Electric Circuits, 9th Edition P73 pdf

696 Two-Port Circuits
If more than two units are connected in cascade, the a parameters of
the equivalent two-port circuit can be found by successively reducing the
original set of two-port circuits one pair at a time.
Example 18.5 illustrates how to use Eqs. 18.74-18.77 to analyze a cas-
cade connection with two amplifier circuits.
Example 18.5
Analyzing Cascaded Two-Port Circuits
Two identical amplifiers are connected in cas-
cade,
as shown in Fig.
18.11.
Each amplifier is
described in terms of its h parameters. The values
are h
u
= 1000 H, h
n
= 0.0015, h
n
= 100, and
h
22
= 100
/xS.
Find the voltage gain
V
2
/V
g
.

10.25 X 10
-6
,
Figure 18.11 •
The
circuit for Example 18.5.
Solution
The first step in finding
V
2
/V
g
is to convert from
h parameters to a parameters. The amplifiers are
identical, so one set of a parameters describes
the amplifiers:
«n =
«12 =
«21 =
-Ah +0.05
-4
-h
n
100
-1000
5 x 10
= -lo a,
h
2]
100

-h
22
-100 x 10~
6
«12 =
«11
«12 + «12«22
= (5 X 10^)(-10) + (-10)(-10
-2
)
= 0.095 a,
«21 = «21«11 + «22«21
= (-KT
6
)(5 X 10~
4
) + (-0.01)(-10
-6
)
= 9.5 x 10~
9
S,
«22
=
«21«12 + «22«22
= (-10^)(-10) + (-10
-2
)
2
= 1.1 X 10

-4
.
From Table 18.2,
V
g
(«n + «2iZ<,)Zi. + a
l2
+
a
22
Z
R
10<
h
2
\
-1
100
•10
-6
S.
1
"* = % - wo = -
,0
Next we use Eqs. 18.74-18.77 to compute the
a parameters of the cascaded amplifiers:
«n = «ii«'n + «i2«2i
= 25 x 10
-8
+ (-10)(-10

-5
)
[10.25 X 10
-6
+ 9.5 X 10
-9
(500)]10
4
+ 0.095 + 1.1 X 10
-4
(500)
=
10
4
^
~ 0.15 + 0.095 + 0.055
_ \tf_
3
= 33,333.33.
Thus an input signal of 150
/JLV
is amplified to an
output signal of 5 V. For an alternative approach to
finding the voltage gain
V
2
IV
g
,
see Problem

18.41.
Practical Perspective 697
I/ASSESSMENT PROBLEM
Objective 3—Know how to analyze a cascade interconnection of two-port circuits
18.7 Each element in the symmetric bridged-tee
circuit shown is a 15 ft resistor. Two of these
bridged tees are connected in cascade between a
dc voltage source and a resistive
load.
The dc
voltage source has a no load voltage of
100
V and
an internal resistance of 8 ft. The load resistor is
adjusted until maximum power is delivered to
the load. Calculate (a) the load resistance, (b) the
load voltage, and (c) the load power.
/l
^-
+
z
a
z
c
v,
•
z
b
—<•—
z

a
h
~*c—
+
Vi
•
Answer:
NOTE: Also try Chapter Problem 18.40.
(a) 14.44 ft;
(b)16V;
(c)
17.73 W.
Practical Perspective
Characterizing
an
Unknown Circuit
We make the following measurements to find the h parameters for our "black
box" amplifier:
With Port 1 open, apply 50
V
at Port 2. Measure the voltage at Port 1
and the current at Port 2:
V
{
=
50 mV;
/
2
= 2.5 A.
With Port 2 short-circuited, apply 2.5 mA at Port 1. Measure the volt-

age at Port 1 and the current at Port 2:
V, = 1.25 V; /
2
= 3.75 A.
Calculate the h parameters according to Eq. 18.14:
hu = —-
h\ =
Vi
h
h
h
1.25
v
2=()
0.0025
3.75
v
2=0
0.0025
= 500 ft; h
12
= 77
V,
= 1500;
^22 = 77
/,=o
7,=0
0.05
50
= 10

