S Parameters

1. Definition of S (Scattering) Parameters and Matrix

The parameters associated to electronic circuits and devices that operate as linear two-
ports are e.g. z, y, h (hybrid) parameters etc. Measuring these parameters require open and
short-circuits at the ports, which is not convenient in high frequency for various reasons
involving permanent damage to the measured device. Therefore, S (scattering) parameters
are used in the latter case since their measurements require matching at ports, which does
not damage the device in general. S parameters are applied to linear two- and multi-ports,
such as linear (pre)amplifiers, attenuators, filters, couplers, power dividers, balancing
("baluns") and matching networks etc.
Consider a linear two-port fed by a generator at one port, through a transmission line and
terminated by a load, connected through another transmission line, Fig. 1. The transmission
lines can be of zero length (absent).

zg i1 i2
1 2
+ + a2
+ b1
eg v1 S v2 zL
a1 b2
_ _ _
1' 2'
l1 l2

Fig. 1. Linear two-port represented for the definition of S parameters

At the ports of the two-port, the complex amplitudes of normalized voltages are v1 and
v2, and the complex amplitudes of normalized currents are i1 and i2, with reference flow
directed towards the two-port, according to the rule for power receptors, at both ports. The
normalized incident waves are a1, a2, and the normalized emerging waves from the two-port
are b1, b2. Note that all electrical quantities in the circuit are normalized. The normalization
impedances can be coincident at the two ports or different.
The sinusoidal waves represented by the complex amplitudes a1, a2, b1 şi b2 are dependent
on the signal generated by the source eg.
As known, two of the signals from the ports are dependent on the other two. In the case
of the a and b waves, this linear dependency is written by convention in the form

b1 = S11a1 + S12 a2

 (1)
b2 = S 21a1 + S 22 a2

The coefficients Sij are called S parameters (from "Scattering"), and they are complex
numbers that are fixed at a given frequency but depend on frequency. The magnitudes of the
S parameters indicate how wave amplitudes change and the arguments indicate how the
phases of the waves change by passing through the two-port.
It can be noticed that, at port 2 (Fig. 1), the wave a2 results from b2 by reflection on the
load impedance zL. Therefore, if the load is matched zL=1, the wave is zero a2=0. This makes
the following relation relevant for the calculation and measurement of the S parameters:

1.1
 b1 b2
S11 = , S 21 = (2)
a1 a2 = 0 ( z L =1)
a1 a2 = 0 ( z L =1)

The S11 parameter has the signification of reflection coefficient at port 1 with port 2
matched, and S21 has the signification of transfer coefficient from port 1 to port 2, with port 2
matched.
Similarly, we can conceive the two-port fed at port 2 (by the generator eg, zg) and
terminated at port 1 by the load impedance zL. In this new configuration of the circuit we have

b2 b1
S 22 = , S12 = (3)
a2 a1 = 0 ( z L =1)
a2 a1 = 0 ( z L =1)

Therefore, the parameter S22 can be considered as the reflection coefficient at port 2 with
port 1 matched, and S12 can be considered as transfer coefficient from port 2 to port 1, with
port 1 matched.
The S parameters are measured by relying on (2) and (3). The instrument that measures S
parameters is called Network Analyzer.

The definition of the S parameters can be readily generalized from linear two-ports to k-
ports, k>2, Fig. 2. The device is connected to transmission lines, every line being connected to
a generator with its own internal impedance. All generators share the same frequency (all
generators are controlled by a pilot). Some of the generators can have a zero e.m.f., such that
the corresponding lines end on passive terminations.

zg1 i1
1
+
a1
eg1 Zc1 v1
b1 1'
_
z1 0
i2
zg2 2
+
a2 linear k-port
eg2 Zc2 v2
b2 2'
_
z2 0
......

zgk ik
k
+
egk ak
Zck vk
bk k'
_
zk 0

Fig. 2. Linear k-port

All electrical quantities in Fig. 2 are normalized either at each port with respect to the
characteristic impedances of the incident line or all to a unique normalization impedance.

1.2
 The "a" normalized waves are incident on, while the "b" normalized waves are emerging
from ports.
Due to linearity, we have

bm = S m1 a1 + S m 2 a 2 + ... + S mk a k , m = 1...k (4)

The S parameters defined above depend only on the multi-port and are fixed at a given
frequency.
The equations in (4) can be combined in matrix form as

b = Sa (5')

 a1   b1 
a  b   S11 S12 S1k 
  2  2 S S 22 S 2 k 
a =  a3  , b =  b3  , S =  21 . (5'')
     
     
 ak  bk   Sk1 Sk 2 S kk 

The S parameters can be conveniently calculated and measured by relying on the following
equations that result from (4):

bi
S im = , i, m = 1...k (6)
am
if
al = 0, l = 1...k , l  m . (7)

That requires the k-port fed at port m and all other ports terminated in matched loads.
The solutions of the transmission lines equations imply that normalized voltages and
currents can be calculated for every port through

v m = a m + bm
(8)
im = a m − bm , m = 1..k

The voltages and currents result from the corresponding normalized quantities by using
the following equations:

Vm
vm =
Z c ,m (9)
im = I m Z c ,m , m = 1..k

(the normalized quantities are denoted by lower case letters, and the un-normalized ones are
denoted by upper-case letters; Zc,m is the characteristic impedance of the line connected at
port m).
In the following, we will consider only short, high-quality lines so that the characteristic
impedance is a real quantity.

