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Fundamentals of Electric Circuits

Charles K. Alexander, Matthew N.O. Sadiku

Chapter 18

Two-Port Networks - all with Video Answers

Educators

Chapter Questions

04:18

Problem 1

Obtain the $z$ parameters for the network in Fig. 18.65.

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04:19

Problem 2

Find the impedance parameter equivalent of the network in Fig. 18.66.

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03:04

Problem 3

Determine the $z$ parameters of the two-ports shown in Fig. 18.67.

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03:40

Problem 4

Calculate the $z$ parameters for the circuit in Fig. 18.68.

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06:09

Problem 5

Obtain the $z$ parameters for the network in Fig 18.69 as functions of $s$.

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05:05

Problem 6

Obtain the $z$ parameters for the circuit in Fig. 18.70.

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09:05

Problem 7

Find the impedance-parameter equivalent of the circuit in Fig. 18.71.

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01:43

Problem 8

Construct a circuit that realizes the following parameters.
$$[\mathbf{z}]=\left[\begin{array}{ll}
10 & 4 \\
4 & 6
\end{array}\right]$$

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02:41

Problem 9

Construct a two-port that realizes each of the following $z$ parameters.
(a) $[\mathbf{z}]=\left[\begin{array}{cc}25 & 20 \\ 5 & 10\end{array}\right] \Omega$
(b) $[\mathbf{z}]=\left[\begin{array}{cc}1+\frac{3}{s} & \frac{1}{s} \\ \frac{1}{s} & 2 s+\frac{1}{s}\end{array}\right] \Omega$

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02:10

Problem 10

For a two-port network
\[
[\mathbf{z}]=\left[\begin{array}{ll}
12 & 4 \\
4 & 6
\end{array}\right] \Omega
\]
find $V_{2} / V_{1}$ if the network is terminated with a $2-\Omega$ resistor.

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03:30

Problem 11

If $[\mathbf{z}]=\left[\begin{array}{ll}50 & 10 \\ 30 & 20\end{array}\right] \Omega$ in the two-port of Fig. 18.72
calculate the average power delivered to the $100-\Omega$ resistor.

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05:03

Problem 12

For the two-port network shown in Fig. $18.73,$ show that
\[
\mathbf{Z}_{\mathrm{Th}}=\mathbf{z}_{22}-\frac{\mathbf{z}_{12} \mathbf{z}_{21}}{\mathbf{z}_{11}+\mathbf{Z}_{s}}
\]
and
\[
\mathbf{V}_{\mathrm{Th}}=\frac{\mathbf{z}_{21}}{\mathbf{z}_{11}+\mathbf{Z}_{s}} \mathbf{V}_{s}
\]

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03:47

Problem 13

For the circuit in Fig. $18.74,$ at $\omega=2 \mathrm{rad} / \mathrm{s}$ $\mathbf{z}_{11}=10 \Omega, \mathbf{z}_{12}=\mathbf{z}_{21}=j 6 \Omega, \mathbf{z}_{22}=4 \Omega .$ Obtain the Thevenin equivalent circuit at terminals $a-b$ and calculate $v_{o}$.

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05:17

Problem 14

Determine the $z$ and $y$ parameters for the circuit in Fig. 18.75.

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04:44

Problem 15

Calculate the $y$ parameters for the two-port in Fig. 18.76.

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04:20

Problem 16

Find the $y$ parameters of the two-port in Fig. 18.77 in terms of $s$.

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02:41

Problem 17

Obtain the admittance parameter equivalent circuit of the two-port in Fig. 18.78.

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05:07

Problem 18

Determine the $y$ parameters for the two-ports in Fig. 18.79.

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01:40

Problem 19

Find the resistive circuit that represents these $y$ parameters:

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05:04

Problem 20

Calculate [y] for the two-port in Fig. 18.80.

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03:54

Problem 21

Find the $y$ parameters for the circuit in Fig. 18.81.

