IEC TR 62152:2009

Transmission properties of cascaded two-ports or quadripols - Background of terms and definitions

IEC/TR 62152:2009(E) 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. This second edition cancels and replaces the previous one and introduces some technical improvements. Important terms and definitions have been added. Some of the terms are better described in the German language and therefore many countries have originally taken terms and definitions from German and translated them into their own language.

General Information

• Status: Published
• Publication Date: 13-Dec-2009
• ICS: 31.020 - Electronic components in general
• Technical Committee: TC 46 - Cables, wires, waveguides, RF connectors, RF and microwave passive components and accessories
• Drafting Committee: WG 5 - TC 46/WG 5
• Current Stage: PPUB - Publication issued
• Start Date: 14-Dec-2009
• Completion Date: 15-Jan-2010

Relations

• Revises: IEC TR 62152:2004 - Background of terms and definitions of cascaded two-ports
• Effective Date: 05-Sep-2023

IEC TR 62152:2009 - Overview

IEC TR 62152:2009 is a Technical Report that establishes a common vocabulary and conceptual bridge for describing the transmission properties of two-ports (quadripoles). It clarifies fundamental terms and definitions used worldwide for cascaded two-ports and links the classical quadripole theory with the scattering matrix (S‑matrix) presentation based on incident and reflected square‑root‑of‑power waves. This second edition (2009) replaces the 2004 edition and adds technical improvements and important terms (some supplied with German equivalents).

Key topics

• Terms and definitions: precise definitions for key quantities such as complex operational attenuation, complex operational insertion loss, image attenuation and related symbols and units.
• Transfer functions: presentation of operational, image and insertion transfer functions for a two‑port and their roles in cascade analysis.
• Complex attenuation/loss: definitions and interpretation of complex attenuation (magnitude in dB and phase in radians) for operational, image and insertion cases.
• Power‑wave formalism: normalized voltage waves and square‑root‑of‑power waves, linking S‑matrix parameters to classical two‑port (quadripole) quantities.
• Measurement context: how reference impedances, matched conditions and terminations affect measured insertion loss and attenuation (including insertion loss deviation).
• Annexes: normative material covering normalized waves, image transmission parameters, two‑port theory and transmission engineering concepts with illustrative figures (Smith chart, reflection, matching, etc.).

Practical applications

• Provides a single reference for engineers and test laboratories to consistently describe and measure transmission behaviour of components inserted into cascaded systems.
• Useful for RF/microwave engineers, telecommunications system designers, cable and connector manufacturers, and conformity assessment bodies who need uniform terminology for:
  • Specifying insertion loss and return loss,
  • Converting between scattering parameters and classical two‑port parameters,
  • Designing and testing matched interfaces and cascade chains,
  • Interpreting measurement results where reference impedances or terminations differ.
• Supports standardization of test procedures and interchange of measurement data between vendors and labs.

Related standards

• IEC 60050‑726 - International Electrotechnical Vocabulary: Transmission lines and waveguides
• IEC 61156‑1 - Multicore and symmetrical pair/quad cables for digital communications - Part 1
• IEC/TR 61156‑1‑2 - Electrical transmission characteristics and test methods for symmetrical pair/quad cables

Keywords: IEC TR 62152, two‑port, quadripole, transmission properties, scattering matrix, transfer functions, complex attenuation, insertion loss, power waves, scattering parameters.

Standards Content (Sample)

IEC/TR 62152 ®
Edition 2.0 2009-12
TECHNICAL
REPORT
colour
inside
Transmission properties of cascaded two-ports or quadripols - Background of
terms and definitions
IEC/TR 62152:2009(E)
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IEC/TR 62152 ®
Edition 2.0 2009-12
TECHNICAL
REPORT
colour
inside
Transmission properties of cascaded two-ports or quadripols - Background of
terms and definitions
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
PRICE CODE
XB
ICS 31.020 ISBN 978-2-88910-636-3
– 2 – TR 62152 © IEC:2009(E)
CONTENTS
FOREWORD.4
1 Scope.6
2 Normative references .6
3 Terms, definitions, symbols, units and abbreviated terms .6
3.1 Definitions .6
3.2 Terms, symbols, units and abbreviated terms.7
4 Transfer functions and complex attenuations or losses of a two-port .10
4.1 General remarks.10
4.2 Operational transfer function ( T ) .11
B
4.2.1 Image transfer function ( T ) .11
′
4.2.2 Insertion transfer function ( T ) .11
B
4.3 Complex attenuation .11
4.3.1 Complex operational attenuation (Γ )(Komplexe Betriebs-Dämpfung) .11
B
4.3.2 Complex image attenuation (Γ )(Komplexe Wellen-Dämpfung) .12
′
4.3.3 Complex insertion attenuation or loss (Γ )(Komplexe Einfüge-
B
Dämpfung).12
Annex A (normative) Concepts of normalized voltage waves, square root of power
waves and operational attenuation and losses .13
Annex B (normative) Image transmission parameters/quantities of a two-port and
transmission line approximations .22
Annex C (normative) Two-port theory and fundamental concepts in transmission
engineering.24
Bibliography.61

