IEC TR 62152:2004
(Main)Background of terms and definitions of cascaded two-ports
Background of terms and definitions of cascaded two-ports
lays the foundation for agreement on the fundamental terms and definitions to be used world wide in describing the transmission properties of a two-port or quadripole end and builds a bridge between the classical quadripole theory and the scattering matrix presentation which is based on incident and reflecting square root of power waves at the input and output of a two-port.
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Standards Content (Sample)
TECHNICAL IEC
REPORT TR 62152
First edition
2004-08
Background of terms and definitions
of cascaded two-ports
Reference number
IEC/TR 62152:2004(E)
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TECHNICAL IEC
REPORT TR 62152
First edition
2004-08
Background of terms and definitions
of cascaded two-ports
” IEC 2004 Copyright - all rights reserved
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mechanical, including photocopying and microfilm, without permission in writing from the publisher.
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– 2 – TR 62152 ” IEC:2004(E)
CONTENTS
FOREWORD.4
1 General .6
2 Operational, image and insertion transfer functions and complex attenuations or
losses .6
3 Terms and definitions .7
Annex A (normative) Concepts of normalized voltage waves, square root of power
waves and operational attenuation and losses.9
A.1 General .9
A.2 Complex operational attenuation or operational propagation coefficient * .9
B
A.3 Impedance .10
A.4 Operational reflection coefficient.10
A.5 Return loss.10
A.6 General coupling transfer function.11
A.7 Benefits of the concept of operational quantities.12
Annex B (normative) Two-port transmission technique – Terms .13
Annex C (normative) Two-port theory and fundamental concepts in transmission
engineering .14
C.1 General .14
C.2 Transfer equations for a passive two-port.14
C.3 Chain matrix.15
C.4 The symmetries and impedances of a two-port .19
C.5 Impedance matching.21
C.6 Level concepts .23
C.7 Attenuation and gain concepts .24
C.8 Concepts related to return loss and matching.28
C.9 Scattering parameter .35
C.9.1 Scattering parameter of a one-port .35
C.9.2 Scattering parameters and scattering matrix of a two-port.38
C.10 Examples .43
C.10.1 Example 1 .43
C.10.2 Example 2 .45
C.11 Reference documents .46
Figure 1 – Defining the transfer functions of a two-port .6
Figure 2 – Constant value A and A curves on a complex plane z = x + jy.8
s r
Figure A.1 – Coupling between two systems.12
Figure C.1 – A quadripole or two-port.14
Figure C.2 – An impedance-unsymmetrical two-port (a) with its equivalent circuit (b).16
Figure C.3 – Two chained two-ports .17
Figure C.4 – An impedance-symmetrical two-port .19
Figure C.5 – An impedance-unsymmetrical two-port for which Z z Z when Z = Z .19
1 2 A B
Figure C.6 – A two-port terminated with an impedance Z .20
B
Figure C.7 – Reflection less matching .22
TR 62152 ” IEC:2004(E) – 3 –
Figure C.8 – Power matching for maximizing the effective power .22
Figure C.9 – Absolute and nominal level in a system .24
Figure C.10 – Definition of the complex image attenuation * of a two-port.24
Figure C.11 – Definition of the complex operational attenuation of a two-port .25
Figure C.12 – Definition of residual attenuation .27
Figure C.13 – Measurement of the sending reference equivalent .27
Figure C.14 – Measurement of the receiving reference equivalent.28
Figure C.15 – Definition of the complex return loss.28
Figure C.16 – Apollonius’ circle.29
Figure C.17 – Return loss .30
Figure C.18 – Curves for constant values of A or A in the complex plane .32
s r
Figure C.19 – Curves for constant values of A or A the complex plane .33
s r in
Figure C.20 – Smith chart for transmission lines .34
Figure C.21 – One-port .35
Figure C.22 – Homogenous transmission line .36
Figure C.23 – One-port fed from a generator with source impedance Zg .37
Figure C.24 – Two-port .39
Figure C.25 – Termination Z by virtue of the stray parameters of the two-port .40
B
Figure C.26 – Ideal transformer.43
Figure C.27 – Determination of a scattering matrix of a passive reciprocal two-port.45
– 4 – TR 62152 ” IEC:2004(E)
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
BACKGROUND OF TERMS AND DEFINITIONS
OF CASCADED TWO-PORTS
FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
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patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
The main task of IEC technical committees is to prepare International Standards. However, a
technical committee may propose the publication of a technical report when it has collected
data of a different kind from that which is normally published as an International Standard, for
example "state of the art".