-3
-7- = 50 mS.
50
Now we use the terminated two-port equations to determine whether or not
it is safe to attach a 2 V(rms) source with a 100 ft internal impedance to
Port 1 and use this source together with the amplifier to drive a speaker
modeled as a 32 ft resistance with a power rating of 100 W. Here we find
the value of /2
fr°
m
TaD
^
e 18
-
2:
hyVg
h
(1 + h
22
Z
L
)(h
n
+ Z
g
) -
h
l2
h
2l

Z
L
1500(2)
" [1 + (0.05)(32)][500 + 100] - (1500)(10^)(32)
= 1.98 A(rms)
Calculate the power to the 32 ft speaker:
P = Rll = (32)(1.98)
2
= 126 W.
The amplifier would thus deliver 126 W to the speaker, which is rated at
100
W,
so it would be better to use a different amplifier or buy a more pow-
erful speaker.
698 Two-Port Circuits
Summary
• The two-port model is used to describe the performance
of a circuit in terms of the voltage and current at its
input and output ports. (See page 678.)
• The model is limited to circuits in which
• no independent sources are inside the circuit between
the ports;
• no energy is stored inside the circuit between the ports;
• the current into the port is equal to the current out of
the port; and
• no external connections exist between the input and
output ports.
(See page 678.)
• Two of the four terminal variables (Vi, /i, V
2

, />)
are
independent; therefore, only two simultaneous equa-
tions involving the four variables are needed to describe
the circuit. (See page 680.)
• The six possible sets of simultaneous equations involv-
ing the four terminal variables are called the z-, y-,
a-,
b-,
h-,
and g-parameter equations. See Eqs. 18.1-18.6. (See
page 680.)
• The parameter equations are written in the s domain. The
dc values of the parameters are obtained by setting s ~ 0,
and the sinusoidal steady-state values are obtained by
setting
$
= jw. (See page 680.)
Any set of parameters may be calculated or measured by
invoking appropriate short-circuit and open-circuit con-
ditions at the input and output ports. See Eqs. 18.7-18.15.
(See pages 681 and 682.)
The relationships among the six sets of parameters are
given in Table
18.1.
(See page 684.)
A two-port circuit is reciprocal if the interchange of an
ideal voltage source at one port with an ideal ammeter
at the other port produces the same ammeter reading.
The effect of reciprocity on the two-port parameters is

given by Eqs. 18.28-18.33. (See page 687.)
A reciprocal two-port circuit is symmetric if its ports
can be interchanged without disturbing the values of
the terminal currents and voltages. The added effect
of symmetry on the two-port parameters is given by
Eqs.
18.38-18.43. (See page 688)
The performance of a two-port circuit connected to a
Thevenin equivalent source and a load is summarized by
the relationships given in Table 18.2. (See page 690.)
Large networks can be divided into subnetworks by
means of interconnected two-port models. The cas-
cade connection was used in this chapter to illustrate
the analysis of interconnected two-port circuits. (See
page 694.)
Problems
Sections 18.1-18.2
18.1 Find the h and g parameters for the circuit in
Example 18.1.
18.2 Find the y parameters for the circuit shown in
Fig. P18.2.
Figure P18.3
I
+
± 1ft
AAA- f WV
4 ft M
12 ft V-,
Figure P18.2
+

V\
8ft
20 ft 4ft
:10ft
-»-
v?
18.3 Find the z parameters for the circuit in Fig.
PI8.3.
18.4 Use the results obtained in Problem 18.3 to calcu-
late the y parameters for the circuit in Fig.
PI8.3.
18.5 Find the h parameters for the circuit in Fig. PI8.5.
Figure P18.5
h
10 ft
-AM,
Problems 699
18.6 Find the b parameters for the circuit shown in 18.11 Find the g parameters for the operational amplifier
Fig. P18.6. circuit shown in Fig. PI
8.11.
Figure P18.6
20
n
Figure P18.ll
l ion
-VW #
+
V^
V,
50

4012
18.7 Select the values of R
h
R
2
, and i?
3
in the circuit
in Fig. P18.7 so that h
u
= 4 ft, h
u
= 0.8,
h
2]
= -0.8, and/*
22
= 0.14 S.
18.12 The operational amplifier in the circuit shown
in Fig. P18.12 is ideal. Find the h parameters of
the circuit.
Figure P18.7
+
V",
!± r
/?i
:Ri i
^3
1
/

