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EA611 – Circuitos II – Turma U – 1o. Semestre de 2015 – Lista de Exercícios 4 FEEC – UNICAMP

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EA611 – Circuitos II – Turma U – 1o. Semestre de 2015 FEEC – UNICAMP Questões e figuras extraídas dos livros:

• Análise de Circuitos em Engenharia, Hayt, Kemmerly e Durbin, 7a. edição, McGraw Hill.

• Fundamentals of Electric Circuits, C.K. Alexander, M.N.O. Sadiku, McGraw Hill.

Lista de Exercícios 3 1. Calcule os parâmetros da matriz de admitância do circuito abaixo.

2. Obtenha os parâmetros da matriz de admitância Y do circuito abaixo. Obtain the y parameters for the circuit in Fig. 19.19.

Answer: y22 ! 0.125 S.y21 ! 0.375 S,y12 ! "0.125 S,y11 ! 0.625 S,

Practice Problem 19.4

3 Ω

io

2 Ω6 Ω

2io

Figure 19.19For Practice Prob. 19.4.

Hybrid ParametersThe z and y parameters of a two-port network do not always exist. Sothere is a need for developing another set of parameters. This third set ofparameters is based on making and the dependent variables. Thus,we obtain

(19.14) I2 ! h21I1 # h22V2

V1 ! h11I1 # h12V2

I2V1

19.4

or

Hence,

Similarly, we get and using Fig. 19.18(b). At node 1,

But ; therefore,

or

Hence,

At node 2,

or

Thus,

Notice that in this case, since the network is not reciprocal.y12 $ y21

y22 !I2V2

!0.625Vo2.5Vo

! 0.25 S

"I2 ! 0.25Vo "14

(2.5Vo) "2Vo

8! "0.625Vo

Vo " V2

4# 2I1 # I2 ! 0

y12 !I1V2

!"Vo!82.5Vo

! "0.05 S

0 ! "Vo # 4Vo # 2Vo " 2V2 1 V2 ! 2.5Vo

0 ! "

Vo8

#Vo2

#Vo " V2

4

I1 !0 " Vo

8

0 " Vo8

! 2I1 #Vo2

#Vo " V2

4

y22y12

y21 !I2V1

!1.25Vo"5Vo

! "0.25 S

"I2 ! 0.25Vo " 1.5Vo ! "1.25Vo

858 Chapter 19 Two-Port Networks

ale29559_ch19.qxd 07/16/2008 05:48 PM Page 858

Resposta: y11 = 0,625S , y12 = −0,125S , y21 = 0,375S , y22 = 0,125S

3. Determine os parâmetros h do circuito abaixo.

1 V40 Ω

+

−

V1

(a)

[h]

I1 I2

+−

60 V

40 Ω

+

−

V1

+

−

V2

(b)

[h]

I1 I2 = 0

+−

Figure 19.26For Example 19.6: (a) finding ,(b) finding .VTh

ZTh

60 V

40 Ω

h11 = 1 kΩh12 = –2h21 = 10h22 = 200 !S

+−

Figure 19.25For Example 19.6.

Determine the Thevenin equivalent at the output port of the circuit inFig. 19.25.

Solution:To find and , we apply the normal procedure, keeping in mind theformulas relating the input and output ports of the h model. To obtain remove the 60-V voltage source at the input port and apply a 1-V voltagesource at the output port, as shown in Fig. 19.26(a). From Eq. (19.14),

(19.6.1)

(19.6.2)

But , and . Substituting these into Eqs. (19.6.1) and(19.6.2), we get

(19.6.3)

(19.6.4)

Substituting Eq. (19.6.3) into Eq. (19.6.4) gives

Therefore,

Substituting the values of the h parameters,

To get , we find the open-circuit voltage in Fig. 19.26(b). At theinput port,

(19.6.5)!60 " 40I1 " V1 # 0 1 V1 # 60 ! 40I1

V2VTh

#104020.21

# 51.46 $

ZTh #1000 " 40

103 % 200 % 10!6 " 20 " 40 % 200 % 10!6

Z Th #V2

I2#

1I2

#h11 " 40

h11h22 ! h21h12 " h2240

I2 # h22 !h21h12

h11 " 40#h11h22 ! h21h12 " h2240

h11 " 40

I2 # h21I1 " h22

!40I1 # h11I1 " h12 1 I1 # !

h12

40 " h11

V1 # !40I1V2 # 1

I2 # h21I1 " h22V2

V1 # h11I1 " h12V2

ZTh,VThZTh

Example 19.6

Determine the h parameters for the circuit in Fig. 19.24.