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06:07

Problem 22

In the circuit of Fig. $18.65,$ the input port is connected to a 1 -A de current source. Calculate the power dissipated by the $2-\Omega$ resistor by using the $y$ parameters. Confirm your result by direct circuit analysis.

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03:35

Problem 23

In the bridge circuit of Fig. $18.82, I_{1}=10 \mathrm{A}$ and $I_{2}=-4 \mathrm{A}.$
(a) Find $V_{1}$ and $V_{2}$ using $y$ parameters.
(b) Confirm the results in part (a) by direct circuit analysis.

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03:09

Problem 24

Find the $h$ parameters for the networks in Fig. 18.83.

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04:55

Problem 25

Determine the hybrid parameters for the network in Fig. 18.84.

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08:23

Problem 26

Find the $h$ and $g$ parameters of the two-port network in Fig. 18.85 as functions of $s$.

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06:34

Problem 27

Obtain the $h$ and $g$ parameters of the two-port in Fig. 18.86.

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02:47

Problem 28

Determine the $h$ parameters for the network in Fig. 18.87.

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03:07

Problem 29

For the two-port in Fig. 18.88
\[
[\mathbf{h}]=\left[\begin{array}{cc}
16 \Omega & 3 \\
-2 & 0.01 \mathrm{S}
\end{array}\right]
\]
Find:
(a) $V_{2} / V_{1}$
(b) $I_{2} / I_{1}$
(c) $I_{1} / V_{1}$
(d) $V_{2} / I_{1}$

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05:23

Problem 30

The input port of the circuit in Fig. 18.76 is connected to a 10 -V de voltage source while the output port is terminated by a $5-\Omega$ resistor. Find the voltage across the $5-\Omega$ resistor by using $h$ parameters of the circuit. Confirm your result by using direct circuit analysis.

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02:26

Problem 31

For the circuit in Fig. $18.89, \mathbf{h}_{11}=800 \Omega$ $\mathbf{h}_{12}=10^{-4}, \mathbf{h}_{21}=50, \mathbf{h}_{22}=0.5 \times 10^{-5} \mathrm{S} .$ Find the input impedance $Z_{\text {in }}$

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04:21

Problem 32

Find the $g$ parameters for the circuit in Fig. 18.90.

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06:16

Problem 33

For the two-port in Fig. 18.73 , show that
\[
\begin{array}{c}
\frac{\mathbf{I}_{2}}{\mathbf{I}_{1}}=\frac{-\mathbf{g}_{21}}{\mathbf{g}_{11} \mathbf{Z}_{L}+\Delta_{g}} \\
\frac{\mathbf{v}_{2}}{\mathbf{V}_{s}}=\frac{\mathbf{g}_{21} \mathbf{Z}_{L}}{\left(1+\mathbf{g}_{11} \mathbf{Z}_{s}\right)\left(\mathbf{g}_{22}+\mathbf{Z}_{L}\right)-\mathbf{g}_{21} \mathbf{g}_{12} \mathbf{Z}_{s}}
\end{array}
\]
where $\Delta_{g}$ is the determinant of $[\mathbf{g}]$ matrix.

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01:24

Problem 34

Find the network which realizes each of the following $g$ parameters:
(a) $\left[\begin{array}{cc}0.01 & -0.5 \\ 0.5 & 20\end{array}\right]$
(b) $\left[\begin{array}{cc}0.1 & 0 \\ 12 & s+2\end{array}\right]$

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03:13

Problem 35

Find the transmission parameters for the single-element two-port networks in Fig. 18.91.

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07:22

Problem 36

Determine the transmission parameters of the circuit in Fig. 18.92.

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05:53

Problem 37

Find the transmission parameters for the circuit in Fig. 18.93.