IEC 62152, which is a technical report, has been prepared by IEC technical committee 46:
Cables, wires, waveguides, r.f. connectors, r.f. and microwave passive components and
accessories.
The text of this technical report is based on the following documents:
Enquiry draft Report on voting
46/129/DTR 46/133/RVC
Full information on the voting for the approval of this technical report can be found in the
report on voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
TR 62152 ” IEC:2004(E) – 5 –
The committee has decided that the contents of this publication will remain unchanged until
the maintenance result date indicated on the IEC web site under "http://webstore.iec.ch" in
the data related to the specific publication. At this date, the publication will be
reconfirmed;
withdrawn;
replaced by a revised edition, or
amended.
A bilingual edition of this document may be issued at a later date.
– 6 – TR 62152 ” IEC:2004(E)
BACKGROUND OF TERMS AND DEFINITIONS
OF CASCADED TWO-PORTS
1 General
It is important and practical that components of a transmission chain can be separated and
tested separately. This means well-defined interfaces and measuring techniques including
agreed terms and definitions. It is advantageous to operate, by the square root of a reference
impedance (normally application impedance of the system), with normalized voltage waves
corresponding to the square root of power waves.
This technical report has two main goals. It lays the foundation for agreement on the
fundamental terms and definitions to be used world wide in describing the transmission
properties of a two-port or quadripole end and builds a bridge between the classical
quadripole theory and the scattering matrix presentation which is based on incident and
reflecting square root of power waves at the input and output of a two-port. Finally, it is shown
that the two concepts are bound together through simple equations and are fundamentally
identical.
The quadripole theory was originally developed for voice- and carrier-frequency technologies
and transmission, and later for microwaves, but both can be used through the whole
frequency range.
2 Operational, image and insertion transfer functions and complex
attenuations or losses
a) Operational transfer function
T is defined as the square root of the power wave into the load (equal to reference
B
impedance R ) of a two-port P compared with an unreflected square root of power
2 2
wave P from the generator with a source impedance equal to the reference impedance
R .
R
1 P
U
Z Z R
E U 01 02 2
0 1
P
R
E
U R
E 1
0 0
P
IEC 1181/04
Figure 1 – Defining the transfer functions of a two-port
TR 62152 ” IEC:2004(E) – 7 –
PU R P
22 2 2
(1)
TS
B21
PU R P
00 1 0
P 0
which is equal to the forward transfer scattering parameter S .
The operational transfer function becomes
b) the image transfer function T when the reference impedance becomes equal to the input
and output characteristic impedances Z and Z of the two-port; and
01 02
c) the insertion transfer function T’ when R = R = R.
B 1 2
Correspondingly, the complex attenuations or losses are as follows.
Complex operational attenuation
>@ > @
* A jB ln 20 log T in dB j˜arg(T ) in rad
B B B B B
T
B
(2)
Complex image attenuation
* A jB ln 20 logT in>@dB j˜arg(T ) in>rad@
(3)
T
Complex insertion attenuation or loss
' ' ' ' '
(4)
* A jB ln 20 logT in>@dB j˜arg(T ) in>rad@
B B B B B
R R R '
1 2
T
B
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
3.1
operational attenuation
quotient of the unreflected square root of the power wave fed into the reference impedance of
the input of the two-port and the square root of the power wave consumed by the load of the
two-port expressed in dB and radians
NOTE By defining a new quantity operational insertion loss in the same way as the operational attenuation, at
least when the reference impedances on both sides of the two-port are the same, the problem of insertion loss and
operational attenuation is solved.
3.2
operational insertion loss
quotient of the unreflected square root of the power wave fed into the reference impedance of
the input of the two-port and the square root of the power wave consumed by the load of the
two-port expressed in dB and radians
– 8 – TR 62152 ” IEC:2004(E)
r
0 0,5 1,0 1,5 2,0 2,5
2,0
1,0
0,5
0,5 10
A
r
1,0
2,5
1,5
5,0
2,0
1,5
1,0 A
s
0,5
2,0
0,5
1,0
jx
2,5
IEC 1182/04
z 1
N
Reflection loss A 20log [dB]
s
2 z
N
z 1
N
Return loss A 20log [dB]
r
z 1
N
Z
z (= normalized impedance) = r + jx
N
Z
Figure 2 – Constant value A and A curves on a complex plane z = x + jy
s r
3.3
operational attenuation and insertion loss
quotient of the unreflected square root of the power wave fed into the reference impedance of
the input of the two-port and the square root of the power wave consumed by the load of the
two-port expressed in dB and radians
NOTE In the IEV, insertion loss is understood as the loss produced by inserting a two-port into a separated point
of the transmission chain. Because of varying terminating impedances of the two-port, this leads to insertion loss
or operational attenuation deviation, that is, depending on where, in the chain, the two-port is inserted.