Figure 1 – Defining the transfer functions of a two-port .11
Figure A.1 – Reflection at a junction .16
Figure A.2 – Constant value A and A curves on a complex plane z = x + jy.16
s r
Figure A.3 – Two-port representation of a transmission line.17
Figure A.4 – Coupling between two systems .19
Figure C.1 – A two-port or quadripole .24
Figure C.2 – An impedance-unsymmetrical two-port (a) with its equivalent circuit (b).26
Figure C.3 – Two chained two-ports.27
Figure C.4 – An impedance-symmetrical two-port with Z = Z , when Z = Z .29
1 2 A B
Figure C.5 – An impedance-unsymmetrical two-port for which Z ≠ Z when Z = Z .29
1 2 A B
Figure C.6 – A two-port terminated with an impedance Z .30
B
Figure C.7 – Reflection loss matching.32
Figure C.8 – Power matching for maximizing the effective power .32
Figure C.9 – Absolute and nominal level in a system .34
Figure C.10 – Definition of the complex image attenuation Γ of a two-port .34
Figure C.11 – Definition of the complex operational attenuation of a two-port .35
Figure C.12 – Definition of residual attenuation.36
Figure C.13 – Measurement of the sending reference equivalent .37
Figure C.14 – Measurement of the receiving reference equivalent .37

TR 62152 © IEC:2009(E) – 3 –
Figure C.15 – Definition of the complex return loss .38
Figure C.16 – Apollonius’ circle.39
Figure C.17 – Return loss .40
Figure C.18 – Curves for constant values of A or A in the complex plane .42
s r
Figure C.19 – Curves for constant values of A or A in the complex plane .43
s r
Figure C.20 – Smith chart for transmission lines .44
Figure C.21 – A one-port .45
Figure C.22 – Homogenous transmission line .46
Figure C.23 – One-port fed from a generator with source impedance Z .47
g
Figure C.24 – A two-port.49
Figure C.25 – Termination Z by virtue of the scattering parameters of the two-port .50
B
Figure C.26 – Ideal transformer .53
Figure C.27 – Scattering matrix of a passive reciprocal two-port .55
Figure C.28 – A two-port.58
Figure C.29 – The insertion of a two-port into a network and the deviations caused to
the operational parameters. .59
Figure C.30 – Power wave schematic for S and T matrix parameters .60

Table B.1 – Transmission quantities of a two-port and homogeneous transmission line .23

– 4 – TR 62152 © IEC:2009(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
international co-operation on all questions concerning standardization in the electrical and electronic fields. To
this end and in addition to other activities, IEC publishes International Standards, Technical Specifications,
Technical Reports, Publicly Available Specifications (PAS) and Guides (hereafter referred to as “IEC
Publication(s)”). Their preparation is entrusted to technical committees; any IEC National Committee interested
in the subject dealt with may participate in this preparatory work. International, governmental and non-
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with the International Organization for Standardization (ISO) in accordance with conditions determined by
agreement between the two organizations.
2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international
consensus of opinion on the relevant subjects since each technical committee has representation from all
interested IEC National Committees.
3) IEC Publications have the form of recommendations for international use and are accepted by IEC National
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the latter.
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8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
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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
46/283/DTR 46/300/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.

TR 62152 © IEC:2009(E) – 5 –
This second edition cancels and replaces the first edition published in 2004 and constitutes
some technical improvements.
Important terms and definitions have been added.
Some of the terms are better described in the German language and also many countries
have originally taken terms and definitions from German and translated them into their own
language.
Therefore important terms have been added in German in the form of a footnote.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
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.

IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates that it
contains colours which are considered to be useful for the correct understanding of its
contents. Users should therefore print this document using a colour printer.

