1
LECTURE 39
Two Port Networks
2
Lecture Outline
1. Two Port Networks
2. Admittance Parameters
Example 1
Example 2
3
2.1 Admittance Parameters
Let us now turn our attention to two-port networks
We will assume in all that follows that the network is composed of
linear elements and contains no independent sources; dependent
sources are permissible
• Further conditions will also be placed on the network in special cases
•
•
•
We will consider the two-port as
it is shown in Fig. 16.8; the
voltage and current at the input
terminals are V1 and I1 , and V2
and I2 are specified at the output
port
•
The directions of I1 and I2 are
both customarily selected as into
the network at the upper
conductors (and out at the lower
conductors)
4
sider the two-port as it is shown in Fig. 17.8; the voltage and
I1
put terminals are V1 and I1, and V2 and I2 are specified at the
+
e directions of I1 and I2 are both customarily selected as into
• Since the network is linear and contains no independent sources
V1
he upper conductors (and out at the lower conductors). Since
within it, I1 may be considered to be the superposition of two –
linear and contains no independent sources within it, I1 may
components, one caused by V1 and the other by V2
o be the superposition of two components,
one caused by V1
■ with
FIGURE
17.8 A g
•
When
the
same
argument
is
applied
to
I
,
we
may
begin
the
2
y V2. When the same argument is applied to I2, we may begin
voltages and currents
set
of
equations
equations
composed of linear el
2.2 Admittance Parameters
I1 = y11 V1 + y12 V2
I2 = y21 V1 + y22 V2
[5]
[6]
where
theproportionality
y’s are no moreconstants,
than proportionality
re no more
than
or unknown coefconstants,
or unknown
thedimensions
present
present.
However,
it should coefficients,
be clear thatfor
their
• Itare
should
be clear
r S. They
therefore
calledthat
thetheir
y (ordimensions
admittance) parambe and
A/V,[6].
or S
efined by must
Eqs. [5]
meters, •as well
other
sets of parameters
will define later
Theyasare
therefore
called the we
y (or
are represented
concisely
as matrices.
we define
admittance)
parameters,
andHere,
are defined
by the
n matrix I,
Eqs. [5] and [6]"
!
dependent sources, b
independent sources.
must
be
A/V,
or
S.
They
areproportionality
therefore called
the yoror
(or
admittance)
where
the
y’s
are
no
more
proportionality
constants,
unknown
coef5coefwhere
are
no
morethan
than
proportionality
constants,
orunknown
unknown
coef- param
where
thethe
y’sy’s
are
no
more
than
constants,
ficients,
for
the
present.
However,
should
be
clear
that
their
dimensions
eters,
and
are
defined
by
Eqs. [5]
and
[6].
ficients,
the
present.
However,
should
beclear
clearthat
thattheir
theirdimensions
dimensions
ficients,
forfor
the
present.
However,
itititshould
be
must
be
A/V,
or
therefore
called
the
(or
admittance)
paramThe
yA/V,
parameters,
asare
well
as other
setsthe
of
parameters
we will
define late
must
A/V,
They
are
therefore
called
theyy
y(or
(oradmittance)
admittance)
parammust
bebe
ororS.
S.S.They
They
are
therefore
called
parameters,
are
defined
by
[5]
and
[6].
eters,
and
defined
Eqs.[5]
[5]and
and
[6].
in
theand
chapter,
are represented
concisely
as matrices. Here, we define the
eters,
and
areare
defined
bybyEqs.
Eqs.
[6].
• The
y
parameters,
as
well
as
other
sets
of
parameters
we will
will define
define later
later
The
y
parameters,
as
well
as
other
sets
of
parameters
we
The
y
parameters,
as
well
as
other
sets
of
parameters
we
will
define
later
The
y
parameters,
as
well
as
other
sets
of
parameters
we
will
define
later
(2 in
× the
1) column
matrix
I,
chapter,
are
represented
conciselyasasmatrices.
matrices Here, we define the
in
the
chapter,
concisely
chapter,are
arerepresented
representedconcisely
concisely
as
matrices.
