IEC TS 61156-1-2:2023

IEC TS 61156-1-2 ®
Edition 1.0 2023-11
TECHNICAL
SPECIFICATION
colour
inside
Multicore and symmetrical pair/quad cables for digital communications –
Part 1-2: Electrical transmission characteristics and test methods of symmetrical
pair/quad cables
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IEC TS 61156-1-2 ®
Edition 1.0 2023-11
TECHNICAL
SPECIFICATION
colour
inside
Multicore and symmetrical pair/quad cables for digital communications –

Part 1-2: Electrical transmission characteristics and test methods of

symmetrical pair/quad cables
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
ICS  21.220 ISBN 978-2-8322-7881-9

– 2 – IEC TS 61156-1-2:2023 © IEC 2023
CONTENTS
FOREWORD . 5
1 Scope . 7
2 Normative references . 7
3 Terms, definitions, symbols, units and abbreviated terms . 7
3.1 Terms and definitions . 7
3.2 Symbols, units and abbreviated terms . 8
4 Basic transmission line formulae. 8
4.1 Overview. 10
4.2 Complex characteristic impedance and propagation coefficient formulae . 11
4.2.1 General . 11
4.2.2 Propagation coefficient . 11
4.2.3 Complex characteristic impedance. 12
4.2.4 Phase and group velocity . 14
4.3 High frequency representation of secondary parameters . 15
4.4 Frequency dependence of the primary and secondary parameters . 17
4.4.1 Resistance . 17
4.4.2 Inductance . 17
4.4.3 Complex characteristic impedance. 17
4.4.4 Attenuation coefficient . 18
4.4.5 Phase delay and group delay . 18
5 Measurement of the complex characteristic impedance . 19
5.1 General . 19
5.2 Open/short circuit single-ended impedance measurement made with a balun
(reference method) . 20
5.2.1 Principle . 20
5.2.2 Test equipment . 21
5.2.3 Procedure . 21
5.2.4 Expression of results . 22
5.3 Function fitting the impedance magnitude and angle . 22
5.3.1 General . 22
5.3.2 Impedance magnitude . 22
5.3.3 Function fitting the angle of the complex characteristic impedance . 24
5.4 Complex characteristic impedance determined from measured phase
coefficient and capacitance . 25
5.4.1 General . 25
5.4.2 Formulae for all frequencies case and for high frequencies. 25
5.4.3 Procedure for the measurement of the phase coefficient . 26
5.4.4 Phase delay . 27
5.4.5 Phase velocity . 28
5.4.6 Procedure for the measurement of the capacitance . 28
5.5 Determination of the complex characteristic impedance using the terminated
measurement method . 28
5.6 Extended open/short circuit method using a balun but excluding the balun
performance . 29
5.6.1 Test equipment and cable-end preparation . 29
5.6.2 Basic formulae . 29
5.6.3 Measurement principle. 29

5.7 Extended open/short circuit method without using a balun . 32
5.7.1 Basic formulae and circuit diagrams . 32
5.7.2 Measurement principle. 34
5.8 Open/short impedance measurements at low frequencies with a balun . 35
5.9 Complex characteristic impedance and propagation coefficient obtained from

modal decomposition technique . 37
5.9.1 General . 37
5.9.2 Procedure . 37
5.9.3 Measurement principle. 38
5.9.4 Scattering matrix to impedance matrix . 40
5.9.5 Expression of results . 42
6 Measurement of return loss and structural return loss . 42
6.1 General . 42
6.2 Principle . 42
7 Propagation coefficient effects due to periodic structural variation related to the