1.3
 Change of Terminal Planes

Ports 11', 22' etc. (Figs. 1 and 2) have clearly defined locations and are called terminal
planes. From various reasons that are related to existence of higher order evanescent modes
at the true locations of the terminals of the devices, measurements of S parameters are
performed at a certain distance from these ports, where the electromagnetic field
corresponding to the higher-order modes becomes negligibly small. This displacement has a
certain impact on the arguments of the measured parameters. The true values of the
parameters are determined from the measured ones by calculation, by tacking into account
the distances between the measurement planes and the terminals.
Suppose the terminal plane of port m is displaced towards the exterior of the device by
a distance lm. Then the phases of the incident am waves increase by a factor
exp( j m l m ) = exp( j m ) , and the phases of the emergent waves bm decrease by the same
factors. We have denoted by  m the phase constant (real wavenumber) on line m, supposed
lossless.
The new waves are a m ' = a m exp( j m l m ) and bm ' = bm exp(− j m l m ) . If all the terminal
planes are displaced towards the exterior of the device, we can write in matrix form

a ' = diag  exp(jm ), m = 1..k  a, m =  m l m

( a = diag exp(-j m ), m = 1..k  a ')
. (10)
b ' = diag  exp(-jm ), m = 1..k  b
( b = diag exp(j m ), m = 1..k  b ')

By diag[xm, m=1..k] we have denoted a diagonal matrix with elements xm on the main
diagonal. There results

b ' = S ' a' (11)

with

S ' = diag exp(-jm ), m = 1..k  S diag[exp(-jm ), m = 1..k ] . (12)

For example, in the case of a two-port, we have

exp(− j1 ) 0   S11 S12  exp( − j1 ) 0 

S' =      =
 0 exp(− j2 )   S 21 S 22   0 exp( − j2 ) 
 exp(− j 21 ) S11 exp[ − j (1 + 2 )]S12 
= (13)
exp(− j 22 ) S 22 
,
exp[− j (1 + 2 )]S 21
1 = 1l1 , 2 =  2l2 .

The S parameters can be easily calculated from S' by using (12) or (13).
The preceding relations hold also in the case of displacement towards the interior of the
device. In this case, negative values for the lm must be considered.

1.4
High Frequency Technique

S Parameters - Definition

Instructor: Aldo De Sabata

2020-2021 1
 Introduction

• The parameters associated to electronic circuits and devices that

operate as linear two-ports are e.g., z, y, h (hybrid) parameters etc.
• Measuring these parameters require open and short-circuits at the
ports, which is not convenient in high frequency for various reasons
involving permanent damage to the measured device.
• Therefore, S (scattering) parameters are used in the latter case since
their measurements require matching at ports, which does not
damage the device in general.
• S parameters are applied to linear two- and multi-ports, such as
linear (pre)amplifiers, attenuators, filters, couplers, power dividers,
balancing ("baluns"), matching networks, etc.
2020-2021 2
 Setting
zg i1 i2
1 2
+ + a2
+ b1
eg v1 S v2 zL
_ a1 b2
_ _
1' 2'
l1 l2

• The transmission lines can be of zero length (absent, l1=l2=0 );

• v1, v2 - complex amplitudes of normalized voltages;
• i1, i2 - complex amplitudes of normalized currents;
• a1, a2 - normalized incident waves;
• b1, b2 - normalized emerging waves from the two-port;
• The normalization impedances can be coincident at the two ports or different.
2020-2021 3
 Definition

b1 = S11a1 + S12 a2


b2 = S 21a1 + S 22 a2

• The coefficients Sij are called S parameters (from "Scattering")

• Complex numbers that are fixed at a given frequency but depend on
frequency.
• The magnitudes of the S parameters indicate how wave amplitudes
change and the arguments indicate how the phases of the waves
change by passing through the two-port.

2020-2021 4
 Calculation and Measurement
zg =1 i1 i2
1 2

+ b1
+ + a2 b1 = S11a1 + S12 a2

b2 = S 21a1 + S 22 a2
eg v1 S v2 zL =1
_ a1 b2
_ _
1' 2'
l1 l2
zL − 1
a2 =  b2 = b2 b1 b2
zL + 1 S11 = , S 21 =
a1 a1
z L = 1  a2 = 0 a2 = 0 ( z L =1) a2 = 0 ( z L =1)

• S11 - reflection coefficient at port 1 with port2matched;

• S21 - transfer coefficient from port 1 to port 2, with port 2 matched.
• The instrument that measures S parameters is called Network Analyzer.
2020-2021 5
 Network Analyzer

https://en.wikipedia.org/wiki/Network_analyzer_(electrical) 2020-2021 6
 Calculation and Measurement
i1 i2 zg =1
1 2
+ + a2 +
b1
zL =1 v1 S v2 eg
a1 b2 _
_ _
1' 2' zin2
• Similarly, we can conceive the two-port fed at port 2 (by the
generator eg, zg) and terminated at port 1 by the load impedance zL;
• S22 - reflection coefficient at port 2 with port 1 matched;
• S12 - transfer coefficient from port 2 to port 1, with port 1 matched.
b2 b1
S 22 = , S12 =
a2 a1 = 0 ( z L =1)
a2 a1 = 0 ( z L =1)