Here,we
wedefine
definethe
the
inin
thethe
chapter,
are
represented
as
matrices.
Here,
! "
•(2
Here,
we
define
the (2
I1 I,
×
1)
column
matrix
I,
column
matrix
(2(2
××
1)1)
column
matrix
I, I,× 1) column matrix
I!!=
[7
! """I
II1I 2
II I=
== 1 1
[7]
[7]
the (2 × 2) square matrix of the y IIparameters,
22I2
2.3 Admittance Parameters
!
"
(2
×
2)
the
square
matrix
parameters,
square
matrixofofthe
they yparameters,
parameters,
(2(2
××
2)2)
thethe
square
y11 y12
!
"""
y!=
y11 yyy1212
y 11
y y== 11 y2112 y22
y21 yy22
y2222
y21
21
and the (2 × 1) column matrix V,
and
columnmatrix
matrixV,V,
(2
×
1)
and
the
column
(2(2
××
1)1)
and
thethe
column
!
!!! """V1
VV
V
V
=
11 1
VV==
V2
V
2
VV
22
"
IyV,
=or
yV,
Thus,
we
may
write
the
matrix
equation
Thus,
we
may
write
matrix
equation
or or
II I=
Thus,
we
may
write
equation
==yV,
yV,
Thus,
we
may
write
thethe
matrix
equation
or
" ""!" "
! !!I" " "! ! y! y """!!! V
I1 1I1
yy12
y11
V111 V1
1111y11
1212 y12V
=
= =
y yy22
y2222 y22V
I2I2I2
y21
VV
222 V2
2121y21
[8
[8]
[8]
[9]
[9]
The
Th
The
is isisstass
[9
pre
prev
pre
qua
quan
qu
cle
clear
cle
!
I1
I2
"
=
!
y11
y21
y12
y22
"!
V1
V2
"
6
2.4 Admittance Parameters
and matrix multiplication of the right-hand side gives us the equality
! " !
"
I1
y11 V1 + y12 V2
=
I2
y21 V1 + y22 V2
•
•
•
•
•
•
These (2 × 1) matrices must be equal, element by element, and thus we are
These (2 × 1) matrices must be equal, element by element, and thus we
led to the defining equations, [5] and [6].
are led to the defining equations, [5] and [6]
The most useful and informative way to attach a physical meaning to the
The
most useful
informative
to attach
physical
meaning
to the
y parameters
is and
through
a direct way
inspection
of aEqs.
[5] and
[6]. Consider
yEq.
parameters
is through
a direct
of Eqs.
[5]that
andy[6]
be zero, then
we see
[5], for example;
if we
let V2inspection
11 must be given
by the ratio
I1 tofor
V1example;
. We therefore
as the
admittance
measured
Consider
Eq.of[5],
if wedescribe
let V2 bey11zero,
then
we see that
y11
at thebeinput
withofthe
output
must
giventerminals
by the ratio
I1 to
V1 terminals short-circuited (V2 = 0).