effects appearing in the structural return loss . 43
7.1 General . 43
7.2 Formula for the forward echoes caused by periodic structural
inhomogeneities . 43
8 Unbalance attenuation . 45
8.1 General . 45
8.2 Unbalance attenuation near end and far end . 46
8.3 Theoretical background . 48
9 Balunless test method . 51
9.1 Overall test arrangement . 51
9.1.1 Test instrumentation . 51
9.1.2 Measurement precautions . 52
9.1.3 Mixed mode S-parameter nomenclature . 52
9.1.4 Coaxial cables and interconnect for network analysers . 54
9.1.5 Reference loads for calibration . 54
9.1.6 Calibration . 55
9.1.7 Termination loads for termination of conductor pairs . 56
9.1.8 Termination of screens . 57
9.1.9 Calibration . 57
9.1.10 Establishment of noise floor . 57
9.2 Cabling and cable measurements . 57
9.2.1 Insertion loss and EL TCTL . 57
9.2.2 NEXT . 59
9.2.3 ACR-F . 61
9.2.4 Return loss and TCL . 63
9.2.5 PS alien near-end crosstalk (PS ANEXT-Exogenous crosstalk) . 64
9.2.6 PS attenuation to alien crosstalk ratio, far-end crosstalk (PS AACR-F-
Exogenous crosstalk . 67
Annex A (informative) Example derivation of mixed mode parameters using the modal
decomposition technique . 70
Bibliography . 74

Figure 1 – Secondary parameters extending from 1 kHz to 1 GHz . 19
Figure 2 – Diagram of cable pair measurement circuit . 21

– 4 – IEC TS 61156-1-2:2023 © IEC 2023
Figure 3 – Determining the multiplier of 2π radians to add to the phase measurement . 27
Figure 4 – Measurement configurations . 30
Figure 5 – Measurement principle with four terminal network theory . 30
Figure 6 – Admittance measurement configurations . 34
Figure 7 – Admittance measurement principle . 34
Figure 8 – Transmission line system . 38
Figure 9 – Differential-mode transmission in a symmetric pair . 46
Figure 10 – Common-mode transmission in a symmetric pair . 46
Figure 11 – Circuit of an infinitesimal element of a symmetric pair . 48
Figure 12 – Calculated coupling transfer function for a capacitive coupling of 0,4 pF/m
and random ±0,4 pF/m ( =100 m; εr1 = εr2 = 2,3) . 51
Figure 13 – Measured coupling transfer function of 100 m Twinax 105 Ω . 51
Figure 14 – Diagram of a single-ended 4-port device . 53
Figure 15 – Diagram of a balanced 2-port device . 53
Figure 16 – Solution for calibration of reference loads . 55
Figure 17 – Resistor termination networks . 56
Figure 18 – Insertion loss and EL TCTL . 59
Figure 19 – NEXT . 60
Figure 20 – FEXT . 62
Figure 21 – Return loss and TCL . 64
Figure 22 – Alien NEXT . 66
Figure 23 – Alien FEXT . 68
Figure A.1 – Voltage and current on balanced DUT. 70
Figure A.2 – Voltage and current on unbalanced DUT . 72

Table 1 – Unbalance attenuation at near end . 47
Table 2 – Unbalance attenuation at far end . 47
Table 3 – Measurement set-up . 47
Table 4 – Mixed mode S-parameter nomenclature . 54
Table 5 – Requirements for terminations at calibration plane . 57

INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
MULTICORE AND SYMMETRICAL PAIR/QUAD
CABLES FOR DIGITAL COMMUNICATIONS –

Part 1-2: Electrical transmission characteristics and
test methods of symmetrical pair/quad cables

FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
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8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) IEC draws attention to the possibility that the implementation of this document may involve the use of (a)
patent(s). IEC takes no position concerning the evidence, validity or applicability of any claimed patent rights in
respect thereof. As of the date of publication of this document, IEC had not received notice of (a) patent(s),
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https://patents.iec.ch. IEC shall not be held responsible for identifying any or all such patent rights.
IEC TS 61156-1-2 has been prepared by subcommittee 46C: Wires and symmetric cables, of
IEC technical committee 46: Cables, wires, waveguides, RF connectors, RF and microwave
passive components and accessories. It is a Technical Specification.
This first edition cancels and replaces the first edition of IEC TR 61156-1-2 published in 2009
and Amendment 1:2014. This edition constitutes a technical revision.
This edition includes the following significant technical changes with respect to the previous
edition (TR):
a) typos and editorial corrections;