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03:54

Problem 38

For a two-port, let $\mathbf{A}=4, \mathbf{B}=30 \Omega, \mathbf{C}=0.1 \mathrm{S}$
and $\mathbf{D}=1.5 .$ Calculate the input impedance $\mathbf{Z}_{\text {in }}=\mathbf{V}_{1} / \mathbf{I}_{1},$ when:
(a) the output terminals are short-circuited,
(b) the output port is open-circuited,
(c) the output port is terminated by a $10-\Omega$ load.

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05:45

Problem 39

Using impedances in the $s$ domain, obtain the transmission parameters for the circuit in Fig. 18.94.

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03:54

Problem 40

Find the $t$ parameters of the network in Fig. 18.95 as functions of $s$.

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03:02

Problem 41

Obtain the $t$ parameters for the network in Fig. 18.96.

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04:15

Problem 42

(a) For the $T$ network in Fig. 18.97 , show that the $h$ parameters are:
\[
\begin{array}{c}
\mathbf{h}_{11}=R_{1}+\frac{R_{2} R_{3}}{R_{1}+R_{3}}, \quad \mathbf{h}_{12}=\frac{R_{2}}{R_{2}+R_{3}} \\
\mathbf{h}_{21}=-\frac{R_{2}}{R_{2}+R_{3}}, \quad \mathbf{h}_{22}=\frac{1}{R_{2}+R_{3}}
\end{array}
\]
(b) For the same network, show that the transmission parameters are:
\[
\begin{array}{c}
\mathbf{A}=1+\frac{R_{1}}{R_{2}}, \quad \mathbf{B}=R_{3}+\frac{R_{1}}{R_{2}}\left(R_{2}+R_{3}\right) \\
\mathbf{C}=\frac{1}{R_{2}}, \quad \mathbf{D}=1+\frac{R_{3}}{R_{2}}
\end{array}
\]

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03:53

Problem 43

Through derivation, express the $z$ parameters in terms of the ABCD parameters.

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03:29

Problem 44

Show that the transmission parameters of a two-port may be obtained from the $y$ parameters as:
\[
\begin{array}{ll}
\mathbf{A}=-\frac{\mathbf{y}_{22}}{\mathbf{y}_{21}}, & \mathbf{B}=-\frac{1}{\mathbf{y}_{21}} \\
\mathbf{C}=-\frac{\Delta_{y}}{\mathbf{y}_{21}}, & \mathbf{D}=-\frac{\mathbf{y}_{11}}{\mathbf{y}_{21}}
\end{array}
\]

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02:35

Problem 45

Prove that the $g$ parameters can be obtained from the
$z$ parameters as
\[
\begin{aligned}
\mathbf{g}_{11}=& \frac{1}{\mathbf{z}_{11}}, & \mathbf{g}_{12}=-\frac{\mathbf{z}_{12}}{\mathbf{z}_{11}} \\
\mathbf{g}_{21} &=\frac{\mathbf{z}_{21}}{\mathbf{z}_{11}}, & \mathbf{g}_{22}=\frac{\Delta_{z}}{\mathbf{z}_{11}}
\end{aligned}
\]

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03:21

Problem 46

Given the transmission parameters
\[
[\mathbf{T}]=\left[\begin{array}{ll}
3 & 20 \\
1 & 7
\end{array}\right]
\]
obtain the other five two-port parameters

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02:30

Problem 47

A two-port is described by
\[
\mathbf{V}_{1}=\mathbf{I}_{1}+2 \mathbf{V}_{2}, \quad \mathbf{I}_{2}=-2 \mathbf{I}_{1}+0.4 \mathbf{V}_{2}
\]
Find: (a) the $y$ parameters, (b) the transmission parameters.