2020-2021 7
 Generalization and Scattering Matrix
bm = S m1 a1 + S m 2 a 2 + ... + S mk a k , m = 1...k
zg1 i1
1
eg1
+
a1 b = Sa
Zc1 v1
b1 1'
_
 a1   b1 
z1 0 a  b   S11 S12 S1k 
i2 S
zg2 2  2  2 S 22 S 2 k 
+ a =  a3  , b =  b3  , S =  21
eg2 Zc2 v2
a2
     
b2 2'      
_ linear k-port
 ak  bk   Sk1 Sk 2 S kk 
z2 0
......
bi
zgk ik
k Sim = , i, m = 1...k ; al = 0, l = 1...k , l  m
+ am
ak
egk Zck vk • S – Scattering Matrix
bk k'
_ • The S parameters defined above depend only on
zk 0 the multi-port and are fixed at a given frequency

2020-2021 8
 Voltage and Current
zg1 i1
vm = am e m zm + bm e − m zm
1
eg
+
a1 im = am e m zm − bm e − m zm , m = 1..k
Zc1 v1 b1 1'
1 _ zm = 0 at ports
z1
v m = a m + bm
i 0
zg2 2 2
+
eg Zc2 v2
a2
b2 2'
im = a m − bm , m = 1..k
2 _ linear k-port
Vm
z2
......
0 vm =
Z c ,m
zgk ik
k
+
ak im = I m Z c ,m , m = 1..k
eg Zck vk
bk k'
k _ The characteristic impedances are
zk 0
supposed real
2020-2021 9
 Change of Terminal Planes

• Ports 11', 22' etc. have clearly defined locations and are called terminal
planes.
• From various reasons ( e.g., existence of higher order evanescent modes at
the true locations of the terminals of the devices) measurements of S
parameters are performed at a certain distance from these ports (where
the electromagnetic field corresponding to the higher-order modes
becomes negligibly small);
• This displacement has a certain impact on the arguments of the measured
parameters.
• The true values of the parameters are determined from the measured ones
by calculation, by tacking into account the distances between the
measurement planes and the terminals

2020-2021 10
 Change of Terminal Planes
l1 l2
1 2
a m ' = a m exp( j m l m )
b1' b1 b2 b2'
a1' a1 S a2 a2' bm ' = bm exp(− j m l m )

 x 0 0
1' 2' diag( x, y, z ) =  0 y 0  ( not.)
 0 0 z 
b = Sa
a ' = diag  exp(jm ), m = 1..k  a, m =  m lm
b ' = S ' a'
( a = diag exp(-j m ), m = 1..k  a ' )
b ' = diag  exp(-jm ), m = 1..k  b
S ' = diag  exp(-jm ), m = 1..k  S diag[exp(-jm ), m = 1..k ]
( b = diag exp(j m ), m = 1..k  b ' )

2020-2021 11
 Example: Two-Port
 S11 S12 
S=
 S 21 S 22 

exp( − j1 ) 0   S11 S12  exp( − j1 ) 0 

S' =       =
 0 exp( − j2 )   S 21 S 22   0 exp( − j2 ) 
 exp( − j 21 ) S11 exp[ − j (1 + 2 )]S12   S11 ' S12 ' 
=  =  ,
exp[ − j (1 + 2 )]S 21 exp( − j 22 ) S 22   S 21 ' S 22 '
1 = 1l1 , 2 =  2l2 .
• The S parameter can easily be calculated from the elements of S';
• The preceding relations hold also in the case of displacement towards
the interior of the device. In this case, negative values for the lm must
be considered.
2020-2021 12
 Examples

https://www.rfpage.com/s-parameter-formats-in-vector-network-analyzers/ 05.05.2021

2020-2021 13
High Frequency Technique

S Parameters - Definition

2020-2021 14
 2. Formulas for Calculation of S Parameters for Two-ports

We will derive some useful equations for calculation of S parameters of two-ports when the
electrical schematic is known, and it contains common circuit elements such as lumped elements
and transmission lines.
Consider a two-port having a matched generator at one port and a matched load to the other
port (zg=zL=1), Fig. 1.
We suppose that all electrical quantities are normalized to the same characteristic impedance
at all ports. Thus, we can apply Kirchhoff's theorems in the same form as for the un-normalized
quantities.

i2
zg = 1 1 i 1 2
+ + a2=0
+ b1
eg v1 S v2 zL = 1
_ a1 b2
_ _

zin,1 1' 2'

Fig. 1. Circuit for calculation of S11 and S21

Equations at port 1 are: the Kirchhoff's theorem for voltage and relations between waves,
voltages and currents as follows.

eg = z g i1 + v1 = i1 + v1

 v1 = a1 + b1 .
 i1 = a1 − b1


We get

e g = 2a1 . (1)

Equations at port 2 are: the Ohm's law for the impedance zL and relations between waves,
voltages, and currents, i.e.

 v 2 = −i2

v 2 = a 2 + b2 .
i = a − b
2 2 2

But
a2 = 0 (2)