Answer: h11 # 2.4 $, h12 # 0.4, h21 # !0.4, h22 # 200 mS.


10 Ω

6 Ω

4 Ω


Thus,

h22 #I2V2

#19

S

19.4 Hybrid Parameters 861

ale29559_ch19.qxd 07/16/2008 05:48 PM Page 861

Resposta: h11 = 2, 4Ω , h12 = 0, 4 , h21 = −0, 4 , h22 = 0,2S ,

4. Determine a matriz de impedância do circuito abaixo.


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5. Para o circuito abaixo, determinar 12h , 12z e 12y .

6. Determine os parâmetros de transmissão do quadripolo abaixo.

Find the transmission parameters for the two-port network in Fig. 19.32.

Solution:To determine A and C, we leave the output port open as in Fig. 19.33(a)so that and place a voltage source at the input port. We have

Thus,

To obtain B and D, we short-circuit the output port so that asshown in Fig. 19.33(b) and place a voltage source at the input port.At node a in the circuit of Fig. 19.33(b), KCL gives

(19.8.1)V1 ! Va

10!Va

20" I2 # 0

V1

V2 # 0

A #V1

V2#

30I117I1

# 1.765, C #I1V2

#I1

17I1# 0.0588 S

V1 # (10 " 20)I1 # 30I1 and V2 # 20I1 ! 3I1 # 17I1

V1I2 # 0

Example 19.8

20 Ω

3I110 ΩI1 I2

+ −


or

(19.27)

The parameters a, b, c, and d are called the inverse transmission, or t,parameters. They are determined as follows:

(19.28)

From Eq. (19.28) and from our experience so far, it is evident that theseparameters are known individually as

(19.29)

While a and d are dimensionless, b and c are in ohms and siemens,respectively.

In terms of the transmission or inverse transmission parameters, anetwork is reciprocal if

(19.30)

These relations can be proved in the same way as the transfer imped-ance relations for the z parameters. Alternatively, we will be able touse Table 19.1 a little later to derive Eq. (19.30) from the fact that

for reciprocal networks.z12 # z21

AD ! BC # 1, ad ! bc # 1

d # Negative short-circuit current gain c # Open-circuit transfer admittance b # Negative short-circuit transfer impedance a # Open-circuit voltage gain

c #I2V1

`I1#0

, d # !

I2I1

`V1#0

a #V2

V1 `I1#0

, b # !

V2

I1 `V1#0

cV2

I2d # c a b

c dd c V1

!I1d # [t] c V1

!I1d

19.5 Transmission Parameters 865

ale29559_ch19.qxd 07/16/2008 05:48 PM Page 865

Resposta: t11 =1, 765 , t12 =15,29Ω , t21 = 0,0588S , t22 =1,176

7. Determine a matriz de admitância do quadripolo abaixo. Use as regras de associação de quadripolos.

8. No circuito abaixo, a matriz Y , em siemens, vale

⎥⎦

⎤⎢⎣

⎡

−

−=

05,080025,01,0

Y .

Determine: • Os ganhos de tensão 12 VV e de corrente 12 II . • As impedâncias de entrada e de saída.


3

9. Determine o equivalente de Thevenin do circuito abaixo.

1 V40 Ω

+

−

V1

(a)

[h]

I1 I2

+−

60 V

40 Ω

+

−

V1

+

−

V2

(b)

[h]

I1 I2 = 0

+−

Figure 19.26For Example 19.6: (a) finding ,(b) finding .VTh

ZTh

60 V

40 Ω

h11 = 1 kΩh12 = –2h21 = 10h22 = 200 !S

+−


Determine the Thevenin equivalent at the output port of the circuit inFig. 19.25.

