RF And Microwave Networks

Professor Bratin Ghosh

Department of Electronics and Electrical Communication Engineering
Indian Institute of Technology, Kharagpur
Lecture 03
One-port network, Two port network
(Refer Slide Time: 00:46)

When equation 39 is applied over the one port network of figure 2, the field vanishes over the
enclosing surface, except where it crosses the input port, as discussed before. The left-hand side
of Equation 39 becomes

 V I *   V I 
   I
*
+
 
( )
V  e h d s = − I*
 
+V * 
 

where V and I are voltages and current at the input reference plane. The Equation 39 can be written
as,

 * V I 
I
 
+V *
 
2 2
(
 = j   H +  E d )
(40)
= j 2( wm + we )
(Refer Slide Time: 02:41)

The input reactance X and the susceptance B are given by

V −j
jX = = (41)
I B

dX − j V
= (42)
d I  I =const .

dB − j I
= (43)
d V  V =const .

From Equations 40 and 42, we get

dX − j  2 j 
=  * ( wm + we ) 
d I I 
(Refer Slide Time: 04:51)

Which can be re written as

dX 2 j
= ( wm + we ) (44)
d I 2

From Equations 40 and 43, we get

dB − j  2 j 
=  * ( wm + we ) 
d V  V 

Which can be re written as

dB 2 j
= ( wm + we ) (45)
d V 2

The equations 44 and 45 give an important inference that is, for a lossless one port network the
slope of the reactance and susceptance are always positive. Which is known as Fosters reactance
theorem.
(Refer Slide Time: 07:16)

From equations 44 and 45, for loss free networks, we have

2
X= 2
( wm − we ) (46a)
I

2
B= 2
( we − wm ) (46b)
V

Solving Equations 44, 45 and 46 for the energies. We get

 dX X  2 
 +  = 2 ( wm + we ) + ( wm − we ) 
 d   I 

2 2
I  dX X  V  dB B 
wm =  + =  −  (47a)
4  d   4  d  
(Refer Slide Time: 10:23)

And similarly, we can obtain

2 2
I  dX X  V  dB B 
we =  − =  +  (47b)
4  d   4  d  

Now, as the energies are positive, we get

dX X dB B
   (48)
d  d 

The Equation 48 shows that the slope of the reactance or susceptance is always greater than the
slope of a straight line from the origin to that point of consideration. Now, from Equations 44, 45
and 46 it can be observed that, for a lossless one port network, all the poles and zeros of the
reactance and susceptance function are simple. Which is another very important criterion. In order
to prove this, let X vanishes at a resonant frequency ω0 i.e X(ω= ω0)=0. Now expanding X about
ω= ω0, using Taylor series, we get

X ( ) = a1 ( − 0 ) + a2 ( − 0 ) 2 +        
X '( ) = a1

Here the X’(ω) must be positive by Foster's reactance theorem. Which necessitates the a1 to be
positive by Foster's reactance theorem and hence X will have a simple zero at ω= ω0. Obviously,
because if other terms are considered like this and other higher order terms, then there is no
guarantee that x prime omega will be positive at a given frequency omega naught, because the
derivative of this term is going to involve 2(ω= ω0) and therefore, X will be dependent on ω, and
it will not be a constant. Therefore, in order for Fosters reactance theorem to be valid X’(ω) = a1,
this slope which is positive by Fosters reactance theorem means that X has a simple zero at omega
naught and similarly.

(Refer Slide Time: 14:57)

And similarly, B =1/ X, has a simple pole at ω0. Similarly, zeros of B are simple, zeros of B are
simple, and hence poles of X are also simple.
(Refer Slide Time: 16:13)

Fig 3 shows the Equivalent circuit for the reactance functions of the Fosters type. The effect of
small losses can be shown in the equivalent circuits by adding large resistances in parallel with the
LC resonators of figure 3a and by adding small resistance in series with the LC resonators of figure
3b.

(Refer Slide Time: 17:48)

Now, let us come to the characterization of two port networks. The primary uses of 2-port
microwave networks are a) the transmission of energy from one place to another and, b) the
filtering of signals from one another. A 2- port network can be expressed in terms of impedance
matrix Z as,

V1   Z11 Z12   I1 

V  =  Z   (49)
 2   21 Z 22   I 2 

Y  =  
−1

(50)

At port-1 V1 = V1i + V1r

(51a)

I1 = I1i + I1r
(51b)
=
1
Z o1
(
V1i − V1r )

(Refer Slide Time: 20:33)

Similar equations can be applied to port 2, you can rewrite similar equations as,

V2 r  T11 T12  V1i 

 i=  r (52)
V2  T21 T22  V1 

Fig 4 depicts a 2-port network. Here we have port-1 voltage V1=V1i+V1r, where V1i voltage of
incident wave and V1r is the voltage of reflected wave. Z01 is a characteristic impedance. Also, we
have I1=I1i+I1r, where I1i current of incident wave and I1r is the current of reflected wave. From a
travelling wave view point a possible matrix for describing two port microwave networks is the
transmission matrix T, which is given by

T  = TN TN −1         T2 T1  (53)

This transmission matrix is convenient when microwave networks are cascaded.

(Refer Slide Time: 23:14)

Here T1, T2 ...TN are the transmission matrices of individual 2-port networks which are cascaded.
And T is the complete transmission matrix of the cascaded2-port network. As can be seen the T
matrix of the overall network is the product of the T matrices of the individual networks. Another
matrix commonly used to describe the microwave networks is the scattering matrix S which is
defined in terms of the incident and reflected port voltages of the 2-port network. The S-matrix of
the 2-port network is expressed as below.

V1r   S11 S12  V1i 

 r =   
S22  V2i 
(54)
V2   S21

where V1r, V2r are the reflected voltages at port-1 and port-2, and V1i, V2i are the incident voltages
at port-1 and port-2, respectively.
(Refer Slide Time: 25:42)

Here the S11 is the reflection coefficient seen at port-1 when port-2 is matched. and S22 is the
reflection coefficient at port-2 when port-1 is matched. The S11 is given by:

V1r
S11 = (55)
V1i V2+ = 0