– 6 – TR 62152 © IEC:2009(E)
BACKGROUND OF TERMS AND DEFINITIONS
OF CASCADED TWO-PORTS
1 Scope
It is important and practical that components of a transmission chain can be separated and
tested separately. To accomplish this, well-defined interfaces and measuring techniques,
including agreed terms and definitions, are required.
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. The report 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 Normative references
The following referenced documents are indispensable for the application of this document.
For dated references, only the edition cited applies. For undated references, the latest edition
of the referenced document (including any amendments) applies.
IEC 60050-726, International Electrotechnical Vocabulary – Chapter 726: Transmission lines
and waveguides
IEC 61156-1, Multicore and symmetrical pair/quad cables for digital communications – Part 1:
Generic specification
IEC/TR 61156-1-2, Multicore and symmetrical pair/quad cables for digital communications –
Part 1-2: Electrical transmission characteristics and test methods of symmetrical pair/quad
cables used for digital communications
3 Terms and definitions, symbols, units and abbreviated terms
For the purposes of this document, the terms and definitions given in IEC 60050-726,
IEC 61156-1, IEC/TR 61156-1-2, as well as the following defintions, apply.
3.1 Terms and definitions
3.1.1
complex operational attenuation
quotient of the unreflected square root of the power wave fed into the reference impedance
R of the input of the two-port and the square root of the power wave consumed by the load
R of the two-port expressed in dB and radians
———————
Komplexe Betriebs-Dämpfung.
TR 62152 © IEC:2009(E) – 7 –
NOTE By defining a new quantity, operational insertion loss, which is the same as the operational attenuation
when the reference impedances on both sides of the two-port are the same R = R , the problem of insertion loss
1 2
and operational attenuation is solved in most usual cases.
3.1.2
complex operational insertion loss
quotient of the unreflected square root of the power wave fed into the reference impedance R
of the measurement system and the square root of the power wave consumed by the load R
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 impedances along the transmission line, it leads to deviation in the
overall losses depending on where in the chain the two-port is inserted. This is called insertion loss deviation (ILD).
In “complex operational insertion loss” the reference impedances at both sides of the two-port are equal.
3.2 Symbols, units and abbreviated terms
3.2.1 Two-port electrical symbols, units and related terms
E generator source voltage (V)
R ,R reference impedance at the two-port input and output, respectively (Ω)
1 2
R reference impedance at the two-port input and output, respectively (Ω)
U ,U voltage at the two-port input and output, respectively (V)
1 2
U voltage at the reference impedance for the condition of matched generator reference
impedance (V)
Z ,Z complex characteristic impedance at the two-port input and output, respectively
01 02
½
P square root of power wave from the two-port (W )
P unreflected square root of power wave from the generator for the condition of
½
matched generator reference impedance (W )
P reflected square root of power wave coming from the reference impedance at the
½
two-port output (W )
T operational transfer function
B
T image transfer function
T′ insertion transfer function
B
S forward transfer scattering parameter
Γ complex operational attenuation
B
A real part of Γ and is the operational attenuation
B B
A = − 20 × log S (dB) or
B 10 21
A = − ln S (Np)
B 21
B imaginary part of Γ and is the operational attenuation phase shift
B B
= − arg()S (rad)
Γ′ complex insertion attenuation or loss
B
A′ real part of Γ′ (dB) or (Np)
B B
B′ imaginary part of Γ′ (rad)
B B
Γ complex image attenuation
———————
Komplexe Betriebs-Einfüge-Dämpfung.

– 8 – TR 62152 © IEC:2009(E)
A real part of Γ (dB) or (Np)
B imaginary part of Γ (rad)
j imaginary denominator
arg argument operator of a complex number
Z
Z complex characteristic impedance, or mean characteristic impedance if the pair is
,
C 0
homogeneous or free of structure (also used to represent a function fitted result) (Ω)
Z
nominal characteristic impedance and resistive part of the mean characteristic
CN
Z
impedance value at a given frequency with tolerance at a given frequency (Ω)
C
Z nominal impedance of the link and/or terminals (the system) between which the two-
N
port is operating (Ω)
Z Z Z
(nominal) reference impedance used in measurements, normally, = . (Ω)
R R N
RL complex operational return loss (dB)