– 6 – IEC TS 61156-1-2:2023 © IEC 2023
b) the scope was updated;
c) Figure 14: ports swapped between port 2 and port 3;
d) new figures for balunless testing.
The text of this Technical Report is based on the following documents:
Draft Report on voting
46C/1247DTS 46C/1259e/RVDTS
Full information on the voting for its approval can be found in the report on voting indicated in
the above table.
The language used for the development of this Technical Specification is English.
This document was drafted in accordance with ISO/IEC Directives, Part 2, and developed in
accordance with ISO/IEC Directives, Part 1 and ISO/IEC Directives, IEC Supplement,
available at www.iec.ch/members_experts/refdocs. The main document types developed by
IEC are described in greater detail at www.iec.ch/publications.
A list of all parts of the IEC 61156 series, under the general title: Multicore and symmetrical
pair/quad cables for digital communications, can be found on the IEC website.
The committee has decided that the contents of this document will remain unchanged until the
stability date indicated on the IEC website under webstore.iec.ch in the data related to the
specific document. At this date, the document will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
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.

MULTICORE AND SYMMETRICAL PAIR/QUAD
CABLES FOR DIGITAL COMMUNICATIONS –

Part 1-2: Electrical transmission characteristics and
test methods of symmetrical pair/quad cables

1 Scope
This part of IEC 61156 specifies symmetrical pair/quad electrical transmission characteristics
and test methods present in IEC 61156-1:2002 (Edition 2) and not carried into
IEC 61156-1:2007 (Edition 3). It details characteristic impedance test methods and function
fitting procedures, the open/short-circuit method and the background of unbalance attenuation
measurement.
It is extended by a description of the balunless measurements technique, which is an
amendment to the former technical report and is improved and incorporated into this new
edition. The complete document is transferred into a technical specification.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their
content constitutes requirements 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 (IEV) – Part 726: Transmission
lines and waveguides
IEC 61156-1:2007, Multicore and symmetrical pair/quad cables for digital communications –
Part 1: Generic specification
IEC TR 62152, Transmission properties of cascaded two-ports or quadripols – Background of
terms and definitions
3 Terms, definitions, symbols, units and abbreviated terms
3.1 Terms and definitions
For the purposes of this document, the terms and definitions given in IEC 60050-726,
IEC 61156-1, IEC TR 62152 and the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following
addresses:
• ISO Online browsing platform: available at https://www.iso.org/obp
• IEC Electropedia: available at http://www.electropedia.org/
3.1.1
single-ended
measurement with respect to a fixed potential, usually ground

– 8 – IEC TS 61156-1-2:2023 © IEC 2023
3.2 Symbols, units and abbreviated terms
For the purposes of this document, the following symbols, units and abbreviated terms apply.
Transmission line equation electrical symbols and related terms and symbols:
R pair resistance (Ω/m)
L pair inductance (H/m)
G pair conductance (S/m)
C pair capacitance (F/m)
α attenuation coefficient (Np/m)
β phase coefficient (rad/m)
γ propagation coefficient (Np/m, rad/m)
v phase velocity of cable (m/s)
P
v group velocity of cable (m/s)
G
τ phase delay time (s/m)
P
τ group delay time (s/m)
G
Z
complex characteristic impedance (Ω)
C
∠Z angle of the complex characteristic impedance in radians
C
Z high frequency asymptotic value of the complex characteristic impedance (Ω)
∞
l length (m)
j imaginary denominator
Re real part operator for a complex variable
Im imaginary part operator for a complex variable
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 DC resistance of a round solid wire with radius r (Ω/m)
R constant with frequency component of resistance which is about 1/4 of the DC
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 =
πf μσ
G
tan δ dissipation factor tanδ =
ωC
q  forward echo coefficient at the far end of the cable at a resonant frequency
p reflection coefficient measured from the near end of the cable at a resonant
−PSRL
ZZ−
CM C
frequency, p 10
ZZ+
CM C
A forward echo attenuation at a resonant frequency (dB) Aq= −20log
Q Q
PSRL structural return loss at a resonant frequency (dB) PSRL = −20log p
K K 2αl− 1 when 2αl 1 (Np)
A A =2× PSRL− 20log 2αl−1 (dB) where 2 αl is in Np
( )
Q Q
Z complex measured open circuit impedance (Ω)
OC
Z complex measured short circuit impedance (Ω)
SC
Z complex characteristic impedance as measured (with structure) (Ω)
CM
Z ZZ×
CM OC SC
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
OUT
Z (Ω)
G
Z nominal (reference) impedance of the link and/or terminals (the system)
N
between which the cable is operating (Ω)
Z
(nominal) reference impedance that is used in measurement. Normally (for
R
Z Z
actual return loss results), = . When using a return loss measurement to
R N
approximate SRL, it is practical to choose Z to give the best balance in the
R
given frequency range (Ω)
Z terminated impedance measurement made with the opposite end of the cable
T
pair terminated in the reference impedance Z (Ω)
R
ς reflection coefficient measured in the terminated measurement method
ZZ−
RC
ς =
ZZ+
RC
Z
termination at the cable input when defining the output impedance of the cable
G
Z (Ω)
OUT
Z
termination at the cable output when defining the input impedance of the cable
L
Z (Ω)
IN
L , L , L , L least squares fit coefficients for angle of the complex characteristic impedance
0 1 2 3
K , K , K , K least squares fit coefficients of the complex characteristic impedance
0 1 2 3
|Z | fitted magnitude of the complex characteristic impedance (Ω)
C
|Z | measured magnitude of the complex characteristic impedance (Ω)
CM
∠ (V ) input angle relative to a reference angle in radians
1N
=
=
==
– 10 – IEC TS 61156-1-2:2023 © IEC 2023
∠ (V ) output angle relative to the same reference angle in radians
1F
k multiple of 2π radians
m indicator of matrix parameter
S
reflection coefficient measured with an S parameter test set