2
+
v
2
Figure P18.12
400 fi
18.8 Find the a parameters for the circuit in Fig. PI8.8.
18.13 The following direct-current measurements were
made on the two-port network shown in Fig. PI8.13.
Figure P18.8
'v l kn
• 'Wv
V, 10"
4
V
1)50/!
140
kH
V
2
Port
2
Open
K, =
20
mV
Is
-5V
/, = 0.25 juA
Port 2 Short-Circuited

l
x
=
200
fiA
h =
50
fiA
V
{
=
10
V
Calculate the g parameters for the network.
18.9 Use the results obtained in Problem 18.8 to calcu-
late the g parameters of the circuit in Fig. PI8.8.
18.10 Find the h parameters of the two-port circuit shown
inFig.P18.10.
Figure P18.10
Figure P18.13
+
v,
•—
I.
ion /20a
20011
I
™
A»„
i

:
—<
+
=:-/ioon v
:
>
•
-/.
+
Vi
g
-*<
h
+
V,
18.14 a) Use the measurements given in Problem 18.13
to find the y parameters for the network.
b) Check your calculations by finding the
y parameters directly from the g parameters
found in Problem
18.13.
18.15 Derive the expressions for the h parameters as
functions of the g parameters.
700 Two-Port Circuits
18.16 Derive the expressions for the b parameters as
functions of the h parameters.
18.17 Derive the expressions for the g parameters as
functions of the z parameters.
18.18 Find the ^-domain expressions for the a parameters
of the two-port circuit shown in Fig. P18.18.

Figure P18.18
*i 1F i „ k
e
:
|(- 1 TVY-Y-V
0
'•'
:4H
18.19 Find the ^-domain expressions for the z parameters
of the two-port circuit shown in Fig. P18.19.
Figure P18.19
18.20 Find the frequency-domain values of the a parame-
ters for the two-port circuit shown in Fig. P18.20.
Figure P18.20
I,
20
n
18.21 Find the h parameters for the two-port circuit
shown in Fig. P18.20.
18.22 a) Use the defining equations to find the s-domain
expressions for the h parameters for the circuit
in Fig. PI 8.22.
b) Show that the results obtained in (a) agree with
the /i-parameter relationships for a reciprocal
symmetric network.
Figure P18.22
18.23 Is the two-port circuit shown in Fig. PI8.23 sym-
metric? Justify your answer.
Figure P18.23
Section 18.3

18.24 Derive the expression for the voltage gain V
2
/V\ of
the circuit in Fig. 18.7 in terms of the y parameters.
18.25 Derive the expression for the input impedance
(Z
in
= Vj//i) of the circuit in Fig. 18.7 in terms of
the b parameters.
18.26 Derive the expression for the voltage gain
V
2
/V
g
°f
the circuit in Fig. 18.7 in terms of the h parameters.
18.27 Derive the expression for the current gain I
2
/l\
of the circuit in Fig. 18.7 in terms of the
g parameters.
18.28 Find the Thevenin equivalent circuit with respect
to port 2 of the circuit in Fig. 18.7 in terms of the
z parameters.
18.29 The b parameters of the amplifier in the circuit
shown in Fig. PI8.29 are
h
n
= 25;
b

lx
= -1.25 S;
b
l2
= 1 kO;
b
22
= -40.
Find the ratio of the output power to that supplied
by the ideal voltage source.
Figure P18.29
loo
a
Problems
701
18.30 The y parameters for the two-port amplifier circuit
in Fig. PI 8.30 are
y
u
= 2mS;
y
n
= -2 /xS;
y
2
\ = 100 mS; y
22
= -
50
JJLS.

The internal impedance of the source is 2500 + /0 ft,
and the load impedance is 70,000 + jO 0 The ideal
voltage source is generating a voltage
v
g
= 80 V2 cos 4000r mV.
a) Find the rms value of V
2
.
b) Find the average power delivered to Z
L
.
c) Find the average power developed by the ideal
voltage source.
Figure P18.30
7
j—Z,
vA
1
V
2
|
L
V22
i
1
1
+
|^2
1

1
_ J
z
L
18.31 For the terminated two-port amplifier circuit in
Fig. P18.30, find
a) the value of Z
L
for maximum average power
transfer to Z
L
b) the maximum average power delivered to Zi
c) the average power developed by the ideal voltage
source when maximum power is delivered to Z
L
.
18.32 The linear transformer in the circuit shown in
Fig. P18.32 has a coefficient of coupling of
0.75.
The
transformer is driven by a sinusoidal voltage source
whose internal voltage is v
g
= 260 cos 4000^ V. The
internal impedance of the source is 25 + ;0 ft.
a) Find the frequency-domain a parameters of the
linear transformer.
b) Use the a parameters to derive the Thevenin
equivalent circuit with respect to the terminals
of the load.