It is obvious that the insertion of a two-port with a certain operational attenuation or operational insertion loss
causes different attenuation increases (or decreases) in separate circuit points of different impedances.
This is called the Insertion Loss Deviation (ILD).
ILD has proved to be a very important subject of discussion in the standardization of a data channel.
TR 62152 ” IEC:2004(E) – 9 –
Annex A
(normative)
Concepts of normalized voltage waves, square root of power waves
and operational attenuation and losses
A.1 General
It is important and practical that components of a transmission chain can be separated and
tested separately. This means well-defined interfaces and measuring techniques including
agreed terms and definitions. It is advantageous to operate, by the square root of a reference
impedance (normally application impedance of the system), with normalized voltage waves
corresponding to the square root of power waves.
In this way, for instance, the scattering parameters are defined. For example, S is the
forward operational transfer function and S is the operational reflection coefficient.
Two of the reasons for using the square root of the impedance normalized voltage waves or
the square root of the power waves are
a) that the network analyser is measuring voltages; and
j˜arg z
b) because the natural logarithm, ln, of a complex quantity z x jy z ˜ e is directly l,
and ln|z| nepers can be expressed in decibels 20˜ log z and the imaginary part still
remains arg(z) in radians, as, for example,
* Aj2BS 0log jarg(z)
BB B 10 21
(see equation (A.1)).
A.2 Complex operational attenuation or operational propagation coefficient *
B
The complex operational attenuation (complex operational loss) introduced by a two-port
component, cascade of components, link, cable assembly etc. into a system is defined by
using the scattering parameter S as
(A.1a)
* A jBS ln(1/ ) lnS j˜ arg(S )
BB B 21 21 21
(A.1b)
* Aj2BS 0log j˜arg(S)
BB B 10 21 21
where
in (A.1a) lnSA Np
> @
21 B
in (A.1b) 20 logSA dB
> @
10 21 B
in (A.1a) and (A.1b)
arg(SB) rad
> @
21 B
where
A is the operational attenuation = 20 log (1/|S |) (dB)
B 10 21
B is the operational attenuation phase constant = –arg(S ) (rad)
B 21
– 10 – TR 62152 ” IEC:2004(E)
NOTE 1 A is equal to the ratio of the unreflected complex power (voltage × current) sent into a two-port, to the
B
complex power consumed by the load of the two-port, in decibels. The load is normally a resistance equal to the
application impedance of the system Z . When the generator and load impedances are the same operational
N
attenuation becomes insertion loss.
NOTE 2 From the theory of complex functions:
ln z ln z j˜arg z
where
j˜arg z
z x jy z ˜ e
and, by using the square root of power waves, we can write, for the natural logarithms of the ratio of two square
root of complex power waves:
§ ·
P P P
1 1 1
¨ ¸
ln ln j˜arg * A jB
¨ ¸
P P P
2 2 2
© ¹
where A is in nepers and B in radians.
When A is expressed in decibels, B will not be affected; it remains in radians.
A.3 Impedance
a) The nominal characteristic impedance Z (of a two-port) is the resistive part of the mean
CN
characteristic impedance Z specified with tolerance at a given frequency.
C
b) Z is the nominal impedance of the system terminals between which the two-port is
N
operating.
c) Z is the (nominal) reference impedance used in measurements. Normally Z = Z .
R R N
A.4 Operational reflection coefficient
The operational reflection coefficient of the two-port is equal to the scattering parameter S
of a two-port. It equals the reflection coefficient r at the input when the two-port is terminated
c
with its reference impedances Z , normally equal to the nominal impedances of the system
R
terminals.
Z Z
in R
(A.2)
Sr
11 B
ZZ
in R
A.5 Return loss
a) Complex operational return loss RL
B
RL ln ln(r ) ln r>@Np j˜ arg(r )>rad@
B B B B
r (A.3)
B
20˜ log |r |>@dB j˜ arg(r )>rad@
10 B B
b) Structural return loss SRL
The return loss where the mismatch effects at the input and output of two-port have been
eliminated (compare with the continuous wave (CW) burst measurement method).