RL return loss (dB)
SRL structural return loss (dB)
CUT cable under test
Unbalance attenuation electrical symbols:
TA transverse asymmetry
LA longitudinal asymmetry
R R resistance of one conductor per unit length (Ω)
1, 2
L L inductance of one conductor per unit length (H)
1, 2
C C capacitance of one conductor to earth (F)
1, 2
G G conductance of one conductor to earth (S)
1, 2
α unbalance attenuation (dB)
u
T unbalance coupling transfer function
u
Z characteristic impedance of the common-mode circuit (Ω)
com
Z characteristic impedance of the differential-mode circuit (Ω)
diff
Z unbalance impedance (Ω)
unbal
x length coordinate (m)
γ propagation factor of the common-mode circuit (Np/m, rad/m)
com
γ propagation factor of the differential-mode circuit (Np/m, rad/m)
diff
α operational differential-mode attenuation of the cable (dB)
diff
α operational common-mode attenuation of the cable (dB)
com
∆R resistance unbalance of the sample length (Ω)
∆L inductance unbalance of the sample length (H)
∆C capacitance unbalance to earth (F)
∆G conductance unbalance to earth (S)
S summing function
U voltage in the differential-mode circuit (V)
diff
U voltage in the common-mode circuit (V)
com
n, f index to designate the near end and far end, respectively
4 Basic transmission line formulae
4.1 Overview
A review of the relationships between the propagation coefficient, the complex characteristic
impedance and the primary parameters R, L, G and C is useful here. Characteristic impedance
is commonly thought of as being a magnitude quantity. While this concept can suffice for high
frequency applications, this quantity is actually a complex one consisting of real and
imaginary components or magnitude and angle. The associated propagation coefficient is

readily viewed as being complex, consisting of the real attenuation and imaginary phase
coefficient components. The four secondary components are readily related to the primary
components. Frequency dependence of these parameters is also developed.
The cable pair parameters are represented as frequency domain dependent quantities. The
measurement methods are based on frequency domain techniques. Measurement methods
based on time domain techniques and combinations of time and frequency while useful in
many cases are not covered here. The present-day availability of excellent frequency domain
equipment such as the network analysers and impedance meters support the frequency
domain approach.
4.2 Complex characteristic impedance and propagation coefficient formulae
4.2.1 General
The frequency domain of the complex characteristic impedance Z relates to the primary
C
parameters as:
Rj+ ωL
Z =
(1)
C
Gj+ ωC
The propagation coefficient, γ, relates to the primary parameters as:
γ=α+ jβ= R+ jωL G+ jωC
( )( ) (2)
4.2.2 Propagation coefficient
4.2.2.1 Attenuation and phase coefficients
Formula (2) is separated into its real and imaginary parts, the attenuation coefficient α and the
phase coefficient β:
2 22 2 22 2
(3)
α=− ωLC−+RG R + ωL G + ω C
( ) ( )( )
2 22 2 22 2
(4)
β ωLC−+RG R+ ωL G+ ω C
( ) ( )( )
Further, by factoring out ω LC we obtain:
  