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04:03

Problem 48

Given that
\[
[\mathbf{g}]=\left[\begin{array}{cc}
0.06 \mathrm{S} & -0.4 \\
0.2 & 2 \Omega
\end{array}\right]
\]
determine:
(a) $[\mathbf{z}]$
(b) $[\mathbf{y}]$
(c) $[\mathbf{h}]$
(d) $[\mathbf{T}]$

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03:21

Problem 49

Let $[\mathbf{y}]=\left[\begin{array}{rr}0.6 & -0.2 \\ -0.1 & 0.5\end{array}\right]$ (S) $.$ Find:
(a) $[\mathbf{z}]$
(b) $[\mathbf{h}]$
(c) $[\mathbf{t}]$

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05:13

Problem 50

For the bridge circuit in Fig. 18.98 , obtain
(a) the $z$ parameters
(b) the $h$ parameters
(c) the transmission parameters

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04:02

Problem 51

Find the $z$ parameters of the op amp circuit in Fig. $18.99 .$ Obtain the transmission parameters.

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06:12

Problem 52

Determine the $y$ parameters at $\omega=1,000 \mathrm{rad} / \mathrm{s}$ for the op amp circuit in Fig. $18.100 .$ Find the corresponding $h$ parameters.

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01:55

Problem 53

What is the $y$ parameter presentation of the circuit in Fig. $18.101 ?$

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05:30

Problem 54

In the two-port of Fig. 18.102 , let $\mathbf{y}_{12}=\mathbf{y}_{21}=0$ $\mathbf{y}_{11}=2 \mathrm{mS},$ and $\mathbf{y}_{22}=10 \mathrm{mS} .$ Find $\mathbf{V}_{o} / \mathbf{V}_{s}.$

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02:51

Problem 55

Figure 18.103 shows two two-ports in series. Find the transmission parameters.

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02:03

Problem 56

Obtain the $h$ parameters for the network in Fig 18.104.

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05:07

Problem 57

Determine the $y$ parameters of the two two-ports in parallel shown in Fig. 18.105.

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04:50

Problem 58

The circuit in Fig. 18.106 may be regarded as two two-ports connected in parallel. Obtain the $y$ parameters as functions of $s$.

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02:53

Problem 59

For the parallel-series connection of the two two-ports in Fig. 18.107 , find the $g$ parameters.

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02:44

Problem 60

A series-parallel connection of two two-ports is shown in Fig. $18.108 .$ Determine the $z$ parameter representation of the network.

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01:57

Problem 61

Find the transmission parameters for the cascaded two-ports shown in Fig. $18.109 .$ Obtain $\mathbf{Z}_{\text {in }}=\mathbf{V}_{1} / \mathbf{I}_{1}$ when the output is short-circuited.

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03:24

Problem 62

Determine the ABCD parameters of the circuit in Fig. 18.110 as functions of $s$ (Hint: Partition the circuit into subcircuits and cascade them using the results of Prob. $18.35 .$

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01:25

Problem 63

Use $P$ Spice to compute the $y$ parameters for the circuit in Fig, 18,111.

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02:03

Problem 64

Using $P$ Spice, find the $h$ parameters of the network in Fig. $18.112 .$ Take $\omega=1 \mathrm{rad} / \mathrm{s}.$

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06:12

Problem 65

Use $P$Spice to determine the $z$ parameters of the circuit in Fig. $18.113 .$ Take $\omega=2 \mathrm{rad} / \mathrm{s}.$

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09:05

Problem 66

Rework Prob. 18.7 using $P$ Spice.

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05:04

Problem 67

Repeat Prob. 18.20 using $P$ Spice.

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04:55

Problem 68

Use $P$ Spice to rework Prob. 18.25.

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02:03

Problem 69

Using $P$ Spice, find the transmission parameters for the network in Fig. 18.114.

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06:12

Problem 70

At $\omega=1 \mathrm{rad} / \mathrm{s},$ find the transmission parameters of the network in Fig. 18.115 using PSpice .

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02:03

Problem 71

Obtain the $g$ parameters for the network in Fig. 18.116 using $P$ Spice .

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04:37

Problem 73

Using the $y$ parameters, derive formulas for $Z_{\text {in }}$ $Z_{\mathrm{out}}, A_{i},$ and $A_{v}$ for the common-emitter transistor circuit.