NOTE It is important to define the structural return loss, although it is not measured direct from the cable
assemblies, because it shows that there are differences between different kinds of return losses.
TR 62152 ” IEC:2004(E) – 11 –
c) Reflection loss of a junction (see Figure 2 )
22 2
(A.4a)
* ln (1SS) ln (1 ) Np j˜ arg( (1S )) rad
>@ >@
r
22 2
or (A.4b)
* ln (1SS) 20˜ log (1 ) dB j˜ arg( (1S ) ) rad
>@ >@
r10
22 2
(A.4c)
* ln (1SS) 10˜ log (1 ) dB j˜ arg(1S ) rad
>@ >@
r10
d) Mismatch loss of a junction (not recommended)
22 2
(A.5a)
* ln(1)SS ln(1) Npj˜arg((1)S)rad
>@ >@
m
or
22 2
(A.5b)
* ln (1SS) 20˜log (1 ) dB j˜arg( (1S ) ) rad
>@ >@
m10
22 12
(A.5c)
* ln (1SS) 10˜log (1 ) dB j˜ arg(1S ) rad
>@ >@
m10
In c) and d) S is the complex reflection coefficient of the junction
P ----->
________________|______________
Z Z
1 2
Z Z
Sr (A.6)
ZZ
A.6 General coupling transfer function
This is distinguished between the near-end and far-end coupling transfer functions T and T .
n f
U
2n,f
P Z
Z U
2n,f 2n,f
1 2n,f (A.7)
T
n,f
U
U
P Z
0 2n,f
Z
– 12 – TR 62152 ” IEC:2004(E)
P P
1f
Z
(1)
U
Z
E
(2)
Z Z
2 2
P P
2n 2f
IEC 1183/04
Key
P is the unreflected power sent into the near end of the system (1).
System (1) is disturbing system (2).
Figure A.1 – Coupling between two systems
Coupling transfer function is a general term valid through the whole frequency range.
It may be expressed in decibels and radians
P
2n,f
(A.8)
TT[dB & rad] 20 log [dB] j˜arg [rad]
n,f 10 n,f
P
and the (complex) operational transfer, coupling screening, unbalance, attenuation, etc. are
* = A + jB = –20 log |T| – j arg(T) (A.9)
x x x 10
where
A is the (operational ) attenuation (dB);
x
B is the (operational ) attenuation phase constant (rad).
x
A.7 Benefits of the concept of operational quantities
Measurements are always taken between well-defined resistive terminations.
This means that the impedances at a reference plane between the cascaded units of the
system are specified.
Individual units can be specified and tested separately and made by different manufacturers.
This makes open systems, networks and cabling possible.
TR 62152 ” IEC:2004(E) – 13 –
Annex B
(normative)
Two-port transmission technique – Terms
P
V
OUT
OUT
a) Image transfer function
T
V
P
IN
IN
b) Image transfer attenuation or loss A = 20 log
T
Z and Z are the image or characteristic impedances of the input or output of the two-port,
C1 C2
equal to the input and output impedances when the opposite port is terminated with its image
impedance.
and are the input and output square root of complex powers.
P V P V
IN IN OUT OUT
When defining the image, there are reflections at the input and output; in other words, the
input and output are terminated with their image impedance.
1 1
c) Complex image attenuation
* 20log >@dB jarg >@rad A jB
T T
d) Image attenuation
A=20 log dB
>@
T
e) Image phase shift
B=arg rad
>@
T
B
f) Image phase propagation time or delay
W
p
Z
dB
g) Image group propagation time or delay
W
g
dZ
h) Image phase velocity
v
p
W
p
i) Image group velocity
v
g
W
g
j) Complex operational attenuation * A jB
BB B
§ ·
V
i1
k) Operational attenuation
¨ ¸
A 20 log
B Vi2 0
¨ ¸
V
© r 2 ¹
§ V ·
i1
l) Operational phase shift
B arg¨ ¸
B Vi2 0
¨ ¸
V
© r 2 ¹
– 14 – TR 62152 ” IEC:2004(E)
Annex C
(normative)
Two-port theory and fundamental concepts in
transmission engineering
C.1 General
This annex has two main goals. It lays the foundation for the fundamental terms and
definitions to be used world wide in describing the transmission properties of a two-port or
quadripole and builds a bridge between the classical quadripole theory and the scattering
matrix presentation, which is based on the incident and reflecting square root of power waves
at the input and output of a two-port. Finally, it is shown that the two concepts are bound
together by simple equations which are fundamentally identical.