2.1
(in accordance with matching at port 2, where the wave a2 results from b2 by reflection on the
load impedance, which is matched); there results

v 2 = b2 , (3)

i2 = −b2 . (4)

The general input-output relations for a two—port are

 b1 = S11 a1 + S12 a 2
 .
b2 = S 21 a1 + S 22 a 2

These relations, together with (2) give:

 b1 = S11 a1
 (5).
b2 = S 21 a1

Calculation of S11

The (normalized) impedance at port 1 is

v1 a1 + b1 (5) a1 (1 + S11 )
z in,1 = = = .
i1 a1 − b1 a1 (1 − S11 )

Thus

1 + S11
z in,1 = (6).
1 − S11

Solving (6) for S11 yields:

z in,1 − 1
S11 = (7)
z in,1 + 1

Calculation of S21

b2
We know that S 21 = (a 2 = 0) ; by substituting in this equation e g = 2a1 (1) and
a1
v 2 = b2 (3), we obtain
2v 2
S 21 = . (8)
eg

2.2
 z in,1
Finally, since v1 = e g , we get another solution
1 + z in,1

2v 2 z in,1
S 21 = . (9)
v1 1 + z in,1

Parameters S12 and S22 can be calculated by the same procedure, by feeding the two-port at
port 1 and terminating the port 2 by a matched load, Fig. 2. We have:

zin 2 − 1
S22 =
zin 2 + 1
, S12 =
2v1
eg
(z L = 1, z g = 1) .

i1 i2 zg = 1
1 2
+ + a2
b1 +
zL = 1 v1 S v2 eg
a1=0 b2 _
_ _
1' 2' zin,2

Fig. 2. Circuit for calculation of S22 and S12

2.3
High Frequency Technique

Formulas for Calculation of S Parameters

Instructor: Aldo De Sabata

2020-2021 1
 Purpose

We will derive some useful equations for calculation of S parameters of

two-ports when the electrical schematic is known, and it contains
common circuit elements such as lumped elements and transmission
lines.
Consider a two-port having a matched generator at one port and a
matched load to the other port (zg=zL=1).
We suppose that all electrical quantities are normalized to the same
characteristic impedance at all ports. Thus, we can apply Kirchhoff's
theorems in the same form as for the un-normalized quantities.

2020-2021 2
 Equations at Port 1
zg =1 i1 i2
1 2
+ + a2
+ b1
eg v1 S v2 zL =1
_ a1 b2
_ _

zin1 1' 2'

eg = z g i1 + v1 = i1 + v1

 v1 = a1 + b1  eg = 2a1
 i1 = a1 − b1


2020-2021 3
 Equations at Port 2
zg =1 i1 i2
1 2
+ + a2
+ b1
eg v1 S v2 zL =1
_ a1 b2
_ _

zin1 1' 2'

 v 2 = −i2
 zL − 1 v2 = b2
v 2 = a 2 + b2 a2 =  L b2 =
zL + 1
b2 = 0 
i = a − b
2 2 2
i2 = −b2

2020-2021 4
 Input-Output Equations
zg =1 i1 i2
1 2
+ + a2
+ b1
eg v1 S v2 zL =1
_ a1 b2
_ _

zin1 1' 2'

 b1 = S11 a1 + S12 a 2
  b1 = S11a1
b2 = S 21 a1 + S 22 a 2 
b2 = S 21a1
a2 =  L b2 = 0
2020-2021 5
 Calculation of S11
zg =1 i1 i2
1 2
+ + a2
+ b1
eg v1 S v2 zL =1
_ a1 b2
_ _

zin1 1' 2'

v1 a1 + b1 (5) a1 (1 + S11 ) z in,1 − 1

z in,1 = = = S11 =
i1 a1 − b1 a1 (1 − S11 ) z in,1 + 1

S11 is the reflection coefficient at port 1 with port 2 matched.

2020-2021 6
 Calculation of S21
zg =1 i1 i2
1 2
+ + a2
+ b1
eg v1 S v2 zL =1
_ a1 b2
_ _

zin1 1' 2'

 b2
S
 21 a=
 1 2v2
S 21 =
eg = 2a1  eg

v2 = b2


S21 is the transmission coefficient from port 1 to port 2 with port 2 matched.

2020-2021 7
 Calculation of S12 and S22
i1 i2 zg =1
1 2
+ 2v1
b1 + a2 + S12 =
eg
zL =1 v1 S v2 eg
a1 b2 _ zin 2 − 1
_ _ S 22 =
zin 2 + 1
1' 2' zin2

S12 is the transmission coefficient from port 2 to port 1 with port 1 matched.
S22 is the reflection coefficient at port 2 with port 1 matched.

2020-2021 8
High Frequency Technique

Formulas for Calculation of S Parameters

2020-2021 9
 Power Transfer Through Matched Two-Ports

Consider a two-port inserted between a matched generator and a matched load, Fig. 1.

zg1(=1) i1 i2
1 2
+ + a1 a2 +
eg1 v1 S v2 zL(=1)
_ _ b1 b2 _

zin,1 1' 2'

Two port inserted between matched generator and load.

