = 13 v

KCL: 22 1 25 2 5

1 2v

i v v= + = + 1v

2 1 1 1 1213 12 5 182 1

i v v v z⎛ ⎞= + = ⇒ =⎜ ⎟⎝ ⎠

8Ω

and

2 2 2 2 2222 5 2.769 0.361

13i v v v z⎛ ⎞= + = ⇒ =⎜ ⎟

⎝ ⎠Ω

3

P17.4-5

KCL: 11 2

1

vi i

R+ =

KVL: 2 2 1 10 0R i bv v− − + − =

Then ( )2 1

2 1 1 12 1 2 1

11 1 1 R R bb bi v and i v vR R R R

⎛ ⎞ + ++ += − = + =⎜ ⎟⎜ ⎟

⎝ ⎠1

2R

so ( )2 1 2

21 111 2 1 1 2

11 andi i Rby yv R v R R

1R b+ ++= = − = =

KVL: 2 1 2 1 22

10R i v i vR

+ = ⇒ = −

KCL: ( )2 3 1 2 3 2 32

1v R i i R v R iR

⎛ ⎞= + = − +⎜ ⎟⎜ ⎟

⎝ ⎠2

Then

1 3 212 2 3 2 22

2 2 2 2 3

1 1and 1i R i

y v R i yv R R v R R

⎛ ⎞= = − + = ⇒ = = +⎜ ⎟⎜ ⎟

⎝ ⎠ 2

1

Finally

( )2 1

1 2 2

2 3

2 2

1 1

1

R R bR R R

3

R RbR R R

⎡ + +−

⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥++⎢ ⎥−⎢ ⎥⎣ ⎦

Y

4

Section 17-5: Hybrid Transmission Parameters P17.5-1

2

1 12 1

2 0

34 56.8 since5 2V

V V4||10 34

B I VI

=

= = = Ω − = =− +

2

12 1

2 0

10 4 101.4 since 10 10 4V

ID II

=

+= = = = −− +

I

2

12 1

2 0

12 10 1.2 since10 10 2I

VA V VV

=

= = = =+

2

1

2 0

1 0.1 S10I

ICV

=

= = =

P17.5-2

2 0V = so

1 i 1 2( ||V R R R= + 1) I therefore

2

111 i 1 2

1 0

|| 600 kV

Vh R R R

I=

= = + = Ω

KVL: 1 i

2 11 2 o

R R1I I A

R R R+ = −

+I

therefore

2

2 i 1 621

1 o 1 20

( )V

I R Rh A

I R R R=

= = − + = −+

10

1

1 i0 0I v A= ⇒ = ⇒ =i 0v

so 2

2o 1 2( )

VI

R R R=

+

therefore

1

2 o 1 2 322

2 o 1 20

10( )

I

I R R Rh

V R R R−

=

+ += = =

+

Next, 1

1 21 2

RV V

R R=

+

therefore

1

1 112

2 1 20

12

I

V Rh

V R R=

= =+

=

P17.5-3

Compare : 2 1

1 2

V nV

I n I

=

=−

to 1 11 1 12

2 21 1 22 2

V h I h V2

I h I h V

= +

= +

11 22 12 211 1Then 0, 0, and h h h hn n= = = = −

P17.5-4

( ) 2 31 1 2 3 1 11 1

2 3

2 22 1 21

2 3 2 3

|| +

R RV R R R I h R

R R

R RI I h

R R R R

= + ⇒ =+

= − ⇒ = −+ +

22 22

2 3 2 3

2 21 2 12

2 3 2 3

1 V

I hR R R R

R RV V h

R R R

= ⇒ =+ +

= ⇒ =+ +R

2

P17.5-5

2 1

2 1

0.1 and 950

so 95

I v v

I I

I= =

=

2

2

111

1 0

221

1 0

50 950 1000

95

V

V

Vh

I

Ih

I

=

=

= = + = Ω

= =

1 0 0I v= ⇒ =

1

1

112

2 0

2 422

2 0

0

10 S

I

I

Vh

V

Ih

V

=

−

=

= =

= =

3

Section 17.6: Relationships between Two-Port Parameters P17.6-1 Start with

1 11 1 121 11 1 12 2

2 21 1 22 2 2 21 1 22 2

Y parameters: and H parameters:

V h I h VI Y V Y V 2

I Y V Y V I h I h V

= +⎧= +⎧ ⎪⎨ ⎨= + = +⎪⎩ ⎩

Solve the Y parameter equations for V to put them in the same form as the H parameter equations.