ρ reflection coefficient
B
SRL structural return loss (dB)
Z
measured input image impedance (Ω)
W
Re real part operator for a complex variable
Im imaginary part operator for a complex variable
R pair resistance (Ω/m)
L pair inductance (H/m)
L pair inductance asymptotic value at high frequencies (H/m)
∞
G pair conductance (S/m)
C pair capacitance (F/m)
v phase velocity of cable (m/s)
P
ω radian frequency (rad/s)
l length (m)
Δf frequency difference between input impedance minima of a short-circuited
transmission line (MHz)
S, ρ complex reflection coefficient of the junction
½
P reflected square root of power wave at the junction (W )
r
½
P incident square root of power wave at the junction (W )
i
Z , Z line impedance to the left and right of the junction, respectively (Ω)
1 2
U ,U incident and reflected voltage at the junction, respectively (V)
i r
V ,V incident and reflected voltage at the junction, respectively (V)
i r
I ,I incident and reflected current at the junction, respectively (A)
i r
Γ complex reflection loss at the junction
s
A reflection loss
s
z + 1
N
A = 20 × log (dB)
s 10
2 × z
N
A return loss
r
z + 1
N
A = 20 × log (dB)
r 10
z −1
N
TR 62152 © IEC:2009(E) – 9 –
Z
z =
z normalized impedance given by = r + jx
N N
Z
r x-axis ordinate
x y-axis ordinate
Γ mismatch loss of a junction (not recommended)
m
3.2.2 Transmission line equation electrical symbols and related terms
α attenuation coefficient (Np/m)
β phase coefficient (rad/m)
γ propagation coefficient (Np/m, rad/m)
νP phase velocity of cable (m/s)
νG group velocity of cable (m/s)
τP phase delay time (s/m)
τG group delay time (s/m)
Z
complex characteristic impedance, or mean characteristic impedance if the pair is
C
homogeneous or free of structure (also used to represent a function fitted result) (Ω)
∠Z angle of the characteristic impedance in radians
C
Z high frequency asymptotic value of the characteristic impedance (Ω)
∞
l length (m)
ω radian frequency (rad/s)
f frequency (Hz)
R’ first derivative of R with respect to ω
C’ first derivative of C with respect to ω
L’ first derivative of L with respect to ω
R d.c. resistance of a round solid wire with radius r (Ω/m)
R constant with frequency component of resistance which is about one-quarter of the
C
d.c. resistance (Ω/m)
R square-root of frequency component of resistance (Ω/m)
S
L external (free space) inductance (H/m)
E
L internal inductance whose reactance equals the surface resistance at high
I
frequencies (H/m)
σ specific conductivity of the wire material (S/m)
ρ resistivity of the wire material (Ω/m )
μ permeability of the wire material (H/m)
r radius of the wire (m)
δ skin depth (not to be confused with the dissipation factor tan δ) (m)
δ =
π f μσ
tan δ dissipation factor
tan δ = G/(ωC)
q forward echo coefficient at the far end of the cable at a resonant frequency

– 10 – TR 62152 © IEC:2009(E)
p reflection coefficient measured from the near end of the cable at a resonant
Z − Z
CM C
−PSRL / 20
frequency, p = 10 =
Z + Z
CM C
A forward echo attenuation at a resonant frequency (dB)
Q
A = −20 × log q
Q 10
PSRL structural return loss at a resonant frequency (dB),
PSRL = −20 × log p
K = 2 × αl – 1 when 2 × α l >> 1 (Np)
A = 2 × PSRL − 20 × log()2 × α l − 1 (dB) where 2 × α l is in Np
Q 10
Z complex measured open circuit impedance (Ω)
OC
Z complex measured short circuit impedance (Ω)
SC
Z characteristic impedance as measured (with structure) (Ω)
CM
Z = Z Z
CM SC OC
Z  complex measured impedance (open or short) (Ω)
MEAS
Z input impedance of the cable when it is terminated by Z (Ω)
IN L
Z output impedance of the cable when the input of the cable is terminated by Z (Ω)
OUT G
Z terminated impedance measurement made with the opposite end of the cable pair
T
terminated in the reference impedance Z (Ω)
R
−
Z Z
R C
ς reflection coefficient measured in the terminated measurement method ς =
+
Z Z
R C
Z
termination at the cable input when defining the output impedance of the cable Z (Ω)
G
OUT
Z
termination at the cable output when defining the input impedance of the cable Z (Ω)
L
IN
L , L , L , L least squares fit coefficients for angle of the characteristic impedance
0 1 2 3
K , K , K , K least squares fit coefficients of the characteristic impedance
0 1 2 3
⎟Z ⎟ fitted magnitude of the characteristic impedance (Ω)
C
⎟Z ⎟ measured magnitude of the characteristic impedance (Ω)
CM
∠ (V ) input angle relative to a reference angle in radians
1N
∠ (V ) output angle relative to the same reference angle in radians
1F
k multiple of 2π radians;
S
reflection coefficient measured with an S parameter test set