The power transferred to the load on an un-normalized, lossless transmission line reads

2 2
A B
P = Pd + Pr = − ,
2Z c 2Z c

where A and B are complex arbitrary constants that result from solving the transmission line
equations, Zc is the characteristic impedance, and Pd, Pr are the direct and reverse powers,
respectively.
A B
Since on a normalized line we have a = and b = , there results
Zc Zc

2 2
a b
P = Pd + Pr = − .
2 2

We recall that, at a zero distance from the load l = 0 , the normalized voltage and current are
v = a + b and i = a − b respectively.
We know that S11 and S22 are the reflection coefficients at ports 1 and 2 when the other port
 zin ,k − 1 bk 
is matched, i.e.,  S kk = =  . There results
 zin ,k + 1 z ak 
 L =1
am = 0, m  k 

| b1 |2 Pr1
| S11 |2 = =
| a1 |2 a2 = 0
Pd 1

where Pd1 is the direct (incident) power at port 1 and Pr1 is the reflected power at port 1. S22 admits
a similar interpretation.

We will show that | S 21 | 2 and | S12 | 2 are the direct and reverse power gains respectively,
when both input and output are matched, i.e., z g = rg + jx g = 1, z L = 1 .

9.1
 The available power of the generator reads, in normalized values

| eg |2 | eg |2
Pa = = (1)
8rg 8

Kirchhoff's theorem for voltage applied on the input loop yields

e g = z g i1 + v1 = i1 + v1 = a1 − b1 + a1 + b1 = 2a1 . (2)

We get
| a1 | 2
Pa = . (3)
2

The power delivered to the load, by tacking into account the references for voltage and current
is

1 1 | b |2
PL = − Re(v 2 i2 *) = − Re[(a 2 + b2 )(a 2 − b2 )*] = 2 , (4)
2 2 2

b2
since a 2 = 0 due to matching (which also implies S 21 = ). Thus, from (3) and (4) we get
a1
PL
| S 21 |2 = . (5)
Pa

The parameter S12 has a similar property.

The following two definitions are frequently used in practice:

-The Return Loss RL of the two-port is

RL = −10log10 S11 = −20log10 | S11 |;

2
(6)

-The Insertion Loss IL of the dipole is defined by

IL = −10log10 | S21 |2 = −20log10 | S21 | . (7)

The Return Loss indicates the fraction of the incident power on the port that is reflected by
the two-port. The Insertion Loss indicates the fraction of the available power of the generator
that is delivered to the load when both the generator and the load are matched.
It can be shown that simple match is not always the best operation mode for an efficient
transfer of power from generator to load. The concepts of Return Loss and Insertion Loss are
applied in general to passive devices that work in matched mode, such as splitters, directional
couplers, connectors, segments of transmission line, filters, baluns etc.

9.2
High Frequency Technique

Power Transfer Through Matched Two-Ports

Instructor: Aldo De Sabata

2020-2021 1
 Objective

• Assess power handling by two ports that connect a matched

generator to a matched load.

2020-2021 2
 Two-Port Inserted Between Matched Generator and Load

zg1(=1) i1 i2
1 2
+ + a1 a2 +
eg1 v1 S v2 zL(=1)
_ b1 b2 _
_
zin,1 1' 2'

The power transferred to the load on an un-normalized, lossless transmission line is

expressed in terms of A and B, which are complex arbitrary constants that result
from solving the transmission line equations. Zc is the characteristic impedance,
and Pd, Pr are the direct and reverse powers, respectively.
2 2
A B
P = Pd + Pr = −
2Z c 2Z c

2020-2021 3
 Power on Normalized Line
A
a= 2 2
Zc a b
P = Pd + Pr = −
b=
B 2 2
Zc

Reminder: at a distance l=0 from the load of a transmission line, when

solutions are expressed in terms of distance to load, we have:
v = a+b
i = a −b
2020-2021 4
 S11 and Power
 z −1 bk 
 Skk = in ,k = 
 zin ,k + 1 z =1 ak 
 L
am = 0, m  k 

2
| b1 | Pr1
| S11 | =
2
=
| a1 |2 a2 = 0
Pd 1

The magnitude of S11 squared is the modulus of the ratio between the
reflected power and the incident power at port 1. A similar results holds for
the other port(s).

2020-2021 5
 S21 and Power

• The magnitude of S21 squared is the power gain from port 1 to port 2,
when both the generator at port 1 and the load at port 2 are matched.
• The power gain is defined as the ratio between the power absorbed by the
load and the available power of the generator.
• Proof follows.