Solution:To find and , we apply the normal procedure, keeping in mind theformulas relating the input and output ports of the h model. To obtain remove the 60-V voltage source at the input port and apply a 1-V voltagesource at the output port, as shown in Fig. 19.26(a). From Eq. (19.14),

(19.6.1)

(19.6.2)

But , and . Substituting these into Eqs. (19.6.1) and(19.6.2), we get

(19.6.3)

(19.6.4)

Substituting Eq. (19.6.3) into Eq. (19.6.4) gives

Therefore,


To get , we find the open-circuit voltage in Fig. 19.26(b). At theinput port,

(19.6.5)!60 " 40I1 " V1 # 0 1 V1 # 60 ! 40I1

V2VTh

#104020.21

# 51.46 $

ZTh #1000 " 40

103 % 200 % 10!6 " 20 " 40 % 200 % 10!6

Z Th #V2

I2#

1I2

#h11 " 40

h11h22 ! h21h12 " h2240

I2 # h22 !h21h12

h11 " 40#h11h22 ! h21h12 " h2240

h11 " 40

I2 # h21I1 " h22

!40I1 # h11I1 " h12 1 I1 # !

h12

40 " h11

V1 # !40I1V2 # 1

I2 # h21I1 " h22V2

V1 # h11I1 " h12V2

ZTh,VThZTh

Example 19.6

Determine the h parameters for the circuit in Fig. 19.24.

Answer: h11 # 2.4 $, h12 # 0.4, h21 # !0.4, h22 # 200 mS.


10 Ω

6 Ω

4 Ω


Thus,

h22 #I2V2

#19

S

19.4 Hybrid Parameters 861

ale29559_ch19.qxd 07/16/2008 05:48 PM Page 861

10. No circuito abaixo, a matriz de impedância Z , em ohms, vale

⎥⎦

⎤⎢⎣

⎡

−=

40020001001000

Z .

Determine as potências dissipadas nos resistores de 200 ohms e 500 ohms.

11. Determine a impedância de entrada Zin do circuito mostrado abaixo.

At the output,(19.6.6)

Substituting Eqs. (19.6.5) and (19.6.6) into Eqs. (19.6.1) and (19.6.2),we obtain

or(19.6.7)

and

(19.6.8)

Now substituting Eq. (19.6.8) into Eq. (19.6.7) gives

or


VTh !60 " 10#20.21

! #29.69 V

!60h21

h12h21 # h11h22 # 40h22VTh ! V2 !

60#(h11 $ 40)h22!h21 $ h12

60 ! c#(h11 $ 40)

h22

h21$ h12 d V2

0 ! h21I1 $ h22V2 1 I1 ! #

h22

h21 V2

60 ! (h11 $ 40)I1 $ h12V2

60 # 40I1 ! h11I1 $ h12V2

I2 ! 0


Find the impedance at the input port of the circuit in Fig. 19.27.

Answer: .1.6667 k%


50 kΩ

h11 = 2 kΩh12 = 10–4

h21 = 100h22 = 10–5 S

ZinFigure 19.27For Practice Prob. 19.6.

Find the g parameters as functions of s for the circuit in Fig. 19.28.

Solution:In the s domain,

To get and , we open-circuit the output port and connect avoltage source to the input port as in Fig. 19.29(a). From the figure,

I1 !V1

s $ 1

V1

g21g11

1 H 1 sL ! s, 1 F 1 1sC

!1s

Example 19.7

1 Ω

1 F1 H


ale29559_ch19.qxd 07/16/2008 05:48 PM Page 862

Resposta: Zin =1666, 7Ω


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12. Para o circuito abaixo:

a) Determinar os parâmetros y do quadripolo. b) Determinar as impedâncias de entrada e de saída do quadripolo. c) Determinar V2(s) para vs(t) = 2u(t) volts.

Vx 1 Ω

2Vx2 Ω

4 Ω

+

−

Figure 19.85For Prob. 19.26.