The two-port theory was originally developed for voice and carrier technologies, transmission
and later for microwaves, but it can be used for the whole frequency range and for various
applications.
In the following Clauses, we will use the term two-port exclusively.
C.2 Transfer equations for a passive two-port
For a passive impedance-symmetrical two-port (see Clause C.4 and Figure C.1), the following
equations are valid.
U U U (C.1)
1 i r
U U
i r
I I I (C.2)
1 i r
Z Z
0 0
ī ī
U U e U e (C.3)
2 i r
ī ī
I I e I e (C.4)
2 1 r
I I
1 2
*
U U
1 2
Z Z
0 0
IEC 1184/04
Figure C.1 – A quadripole or two-port
———————
L.HALME: CHAPTER L4, part of English version of the L. Halme´s book (Halme, L.K.: Johtotransmissio ja
sähkömagneettinen suojaus, (Transmission on lines and electromagnetic screening, in Finnish), Parts A and B,
Otakustantamo 2nd Eddition Helsinki 1989, 605 pages), corrected by J. Walling (2000-09-27).
TR 62152 ” IEC:2004(E) – 15 –
Where Z is the image impedance of the two-port, * = A + jB is the complex image attenuation
or the image transfer constant. It equals the complex image attenuation of a two-port
terminated in its image impedance (see Clause C.7). U and I represent the incident voltage
i i
and current waves fed to the input of the two-port, while U and I represent the voltage and
r r
current waves reflected back to the input from the output of the two-port. By solving equations
(C.1) and (C.2) for U , U , I and I and by substituting these into equations (C.3) and (C.4), we
i r i r
obtain the actual voltage and current at the output terminals:
U U coshī Z I sinhī (C.5)
2 1 0 1
U
I I coshī sinhī (C.6)
2 1
Z
By solving equations (C.5) and (C.6) for U and I we obtain
1 1
U U coshī Z I sinhī (C.7)
1 2 0 2
U
I I coshī sinhī (C.8)
1 2
Z
From which we can deduce that equations (C.7) and (C.8) for input terminals can be obtained
from the equations (C.5) and (C.6) for output terminals by interchanging the voltages, by
interchanging the currents and by replacing * with –*.
From equations (C.5), (C.6), (C.7) and (C.8), we can also solve the currents expressed by
means of the voltages, as well as the voltages expressed by means of the currents:
U U
1 2
I cothī (C.9)
Z Z sinhī
0 0
U 1 U
1 2
I cothī (C.10)
Z sinhī Z
0 0
U Z I cothī Z I (C.11)
1 0 1 0 2
sinhī
U Z I Z I cothī. (C.12)
2 0 1 0 2
sinhī
C.3 Chain matrix
Equations (C.7) and (C.8) can be presented in matrix form
coshī Z sinhī
ª º
U U
ª º ª º
1 2
« »
(C.13)
« » « »
sinhī coshī
« »
I I
¬ 1¼ ¬ 2¼
Z
¬ 0 ¼
– 16 – TR 62152 ” IEC:2004(E)
Here, the multiplier matrix is called the chain matrix and is generally expressed as:
U A B U
ª º ª ºª º
1 2
(C.14)
« » « »« »
I C D I
¬ 1¼ ¬ ¼¬ 2¼
Where the constants A, B, C and D forming the chain matrix are called the transfer
parameters. They are bound to each other by the relation
AD BC 1 (C.15a)
The transfer parameters can be calculated by alternately considering the output of the two-
pole either as short-circuited or open-circuited, whereby
§ · § ·
U U
1 1
¨ ¸ ¨ ¸
A B
¨ ¸ ¨ ¸
U I
© 2¹ © 2 ¹
I 0 U 0
2 2
(C.15b)
§ I · § ·
I
¨ ¸
C D ¨ ¸
¨ ¸ ¨ ¸
U I
2 © 2¹
© ¹
I 0 U 0
2 2
The chain matrix is well suited for the examination of cascaded two-ports.
An impedance-unsymmetrical two-port (see Clause C.3) can be treated as a symmetrical one
by cascading it (as shown by Figure C.2) with an ideal transformer with a turns ratio of
N 1 Z
1 01
K (C.16)
N n Z
2 02
I I
1 2
*
U U
1 Z Z 2
0 0
a)
I I I' I
1 2 2 2
1:n
*
U U' U
1 2 2
Z Z Z
01 01 01
b)
IEC 1185/04
Figure C.2 – An impedance-unsymmetrical two-port (a) with its equivalent circuit (b)
We are here concerned with the cascading (or chaining) of two-ports, whereby the
calculations can be appropriately carried out by means of chain matrices.