4 Transfer functions and complex attenuations or losses of a two-port
4.1 General remarks
Figure 1 indicates the variables and their relationships for defining the transfer functions of a
two-port. E is the generator source voltage in Figure 1.
TR 62152 © IEC:2009(E) – 11 –
R
1 P
U
Z Z 2 R
E U 01 02
0 1
P
R
E
R
E U
0 1
P
IEC  1181/04
Figure 1 – Defining the transfer functions of a two-port

4.2 Operational transfer function ( T )
B
Referring to Figure 1, the operational transfer function T is defined as the ratio of the square
B
root of the power wave into the load (equal to reference impedance R ) of a two-port P
2 2
with the unreflected square root of power wave P from the generator with a source
impedance equal to the reference impedance R . See Equation (1).
P U R P
2 2 2 2
T = = = S = (1)
B 21
P U R P
0 0 1 0
P =0
4.2.1 Image transfer function ( T )
The operational transfer function becomes the image transfer function T when the reference
impedance becomes equal to the input and output characteristic impedances Z and Z of
01 02
the two-port.
4.2.2 Insertion transfer function ( T ′ )
B
The operational transfer function becomes the insertion transfer function T ′ when
B
R = R = R .
1 2
4.3 Complex attenuation
4.3.1 Complex operational attenuation (Γ )
B
The complex operational attenuation is given by Equation (2):
———————
Komplexe Betriebs-Dämpfung.
– 12 – TR 62152 © IEC:2009(E)
()
Γ = A + j ⋅ B = ln = −20 × log T − j ⋅ arg T (2)
B B B 10 B B
T
B
4.3.2 Complex image attenuation (Γ )
The complex image attenuation is given by Equation (3):
Γ = A + j ⋅ B = ln = −20 × log T − j ⋅ arg()T (3)
T
′
4.3.3 Complex insertion attenuation or loss (Γ )
B
The complex insertion attenuation or loss is given by Equation (4):
Γ ′ = A′ + j ⋅ B′ = ln = −20 × log T ′ − j ⋅ arg()T ′ (4)
B R1 =R2 = R B B 10 B B
′
T
B
———————
Komplexe Wellen-Dämpfung.
Komplexe Einfüge-Dämfung.
TR 62152 © IEC:2009(E) – 13 –
Annex A
(normative)
Concepts of normalized voltage waves, square root of power waves
and operational attenuation and losses

