1
FOUR - TERMINAL NETWORKS
Electrical circuit having four terminals is called four-terminal network or two-port. In some
cases, when one takes four-terminal network into consideration as separated element, as example
acoustic, electromechanical, optic devices etc., such network can be called as quadripole. Two
terminals or poles are input, and another two - output. Each four-terminal network can be
investigated using common circuit theory laws such as Ohm’s, Kirchoff’s and calculating mesh
currents or nodal voltages. Four-terminal networks theory is the case of adaptation of these laws
to take into consideration only input and output variables.
Primary parameters of the four-terminal networks. Symbol of the four-terminal
network and possible directions of the voltages and currents are shown in the Fig. 3.1. Properties
of the four-terminal network depend on relation between input and output voltages and currents.
Usually two of voltages or currents are known and
are called excitation or action. Another two
voltages or currents are called response or reaction
into these excitations. Relation between these
voltages and currents depends on four-terminal
properties. To build equations for these relations
means to express reaction voltage or current in
Fig. 3.1. Standard four-terminal network
term of action voltage or current. There are six
configuration having four external variables
combinations of designation off excitations and
reactions in the four-terminal networks. So
voltages and currents may be expressed using six equation systems. Coefficients in these
equations are called primary parameters of the network. All these parameters are shown in the
table 3.1. Equations whose relates action, reaction variables and primary parameters are shown
below table.
Table3.1
Version
1
2
3
4
5
6
&
&
&
&
&
&
&
&
&
&
&
Action variables
I ,V
V, I
V , I&
I, I
V ,V
V, I
Reaction variables
V& , I&
V& , I&
V& ,V&
I& , I&
I& ,V&
V& , I&
1
2
1
System of parameters
Z
2
1
1
2
2
Y
1
2
H
2
1
2
2
1
1
A
1
2
1
1
2
1
2
2
G
⎧⎪V&1 = z11 I&1 + z12 I&2 ,
⎨&
⎪⎩V2 = z21 I&1 + z22 I&2 .
⎧⎪ I&1 = y11V&1 + y12V&2 ,
⎨&
⎪⎩ I 2 = y21V&1 + y22V&2 .
⎧⎪V&1 = h11I&1 + h12V&2 ,
⎨&
⎪⎩ I 2 = h21I&1 + h22V&2 .
⎧⎪V&1 = t11V&2 + t12 I&2 ,
⎨&
⎪⎩ I1 = t21V&2 + t22 I&2 .
⎧⎪V&2 = g11V&1 + g12 I&2 ,
⎨&
⎪⎩ I1 = g 21V&1 + g 22 I&2 .
⎧⎪V&2 = b11V&1 + b12 I&1 ,
⎨&
⎪⎩ I 2 = b21V&1 + b22 I&1.
B
(3.1)
From these equations becomes clear mean of primary parameters. First two sets of equations
relate network voltages and currents. Their coefficients as in Ohm’s law have dimension of
impedance or admittance. So why coefficients in the first equation are marked as z-parameters
and can be called impedance parameters. In the second equation coefficients are denoted as yparameters and can be called as admittance parameters and they are inverse of impedance
parameters in most cases. In the third equation we have mixture of variables. Such parameters
were called h-parameters because of they are hybrid. As you had learned in the course of
electronics fundamentals such mixture of variables actually arises as a simplification of midfrequency model of a common emitter configuration of bipolar transistor. Coefficients of the
fourth equation relate output variables with input variables and are called transmission
parameters or t-parameters. In some textbooks these parameters can be called as a-parameters or
ABCD parameters. Last two types of equations also are mixtures of variables. As you can see
2
they are similar to third and fourth equations and have exchanged action and reaction variables:
V&1 ⇔ V&2 , I&1 ⇔ I&2 .
We know that voltage and current phasors are complex quantities. So coefficients in the
equations above could be complex values also. As we’ll see later, in some cases these
coefficients could be real values.
Usually only four types of primary four-terminal network parameters – impedance,
admittance, hybrid and transmission - are used in circuit theory.