Let us suppose two two-port with the chain matrices A and A being interconnected as shown
1 2
by Figure C.3.
TR 62152 ” IEC:2004(E) – 17 –
I I' I
1 2 2
U U' U
1 2 2
[A ] [A ]
1 2
IEC 1186/04
Figure C.3 – Two chained two-ports
The matrix equations, with the direction arrows as indicated in Figure C.3, are
c c
U U U U
ª º ª º ª º ª º
1 2 2 2
>@A >A@ (C.17)
« » 1 « » « » 2 « »
c c
I I I I
¬ 1¼ ¬ 2¼ ¬ 2¼ ¬ 2¼
The combining of equations (C.17) yields
U U U
ª º ª º ª º
1 2 2
>@A>A@ >A@ (C.18)
1 2
« » « » « »
I I I
¬ 1¼ ¬ 2 ¼ ¬ 2 ¼
where >@A >A@>A@.
1 2
The matrix A is hence obtained as a product between the chain matrices of the two-ports to
be chained.
The turns ratio of the transformer in Figure C.2 can be rewritten as
c
1 U I Z
2 2 01
K
'
n U Z
I
2 02
and the transfer equation of the transformer is obtained in the matrix form
ª º
Z
« »
c
U U U
ª º Z ª º ª º
2 2 2
« »
>@A (C.19)
« » « » 2 « »
« »
c
I I I
Z
¬ 2 ¼ ¬ 2 ¼ ¬ 2 ¼
« »
Z
« »
¬ ¼
In accordance with equation (C.13), the chain matrix A of a symmetrical two-port is equal to
coshī Z sinhī
ª º
« »
>@A (C.20)
sinhī coshī
« »
Z
¬ 01 ¼
– 18 – TR 62152 ” IEC:2004(E)
The matrix A thus becomes
ªºª º
ZZ Z
01 01 02
0 coshī Z sinhī
ªºcoshī Z sinhī«»« 01 »
ZZ Z
(C.21)
02 02 01
«»« »
«»
>@AA > @>A@ 1
«»« »
«»
sinhīīcosh
ZZ1 Z
02 01 02
«»« »
«»Z
0sinhīcoshī
¬¼01
«»ZZ« Z Z »
01 01 02 01
¬¼¬ ¼
The transfer equations of an impedance-unsymmetrical two-port can be written in the matrix
form as follows
ª º
Z Z
01 02
coshī Z sinhī
« »
U U
ª º Z Z ª º
1 2
02 01
« »
(C.22)
« » « »
« »
I I
1 Z Z
¬ 1¼ ¬ 2 ¼
01 02
sinhī coshī
« »
Z Z Z
« »
01 02 01
¬ ¼
This matrix equation can also be solved for U and I .
2 2
U U
ª º ª º
1
2 1
>@A (C.23)
« » « »
I I
¬ 2 ¼ ¬ 1¼
ª º
Z Z
02 02
coshī Z sinhī
« 01 »
U U
ª º Z Z ª º
2 1
01 01
« »
(C.24)
« » « »
« »
I 1 Z Z I
¬ 2¼ 01 01 ¬ 1¼
sinhī coshī
« »
Z Z Z
01 02 02
¬ ¼
From the matrix equations (C.22) and (C.24), we can obtain the following transfer equations
for an impedance-unsymmetrical two-port:
Z Z
01 02
U U coshī Z I sinhī (C.25)
1 2 01 2
Z Z
02 01
Z 1 Z
02 01
I I coshī U sinhī (C.26)
1 2 2
Z Z Z
01 01 02
Z Z
02 02
U U coshī Z I sinhī (C.27)
2 1 01 1
Z Z
01 01
Z Z
01 01
I I coshī U sinhī (C.28)
2 1 1
Z Z Z
02 01 02
The end results obtained can also be obtained direct from the transfer equations of an
impedance-symmetrical two-port on the basis of Figure C.2.
TR 62152 ” IEC:2004(E) – 19 –
By solving equations (C.25), (C.26), (C.27) and (C.28,) currents can be expressed by means
of voltages or vice-versa, resulting in the following expressions:
U U Z
1 2 01
I cothī (C.29)
Z Z Z sinhī
01 01 02
U Z U
1 01 2
I cothī (C.30)
Z Z sinhī Z
01 02 02
Z
U Z I cothī Z I (C.31)
1 01 1 01 2
Z sinhī
Z 1
U Z I Z I cothī (C.32)
2 01 1 02 2
Z sinhī
NOTE A short reminder on matrices:
M K * M
2 1
1
M K * M
1 2
When M = K *M , where M , M and K are matrices,
2 1 1 2
-1 -1
then M = K *M where K is the inverse matrix of K.