c) Derive the steady-state time-domain expression
for ih.
Figure P18.32
25 a
^vw-
'-'l
50
O o.75
400
.°
"k
12.5
mH
18.33 The g parameters for the two-port circuit in
Fig. PI 8.33 are
1
!
Q
^
=
6"
;
6
S
'
£12 = -0.5 + /0.5;
-/0.5;
g
22
= 1.5 + /2.5ft.

The load impedance Z
L
is adjusted for maximum
average power transfer to Z
L
. The ideal voltage
source is generating a sinusoidal voltage of
v
g
= 42V2" cos 5000* V.
a) Find the rms value of V
2
.
b) Find the average power delivered to Z
L
.
c) What percentage of the average power developed
by the ideal voltage source is delivered by Z
L
?
Figure P18.33
18.34 The following dc measurements were made on the
resistive network shown in Fig. P18.34.
Measurement 1
V, = 4V
/, - 5 raA
h = -200 mA
Measurement 2
V! = 20 mV
/, =

20 juA
V
2
=
40
V
/
2
= 0A
A variable resistor R
0
is connected across port
2 and adjusted for maximum power transfer to R
()
.
Find the maximum power.
Figure P18.34
1
5.25
mvf
+
T
/.
250 O
+
) ".
Resistive
network
U
+

V-, i
A
ikn
702 Two-Port Circuits
18.35 The following measurements were made on a resis-
tive two-port network:
Condition 1 - create a short circuit at port 2 and
apply 20 V to port 1:
Measurements: I
:
=
1
A; h = -1 A.
Condition 2 - create an open circuit at port 1 and
apply 80 V to port 2:
Measurements: V
{
= 400 V; I
2
= 3 A.
Find the maximum power that this two-port circuit
can deliver to a resistive load at port 2 when port 1
is driven by a 4 A dc current source with an internal
resistance of 60 O.
Figure P18.38
soon
jf
a
c
[h]

b
1 d
c
d
2
e
f
t
c»-
(a)
R
R
R
•^WV f -VS-V *
-•c
18.36 a) Find the s-domain expressions for the g parame-
ters of the circuit in Fig. PI 8.36.
b) Port 2 in Fig. P18.36 is terminated in a resistance
of 400 O, and port 1 is driven by a step voltage
source v
x
(t) = 30u(t) V. Find v
2
(t) for t > 0 if
C = 0.2 /xF and L = 200 mH.
R = 72 kH IR
Figure P18.36
• •-
sL
1/rC

1/sC
+
(b)
18.39 The networks A and B in the circuit in Fig. PI8.39
are reciprocal and symmetric. For network A, it is
known that a'
n
= 5 and a\
2
= 24 O.
a) Find the a parameters of network B.
b) Find V
2
when V
g
= 75/CT V,
Z
g
= 1/tT n, and Z
L
= 10/0° a.
Figure P18.39
18.37 a) Find the y parameters for the two-port network
in Fig. PI8.37.
b) Find v
2
for t > 0 when v
8
= 50u(t) V.
Figure P18.37

Section 18.4
18.38 The h parameters of the first two-port circuit in
Fig. PI8.38(a) are
h
n
= 1000 O; h
n
= 5 x 10
-4
:
/i
2
i = 40; h
22
=
25
fiS.
The circuit in the second two-port circuit is shown
in Fig. P18.38(b), where R = 72 kH. Find v
a
if
v
n
= 9 mV dc.
1 r
50 j is n
A
j
5
a,j is a 512