A.1 General
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 the scattering parameters are defined. For example, S is the forward operational
transfer function and S is the operational reflection coefficient.
Two primary 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 + j ⋅ y = z ⋅ e is directly
ln()z = ln z + j ⋅ arg()z and ln z , in nepers, can be expressed in decibels 20 × log z and
the imaginary part still remains arg(z) in radians, as, for example,
Γ = A + j × B = −20 × log S − j × arg()S
B B B 10 21 21
(see Equations (A.1) and (A.2)).
Furthermore, usage of operational quantities means the measurements are always made
between resistive terminations in well-defined circumstances.
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.
A.2 Complex operational attenuation or operational propagation
coefficient (Γ )
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
Γ = A + j ⋅ B = ln()1 S = −ln S − j ⋅ arg()S (A.1)
B B B 21 21 21
Γ = A + j ⋅ B = −20 × log S − j ⋅ arg()S (A.2)
B B B 10 21 21
– 14 – TR 62152 © IEC:2009(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, complex
N
operational attenuation becomes complex operational insertion loss.
NOTE 2 From the theory of complex functions:
ln z = ln z + j ⋅arg()z
where
j⋅arg()z
z = x + j ⋅ y = 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 =Γ = A + j ⋅ B
⎜ ⎟
P P P
2 2 2
⎝ ⎠
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 Impedances
The different kinds of impedances are defined as follows:
a) the nominal characteristic impedance Z (of a two-port) is the resistive part of the mean
CN
characteristic impedance Z specified with a 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 (S )
The operational reflection coefficient of a two-port is equal to the scattering parameter S of
the two-port. It equals the reflection coefficient ρ at the input when the two-port is
B
terminated with its reference impedances Z normally equal to the nominal impedance of the
R1
system terminals.
Z − Z
in R1
S = ρ = (A.3)
11 B
Z + Z
in R1
A.5 Return loss
A.5.1 Complex operational return loss of a transmission line (RL )
B
The complex operational return loss, RL of a transmission line is given in Equation (A.4):
B
———————
Nominale Wellen-Widerstand.
TR 62152 © IEC:2009(E) – 15 –
RL = ln = −ln()ρ = −ln ρ − j ⋅arg()ρ
B B B B
ρ
B
(A.4)
= −20 ×log ρ − j ⋅arg()ρ
10 B B
A.5.2 Structural return loss of a transmission line ( SRL )
SRL is the return loss where the mismatch effects at the input and output of transmission line
have been eliminated (compare with the continuous wave (CW) burst measurement method).
This quantity is obtained by calculation (Equation (A.5)) using the measured input image input
impedance and measured, calculated (Equation (A.6)) and curve fitted mean characteristic
impedance where both are complex quantities. The structural return loss is as follows:
⎛ ⎞
Z − Z Z − Z
W 0 W 0
⎜ ⎟
SRL = −20 ×log − j ⋅ arg (A.5)
⎜ ⎟
Z + Z Z + Z
W 0 ⎝ W 0⎠
See Clauses A.6 and A.7.
The complex characteristic impedance of a homogeneous transmission line is as follows:
L R L R
⎛ ⎞
∞ ∞
Z = Re()Z + j ⋅Im()Z ≈ ⎜1+ ⎟ − j ⋅ (A.6)
0 0 0
C 2ωL C 2ωL
⎝ ⎠
L R 1 1
⎛ ⎞
∞
Re()Z ≈ ⎜1+ ⎟ ≈ = (A.7)
C 2ωL v C 2Δf l C
⎝ ⎠
p
L R L
⎛ ⎞
∞ ∞
-Im()Z ≈ ⎜ ⎟ = Re()Z − (A.8)
0 0
C 2ωL C
⎝ ⎠
where v and l are the phase velocity and length of the transmission line and C the
p
capacitance of the low dielectric loss line measured at such a low frequency that the length is
electrically short, l < λ/40. Δf is the distance in frequency between two input impedance
minima of the short-circuited measured transmission line; and L the asymptotic value of the
∞
inductance reached at high frequencies. See Annex B.
NOTE It is important to distinguish between the two return losses RL and SRL although they are normally not
measured separately.
A.5.3 Reflection loss of a junction
The quantities that determine the reflection loss of a junction are shown in Figure A.1.
Normalized plots of reflection loss and return loss are given in Figure A.2.

P →
i
← P
r
———————
Z .
Eingangs- Wellen-Widerstand,
W
Mitlerer Wellen-Wiederstand, Z .
Komplexe Wellen-Wiederstand.
– 16 – TR 62152 © IEC:2009(E)
________________|______________
Z                 Z
1 2
Figure A.1 – Reflection at a junction
P V U I Z − Z
r r r r 2 1
S = ρ = = = = − = (A.9)
P V U I Z + Z
i i i i 2 1
S is the complex reflection coefficient of the junction. See Annex C.
2 2 ⎛ 2 ⎞
Γ = −ln()1− S = −ln()1− S − j ⋅arg()1− S (A.10)
⎜ ⎟
S
⎝ ⎠
or
2 2 ⎛ 2 ⎞
Γ = −ln()1− S = −20 ×log ()1− S − j ⋅ arg()1− S (A.11)
⎜ ⎟
S 10
⎝ ⎠
2 2 2
Γ = −ln()1− S = −10 ×log()1− S − j× × arg()1− S (A.12)
S 10
r
0 0,5 1,0 1,5 2,0 2,5
2,0
1,0
0,5
−0,5
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
Key
z +1
N
A is the reflection loss given by A = 20 ×log expressed in dB;
s s 10
2 × z
N
z +1
N
A is the return loss given by A = 20 ×log expressed in dB;
r r 10
z −1
N
Z
z is the normalized impedance given by z = = r + jx  (see Figure A.1);
N N
Z
r is the x-axis ordinate;
x is the y-axis ordinate.
Figure A.2 – Constant value A and A curves on a complex plane z = x + jy
s r
TR 62152 © IEC:2009(E) – 17 –
A.5.4 Mismatch loss of a junction (Γ )(not recommended)
m
Mismatch loss of a junction is expressed as a function of S the complex reflection coefficient ,as
follows:
2 2 ⎛ 2⎞
Γ = −ln()1− S = −ln()1− S − j⋅arg()1− S (A.13)
⎜ ⎟
m
⎝ ⎠
or
2 2 2
⎛ ⎞
Γ = −ln()1− S = −20 ×log ()1− S − j⋅arg()1− S (A.14)
⎜ ⎟
m 10
⎝ ⎠
2 2 1 2
Γ = −ln()1− S = −10 ×log()1− S − j× × arg()1− S (A.15)
m 10
A.6 Definition of the characteristic input impedance of a transmission line
(cable pair)
The important variables associated with the two-port representation of a transmission line are
given in Figure A.3. V and V are the incident and reflected square root of power waves.
i r
Z         Z
01 02
R
R
I I
1 2
Two-port
Z Z
U 1 2 U
E 1 2 E
1 2
V V
i1 i2
IEC  1207/04 A
V V
r1 r2
Key
E1, E2 network analyser at input, output, respectively Vi1, Vi2 incident square root of power waves at input
and output, respectively
R reference impedance at input and output V , V reflected square root of power waves at input
1 r1 r2
and output, respectively
I , I current at input and output, respectively Z , Z impedance at input and output, respectively
1 2 1 2
U , U voltage at input and output, respectively Z , Z characteristic input impedance or complex
1 2 01 02
image input impedance
Figure A.3 – Two-port representation of a transmission line
The characteristic input impedance or complex image input impedance is given by
Equation (A.16):
1+ S
W11
Z = R = Z ⋅ Z (A.16)
01 1 shortc openc
1− S
W11
———————
Komplexe eingangs-wellenwiderstand.