Scalar notation of equations in terms of primary parameters, voltages and currents in the fourterminal networks it is possible to change into matrix form:
V&1
I&1
z
= Z ⋅ , Z = 11
z 21
V&2
I&2
z12
.
z 22
I&1
V&1
y
= Y ⋅ , Y = 11
y 21
I&2
V&2
y12
.
y 22
(3.2)
V&1
I&
h
= H ⋅ 1 , H = 11
h21
I&2
V&2
h12
.
h22
V&1
V&
t
t
= T ⋅ 2 , T = 11 12 .
t 21 t 22
I&1
I&2
Fig. 3.2. Typical connection of two four-terminals in series
and external variables notation
Matrix form is more convenient in the
analysis of the complicated network,
consisting on some simple four-terminals.
Let’s consider two four-terminals in series
(Fig. 3.2) as example. If we’ll choose
directions of the currents and voltages as
shown in the Fig. 3.2, then
V&1
V&
V&12
V&
= T1 ⋅ 12 ,
= T2 ⋅ 2
I&
I&
I&
I&
1
12
12
2
Because of left-side network’s output
voltage and current phasors are the same as for the right-side network’s input voltage and current
phasors
V&1
V&
V&
= T1 T2 ⋅ 2 = TΠ ⋅ 2 .
I&1
I&2
I&2
Here TΠ = T1 T2 - transmission coefficient’s matrix of the equivalent four-terminal network,
which represents two in series,
connected networks.
Now let’s consider two fourterminal networks connected in parallel
(Fig.3.3a). If they are described using
y-parameters, input and output currents
and voltages in such network have
following relationships:
I&1
V&
= YΣ ⋅ 1 , YΣ = Y1 + Y2 .
I&2
V&2
When two parallel four-terminal
networks are connected as shown in the Fig. 3.3b, use of the z-parameters yields to relationships
Fig. 3.2. Typical connection of two four-terminals in parallel
and external variables notation
3
V&1
I&
= ZΣ ⋅ 1 ,
V&2
I&2
Z Σ = Z1 + Z 2 .
General relations between primary parameters of the four-terminal networks. All types of
four-terminal network’s primary parameters are interrelated. It is rather simple to express one
type of parameters in the terms of another type of parameters. Let’s try to express z-parameters
through t-parameters as example. At the beginning it is necessary to note that the goal of such
expression is to obtain V&1 and V&2 in terms of I&1 and I&2 . So it is necessary to do some transforms
of initial equations whose relates external variables in four-terminal network and network tparameters. At first let’s recall initial system of equations with t-parameters:
⎧⎪V&1 = t11V&2 + t12 I&2 ,
⎨&
⎪⎩ I 1 = t 21V&2 + t 22 I 2 .
Transforming second equation and substituting value of V&2 into the first equation we are
obtaining:
I& − t I&
V&2 = 1 22 2 ,
t 21
From these equations follows:
I& − t t I&
V&1 = t11 1 11 22 2 + t12 I&2 .
t 21
⎧
⎛
t
t t ⎞
⎪V&1 = 11 I&1 + ⎜⎜ t12 − 11 22 ⎟⎟ I&2 = z11 I&1 + z12 I&2
t 21
t 21 ⎠
⎪
⎝
⎨
⎪V& = 1 I& − t 22 I& = z I& + z I&
21 1
22 2
⎪ 2 t 1 A 2
21
21
⎩
Therefore,
z11 =
t11
t t
t
1
; z12 = t12 − 11 22 ; z 21 = ; z 22 = − 22 .
t 21
t 21
t 21
t 21
So, in the same manner it is possible to relate all systems of primary parameters.
Basic simple four-terminal networks and their primary parameters. Four-terminal network
primary parameters depend on real circuit parameters. In network theory are well known some
basic simple circuits, shown in the figure 3.4.
Parameters at these circuits could be calculated in easy way. As example we’ll show calculation
of the T parameters for first three circuits. For this we shall denote directions of the currents
and voltages as shown in the Fig. 3.4.
At first let’s try to consider first circuit (Fig.3.4a). In this circuit
⎧⎪ I&1 = I&2 = t21V&2 + t22 I&2 ,
⎨&
⎪⎩V1 = I 2 Z + V&2 = t11V&2 + t12 I&2 .
4
Figure 3.4. Basic simple circuits of the four-terminal network
Therefore t11 = 1; t12 = Z ; t21 = 0; t22 = 1.
For the second circuit
V&1 = V&2 ,
I& = V& Z + I& .
1
2
2
So, from these equations follows that t11 = 1; t12 = 0; t21 = 1 Z ; t22 = 1.