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01:28

Problem 74

A transistor has the following parameters in a common-emitter circuit
\[
\begin{array}{c}
h_{i e}=2640 \Omega, \quad h_{r e}=2.6 \times 10^{-4} \\
h_{f e}=72, \quad h_{o e}=16 \mu \mathrm{S}, \quad R_{L}=100 \mathrm{k} \Omega
\end{array}
\]
What is the voltage amplification of the transistor? How many decibels gain is this?

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03:18

Problem 75

A transistor with
\[
\begin{array}{cl}
h_{f e}=120, & h_{i e}=2 \mathrm{k} \Omega \\
h_{r e}=10^{-4}, & h_{o e}=20 \mu \mathrm{S}
\end{array}
\]
is used for a CE amplifier to provide an input resistance of $1.5 \mathrm{k} \Omega$
(a) Determine the necessary load resistance $R_{L}.$
(b) Calculate $A_{v}, A_{i},$ and $Z_{\text {out }}$ if the amplifier is driven by a $4 \mathrm{mV}$ source having an internal resistance of $600 \Omega.$
(c) Find the voltage across the load.

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02:33

Problem 76

For the transistor network of Fig. 18.118
\[
\begin{array}{c}
h_{f e}=80, \quad h_{i e}=1.2 \mathrm{k} \Omega \\
h_{r e}=1.5 \times 10^{-4}, \quad h_{o e}=20 \mu \mathrm{S}
\end{array}
\]
Determine the following:
(a) voltage gain $A_{v}=V_{o} / V_{s}$
(b) current gain $A_{i}=I_{o} / I_{i}$
(c) input impedance $Z_{\text {in }}$
(d) output impedance $Z_{\text {out }}$

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02:29

Problem 77

Determine $A_{v}, A_{i}, Z_{\text {in }},$ and $Z_{\text {out }}$ for the amplifier shown in Fig. $18.119 .$ Assume that
\[
\begin{aligned}
h_{i e} &=4 \mathrm{k} \Omega, & h_{r e} &=10^{-4} \\
h_{f e} &=100, & h_{o e} &=30 \mu \mathrm{S}
\end{aligned}
\]

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02:03

Problem 78

Calculate $A_{v}, A_{i}, Z_{\text {in }},$ and $Z_{\text {out for the transistor }}$ network in Fig. $18.120 .$ Assume that
\[
\begin{array}{cc}
h_{i e}=2 \mathrm{k} \Omega, & h_{r e}=2.5 \times 10^{-4} \\
h_{f e}=150, & h_{o e}=10 \mu \mathrm{S}
\end{array}
\]

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03:33

Problem 79

A transistor in its common-emitter mode is specified by
\[
[\mathbf{h}]=\left[\begin{array}{cc}
200 \Omega & 0 \\
100 & 10^{-6} \mathrm{S}
\end{array}\right]
\]
Two such identical transistors are connected in cascade to form a two-stage amplifier used at audio frequencies. If the amplifier is terminated by a $4-\mathrm{k} \Omega$ resistor, calculate the overall $A_{v}$ and $Z_{\text {in }}$

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03:32

Problem 80

Realize an $L C$ ladder network such that
\[
y_{22}=\frac{s^{3}+5 s}{s^{4}+10 s^{2}+8}
\]

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04:00

Problem 81

Design an $L C$ ladder network to realize a lowpass filter with transfer function
\[
H(s)=\frac{1}{s^{4}+2.613 s^{2}+3.414 s^{2}+2.613 s+1}
\]

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04:45

Problem 82

Synthesize the transfer function
\[
H(s)=\frac{V_{o}}{V_{s}}=\frac{s^{3}}{s^{3}+6 s+12 s+24}
\]
using the $L C$ ladder network in Fig. 18.121

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05:28

Problem 83

Assume that the two circuits in Fig. 18.122 are equivalent. The parameters of the two circuits must be equal. Using this factor and the $z$ parameters, derive Eqs. (9.67) and (9.68).

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