– 18 – TR 62152 © IEC:2009(E)
where
S is the the complex image reflection factor at the input when there are no reflections
W11
from the far-end, V = 0 (see note);
i2
Z is the the measured impedance at the input when the output is terminated in a short-
shortc
circuit;
Z is the the measured impedance at the input when the output is terminated in an open
openc
circuit.
The complex image reflection factor at the input may also be expressed by Equation (A.17):
Z − R
01 1
ρ = S = (A.17)
W11 W11
Z + R
01 1
where
ρ = S is the complex image reflection factor at the input when there are no reflections
W11 W11
from the far-end, ( V = 0 ) (see note).
i2
NOTE The condition V = 0 can be simulated by a long line terminated with its nominal impedance. When the
i2
roundtrip attenuation of the line added with the return loss at the far end is not less than 40 dB (see
Equation (A.18)) the maximum uncertainty in Z is less than 2 %. With more than 60 dB round-trip attenuation the
maximum uncertainty in Z is less than 0,2 %.
()2 ×α ⋅ L − 20 ×log ρ ≥ 40 (A.18)
10 W22
A.7 General coupling transfer function, crosstalk and echoes
A.7.1 General
Figure A.4 indicates the key variables and their relationships for defining the coupling transfer
function between two systems.
———————
Komplexe Eingangs-Wellen-Reflexions-Faktor.

TR 62152 © IEC:2009(E) – 19 –
P
P 1f
Z
(1)
U
Z
E
(2)
Z Z
2 2
P P
2n
2f
IEC  1183/04
Key
(1), (2) disturbing and disturbed systems, respectively P unreflected power sent into the near end
of the system (1)
E generator source voltage in system (1) U0 input voltage, system (1)
Z , Z terminations
1 2
P , P , P power in systems (1) and (2)
1f 2n 2f
Figure A.4 – Coupling between two systems
Coupling transfer functions T and T may be defined at the near and far ends, respectively
n f
(see Equation (A.5)).
P U Z
Z U
2n,f 2n,f 2n,f
1 2n,f
T = = = (A.5)
n,f
U
P U Z Z
0 0 1 2n,f
where
T is the complex coupling transfer function at near or far end;
n,f
n, f are the near end and far end, respectively;
U is the voltage at the near or far end of system (2);
2n,f
Z is the input impedance at the near or far end of system (2).
2n,f
The coupling transfer function is a general term that is valid through the whole frequency
range.
A.7.2 Transfer function for near-end and far-end crosstalk
This may be expressed in decibels and radians, e.g. near-end and far-end crosstalk
attenuation (NEXT and FEXT) as given in Equation (A.6).
⎛ ⎞
P P
2n,f 2n,f
⎜ ⎟
T = 20 ×log + j ⋅arg (A.6)
n,f 10
⎜ ⎟
P P
0 0
⎝ ⎠
where
– 20 – TR 62152 © IEC:2009(E)
P
2n,f
20 ×log is expressed in decibels;
P
⎛ ⎞
P
2n,f
⎜ ⎟
j ⋅ arg is expressed in radians.
⎜ ⎟
P
⎝ ⎠
A.7.3 Complex operational transfer attenuation
The complex operational transfer coupling function may be expressed for screening,
unbalance or crosstalk attenuations. See Equation (A.7).
Γ = A + jB = –20 log |T | – j arg(T ) (A.7)
x x x 10 x x
where
Γ is the complex operational attenuation;
x
A is the (operational ) attenuation (dB);
x
B is the (operational) attenuation phase shift (rad).
x
EL FEXT = Equal Level Far-End Crosstalk
= FEXT – Γ
B1
ACR-F =  Attenuation to Crosstalk Ratio in the Far-end (compare Signal to Crosstalk
Ratio and EL FEXT)
= FEXT – Γ
B2
Γ = (complex) operational attenuation of system (1)
B1
Γ = (complex) operational attenuation of system (2)
B2
AACR-F = Alien (exogenous) Attenuation to Crosstalk Ratio in the Far-end
PS = Power-Sum.
A.7.4 Complex image backward echo attenuation
Complex image backward echo attenuation is the quotient of the (unreflected) square root of
the power wave sent into the input of the two-port and the square root of the power wave due
to reflections received from the input expressed in dB and radians.
Compare with Clause A.6
A.7.5 Complex image forward echo attenuation
Complex image forward echo attenuation is the quotient of the (unreflected) square root of the
power wave (main signal) received from the output of a two-port and echo square root of the
power waves of the main signal from multiple reflection points following the main signal
expressed in dB and radians.
A.7.6 Backward and forward echo attenuation or loss of a transmission line
Γ = A + jB = Complex image backward echo attenuation or loss (structural return loss)
wr wr wr
(Complex image backward echo attenuation )
———————
Komplexe Wellen-Rückfluss-Dämpfung.
Komplexe Wellen-Mitfluss-Dämpfung.

