RF and Microwave Networks

Professor Bratin Ghosh

The incident and reflected voltages can be expressed in terms of the port currents using Z matrix
as below.

 Z  I  =  Z  V i  −  Z  V r 
= V  = V i  + V r 

V i  ( Z  −  I ) = V r  ( Z  +  I )
(Refer Slide Time: 01:29)

Or we can easily obtain from there,

V r 
  (     ) ( Z  −  I )
−1
= S = Z + I (56)
V i 

V1r = S11V1i + S12V2i (57a)

V2 r = S21V1i + S22V2i (57b)

From Equation 57a we get

V1r S11 i
V2i = − V1 (58)
S12 S12

By substituting V2i in 57b, we get

Vr S 
V2 r = S21V1i + S22  1 − 11 V1i 
 S12 S12 
(Refer Slide Time: 04:24)

Rearranging the above Equation, we get

S22  S S 
V2 r = V1r +  S21 − 22 11 V1i (59)
S12  S12 

Comparing the Equations 52, 58 and 59 we can express the T metrics in terms of the scattering
metrics as

 S 22 S11 S 22 
 S 21 − S S12 
T  =  12
 (60)
 S11 1 
 −S S12 
 12
For any particular network it is important to realize that an infinite number of equivalent circuits
will exist. For loss free network we can develop the symbolism which we are going to use in the
equivalent circuits and that is all too common.

(Refer Slide Time: 06:17)

However, for the sake of completeness we can mention that for an inductor it can be represented
by jX and that signifies positive reactance and jB which signifies negative reactance. Similarly,
for a capacitor we can develop jX which is negative reactance and jB which is positive reactance
and we can have an ideal transformer which represents a change in impedance level like a change
from one guided structure to another guided structure or guided structure with one kind of
characteristic impedance to another kind of characteristic impedance. Then our transmission line
with characteristic impedance Z0 and length l represents a waveguide section or guided booth
section.
(Refer Slide Time: 09:13)

Then we can have an equivalent circuit or typical equivalent circuit for a loss free two port
microwave network as shown in the above figure. In the case of dissipative networks resistors in
series with X or in parallel with B can be used to signify the losses. Similarly, the characteristic
impedance and propagation constants of equivalent transmission line can be assumed to be
complex to account for losses.

(Refer Slide Time: 11:38)

Now, we consider a transition between a coaxial line and a microstrip line. Here, the characteristic
impedance of the microstrip line Z0m and Z0c is the characteristic impedance of the coaxial line.
Here, the outer conductor of the coax is connected to the ground plane of the microstrip the center
conductor of the coax is connected to the signal line of the microstrip. For this coaxial line to
microstrip line transition we can also draw the equivalent circuit or find the equivalent
representation of this as well.

(Refer Slide Time: 14:27)

Now because of the physical discontinuity in the transition from the coaxial to the microstrip line,
the electric or magnetic energy can be stored in the vicinity of the junction wherever there is a
junction between two transmission line the electrical magnetic energy can be stored in the
transition and that will lead to reactive effects. Characterization of such effects can be either done
by measurements or by theoretical analysis. This is called the transition problem which represent
the transition between two types of transmission lines.
Now, we come to the topic of signal flow graph (SFG). SFG it is an additional technique that is
quite useful for the analysis of microwave networks in terms of transmitted and reflected waves.
The basic elements of SFG are nodes and branches. The nodes represent each port of a microwave
network. The ith port of microwave network has two nodes ai and bi. The nodes ai is defined or
identified with a wave entering port i while node bi is identified with a wave reflected from port i.
Now, the branch of an SFG is a directed path between two nodes. Branch represents the signal
flow from one node to the other. Every branch has an associated S-parameter or reflection
coefficient.

(Refer Slide Time: 15:38)

Now, we consider an ordinary two port network (as shown in the figure 8). Here, ai=a1 is the wave
entering port 1 and bi=b1 is the wave which is reflected from port 1. The wave of amplitude a1
which is incident at port 1 is the split with part going through S11 and output from port 1 as a
reflected wave. The 2-port network can be replaced or represent this by signal flow as shown in
the figure. From the SFG it can be seen that the wave entering a1 it gets multiplied by S11 to form
the reflected wave b1 and the remaining part is transmitted through S21 to node b2 from where the
wave goes out through port 2. If a load with non-zero reflection coefficient is connected at port 2,
this wave will be at least partly reflected or partially reflected and reentered the 2-port network at
node a2. So, this part of the wave can be reflected back. So, the part of the wave can be reflected
back out via port 2 via S22 and part can be transmitted out port-1 through S12.
(Refer Slide Time: 22:51)