1 and 2I

I

111 1 12 1 1 11 12 111 1 1 12 2

21 1 2 22 2 22 2221 2 2 2 21 2

Y 0 01 1

Y 1 0 1 0 V I V YY YY V I Y V

Y V I Y V I Y V I Y Y V

−− −− −− + = ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⇒ = ⇒ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− + = − −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

121

11 12 11 12 11 11

21 22 21 22 21 12 2122

11 11 11

1 1 0 0 1 1

= 1 0 0 1

YY Y Y Y Y YY Y Y Y Y Y

Y Y

−

Y YY

⎡ ⎤ ⎡− − ⎤⎢ ⎥ ⎢− − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎥

⎢ ⎥ ⎢ ⎥∴ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦− − ⎥

⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣

H

⎦ P17.6-2

First Δ = . Then (3)(6) (2)(2) 14− =Z

22 12

21 11

6 2 14 14= =

2 3 14 14

Z Z

Z Z

⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥Δ Δ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ Δ Δ ⎦

Z ZY

Z Z

.

P17.6-3

First Δ = . Then (0.1)(0.5) (0.4)(0.1) .01 S− =Y

12

11 11

21

11 11

1 10 1

=4 0.1

YY YYY Y

⎡ ⎤−⎢ ⎥ −⎡ ⎤⎢ ⎥= ⎢ ⎥Δ⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

HY

.

P17.6-4

First Δ = . Then (0.5)(0.6) ( 0.4)( 0.4) S − − −Y

12

11 11

21

11 11

1 Y -Y Y 2 0.8

Y Δ 0.8 0.28 Y Y

⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥= = ⎢ ⎥−⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

HY

1

Section 17.7: Interconnection of Two-Port Networks P17.7-1

12 21

22 21

11 12 11

1 S310 S3

41 S S3

Y Y

Y Y

Y Y Y

= = −

= − =

+ = ⇒ =

a

4 1 3 31 1 3 3

⎡ ⎤−⎢ ⎥=⎢ ⎥−⎣ ⎦

Y

12 21

11 12 11

21 22 22

1 S31 S S2 2

1 4 S S3 3

Y Y

Y Y Y

Y Y Y

= = −

+ = ⇒ =

+ = ⇒ =

b

3 12 41 3

⎡ ⎤−⎢ ⎥=⎢ ⎥−⎣ ⎦

Y

a b

3 174 4( ) 3 2 3 6 35 54 4 3 3 3

−4

3

⎡ ⎤ ⎡ ⎤+ −⎢ ⎥ ⎢ ⎥= + = =⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

Y Y Y

P17.7-2

Admittance parameters:

10 6 44 44

6 8 44 44

−⎡ ⎤⎢ ⎥

= ⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

Y

Transmission parameters:

8 44 6 61 10 6 6

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

T

p

20 12 44 44

12 16 44 44

−⎡ ⎤⎢ ⎥

= + = ⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

Y Y Y

1

C

108 792 36 3618 144 36 36

⎡ ⎤⎢ ⎥

= ⋅ = ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

T T T

P17.7-3

1 2 2a b

2 2

1

1

sC sC G G Gs LG G GsC sC

s L

⎡ ⎤+ −⎢ ⎥ + −

3

⎡ ⎤⎢ ⎥= + = + ⎢ ⎥− +⎢ ⎥ ⎢ ⎥⎣ ⎦− +⎢ ⎥⎣ ⎦

Y Y Y

2

Section 17.8 How Can We Check…? P17.8-1

1 27550 15

175 75V I⎛ ⎞= =⎜ ⎟+⎝ ⎠

2I

1

112

2 0

15 I

VZI

=

= = Ω

1 11 1 0.02850 125 1I V V⎛ ⎞= + =⎜ ⎟

⎝ ⎠

2

111

1 0

28 mSV

IYV

=

= =

11 24 mS, so the report is not correct.Y ≠

P17.8-2

111 1

212 1

2 0.2 0.2 ( 10) (2 0.2 ) 0.1 (0.1 )

Z s sV s IZ sV s I

= + = += + ⎫⇒⎬ == ⎭

22 122 0.2 and 0.1Z s Z s= + =

2(2 0.2 )(2 0.2 ) (0.1 )(0.1 ) 0.01(3 80 400)s s s s s sΔ = + + − = + +Z

1

211

21 21

22

21 21

2( 10) 0.1(3 80 400)

= =1 10 2( 10)

Z s s sZ Z s s

Z sZ Z s s

Δ⎡ ⎤ ⎡ ⎤+ + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ +⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

Z

T

This is the transmission matrix given in the report.

2