TR 62152 © IEC:2009(E) – 21 –
Γ = A + jB = Complex image forward echo attenuation or loss
wq wq wq
(Complex image forward echo )
The relation between structural return loss A and forward echo loss A of regular and
wp wq
periodic reflections of a transmission line is
A ≈ 2 A – 20 log ( 2αL – 1 ) [dB]
wq wr
When 2 × (attenuation of the transmission line) = 2αL >> 1
2αL in neper [Np]. 1 dB = 0,115 Np
See [6] .
———————
References in square brackets refer to the Bibliography.

– 22 – TR 62152 © IEC:2009(E)
Annex B
(normative)
Image transmission parameters/quantities
of a two-port and transmission line approximations

B.1 General
The image transmission parameters/quantities of a two-port are defined for the condition of no
reflections at the input and output. This condition is achieved by terminating the input and
output with their image or characteristic impedance.
B.2 Image transfer function
The image transfer function and associated terms are given below:
P Z
U
OUT 01
OUT
a) Image transfer function T = = ⋅
U
P Z
IN
IN 02
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,
01 02
equal to the input and output impedances when the opposite port is terminated with its image
impedance.
P = U Z and P = U Z are the square roots of the complex input and output
IN IN 01 OUT OUT 02
powers.
B.3 Image quantities
The transmission quantities of a two-port corresponding to the secondary parameters of a
transmission line are shown in Table B.1. The corresponding high and low frequency
approximations for the secondary parameters of a homogenous transmission line are also
given.
When R/ωL and G/ωC are smaller than 0,1 at high frequencies, the deviations are smaller
ωL/R instead of R/ωL is smaller than
than 1 %. The same is valid for low frequencies when
0,1.
R, L, G and C are the primary parameters – resistance, inductance, conductance and
capacitance – per unit length of a homogenous transmission line. The secondary parameters
are then also per unit length
See [6].
TR 62152 © IEC:2009(E) – 23 –
Table B.1 – Transmission quantities of a two-port and homogeneous transmission line
Transmission lines Transmission lines
Transmission
at high frequencies at low frequencies
Two-port
quantity
R/ωL and G/ωC< 0,1 ωL/R and G/ωC< 0,1
Complex image
attenuation (Komplexe
Γ = 20 ×log []dB
T
Wellen-Dämpfung) γ = α + j β
γ = α + j β
ω RC
+ arg []rad = A + j B
α ≈ β ≈
T
a
Image attenuation
R 2 L ω RC
A = 20 ×log []dB
α = + G α ≈
T
C 2
L C
b
Image phase shift
⎛ ⎞
R ω RC
B = arg []rad
⎜ ⎟
β ≈ L C 1+ β ≈
∞
⎜ ⎟
T
2ω L 2
⎝ ⎠
Image phase
B
...