Summary Don't judge a book of cover

current through the inductor is 1. ApDlving KVL V.(c) = 2s + 4and Vz(3)
Since, for unity excitation, the resDonse V.(s) eauals the network function Zi1?
and the response V(s) equals z.(s). we conchude that Z.(s) equals 2s + 4 ad
Z21(S) equals 4. A similar procedure is used to fnd the
remaining
namely, Z12(s) and zT2(s). For this case. we apply an excitation z parne
unity value at port 2as shown in Fig. 11-2.2(c), Since current sour
for the ICIS is zero, in this case we the controlling cu
fnd that V.(s) = V,(s) = 1and
Z22(S) F I. The z-paramneter matrix for the network shown in Fig. 11-2.2(a)Zi2S)
15




EXERCISE
11-2.1 Find the z parameters for the circuit shown in Fig. 11-2.2 and discussed in Example
11-2.1 for each of the following sets of values of the resistor R(shown as 1
the figure) and the ICIS gain K (shown as 3 in the
figure).
Answers
R K Z|(s) Zz1(s) Z(s) Z2(s)
4 2s + 12 12 4
2 4 2s + 10 10 2 2
3 -3 2,.-6 -6 3
5 2.s 5


SUMMARY 11-2.1
The z Parameters of a Two-Port Network. The z parameters of a two-port network
are the functions zs) which specify the port voltages as functions of the port
currents. Two of the four z parameters may be found by applying a unit test excitation
current at port 1 and measuring the resulting voltage response at both ports. The
other two zparameters may be found by similarly applying a unit test excitation
current at the other port. The various test conditions used todefine each parameter
are summarized in Fig. 11-2.3.


Z1
V
lI,=0



212 1L, =0

Figure 11-2.3 Test conditions used to define the z parameters.


rameters 591

, Reciprocal Network
Now let us investigate sonme
that if thc points of excitation
properties of the z
and
of a reciprocal network will be response arc
aramelers, hc
reciprocal network will be modiicd
is exactly what is done in
Fig. I1-2.3. We may conclude
Mnc
detIi.e., lan ge d,
Such an while the
of
In

crmthatining the parianmettenrkcehercarhnsmgegtreu,nsfer lransfer \mped
a
Z12(rsc) cipwillrocalcqualnctwork CHCILalion and
SYmmetric 2-parameter matrix
network will have a non-symmetric
11-12) forZ-theparameter matrzz)) andWil , in
nctwork shown an hatageneral, resDhOanve
parameter matrix given in
ix. NS
Fig. CXaniplc nonrecinoKhis
network clements and such networks n
symmetric z-parameter matrix is givenarein always of
be anticipated, since the network contains anExamplreecip| -rO2.1cal. \ -2.\(0) sy is
active This
Other Network Functions element, Le.,resanul\CISagan Co
An important property of the z
parameters are known, we may find pararneters is that if.
from them. For example, consider theany other desiredfor a
transfer function V,(s)/V,(s). By the term
port (i.e., the port where V,(s) is detopen-cieisrmricuinattionhere,of newethetworgvenk Îuncinctownork,direthcte
h
transfer function, let us rewrite themeasured| ofopen-circuited.implToypenth-catircuheil voltac
that 1,(s) = 0. We obtain equations (11-6) with the find thisrespons
V,(s) =z(s)/,(9) added restrvolicttiag
V,(s) = z,|()7(9)i
Taking the ratio of these two equations and
V,()
inverting the result, we obtain
oaoban o e i s ) 2()
This is the desired voltage transfer function
In a similar fashion we may obtain the expressed in terms of the 2
I(s). By the term short-circuit here, we short
imply-cirthat
cuit thecurrent transfer functparioanmetI,(ser
at which ,(9) is measured] is short-circuited. To find thisresponse port (i.¬., the op
write the second equation of (11-6) with the added functthation, need simp
restriction
we
we obtain V,(s) = 0. Thr
0= Z1(s)/,(s) +z2(s)l,(6)
Rearranging the terms of the equation above, we see that
I,(5) -2,(s)
1,(6) Z26)

t MANI
SHANKA
NOTE
Two-Port Network Parametere (nap.

MI
ORED

, ,()
(b




(a)

Figure 11-2.7 Series connection of
do NOT add.
In this case, the z parameters of the
two-port networks in which
the z parameterS of the component Overall network are not the z

path of the test current Z,(s) shownnetworks.
in Fig. To see why this equal to parameters
he
node this current branches into two equal 11-2.7(a). At
current currents /(s)/2 as the
is
so, sunn theof
indicatconsed idierntuppererior
two-port component network, the
1(s), and the current out of the other into one of the shown.
violates the port curent condition. and,terminal
as
of
this terminals
port ofForits the
component networks.
a
it is
result, invalidatesI{(s)/2. Thisport I is
for the
The results given above may be
summarized the zcleary
SUMMARY 11-2.2
as
follows: parameters
Properties of the z Parameters of a Two-Port
of a given two-port network completly determine Network. The four z parameters
the
the sense that they may be used to find any
for the two-port network. If other
two two-port networks propert
network
are
ie s
function
of the
that net
iswork in
individual
sum of their z
parameters gives the z
two-port network (unless the connection violatesparameters
connect
for ed in series, defintheed
the
current resulting overalon
the port currents of the component two-port networks) the port

Models of z Parameters
requirement
In the preceding paragraphs descriptions of various techniques
parameters of a specific network have been given. Now let us for finding the z
nroblem: Given a set of z parameters, nOW do we find a consider the whish
two-port network inverse
has such a set of parameters? This problem is basically a synthesis problem wnd
there is, in general, no unique solution to synthesis problems. There are. howear
some useful network forms which may be used as circuit
representations for a
given set of z parameters. One of the most common of these forms is the spy

Two-Port Network Parameters Chap. 11
MANI
SHANKAR
DMINOTE