2020-2021 6
 S21 and Power

zg1(=1) i1 i2
1 2
+ + a1 a2 +
eg1 v1 S v2 zL(=1)
_ b1 b2 _
_
zin,1 1' 2'

| a1 |2
z g = rg + jx g = 1, z L = 1 eg = z g1i1 + v1 = i1 + v1 = a1 − b1 + a1 + b1 = 2a1  Pa =
2
1 1 | b2 |2
( a2 = 0 )
2 2
| eg | | eg | PL = − Re(v2 i2 *) = − Re[( a2 + b2 )( a2 − b2 )*] =
Pa = = 2 2 2
8rg 8 The parameter S12 has a similar property
P
| S 21 | = L
2
when the two-port is fed at port 2 and
b2 Pa
S 21 = terminated in a matched load at port 1.
a1 a2 = 0
2020-2021 7
 Return Loss (RL) and Insertion Loss (IL)

RL = −10log10 S11 = −20log10 | S11 |;

2

IL = −10log10 | S21 |2 = −20log10 | S21 |

• The Return Loss indicates the fraction of the incident power on the port
that is reflected by the two-port.
• The Insertion Loss indicates the fraction of the available power of the
generator that is delivered to the load, when both the generator and the
load are matched

2020-2021 8
 Remark

• It can be shown that simple match is not always the best operation
mode for an efficient transfer of power from generator to load.
• The concepts of Return Loss and Insertion Loss are applied in general
to passive devices that work in matched mode, such as splitters,
directional couplers, connectors, segments of transmission line,
filters, baluns etc.

2020-2021 9
High Frequency Technique

Power Transfer Through Matched Two-Ports

2020-2021 10
High Frequency Technique

Examples of S Parameters Calculation

Instructor: Aldo De Sabata

2020-2021 1
 Outline

• Passive, reciprocal power splitter

• Junction of three coax cables with different characteristic impedances

2020-2021 2
 Power Splitters

• Sometimes, signals that propagate on transmission lines must be directed

to different receivers.
• For example, such situation occurs inside buildings with a single TV input
and several receivers, and in computer networks. The devices that realize
this operation are called signal (power) splitters.
• Splitters can be active or passive, reciprocal or non-reciprocal.
• Active splitters, using e.g., transistors, require a power supply but can
achieve signal amplification.
• Passive splitters attenuate signals but are cheaper than active splitters.
• Non-reciprocal splitters use magnetic materials such as ferrites or
amplifiers.

2020-2021 3
 Properties of an "Ideal" Splitter

• Matched at all ports (a port i is called matched if the Sii=0, i.e., the
input impedance is equal to the characteristic impedance of the input
line when all the other ports are matched. Signal reflection do not
occur at a matched port when all other ports are terminated on
matched loads) ;
• Does not introduce power loss (the device is lossless).
• In these conditions, the power incident at one port is split between
the other ports when these ports are terminated in matched loads.

2020-2021 4
 Three-Port Splitter
b2
2
a1 a2=0 1
zL=1 P2 =
2
1 2 b2
P1 = a1 Splitter 2
1 Pd=0 b3
2 3
b1=0
a3=0 1 2
zL=1 P3 = b3
2

P1 = P2 + P3 ( Ideal Splitter )

Such a splitter can be realized only with non-reciprocal devices, such as ferrites or
amplifiers, but it cannot be realized with reciprocal, lossless devices, such as capacitors
and inductors. It can be shown mathematically that lossless, reciprocal three-ports
cannot be matched at all ports.

2020-2021 5
 Lossy Splitters
• Passive, reciprocal devices are however necessary, being cheaper and easier to
construct than non-reciprocal splitters.
• Therefore, the lossless condition is dropped in general, and only the condition of
matching at all ports is kept.
• A possible solution is represented here. All electrical quantities are supposed
normalized to the common characteristic impedance of the lines connected to
the ports.
r
1 r 2
r
3

2020-2021 6
 S Parameters for 3-Port Splitter
Sij = S ji , i, j = 1...3 (Symmetry )
r i2 2
1 1 r v2 r + 1 3r + 1
v zin1 = r + ( r + 1) || ( r + 1) = r + =
+ 2 2
r 3v 1
eg 3
i3 1 zin1 − 1 3r − 1
S11 = =
zin1 + 1 3r + 3
zin1

( r + 1) || ( r + 1) 1
v = eg = eg
r + 1 + ( r + 1) || ( r + 1) 3 2v2 2
S 21 = =
1 1 eg 3r + 3
v2 = v = eg
r +1 3 ( r + 1)

2020-2021 7
 Scattering Matrix
3r − 1 2 2 
1  
S= 2 3r − 1 2
3r + 3  
 2 2 3r − 1

1
Sii = 0 ( matched ports )  r=
3
0 1 1 
S = 1 0 1 
1 scattering matrix for a matched, reciprocal splitter
2
1 1 0 
• If power incident at port 1 is P (which is also the available power of the matched
generator) then P/4 will reach each of ports 2 and 3, and P/2 will be dissipated inside the
device (because S11=0 and S21=S31=1/2).
2020-2021 8
 Practical Splitters
• In practice, if symmetrical transmission lines are used (such as two-
wire cables), the resistance is split between the two wires.
• The resistance is mounted on the central conductor of non-
symmetrical lines, such as coaxial cables.
12.5 Ω 25 Ω
75 Ω cables

12.5 Ω
Coaxial cable
Two-wire line

2020-2021 9
 Junction of Three Coaxial Cables

• In the preceding example, the transmission lines connected to the

device ports had the same characteristic impedance, which allowed
for the application of the Kirchhoff's theorems to normalized voltages
and currents.
• If the characteristic impedances are not equal, Kirchhoff's theorems
can be applied to non-normalized quantities only.
• Normalization is accomplished after performing the calculations, as
presented in the next example.