For the third circuit
⎧⎪V&1 = I&1Z1 + V&2 ,
⎨& &
⎪⎩ I1 = I 2 + V&2 Z 2 .
⎧
⎛
⎛
V&2 ⎞
Z1 ⎞
⎪V&1 = ⎜⎜ I&2 + ⎟⎟ Z1 + V&2 = V&2 ⎜⎜1 + ⎟⎟ + I 2 Z1 ,
Z2 ⎠
⎨
⎝
⎝ Z2 ⎠
⎪& &
&
⎩ I1 = I 2 + V2 Z 2 .
So t11 = 1 + Z1 Z 2 ; t12 = Z1; t21 = 1 Z 2 ; t22 = 1.
Parameters of the third circuit could be obtained multiplying transmission matrixes T of the
first and second circuits also:
1 Z1 1
T =
⋅
0 1 1 Z2
0
=
1
Z1
Z2
1
Z2
1+
Z1
.
1
Let’s try to derive t-coefficients for the fifth circuit. At first we can write two equations, which
relate voltages and current in the circuit in terms of circuit impedance:
5
⎧
⎛ V&2
⎞
⎪V&1 = V&2 + I&12 Z 2 = V&2 + ⎜⎜ + I&2 ⎟⎟ Z 2 ,
⎨
⎝ Z3
⎠
⎪
&
&
&
⎩ I1 = V1 Z1 + I12 = V1 Z1 + I 2 + V2 Z 3 .
Simplifying input voltage phasors expression
⎛ Z ⎞
V&1 = V&2 ⎜⎜1 + 2 ⎟⎟ + I&2 Z 2
⎝ Z3 ⎠
and substituting it into input current phasor expression
⎛ Z ⎞
⎛ 1
⎛ Z ⎞
V&
1
Z ⎞
I&1 = V&2 ⎜⎜1 + 2 ⎟⎟ + I&2 Z 2 + 2 + I&2 = V&2 ⎜⎜ + + 2 ⎟⎟ + I&2 ⎜⎜1 + 2 ⎟⎟
Z3
Z1 ⎠
⎝
⎝ Z3 ⎠
⎝ Z 3 Z1 Z1Z 3 ⎠
we are able to obtain final expressions of t-parameters:
Z
Z
1
1
t11 = 1 + Z 2 Z 3 ; t12 = Z 2 ; t21 =
+ + 2 ; t22 = 1 + 2 .
Z 3 Z1 Z1Z 3
Z1
In the same manner we can write equations for the sixth network
V&1 = I&1Z1 + I&2 Z 3 + V&2 ;
I&1 = I&2 + (V&2 + I&2 Z 2 ) / Z 3 .
Transforming these equations yields
I&1 = V&2 / Z 3 + I&2 (1 + Z 2 / Z 3 );
V&1 = (V&2 / Z 3 + I&2 (1 + Z 2 / Z 3 )) Z 1 + I&2 Z 3 + V&2 = V&2 (1 + Z1 / Z 3 ) + I&2 ( Z 1 + Z 3 + Z1 Z 2 / Z 3 ).
So, t-coefficients could be found as follows:
t11 = 1 / Z 3 ;
t12 = Z 1 + Z 3 + Z 1 Z 2 / Z 3 ;
.t 21 = 1 / Z 3 ;
1 + Z 2 / Z3.
Experimental definition of the primary parameters. Primary parameters of the fourterminal networks could be found experimentally, then one of the excitations (input variables) is
equal to zero. Such conditions can be obtained in open circuit (then current is equal to zero) or
shorted circuit (then voltage is equal to zero). Let’s try to find in such way primary parameters of
the four-terminal network.
If network output is open, then I&2 = 0 and network equations could be written in more
simple form:
V&1 = t11V&2 ,
From these equations follows:
V&1 = z11 I&1 ,
I&1 = t 21V&2 ,
V&2 = z 21 I&1 .
6
V&1
V&
t11 =
;
z11 =
2 I&2 =0
V&1
I&
t 21 =
;
2 I&2 =0
I&2
V&
z 21 =
;
2 I&2 = 0
V&2
I&
.
1 I&2 =0
When network output is shorted ( V&2 = 0 ), then
V&1 = t12 I&2 ,
I&1 = y11V&1 ,
V&1 = h11 I&1 ,
I&1 = t 21 I&2 ,
I&2 = y 21V&1 ,
I&2 = h21 I&1 .