1 2 ,
When
A B
ª º
K
« »
C D
¬ ¼
the inverse is
1
ªA Bº 1 ª D Bº
1
K *
« » « »
C D ' C A
¬ ¼ ¬ ¼
where the determinant is ' AD BC .
C.4 The symmetries and impedances of a two-port
Let us examine the two two-ports illustrated in Figures C.4 and C.5.
I I
1 2
I I
1 2
U U
1 Z Z Z Z 2
U U
A 01 02 B
1 Z Z Z Z 2
A 0 0 B
Z Z
1 2
Z Z
1 2
IEC 1188/04
IEC 1187/04
Figure C.4 – An impedance-symmetrical Figure C.5 – An impedance-unsymmetrical
two-port with Z = Z , when Z = Z
two-port for which Z z Z when Z = Z
1 2, A B
1 2 A B
– 20 – TR 62152 ” IEC:2004(E)
The two-port in accordance with Figure C.4 is referred to as impedance-symmetrical or port-
symmetrical, while the two-port of Figure C.5 is called an impedance-unsymmetrical or port-
unsymmetrical.
If the complex composite loss (see Clause C.7) in the direction A->B is equal to that in the
direction B->A for any values of generator and terminating impedance, then the two-port is
referred to as transfer-symmetrical or reciprocal. Two-ports that consist of passive
components (except gyrators) are always reciprocal. A two-port with none of its properties
depending on the direction of transmission is both reciprocal and impedance-symmetrical.
Such a two-port is referred to as longitudinally symmetrical. The input terminals of a two-port
are earth-symmetrical, if the admittances measured at each input terminal relative to earth are
equal. In this case we speak of transversal symmetry of the two-port [3] .
In addition to the complex image attenuation * = A + jB, there is another characteristic
quantity for a two-port, that is, the image impedance. The image impedances Z and Z of a
01 02
two-port in accordance with Figure C.5 can be determined by means of short-circuit and open-
circuit measurements:
Z Z Z Z Z Z
01 1k 1t 02 2k 2t
where the subscripts k and t refer to the short-circuit and open-circuit conditions, respectively.
Let us recall the equations (C.7) and (C.8) valid for a longitudinally symmetrical two-port:
U U coshī Z I sinhī
1 2 0 2
(C.33)
U
I I coshī sinhī
1 2
Z
Taking into account that U = Z I (see Figure C.6), we obtain with equations (C.33) the input
2 B 2,
impedance of the two-port:
U Z Z tanhī
1 B 0
Z Z (C.34)
1 0
I Z Z tanhī
1 0 B
I I
1 2
*
U
1 Z U
B 2
Z
Z
IEC 1189/04
Figure C.6 – A two-port terminated with an impedance Z
B
Hence, the input impedance Z depends on the properties of the two-port as well as on the
terminating load impedance Z . It can be shown that when the attenuation A is high, Z is only
B 1
slightly affected by Z . From equation (C.34), we see that Z | Z , when tanh |* | |1, i.e. when
B 1 0
A >2 Np. The input impedance is then solely determined by the properties of the two-port. A
two-port is called electrically short, when A << 2 Np and B << S/2, and correspondingly
electrically long, when A t 2 Np and BtS/2.
.
———————
Numbers in square brackets refer to the reference documents at the end of this Annex.
TR 62152 ” IEC:2004(E) – 21 –
When the output is short-circuited (Z = 0), we have
B
Z Z tanhī (C.35)
1k 0
If Z = 0, but AÆ 0 and B = S/2, then Z Æ f. This applies, for example, to a lossless short-
B 1k
circuited line with length O/4. When the output is open (Z = f), we have
B
Z Z (C.36)
1t 0
tanhī
Further, when AÆ 0 and BÆ S/2, then Z Æ 0. Additionally we have * = jB. Replacing B by
1t
S/2, the equation (C.34) can then be written in the following form:
Z
Z (C.37)
Z
B
Using the conversion (C.37), the impedance Z can be transformed into an impedance Z .
B 1
This is only feasible at the exact frequency for which the length of the lossless line is O/4,
corresponding to a so-called quarter-wavelength transformer. Equations (C.35) and (C.36)
reveal that also the Z and * of a longitudinally symmetrical two-port can be determined from
the short-circuit and open-circuit impedances.