11RG R G
(5)
β ω LC 1−−+ 11+ + 

22 2 2
  
22ωL ωC

ω L ωC
  
=
=
– 12 – IEC TS 61156-1-2:2023 © IEC 2023
It can be shown that:

RC G L
αβ ω LC +
 (6)

22LC

4.2.2.2 Formulae useful at high frequencies
From Formulae (5) and (6) we can solve for α and thus obtain for α and β the following
expressions, valid within the entire frequency range:
RC G L
+
22LC
α =
(7)
  
11RG R G

1− − + 11+ +
  

22 2 2
  
22ωL ωC

ω L ωC
  
  
11RG R G

(8)
β ω LC 1−−+ 1+ 1+ 

 22  2 2 
22ωL ωC
 ω L ωC
  
Formulae (7) and (8) are well suited for evaluation of high frequencies.
4.2.2.3 Formulae useful at low frequencies
For low frequency evaluations, the expressions given by Formulae (9) and (10) are suitable.
22 2
  
ωRC G ωL ω L G
 
α −++11+ (9)
  
 
 2  22 
2 ωC R
 
R ωC
  
22 2
  
ωRC ωL G ω L G
 
(10)
β −++11+ 
 
 2  22 
2 R ωC
  R ωC
  
4.2.3 Complex characteristic impedance
4.2.3.1 Real and imaginary parts
Z
The complex characteristic impedance can also be separated into its real and imaginary
C
parts as developed in Formulae (11) and (12).
=
=
=
=
Rj+ ωL α + jβ
Z=Re Z+=jZ I m = (11)
CC C
Gj++ωC G jωC
1 GG
   
β + α −−jα β
   

ωC ωC ωC
   
Z =
(12)
C
G
1+
ωC
4.2.3.2 Formulae useful at high frequencies
After substituting Formulae (7) and (8) into Formula (12), the real and imaginary parts of the
complex characteristic impedance are obtained as given in Formulae (13) and (14)
respectively. These are well suited for simplification (see 4.3) at high frequencies:

 
  
L 11 RG  R G
 
1− + 11+  + 
 
 22  2 2 
C 22ωL ωC 
  ω L ωC
  
 
Re Z =
(13)
C
2 22
  
G 11RG R G
 
1+ 1− + 11+ +
 
22  2 2 22
22ωL ωC
ωC   ω L ωC
  
 22 
  
R G L G L 11RG R G
 
 
+ − 1− + 11+ +
  
 
22 2 2
  
2ωC C ωC C 22ωL ωC 
2ω LC  
ω L ωC
  
 
−Im Z =
(14)
C
2 22
  
G 11 RG  R G
1+ 1− + 11+ +
  
 
22 2 2 22
  
22ωL ωC
 
ωC ω L ωC
  
4.2.3.3 Formulae useful at low frequencies
On the other hand, by substituting Formulae (9) and (10) into Formula (12), the real and
imaginary parts given in Formulae (15) and (16) respectively are obtained. These are useful
for simplification in the low frequency range:

22 2 22 2
     
R ωL G ω L G G G ωL ω L G

− ++11+ + − ++11+
     
2 22 2 22
     
2ωC R ωC ωC ωC R
R ωC R ωC
     


(15)
Re Z =
C

G
1+
22
ωC

– 14 – IEC TS 61156-1-2:2023 © IEC 2023
 
22 2 22 2
     
RG ωL ω L G G ωL G ω L G
 
− ++11+ − − ++11+
     
 2 22 2 22 
     
2ωC ωC R ωC R ωC
R ωC R ωC
     
 
  (16)
−Im Z =
C

G
1+

ωC

4.2.4 Phase and group velocity
The phase propagation time (per unit length) is:
β
τ =
(17)
P
ω
By introducing β from Formulae (8) and (10), we obtain:
  