2020-2021 10
 Junction of Three Coaxial Cables
2
Zc2
a1
_
1 V12
Zc1 +
i1
Zin,1
b1 Zc3
3

We suppose that each line is terminated in its characteristic impedance (terminations are
not represented)

2020-2021 11
 Calculation of S11

2 1
Zc2 Yin,1 = = Yc 2 + Yc 3
Z in1
a1
_
V12 Z in ,1 Yc ,1
1 −1 −1
Zc1 + zin ,1 − 1 Z c1 Yin ,1 Y −Y Y −Y −Y
i1 S11 = = = = c1 in1 = c1 c 2 c 3
zin ,1 + 1 Z in ,1 Yc ,1 Y +Y Yc1 + Yc 2 + Yc 3
Zin,1 +1 + 1 c1 in1
Z c1 Yin ,1
b1 Zc3
3

The input impedance of the lines as seen from the junction are equal to their characteristic
impedances due to the matched terminations. Since lines are connected in parallel, it is
convenient to use admittances Y rather than impedances Z

2020-2021 12
 Calculation of S21
V12 is common to all three cables.

b1
V12 = (a1 + b1 ) Z c1 = a1 (1 + ) Z c1 = a1 (1 + S11 ) Z c1 ( cable 1)
a1
2
Zc2
V12 = b2 Z c 2 (a2 = 0) ( cable 2 )
a1
_
1 V12 b2 V12 Yc 2 2 Yc1Yc 2
Zc1 + S 21 = = =
i1 a1 V12 Yc1 Yc1 + Yc 2 + Yc 3
Zin,1
(1 + S11 )
b1 Zc3
3

2020-2021 13
 Scattering Matrix
Due to the symmetry of the structure, the other parameters can be
obtained from the calculated ones by circular permutations:
1→2 →3 →1

Yc1 − Yc 2 − Yc 3 2 Yc1Yc 2 2 Yc1Yc 3 

1  
S=  2 Yc1Yc 2 Yc 2 − Yc 3 − Yc1 2 Yc 2Yc 3 
Yc1 + Yc 2 + Yc 3 
 2 Yc1Yc 3 2 Yc 2Yc 3 Yc 3 − Yc1 − Yc 2 

2020-2021 14
 Scattering Matrix (Identical Lines)

Yc1 = Yc 2 = Yc 3

− 1 2 2 
S =  2 − 1 2 
1
3
 2 2 − 1

Note that all ports are not matched, since the elements on the main
diagonal are not zero.

2020-2021 15
High Frequency Technique

Examples of S Parameters Calculation

2020-2021 16
 Examples of S Parameters Calculation

1. Splitters

In many situations, signals that propagate on transmission lines must be directed to different
receivers. For example, such situation occurs inside buildings with a single TV input and several
receivers, and in computer networks. The devices that realize this operation are called signal
splitters.
Splitters can be active or passive, reciprocal, or non-reciprocal. Active splitters, using e.g.,
transistors, require a power supply but can achieve signal amplification. Passive splitters
attenuate signals but are cheaper than active splitters. Non-reciprocal splitters use magnetic
materials such as ferrites or amplifiers.
An ideal splitter would have the following properties:
• it is matched at all ports (a port i is called matched if the Sii=0, i.e., the input impedance is
equal to the characteristic impedance of the input line when all the other ports are matched.
Signal reflection do not occur at a matched port when all other ports are terminated on matched
loads) ;
• it does not introduce power loss (the device is lossless).
In these conditions, the power incident at one port is split between the other ports when these
ports are terminated in matched loads.
We consider the problem of designing a passive splitter with three ports, Fig. 1. The power
incident at one port is split between the other two ports.

b2
2

a1 a2=0 zL=1
Splitter
1 Pd=0 b3
b1=0 3
a3=0 zL=1

Fig. 1. Ideal splitter.

1 2
In Fig. 1, the power incident at port one is P1 = a1 , and the power exiting ports 2 and 3 are
2
1 2 1 2
P2 = b2 and P3 = b3 respectively. For an ideal power splitter, we have P1 = P2 + P3 . The
2 2
reflected wave at port 1 is zero (b1=0) since port 1 is matched.
Such a splitter can be realized only with non-reciprocal devices, such as ferrites or amplifiers,
but it cannot be realized with reciprocal, lossless devices, such as capacitors and inductors. It can
be shown mathematically that lossless, reciprocal three-ports cannot be matched at all ports.
Passive, reciprocal devices are however necessary, being cheaper and easier to construct than
non-reciprocal splitters. Therefore, the lossless condition is dropped in general, and only the
condition of matching at all ports is kept. A possible solution is represented in Fig. 2. All electrical
quantities are supposed normalized to the common characteristic impedance of the lines
connected to the ports.

r 2
1 r
r 3

Fig. 2. Passive, reciprocal

splitter

We calculate the S parameters of the three-port in Fig. 2. We note that the circuit is symmetric,
since it remains unchanged if port numbers are renumbered. Thus, the S parameters must satisfy
the following symmetry condition:

Sij = S ji , i, j = 1...3

For calculating S11 and S21, we use the standard procedure outlined in the preceding lectures.
We add a matched generator at port 1 and terminate the other ports in matched loads, Fig. 3.

r i2 2
1 1 r v2
v
+ r 3 1
eg v3
i3 1

zin1

Fig. 3. Splitter: S parameters calculation.