So
t12 =
V&1
I2
;
y11 =
V&2 = 0
I&1
V&1
h 11 =
;
V&2 = 0
V&1
I&1
.t 21 =
;
V&2 = 0
I&1
I&2
;
y 21 =
V&2 = 0
I&2
V&1
;
V&2 = 0
h 21 =
I&2
I&1
V&2 = 0
When network input is open ( I&1 = 0 ), then
V&1 = z12 I&2 ;
V&1 = h12V&2 ;
V&2 = z 22 I&2 ;
I&2 = h22V&2 .
Therefore
z12 =
V&1
I&2
,
h12 =
I&1 = 0
V&1
V&2
z 22 =
,
I&1 = 0
V&2
I&2
,
I&1 = 0
h22 =
I&2
V&2
I&1 = 0
.
.
And when network input is shorted ( V&1 = 0 ), then
I&1 = y12V&2 ;
I&2 = y22V&2 .
So
y12 =
I&1
;
V&2 V& = 0
1
y22 =
I&2
.
V&2 V& = 0
1
All these formulas of the primary parameters could be used as mathematical definitions. These
formulas also show physical meaning of the primary parameters.
For example, four-terminal network primary parameter
t11 =
V&1
V&2
=
I&2 = 0
1
V&2
V&1
=
1
H& V
.
I&2 = 0
I&2 = 0
is reverse function of the voltage transfer function in the open four-terminal network. Parameter
t21 relates input current and output voltage in the open network. It can be called feedbackcoupling admittance. t12 – shorted network crossing (direct coupling) impedance, t22 – reverse
function of the current transfer in the shorted network. Also we can see that all z-parameters has
impedance dimension and y-parameters – admittance dimension. For example z11 – input
impedance of the open network, z21 – direct coupling impedance, z12 – network with open input
7
feedback coupling resistance, z22 – output impedance of the network with the open input. For the
networks described using h-parameters - h11 – input impedance of the open network, h21 – current
transfer coefficient for the open circuit, h22 – output admittance of the circuit with the open input
and h12 – voltage feedback coefficient in the network with open input.
Equivalent circuits of the four-terminal networks. Usually fourterminal network represents complicated circuit consisting on a lot of
various components. Using various transforms such circuit can be
simplified and expressed as canonic circuit with minimal amount of
passive components. Minimal amount of independent components in
linear circuit is equal to three. So, each four-terminal network could
be represented as T or Π circuit (Fig. 3.5), consisting on three
impedances. These impedances can be expressed in terms of fourterminal network primary parameters. Let’s try to express
impedances of the T circuit through primary t- parameters as
example. At first let’s recall equations which relates t-parameters and
circuit impedances:
Fig. 3.5. T-type
equivalent circuits
terminal networks
Z1 =
t11 − 1
;
t21
ZZ
Z1
; t12 = Z 1 + Z 3 + 1 3 ;
and Π-type
Z2
Z2
of the fourFrom these equations follows, that
t −1
1
Z 2 = ; Z 3 = 22 .
t21
t21
t11 = 1 +
t 21 =
1
;
Z2
t 22 = 1 +
Z3
Z2
In the same manner we can write expressions of t-parameters for the Π-type circuit:
t11 = 1 +
Z2
;
Z3
t12 = Z 2 ;
t 21 =
Z
1
1
+
+ 2 ;
Z1 Z 3 Z1 Z 3
t 22 = 1 +
Z2
.
Z1
Therefore
Z1 =
t12
;
t22 − 1
Z 2 = t12 ;
Z3 =
t12
.
t11 − 1
So, having primary parameters we always can to construct
equivalent T or Π circuit.
Now let’s consider another type of the fourterminal network equivalent circuits. As you learned in
electronics fundamentals course, model of a common
emitter configuration of bipolar transistor consist on two
depended sources, impedance and admittance (Fig.3.7a).
These elements could be described using h-parameters.
Such type of model rests with the interpretation of
appropriate mathematical equations:
⎧⎪V&1 = h11I&1 + h12V&2 ,
⎨&
⎪⎩ I 2 = h21I&1 + h22V&2 .