11 RG  R G
τ LC 1− + 11+ +  (18)
P  
22 2 2
  
22ωL ωC
 
ω L ωC
  
and
22 2
  
RC  ωL G  ω L G
(19)
τ −++11+ 
P  
2 22
  
2ω R ωC
 
R ωC
  
The group propagation time (per unit length) is:
dβ
τ = (20)
G
dω
 


R G
 R 

1+

d
2    

ωL
′′ ωC
β 1 L C ω LC G ωL
  
 
τβ=+ + + −+
(21)
G 
ω 24L C β  ωC 22 dω 
 
  
RG
 
11+ +
  
22 2 2
  

 
ω L ωC
  
  
=
=
 
 

GR
   G 
1+

d
     
ωC
R ωL
ωC
  
   
+− +
  ωL 22  dω 
  
RG
   
11+  + 
 22  2 2 
 
 
ω L ωC
  
 
 
The phase and group velocities are, respectively,
v =
(22)
P
τ
P
v =
(23)
G
τ
G
The above expressions are accurate and valid within the whole frequency range. If C and
G/(ωC) can be regarded as frequency independent coefficients, then we obtain:
 

R G
 
1+


 
ωL
′ ωC ′
β β L C G  LR 

τ = + + − + −+R Rω′ − ω
(24)
G  
ω 24L β ωC 22 L
   
  
RG
 11+ + 
  
22 2 2
  
 
ω L ωC

 
The above expressions, which are valid within the entire frequency range, can be simplified
into approximate expressions, which are valid at high or low frequencies only.
4.3 High frequency representation of secondary parameters
The high frequency representations of the formulae are useful over a broad range of
frequencies extending from voice frequency on up because of the range of values for the
dissipation factor. G/(ωC) = tan δ < 0,03 (< 3 %) even for PVC insulated cables up to 1,5 MHz
and for the polyethylene (PE), insulation is exceedingly small at about 0,000 1 (0,01 %). This
results in approximations, which in practice are valid for the whole frequency range as follows:
LR11
(25)
Re Z ≈ ++1
C
C 22
ωL
GL
R G
2ωC C
(26)
−Im ZZ≈ −Re +
CC
2ωC Re Z ωC Re Z
CC
– 16 – IEC TS 61156-1-2:2023 © IEC 2023

 
L
G
 
 
(27)
C
R
 
α ≈+
2 Re ZZ2Re
CC
β ≈ ωC Re Z
(28)
C
τ ≈ LC (29)
P
 
 
R
 
β β L′′C G LR
 
ωL
  ′
τ ≈ + + − + −+R Rω − ω
G   (30)
ω 24L β  ωC 2  L
 

R
 
1+
22
 
ωL
 
 
When also R/(ωL) < 0,1, which is true for high frequencies (f > 1 MHz for 0,5 mm wire), the
formulae holding better than about 1 % accuracy can be further simplified as shown below.
L
Re Z ≈ (31)
C
C
R G LR G

−Im ZZ≈ −Re ≈ −
(32)
CC 
2ωC Re Z ωC C 2ωL 2ωC

C
R G RC G L
αZ≈ + Re ≈+ (33)
C
2 Re Z 2 2 LC2
C
β ≈≈ωC Re Z ω LC (34)
C
τ ≈ LC (35)
P
β L′′C G R LR
  
′
τ ≈ τ + + − + −+R Rω − ω
(36)
G P
  
24L β ωC ωL L
 
4.4 Frequency dependence of the primary and secondary parameters
4.4.1 Resistance
The high frequency resistance (surface resistance) of a solid round wire for frequencies where
the wire radius r is greater than twice the skin depth δ can be regarded as consisting of two
0,5
parts where one is constant and the other f dependent.
1 r