We have

r + 1 3r + 1
zin1 = r + ( r + 1) || ( r + 1) = r + =
2 2

Thus
 zin1 − 1 3r − 1
S11 = = .
zin1 + 1 3r + 3

Due to symmetry, we have i2 = i3 , and by using Ohm's law and the voltage divider theorem we
get:

r +1
v 1 2 eg
i2 = i3 = − =− eg =−
r +1 r +1 1+ r + r +1 3r + 3
2

where v is the voltage referred to the ground defined in Fig. 3.

Since v2 = −i2 (Ohm's law), we get

2v2 2
S 21 = = .
eg 3r + 3

Since all other S parameters have the same expressions due to symmetry, we obtain for the
scattering matrix the expression:

3r − 1 2 2 
1 
S= 2 3r − 1 2  .
3r + 3 
 2 2 3r − 1

1
To have a match at all ports, we must impose the condition Sii = 0 . Therefore, we get r = .
3
The scattering matrix for a matched, reciprocal splitter results as

0 1 1 
S = 1 0 1  .
1
2
1 1 0 

If power incident at port 1 is P (which is also the available power of the matched generator)
then P/4 will reach each of ports 2 and 3, and P/2 will be dissipated inside the device (because
S11=0 and S21=S31=1/2).
Due to its simple structure, the presented solution can be used for the construction of passive
splitters with n>3 ports, although lossless solutions exist in this case. By following the above
presented procedure, the following S parameters are obtained in the general case:

nr − n + 2 2
S11 = , S21 = .
n ( r + 1) n ( r + 1)

To provide match at all ports we must take

 n−2 1
r=  Sii = 0, Sij = , i, j = 1...n, i  j .
n n −1

For n=3, we obtain again the results calculated above.

In practice, if symmetrical transmission lines are used (such as two-wire cables), the
resistance is split between the two wires, Fig. 4. The resistance is mounted on the central
conductor of non-symmetrical lines, such as coaxial cables.

12.5 Ω
25 Ω

12.5 Ω
Two-wire line Coaxial cable

Fig. 4. Mounting resistances for splitters on

75 Ω cables.

2. Junction of Three Coaxial Cables

In the preceding example, the transmission lines connected to the device ports had the same
characteristic impedance, which allowed for the application of the Kirchhoff's theorems to
normalized voltages and currents. If the characteristic impedances are not equal, Kirchhoff's
theorems can be applied to non-normalized quantities only. Normalization is accomplished after
performing the calculations, as presented in the next example.
Consider three coaxial cables having different characteristic impedances Zc,i, i=1..3, connected
as presented in Fig. 5.

Zc2
a1

V12
1 +
Zc1 i1
Zin,1
b1 Zc3
3

Fig. 5. Junction of three coax cables

 We suppose that each line is terminated in its characteristic impedance (terminations are not
represented in Fig. 5).
To calculate S11, we determine firstly the input impedance Zin,1 (Fig. 5). The input impedance of
the lines as seen from the junction are equal to their characteristic impedances due to the
terminations. Since lines are connected in parallel, it is convenient to use admittances Y rather
1
than impedances Z, Y = . We have
Z

1
Yin,1 = = Yc 2 + Yc 3
Z in1
Thus

Z in ,1 Yc ,1
−1 −1
zin ,1 − 1 Z c1 Yin ,1 Y −Y Y −Y −Y
S11 = = = = c1 in1 = c1 c 2 c 3
zin ,1 + 1 Z in ,1 Yc ,1 Y +Y Yc1 + Yc 2 + Yc 3
+1 + 1 c1 in1
Z c1 Yin ,1

(normalization is made with respect to the characteristic impedance at each port).

To calculate S21, we note that the un-normalized voltage between the common point of the
central conductors and the common ground (at the common point of the shields), denoted by V12
in Fig. 5, is the same for all three cables. Following definitions of the voltage waves and the
normalization procedure for voltage that is made by division with the square root of the
characteristic impedance, we have (for cable 1):

b1
V12 = (a1 + b1 ) Z c1 = a1 (1 + ) Z c1 = a1 (1 + S11 ) Z c1 .
a1

For cable 2, we have due to the (unrepresented) matched termination:

V12 = b2 Z c 2 ( a 2 = 0)

We get
b2 V12 Yc 2 2 Yc1Yc 2
S 21 = = =
a1 V12 Yc1 Yc1 + Yc 2 + Yc 3
(1 + S11 )

Due to the symmetry of the structure, the other parameters can be obtained from the
calculated ones by circular permutations. The following scattering matrix is obtained:

Yc1 − Yc 2 − Yc 3 2 Yc1Yc 2 2 Yc1Yc 3 

1  
S=  2 Yc1Yc 2 Yc 2 − Yc 3 − Yc1 2 Yc 2Yc 3  .
Yc1 + Yc 2 + Yc 3 
 2 Yc1Yc 3 2 Yc 2Yc 3 Yc 3 − Yc1 − Yc 2 
If the lines are identical, Yc1 = Yc 2 = Yc 3 , then

− 1 2 2
S =  2 − 1 2  .
1
3
 2 2 − 1

Note that all ports are not matched, since the elements on the main diagonal are not zero.