C.5 Impedance matching
If the image impedances of the two-ports to be cascaded differ from each other, reflections
will be generated at the interconnection points, and those reflections then affect the uniformity
of transmission. In telecommunication engineering, to avoid reflections in transmission, it is
important that the impedances of the consecutive sections included in a transmission path are
carefully matched to each other, i.e. the characteristic impedances of the devices to be
cascaded shall very closely equal each other. A non-distorted transmission will only be
possible under such conditions. However, it should be noted that one single major mismatch
can be allowed within each repeater section; for example, provided that all other mismatches
are small enough, because at least two mismatches are required for the generation of a
propagating, signal-distorting forward-echo.
By substituting the quantities U = I Z which correspond to a proper matching (Z = Z ) into
2 2 0, B 0
equations (C.33), we obtain
* *
U U e I Z e (C.38)
1 2 2 0
*
I I e (C.39)
1 2
from which it follows that the input impedance is
*
U I Z e
1 2 0
Z Z (C.40)
1 0
*
I I e
1 2
Hence the input impedance is under these conditions independent of *.
– 22 – TR 62152 ” IEC:2004(E)
Correct matching enables the greatest possible complex power to be transmitted from a
generator to the load. (In the literature, the term complex power often refers to the quantity
*
UI , while the quantity UI is called the apparent power. In transmission engineering, it is
logical to use the term complex power to denote the product of voltage and current phasors.)
Hence,
P UI (C.41)
Z
g
Z = R + jX
g g g
E U Z
1 p
Z = R + jX
E U
1 p p p
IEC 1190/04 IEC 1191/04
Z = Z Z = Z * or R = R with X = X = 0
g p g p g p g p
Figure C.7 – Reflection less Figure C.8 – Power matching for
matching maximizing the effective power.
The complex power obtained with the load Z is
B
E Z
p
P (C.42)
(Z Z )
g p
which reaches a maximum when Z = Z , which yields
P g
E
P (C.43)
max
4Z
g
With Z = R + jX and Z = R + jX the greatest possible effective power is absorbed by the
g g g p p p
load when R = R and j(X + X ) = 0. The condition is met when both imaginary parts are
g p g p
*
zeros, or when the impedances are complex conjugates, i.e. Z = Z . This kind of matching is
g p
called power matching. It is commonly used when matching transmitters to antennas, but,
being normally valid at a single frequency only (the tuning frequency), it has found no
applications in broad-band transmission techniques. Even a two-port (its output or input,
respectively) can be considered as a power source or a load.
The input or output impedance of a two-port can be built out in such a way as to be resistive
while being independent of frequency, under the condition that it is represented by a series
combination of R and L, or R and C. For example, if an impedance R + 1/jZC is connected in
parallel with an impedance R + jZL by choosing C = L/R , a frequency-independent resistive
impedance R will be obtained.
TR 62152 ” IEC:2004(E) – 23 –
C.6 Level concepts
The term level is used to indicate a relative or an absolute value. If the power, voltage or
current along a transmission system is concerned, one speaks of power, voltage or current
levels.
When comparing the power, voltage or current at a measuring point with the respective
quantity at the feeding point of the transmission system, we are concerned with a relative
level, whereas, when the comparison is made to a standardized reference value, an absolute
level will be obtained.
Levels are commonly expressed in decibels (dB), more seldom in nepers (Np). The use of
nepers is actually restricted to some theoretical calculations. The units are related by
1 dB = 0,05/lg e = 0,1151 Np or 1 Np = 20 lg e = 8,686 dB
If P and V denote the power and the voltage at the measuring point, while P and V are the
x x A A
corresponding values at the feeding point (input) of the system, the relative power level is
P 1 P
x x
N 10 lg >@dB ln >@Np (C.44)
P 2 P
A A
and the relative voltage level is
V V
x x
N 20 lg >@dB ln >@Np (C.45)
v
V V
A A
The relative level at the input of the system is always zero.
If P and V are the standardized reference values, the absolute power level is given by
1 1
P 1 P
x x
N 10lg >@dB ln >@Np (C.46)
P 2 P
1 1
while the absolute voltage level is
V V
x x
N 20lg >@dB ln >@Np (C.47)
v
V V
1 1
In telecommunication engineering, the reference for absolute power levels is 1 mW and the
reference for absolute voltage levels is 0,775 V, which corresponds to 1 mW in a 600 : load.
Nowadays, voltage levels are seldom
...
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