;-/ion
J L
18.40 Tlie z and
_y
parameters for the resistive two-ports
in Fig. P18.40 are given by
z\\
= -r-U; )
;
n = 200/xS;
3
100
<12
ft; y
12
= 40
/x.S;
z
21
= -kO; y
2l
= -800/xS
Z22 = v ^
ii;
-
y
22
= 4W
A<S;
Calculate

t>
0
if
1?»
= 30 mV dc.
Problems
703
Figure P18.40
ion
?
[z]
[>']
+
vn<
Sections 18.1-18.4
18.41 a) Show that the circuit in Fig. P18.41 is an equiva-
lent circuit satisfied by the //-parameter equations.
b) Use the /i-parameter equivalent circuit of (a) to
find the voltage gain V^Vg in the circuit in
Fig.
18.11.
Figure P18.41
/,
18.42 a) Show that the circuit in Fig. PI 8.42 is an equiva-
lent circuit satisfied by the z-parameter equations.
b) Assume that the equivalent circuit in Fig. PI 8.42
is driven by a voltage source having an internal
impedance of Z
?
ohms. Calculate the Thevenin

equivalent circuit with respect to port 2. Check
your results against the appropriate entries in
Table 18.2.
Figure P18.42
/,
I-,
+
-o-
h(Z[2 ~ Zl\)
-•II
tl\
^22
—
^21
221
+
V,
18.43 a) Show that the circuit in Fig. P18.43 is also an
equivalent circuit satisfied by the z-parameter
equations.
b) Assume that the equivalent circuit in Fig. PI8.43
is terminated in an impedance of Z
L
ohms at
port 2. Find the input impedance
V\jl\.
Check
your results against the appropriate entry in
Table 18.2.
Figure P18.43

/.
•*-
+
v.
•
?11 - *12
z
12
—II—
^22
~~
-¾
l
2
\^ +
A(*21
_
^12)
•
18.44 a) Derive two equivalent circuits that are satisfied
by the y-parameter equations. Hint: Start with
Eqs.
18.2. Add and subtract
y2\V
2
to the first
equation of the set. Construct a circuit that satis-
fies the resulting set of equations, by thinking in
terms of node voltages. Derive an alternative
equivalent circuit by first altering the second

equation in Eq. 18.2.
b) Assume that port
1
is driven by a voltage source
having an internal impedance Z
?
, and port 2 is
loaded with an impedance Z
L
. Find the current
gain
/2//1.
Check your results against the appro-
priate entry in Table 18.2.
18.45 a) Derive the equivalent circuit satisfied by the
^-parameter equations.
b) Use the g-parameter equivalent circuit derived
in part (a) to solve for the output voltage in
Problem
18.38.
Hint: Use Problem 3.65 to simplify
the second two-port circuit in Problem 18.38.
18.46 a) What conditions and measurements will allow
you to calculate the b parameters for the "black
box" amplifier described in the Practical
Perspective?
b) What measurements will be made if the result-
ing b parameters are equivalent to the h param-
eters calculated in the Practical Perspective?
18.47 Repeat Problem 18.46 for the z parameters.

Appendix m\
r\
The Solution of Linear
Simultaneous Equations
Circuit analysis frequently involves the solution of linear simultaneous
equations. Our purpose here is to review the use of determinants to solve
such a set of equations. The theory of determinants (with applications) can
be found in most intermediate-level algebra texts. (A particularly good
reference for engineering students is Chapter 1 of E.A. Guillemin's The
Mathematics of Circuit Analysis [New York: Wiley, 1949]. In our review
here,
we will limit our discussion to the mechanics of solving simultaneous
equations with determinants.
A.l Preliminary Steps
The first step in solving a set of simultaneous equations by determinants is
to write the equations in a rectangular (square) format. In other words, we
arrange the equations in a vertical stack such that each variable occupies
the same horizontal position in every equation. For example, in Eqs. A.l,
the variables i
h
/
2
, and /
3
occupy the first, second, and third position,
respectively, on the left-hand side of each equation:
21/!
- 9/
2

- 12/3 = -33,
-3/j + 6/
2
- 2/
3
= 3, (A.l)
-8/,
- 4*
2
+ 22/3 = 50.
Alternatively, one can describe this set of equations by saying that i\
occupies the first column in the array, i
2
the second column, and f
3
the
third column.
If one or more variables are missing from a given equation, they can
be inserted by simply making their coefficient zero. Thus Eqs. A.2 can be
"squared up" as shown by Eqs. A3:
2v
x
- v
2
= 4,
4v
2
+ 3¾ = 16, (A.2)
lv\ + 2?;
3

= 5;
2V] - v
2
+ 0v
3
= 4,
Ovi + 4v
2
+ 3¾ = 16, (A.3)
7y, + 0v
2
+ 2v
3
= 5.