8
Model’s input voltage V1 equals the sum of two
voltages; h11I1 plus the voltage due to a voltagecontrolled voltage source given by h12V2 . This is
precisely the left-hand portion of figure 3.7a. A
similar interpretation follows for the right-hand side
of figure 3.7a, here current I2 equals to current due to
current controlled current source h21I1 plus current
h22V2 . Such type of four-terminal representation is
called as two-depended source equivalent circuit. The
similar two-depended source equivalent circuits can
be build for the four-terminal networks equated by z,
y or t parameters (Fig.3.7).
It is possible to build one-depended source
equivalent circuits of the four-terminal circuits also.
Transforming one depended source of the earlier
discussed circuits in to the direct coupling impedance
we can obtain two types of one-depended source equivalent circuits shown on the Fig.3.8.
Let’s try to relate parameters of these circuits with z-parameters and y-parameters of the
four-terminal networks. Let’s start from the circuit shown in the Fig. 3.8a. The mesh equations
for this circuit yields
⎧⎪V&1 = Z1I&1 + Z 2 ( I&1 + I&2 ),
⎨&
⎪⎩V2 = Z 3 I&2 + Z 2 ( I&1 + I&2 ) + E& .
The next phase is to transform initial z-parameters equations:
V&1 = z11 I&1 + z12 I&2 = z11 I&1 + z12 I&2 + z12 I&1 − z12 I&1 = ( z11 − z12 ) I&1 + z12 ( I&1 + I&2 ),
V&2 = z 21 I&1 + z 22 I&2 = z 21 I&1 + z 22 I&2 + z12 ( I&1 + I&2 ) − z12 ( I&1 + I&2 ) =
= ( z − z ) I& + z ( I& + I& ) + ( z − z ) I& .
22
12
2
12
1
2
21
12
1
Here, comparing these equations with the previous formula, we see that
Z 1 = z11 − z12 ;
Z 2 = z12 ;
Z 3 = z 22 − z12 ;
E& = ( z 21 − z12 ) I&1 .
In the same manner it is possible to derive that circuit’s on Fig.3.8b admittance and voltage
controlled current source may be expressed through y-parameters. Nodal equations
I&1 = Y1V&1 + Y2 (V&1 − V&2 );
I& = Y V& + Y (V& − V& ) + J&
2
3 1
2
1
2
I&1 = y11V&1 + y12V&2 = y11V&1 + y12V&2 + y12V&1 − y12V&1 = ( y11 + y12 )V&1 − y12 (V&1 − V&2 );
As in previous equivalent circuit let’s transform initial y-parameters equations:
9
⎧ I&1 = y11V&1 + y12V&2 = y11V&1 + y12V&2 + y12V&1 − y12V&2 = ( y11 + y12 )V&1 − y12 (V&1 − V&2 ),
⎪
+ ⎨ I&2 = y 21V&1 + y 22V&2 = y 21V&1 + y 22V&2 + y12 (V&1 − V&2 ) − y12 (V&1 − V&2 ) =
⎪
&
& &
&
⎩= ( y 22 + y12 )V2 + y12 (V1 − V2 ) + V2 ( y 21 − y 22 )
Y1 = y11 + y12 ;
Y2 = − y12 ;
J& = ( y 21 − y12 )V&2 .
Y3 = y 22 + y12 ;
Secondary parameters of the four-terminal networks. Such parameters are useful and
important for determining power transfer and various gain computations. In four-terminal
networks theory are known following secondary parameters:
Input impedance
Z in =
V&1 1
= .
I&1 Yin
V&2 o
.
I&2 s
Here V20 – output voltage phasor in the open circuit and I2s - output current phasor in the shorted
circuit.
V&
Voltage gain or complex voltage transfer function H& V = 2 .
V&
Output impedance
Z out =
1
I&
Current gain or complex current transfer function H& I = 2 .
I&1
Transfer (direct coupling) impedance
Z=
V&2
= H& I Z L .
I&
1
I&2 H& V
.
=
V&1 Z L
Characteristic and wave impedances. Definition of these impedances will be given below.
Transfer (direct coupling) admittance
Y=
Propagation constant is parameter used in circuit theory as alternative for four-terminal network
voltage or current gain. This constant can be denoted as g. Four-terminal network output voltage
or current using this parameter could be calculated as follows:
V&2 = V&1e − g ,
I& = I& e − g .
2
1
It is necessary to mention that equality between voltage and current expressions can be reached
only in particular cases.
Let’s consider all these parameters in more detailed form. Expression of the four-terminal
network secondary parameters depends on type of its primary parameters and network equivalent
circuit. Let’s try to show methods of determination of secondary parameters using t-parameters
and two-depended source equivalent circuits.