Figure 9 shows the figures for a 1-port network and a voltage source. The SFG of the1 port network
has a branch with multiplier Γ1 and has nodes a and b. here, for this SFG the incident wave
multiplied by gamma 1 to form the reflected wave. Figure 9b shows the network for a voltage
source Vs with a load Zs or with a source impedance. The corresponding SFG contains a branch
with multiplier Γs and common nodes a and b. Here, due to the source Vs, the amplitude of the
wave coming from the right side gets multiplied by the source reflection coefficient Γs to form the
reflected amplitude b. The SFG facilitates a very simple representation of a network which allows
to solve for the ratio of any combination of wave amplitudes in a simpler manner.
(Refer Slide Time: 26:08)

Now, we discuss the decomposition of SFGs. We have four rules relating to this decomposition.

Rule-1 is called as the Series rule. It states that two branches are said to be in series if their common
node has only one incoming and one outgoing way. As seen in the above figure the branches with
multiplier S21 and S32 are in series as they have only common node V2 and the node V2 has only
one way in and one way out. As explained earlier two branches whose common node has only one
incoming and one outgoing wave which are branches in series. For these two branches in series, it
may be combined to form a single branch whose coefficient is the product of the coefficients of
the original branches.

i.e. V3 = S32 V2 = S32S21 V1.

Rule 2: The parallel rule: it states, that for any two nodes V1 and V2. Having multiple branches in
between it can be replaced with a single branch with the effective multiplier given by the sum of
multipliers of the individual branches.

(Refer Slide Time: 29:54)

Rule 3: the self-loop rule: A self-loop is a branch which has only a single node. From the above
figure it can be observed that the branch with multiplier S22 is a self-loop as it has only one node
V2. For the node V2 we can write
 V2 = S21 V1 + S22 V2

and V3 = S32

if we combine the above two relationships we can get

𝑆32 𝑆21
𝑉3 = 𝑉
(1 − 𝑆22 ) 1

So, the SFG with self-loop can be simplified as the SFG with series branches with modified branch
multipliers or coefficients as shown in the figure.

(Refer Slide Time: 32:50)

Rule 4: the splitting rule: it says that “When a node has a single incoming branch and one or more
exiting branches, it is possible to divide the incoming branch and merge it with each of the exiting
branches”. Consider the SFG as shown in the above figure which has four nodes V1, V2, V3 and
V4. Here the nodes V3, V4 are the nodes for the branches which are split from the node V2.

Here V4=S42V2=S21S42V1

considering this we can re draw the SFG as shown in the figure. the new SFG has two paths one
with nodes V1 V2 V3 with branch multipliers S21 and S32 respectively, and the other with V1 V2’
V4 with branch multipliers S21 and S42 respectively. Here a node may be separated into two
separate nodes as long as the resulting flow graph contents once and only once each combination
of separate input and output branches that connect to the original node not self-loops.

(Refer Slide Time: 36:10)

The above figure gives an illustration of a SFG. The Γs is source reflection coefficient, [S] is the S
matrix for the 2-port with characteristic impedance Z0. Γo is the output reflection coefficient. He
corresponding SFG is also shown in the figure.

So, we have VS it goes to 1 then the wave which is going towards the right side entering the
network with the scattering parameter S that is a1. So, the incident wave it gets reflected S11 to
form the deflected amplitude b1 and b1 gets reflected this wave which is traveling towards the left
gets reflected by the source reflection coefficient gamma S to form a1 the reflected field b1
multiplied by Γs should give the incident field a1 and then a1 goes through S21 to form the
outgoing wave at this port which is b2. So, this is port 1 port 2 and this b2 which is outgoing from
port 2 it gets reflected by Γl to form the wave traveling towards the left side from the right to the
left and then the wave entering port 2 get multiplied by S22 to form the wave going towards the
right side, which is b2. Because this gets reflected the wave going towards the left gets reflected
at this port to form S22, which is the wave going out and then the other part of a2 gets transmitted
as the wave b1 through S12.
Because the wave entering going from the right to the left which is incident on port 2 that travels
all the way down the network to become the outgoing wave the outgoing wave at port 1. So, this
is the complete representation of this network through the signal flow graph.