V1

I1 I2

+

−

+

−

V210 Ω

4 Ω

0.1V2 20I1+−


I1 V21 Ω

3 Ω

3 Ω

3 Ω

+

−

V1

+

−

I2


60 Ω

40 Ω

(a)

20 Ω

(b)

10 Ω


10 Ω V2V1

0.2V1

5 Ω

+

−

+

−


0.5V2V1 5 Ω

5 Ω+

−

V2

+

−

2 Ω

I1 I2


vs −+

1

1

1 2s

1s

V1 V2− −

+ +


19.21 Obtain the admittance parameter equivalent circuitof the two-port in Fig. 19.82.

19.27 Find the y parameters for the circuit in Fig. 19.86.

Problems 893

19.22 Obtain the y parameters of the two-port network inFig. 19.83.

19.23 (a) Find the y parameters of the two-port in Fig. 19.84.(b) Determine for .vs ! 2u(t) VV2(s)

19.24 Find the resistive circuit that represents these yparameters:

19.25 Draw the two-port network that has the following yparameters:

19.26 Calculate for the two-port in Fig. 19.85.[y]

[y] ! c 1 "0.5"0.5 1.5

d S

[y] ! ≥ 14

"

18

"

18

316

¥ S

19.28 In the circuit of Fig. 19.65, the input port isconnected to a 1-A dc current source. Calculate thepower dissipated by the resistor by using the yparameters. Confirm your result by direct circuitanalysis.

19.29 In the bridge circuit of Fig. 19.87, and

(a) Find and using y parameters.(b) Confirm the results in part (a) by direct circuit

analysis.

V2V1

I2 ! "4 A.I1 ! 10 A

2-#

Section 19.4 Hybrid Parameters

19.30 Find the h parameters for the networks in Fig. 19.88.

ale29559_ch19.qxd 07/16/2008 05:49 PM Page 893

13. Determine as impedâncias de entrada e de saída do quadripolo no circuito abaixo,

sabendo que?

• A matriz dos parâmetros híbridos do quadripolo vale h = 600Ω 0,0430 2mS

"

#$$

%

&''

• Zs = 2kΩ , ZL = 400Ω .

I1 I2

+

−

V2

+

−

V1

4 Ω

25 Ω[h]+−10 V


[h]+−

Zs

ZLVs

Zin Zout

+

−

V1

+

−

V2


1:21 Ω 4 Ω


V1

I1 I2

+

−

+

−

V2

1 Ω

2 Ω 4I1

2 Ω 1 Ω


R L1 L2

C


5 Ω − j3 Ω

j6 Ω4 Ω


10 Ω

100 ΩVx 10Vx+−

50 Ω

300 Ω

+

−


19.31 Determine the hybrid parameters for the network inFig. 19.89.

19.36 For the two-port in Fig. 19.94,

Find:

(a) (b) (c) (d) V2!I1I1!V1

I2!I1V2!V1

[h] ! c 16 " 3#2 0.01 S

d


19.32 Using Fig. 19.90, design a problem to help otherstudents better understand how to find the h and gparameters for a circuit in the s-domain.

19.33 Obtain the h parameters for the two-port of Fig. 19.91.

19.34 Obtain the h and g parameters of the two-port inFig. 19.92.

19.35 Determine the h parameters for the network inFig. 19.93.

19.37 The input port of the circuit in Fig. 19.79 isconnected to a 10-V dc voltage source while theoutput port is terminated by a resistor. Find thevoltage across the resistor by using h parametersof the circuit. Confirm your result by using directcircuit analysis.

19.38 The h parameters of the two-port of Fig. 19.95 are:

Given the and , find and.Zout

ZinZL ! 400 "Zs ! 2 k"

[h] ! c 600 " 0.0430 2 mS

d5-"

5-"

19.39 Obtain the g parameters for the wye circuit ofFig. 19.96.

ale29559_ch19.qxd 07/16/2008 05:49 PM Page 894

14. No circuito abaixo, as matrizes de parâmetros híbridos ah e bh valem

⎥⎦

⎤⎢⎣

⎡ Ω==

S500200004,02000µaa hh .

Com Ω= kZL 20 , determine o valor de SV para produzir V160 =V . Dica: determine primeiramente o equivalente de Thevenin do circuito formado pela fonte SV , pelo resistor e pelo quadripolo de matriz ah .

EA611 Exercicio 03 - DECOM | FEEC - Faculdade de ...cardieri/NotasdeAula_EA611...• Fundamentals of...

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