Since
there can
be noyquestion
which terminals are short-circuited, y11 is
We
therefore
describe
11 as the admittance measured at the input
best described
as the
short-circuit
admittance.(V
Alternatively,
we might
terminals
with the
output
terminalsinput
short-circuited
=
0)
2
describe y11 as the reciprocal of the input impedance measured with the outSince
there canshort-circuited,
be no questionbut
which
terminalsasare
short-circuited,
y11 is
put terminals
a description
an admittance
is obviously
best
input
admittance
moredescribed
direct. Itas
is the
not short-circuit
the name of the
parameter
that is important. Rather, it
Alternatively, we might describe y11 as the reciprocal of the input
impedance measured with the output terminals short-circuited, but a
description as an admittance is obviously more direct
7
2.5 Admittance Parameters
•
It is not the name of the parameter that is important
Rather, it is the conditions which must be applied to Eq. [5] or [6], and
hence to the network, that are most meaningful
• When the conditions are determined, the parameter can be found directly
from an analysis of the circuit (or by experiment on the physical circuit)
• Each of the y parameters may be described as a current-voltage ratio with
either V1 = 0 (the input terminals short-circuited) or V2 = 0 (the output
terminals short-circuited):
•
8
2.5 Admittance Parameters
Because each parameter is an admittance which is obtained by
short-circuiting either the output or the input port, the y
parameters are known as the short-circuit admittance
parameters
• The specific name of y11 is the short-circuit input admittance,
y22 is the short-circuit output admittance, and y12 and y21 are the
short-circuit transfer admittances
•
9
2.6.1
Admittance Parameters
Find the four short-circuit admittance parameters for the
resistive two-port shown in Fig. 16.9.
• The values of the parameters may be
easily established by applying Eqs.
[10] to [13], which we obtained
directly from the defining equations,
[5] and [6]
• To determine y11 , we short-circuit the output and
find the ratio of I1 to V1
• This may be done by letting V1 = 1 V, for then
y11 = I1
• By inspection of Fig. 16.9, it is apparent that 1 V
applied at the input with the output short-circuited
will cause an input current of ( 1/5 + 1/10 ), or 0.3 A
• Hence,
𝐈
#
• We can also find y21 = − ! = − , because V! = −10I"
𝐕"
#$
10
2.6.2
Admittance Parameters
Find the four short-circuit admittance parameters for the
resistive two-port shown in Fig. 16.9.
• In order to find y12 , we short-circuit the input terminals and
apply 1 V at the output terminals
• The input current flows through the short circuit and is
#
I1 = − A
• Thus
#$
11
2.6.3
Admittance Parameters
Find the four short-circuit admittance parameters for the
resistive two-port shown in Fig. 16.9.
The four Admittance parameters found for the netwrok are:
The describing equations for this two-port in
terms of the admittance parameters are,
therefore,
and
𝑦
0.3 −0.1
−0.1 0.15
(all S)
It is not necessary to find these parameters one at a time by
using Eqs. [10] to [13], as we see in next example;
12
2.7 Admittance Parameters
Assign node voltages V1 and V2 in the two-port of Fig. 16.9
and write the expressions for I1 and I2 in terms of them
V1
V2
In general, it is easier to use Eq. [10], [11], [12], or [13] when only one
parameter is desired
• If we need all of them, however, it is usually easier to assign V1 and V2
to the input and output nodes, to assign other node-to-reference voltages
at any interior nodes, and then to carry through with the general solution
• In order to see what use might be made of such a system of equations,
let us now terminate each port with some specific one-port network
•
13
2.8.1 Admittance Parameters
•
•
•
•
Consider the simple two-port network of Example 16.4, shown in Fig. 16.11
with a practical current source connected to the input port and a resistive load
connected to the output port
A relationship must now exist between
V1 and I1 that is independent of the
two-port network
This relationship may be determined
solely from this external circuit
If we apply KCL (or write a single
nodal equation) at the input,
14
2.8.2
•
Admittance Parameters
The input and output currents are
also easily found:
𝐼! = 11 A and 𝐼" = -2.5A
•
Thus, the complete terminal
characteristics of this resistive
two-port are then known
•
The advantages of two-port analysis do not show up very strongly for such a
simple example, but it should be apparent that once the y parameters are
determined for a more complicated two-port, the performance of the two-port
for different terminal conditions is easily determined; it is necessary only to
relate V1 to I1 at the input and V2 to I2 at the output
In the example just concluded, y12 and y21 were both found to be −0.1 S
It is not difficult to show that this equality is also obtained if three general
impedances ZA, ZB , and ZC are contained in this 𝜋 network
It is somewhat more difficult to determine the specific conditions which are
necessary in order that y12 = y21 , but the use of determinant notation is of
some help
•
•
•
15
2.8.3
•
•
•
•
•
•
•
•
Admittance Parameters
Let us see if the relationships of Eqs. [10] to [13] can be expressed in terms of
the impedance determinant and its minors
Since our concern is with the two-port and not with the specific networks with
which it is terminated, we will let V1 and V2 be represented by two ideal
voltage sources
Equation [10] is applied by letting V2 = 0 (thus short-circuiting the output) and
finding the input admittance
The network now, however, is simply a one-port, and the input impedance of a
one-port was found in Sec. 16.1
We select loop 1 to include the input terminals, and let I1 be that loop’s
current; we identify (−I2) as the loop current in loop 2 and assign the
remaining loop currents in any convenient manner
Thus,
and, therefore,
Similarly,
!Z
! 31
! ···
!