Input impedance can be easy calculated in terms of t-parameters:
10
Z in =
V&1 t11V&2 + t12 I&2
.
=
I&1 t21V&2 + t22 I&2
Using the terminal condition by the load impedance Z L we obtain I&2 = V&2 / Z L . Incorporating this
terminal condition into previous equation yields
Z in =
t11Z L + t12
.
t21Z L + t22
If network output is shorted ( Z L = 0 ), then input impedance Z ins =
If network output is open ( Z L = ∞ ), then input impedance Z ino =
t12
.
t22
t11
.
t21
Output impedance. At first we must find V&2 o and I&2 o . In the open network ( Z L = ∞, I&2 = 0 )
V&1 = t11V&2 o ;
I&1 = t 21V&2 o
.
Input phasor voltage could be found from following equation:
V&1 = V&S − I&1 Z S .
Here V&S – source voltage phasor, Z S – source internal resistance.
Comparison of these equations produces
t11V&2 o = V&S − Z S t 21V&2 o and V&2o =
V&S
.
t11 + Z S t 21
In the shorted network Z L = 0, V&2 = 0 and
V&1 = t12 I&2 s = V&S − I&1 Z S ;
I&1 = t 22 I&2 s .
From these equations follows that
t12 I&2 s = V&S − Z S t 22 I&2 s and I&2 s =
V&S
.
t12 + Z S t22
Incorporating of V&20 and I&2 S in to primary expression of output impedance yields
Z out =
t22 Z S + t12
.
t21Z S + t11
If voltage source is ideal ( Z S = 0 ), such network output impedance Z out s =
t12
.
t11
In the same manner it is possible to prove that four-terminal network output impedance
expressed in terms of t-parameters
11
t12
.
t11
if ideal current source ( Z i = ∞ ) is acting in the input of such network.
Z out o =
Voltage gain or transfer coefficient.
Equation, which relates input voltage phasor and
output voltage phasor in terms of t-coefficients
V&1 = t11V&2 + t12 I&2
we can transform incorporating of terminal condition
I&2 = V&2 / Z L :
V&1 = (t11 + t12 Z L )V&2 .
So, voltage transfer function
1
V&
.
K& V = 2 =
V&1 t11 + t12 Z L
Current gain or current transfer coefficient can be derived in terms of network t-parameters in
the same manner as voltage gain:
K& I =
1
.
t21Z L + t22
Now let’s consider voltage gain as ratio V&2 V&S (Fig. 3.9). From t-parameters equation with
terminal conditions
⎧V&1 = t11V&2 + t12 I&2 = V&2 (t11 + t12 Z L );
⎪&
⎨ I1 = t21V&2 + t22 I&2 = V&2 (t21 + t22 Z L );
⎪& & &
⎩V1 = VS − I1Z S .
follows:
V&2 (t11 + t12 Z L ) = V&S − Z S (t 21 + t 22 Z L );
V&2 (t11 + t12 Z L + Z i t 21 + t 22 Z S Z L ) = V&S
and
V&
ZL
ZL
K& V = 2 =
=
&
VS t12 + t11Z L + t22 Z S + t21Z S Z L t12 + t11Z L + Z S (t22 + t21Z L )
Characteristic and wave impedance. It is known in the four-terminal networks theory that there
is two impedance Z 01 and Z 02 with following properties:
1. Network input impedance is equal to Z 01 then network is loaded with impedance Z 02 .
2. Network output impedance is equal to Z 02 then source internal resistance is equal to Z 01 .
12
These impedances are called characteristic impedances. With respect to definition of input and
output impedances we can write:
Z 01 =
t11Z 02 + t12
;
t21Z 02 + t22
Z 02 =
t22 Z 01 + t12
.
t21Z 01 + t11
t11 t12
;
t21 t22
Z 02 =
t12 t22
.
t11 t11
Solving these equations yields
Z 01 =
These equations can be rewritten in following manner:
Z 01 = Zin s Zin o ;
Z 02 = Z out s Z out o .
Therefore, these equations can be used to define characteristic impedances in experimental way.
In the symmetric networks parameters t11 = t22 and both characteristic impedances are equal also:
Z 01 = Z 02 =
t12
= Z0.
t21
Characteristic impedance Z 0 is called wave impedance in such type of the four-terminal
networks.