R= R+=R R+ ρω≈ R + (37)
CS C 0 
42δ

r
R
R
S
(38)
ρ 2μσ
ω
The above is true for a solid wire alone. In a pair, the proximity effects and the presence of
other pairs and screen contribute both to the resistance and inductance. These effects can
increase the R by about 15 % at 1 MHz and follow also approximately the square-root of
frequency law. Also, the constant component of resistance while often neglected is about
15 % of the frequency dependent component at 1 MHz for a 0,5 mm diameter copper pair.
4.4.2 Inductance
The total inductance consists also of two main components such that
R
ρ
S
LL≈ +=L L+ = L+
(39)
EI E E
ω
ω
The external free space inductance is reduced by the proximity effect of the pair and the free
space limiting effects of the nearby shield and/or other pairs. These inductive components are
negative and fairly frequency independent at high frequencies.
4.4.3 Complex characteristic impedance
The complex characteristic impedance at high frequency asymptotic value Z is given by
∞
Formula (40).
L
E
(40)
Z =
∞
C
The high frequency impedance formulae are given by Formulae (41) and (42):
==
– 18 – IEC TS 61156-1-2:2023 © IEC 2023
L R ρ
S
Re ZZ≈≈1+ =Z+
(41)

C ∞∞
C 2ωL
2 LC ω
E
E
R + ρω
LR G  L ρ tanδ
C E
−Im Z ≈ − ≈ − 1+

C

C 22ωL ωC ρ C 2
 2L ω
E
21ω LC +
E

2L ω
(42)
E
R ρρZL Z
 
C ∞∞I
≈+ − 1+≈tanδδ− tan
 
22L
22ω LC LC ω 2 LC ω
 E 
EE E
4.4.4 Attenuation coefficient
Using the above approximations with Formulae (31) through (36) results in the remaining
formulae of this subclause:

ρ
R −
C

2L (43)
ω LC tanδ
E ρω ρω tanδ
 E
α ≈ ++ +
2Z 24ZZ 2
∞ ∞∞
which is of the form:
(44)
α ≈+ A B ω + Cω
where A, B and C are constants.
The first term of Formula (44) indicates that at the low end of the high frequency range the
0,5
attenuation increases a little more slowly than the square-root-law. The first ω term in
Formula (43) which is dominant in the high frequency attenuation formula also appears in the
phase coefficient, Formula (45).

R ρω
β ≈≈ωLC ωL C 1+ ≈ ωL C +
(45)

EE
22ωL Z
E ∞
4.4.5 Phase delay and group delay
The phase and group delay are given in Formulae (46) and (47) respectively:

R ρ
τ≈=LC LC 1+ ≈ LC+
(46)

PE E
2ωL
2Z ω
E
∞
′′
β L C  G R  LR   R  R  R G 
′
τ τ+ + − + −+R Rω− ω≈ 1− − − ≈
     
G P
24L β  ωC ωL  L   4ωL 8ωL ωL ωC 

(47)
 RR   ρ
τ 11−≈ LC + ≈ LC +
 
PE  E
 44ωL  ωL
4 ωZ
E
∞
Figure 1 – Secondary parameters extending from 1 kHz to 1 GHz
Figure 1 shows the secondary parameters of a UTP pair with 0,5 mm conductors versus
frequency. At voice frequencies, the attenuation and phase coefficients are equal. At these
frequencies, the absolute value of the complex characteristic impedance and the real part of
the complex characteristic impedance differ by the square root of 2. At frequencies above
100 kHz, attenuation is much less than the phase coefficient on the Nepers and radians scale,
and the complex characteristic impedance is mostly real. The total attenuation (Alpha) differs
from the conductor attenuation (Alpha-R) by the dielectric component of attenuation for this
example, where the dissipation factor is assumed to be 0,01.
5 Measurement of the complex characteristic impedance
5.1 General
Z
The complex characteristic impedance of a homogeneous cable pair is defined as the
C
quotient of a voltage wave and current wave which are propagating in the same direction,
either forwards (f) or backwards (r). For homogeneous cables (with no structural variations),
the complex characteristic impedance can be measured directly as the quotient of voltage U
and current I at the cable ends.
=
– 20 – IEC TS 61156-1-2:2023 © IEC 2023
UU
f r
Z
(48)
C
II
f r
A number of methods for obtaining complex characteristic impedance are described. Some of
these methods offer convenience (at the cost of accuracy in portions of the frequency range).
Others offer capability beyond what is currently needed for routine product inspection but are
useful in laboratory evaluation where measurement throughput is not as critical.
The open/short circuit single-ended impedance measurement made with a balun in 5.2 is
viewed as the reference method for obtaining the data. Alternative methods are listed below:
a) Complex characteristic impedance determined from phase coefficient and capacitance
mea
...