ZN 1
2.8.4
•
•
Z32
···
ZN 2
···
···
···
Z3N !!
· · · !!
ZN N
16
Admittance Parameters
In orderThus,
to find y12, we let V1 = 0 and find I1 as a function of V2
We find that I1 is given by the ratio
(−V2 )!21
and
I1 = −
Thus,
and,
y12 =
!Z
!21
!Z
may show that
In a similar manner, weSimilarly,
may showwethat
!12
y21 =
!Z
The equality of y12 and y21 is thus contingent on the equality of th
• The equality
12 and y21 is
minorsof
ofy!
Z —!12 and !21 . These two minors are
thus contingent on the
equality of the two minors of
!Z
! 12
!
! Z32
ΔZ - Δ12 and Δ21
!
!21 = !
• These two minors are shown: ! Z42
! ···
!
Z13
Z33
Z43
···
Z14
Z34
Z44
···
···
···
···
···
Z1N !!
!
Z3N !!
Z4N !!
· · · !!
17
2.8.5
Admittance Parameters
•
Their equality is shown by first interchanging the rows and columns
of one minor (for example, Δ21), an operation which any college
algebra book proves is valid, and then letting every mutual impedance
Zij be replaced by Zji
• Thus, we set
•
This equality of Zij and Zji is evident for the three familiar passive
elements—the resistor, capacitor, and inductor—and it is also true for
mutual inductance
• However, it is not true for every type of device which we may wish to
include inside a two-port network
• Specifically, it is not true in general for a dependent source, and it is
not true for the gyrator, a useful model for Hall-effect devices and for
waveguide sections containing ferrites
18
2.8.5
Admittance Parameters
•
Over a narrow range of radian frequencies, the gyrator provides an
additional phase shift of 180° for a signal passing from the output to
the input over that for a signal in the forward direction, thus
y12 = −y21
• A common type of passive element leading to the inequality of Zij and
Zji, however, is a nonlinear element
• Any device for which Zij = Zji is called a bilateral element, and a
circuit which contains only bilateral elements is called a bilateral
circuit
• We have therefore shown that an important property of a bilateral
two-port is
y =y
12
21
• This property is glorified by stating it as the reciprocity theorem:
19
2.8.6
Admittance Parameters
Reciprocity Theorem:
• In any passive linear bilateral network, if the single voltage source Vx
in branch x produces the current response Iy in branch y, then the
removal of the voltage source from branch x and its insertion in
branch y will produce the current response Iy in branch x.
• If we had been working with the admittance determinant of the circuit
and had proved that the minors Δ21 and Δ12 of the admittance
determinant ΔY were equal, then we should have obtained the
reciprocity theorem in its dual form:
• In any passive linear bilateral network, if the single current source Ix
between nodes x and x’ produces the voltage response Vy between
nodes y and y’, then the removal of the current source from nodes x
and x‘ and its insertion between nodes y and y’ will produce the
voltage response Vy between nodes x and x’.
20
2.8. Admittance Parameters