Network loaded by impedance equal to its wave impedance is called matched network. Input
impedance in the matched network is equal to its wave impedance. So matched networks are
acting as impedance transformers.
Let’s consider voltage and current transfer functions in the four-terminal network. Suppose that
network is matched. Inspecting terminal conditions and characteristic impedance definition
yields
ZL =
V&2
t t
= Z 02 = 12 22 .
I&2
t11t21
Recalling network equations with t-parameters
V&1 = t11V&2 + t12 I&2 ,
I& = t V& + t I&
1
21 2
22 2
and substituting previous equation we obtain:
V&
I&
t t t
t
H& V = 1 = t11 + 2 t12 = t11 + 11 21 12 = 11
&
&
V2
V2
t 22
t 22
(
t11t 22 + t12t 21 .
)
t t t
t
I&
V&
H& I = 1 = t 22 + 2 t 21 = t 22 + 12 21 22 = 22
I&2
I&2
t11
t11
(
t11t 22 + t12 t 21 .
)
13
In the symmetric networks t11 = t22 and
V&1 I&1
= = t11 + t12t21 .
V&2 I&2
This ratio could be marked as value e g , where g – network propagation constant.
From this definition follows that
eg =
V&1 I&1
=
V&2 I&2
and
g = ln
V&1
I&
= ln 1
V&2
I&2
Propagation constant is complex value and consists on slope (attenuation) coefficient α and
phase coefficient β:
g = α + jβ .
In some textbooks propagation constant real part α is called damping constant and imaginary
part β – phase constant.
V&
I&
If four-terminal network is non-symmetric when 1 ≠ 1 . Propagation constant in such network
V&2 I&2
could be found in following way:
(
1 V& 1 I& 1 V& I&
g = ln 1 + ln 1 = ln 1 1 = ln t11t22 + t12t21
2 V&2 2 I&2 2 V&2 I&2
)
Production V&I& do not have physical mean in this equation. Earlier we had learned that its
modulus is equal to total power:
V&I& = S .
Substituting this expression into previous equation yields
1
V& I&
1
VI
1
S
1
S
1
α + jβ = ln &1 &1 = ln 1 1 e jϕ = ln 1 e jϕ = ln 1 + j ϕ .
2 V2 I 2 2 V2 I 2
2 S2
2 S2
2
Here - difference in phases between voltage and current phasors. Damping constant
α=
1 S1
ln
2 S2
could be measured in nepper units (Np). It is known that in the symmetric networks
14
1
2
α = ln
V1I1
V
I
= ln 1 = ln 1 .
V2 I 2
V2
I2
In practice more convenient is to use logarithms with decimal base. Damping constant can be
measured in decibels as magnitude response:
α = 10 ⋅ lg
S1
S2
or
α = 20 ⋅ lg
V1
I
= 20 ⋅ ln 1 .
V2
I2
Primary parameters of the network may be expressed through characteristic impedance and
propagation constant. Recalling previous equations
e g = t11t22 + t12t21 ;
and doing simple transforms
e− g =
t t − t12t21
1
= 11 22
= t11t22 − t12t21 ;
t11t22 − t12t21
t11t22 + t12t21
we are obtaining formulas necessary to express primary four-terminal networks parameters using
their secondary parameters:
e g + e− g
= chg = t11t22 ;
2
e g − e− g
= shg = t12t21 ;
2
Z 01Z 02 =
t12
;
t21
Z 01 t11
= .
Z 02 t22
From the analysis of these equations we can obtain following expressions of the t-parameters:
t11 =
Z 01
chg; t12 = Z 01 Z 02 shg;
Z 02
t 21 =
1
Z 01 Z 02
shg;
t 22 =
Z 02
chg.
Z 01
Therefore, four-terminal network equations can be rewritten in following form:
V&1 =
Z 01 &
(V2chg + Z 02 I&2shg ),
Z 02
I&1 =
⎞
Z 02 ⎛ V&2
⎜⎜
shg + I&2chg ⎟⎟.
Z 01 ⎝ Z 02
⎠
If characteristic impedances Z01 and Z02 are the same (Z01= Z02 = Z0), then
15
V&1 = V&2chg + Z 0 I&2shg,
V&
I&1 = 2 shg + I&2chg.
Z0
Later, in the section ___, we’ll show the same nature of equations for the networks with
distributed parameters.