Calculation in Decibels in the Scope of Electromagnetic Compatibility

Electromagnetic Compatibility
Calculation in Decibels
Mathias Magdowski
Chair for Electromagnetic Compatibility
Institute for Medical Engineering
Otto von Guericke University Magdeburg
License: cba CC BY-SA 3.0 (Attribution + ShareAlike)
Magdowski (Chair for EMC) Electromagnetic Compatibility
License: cba CC BY-SA 3.0 (Attribution + S
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Measurement quantities
most important measurement quantities are:
current I
voltage U
power P
electric ﬁeld strength E
magnetic ﬁeld strength H
frequency f
measurement according to the task in the frequency domain
and/or in time domain
typical f range for EMC measurements: DC to 6 GHz
clear increase of the upper frequency limit to 18 GHz in the next
years
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Source:
Klaus H. Blankenburg: “Standard-compliant usage of quantities, units
and equations”, Application note from Rohde&Schwarz
Standard-
compliant usage
of quantities, units
and equations
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International system of units and quantities
International system of units or SI:
from french Système international d’unités
deﬁnes 7 base units
adopted as the legal units in almost all countries worldwide
editor: International Bureau of Weights and Measures (BIPM)
International system of quantities or ISQ:
deﬁnes 7 base quantities
editor: International Organization for Standardization (ISO)
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ISQ base quantities and SI base units
Table: ISQ base quantities and SI base units
ISQ base quantity SI base unit
Name Letter symbol Name Unit symbol
Length l meter m
Mass m kilogram kg
Time t second s
Electric Current I ampere A
Temperature T kelvin K
Amount of substance n mole mol
Luminous intensity Iv candela cd
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Derived electrical quantities and units
Table: Derived electrical quantities and units
ISQ quantity Derived SI unit
Voltage U volt V = kg m2
A s3
Charge Q coulomb C = A s
Capacitance C farad F = A s
V
Resistance R ohm Ω = V
A
Conductance G siemens S = A
V
Inductance L henry H = V s
A
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Energy W joule J = kg m2
s2
Real power P watt W = J
s
Reactive power Q var var = W
Apparent power S voltampere VA = W
Frequency f hertz Hz = 1
s
Angular frequency ω - 1
s
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Name Letter Symbol Name Unit symbol
Magn. flux Φ weber Wb = V s
Magn. flux density B tesla T = V s
m2
Magn. field strength H - A
m
Elec. flux Ψ - A s
Elec. flux density D - A s
m2
Elec. field strength E - V
m
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Prefixes and prefix symbols
Table: Prefixes and prefix symbols for decimal submultiples and multiples of
units
Prefix Symbol Factor
yocto y 10−24
zepto z 10−21
atto a 10−18
femto f 10−15
pico p 10−12
nano n 10−9
micro µ 10−6
milli m 10−3
centi c 10−2
deci d 10−1
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Prefixes and prefix symbols
Table: Prefixes and prefix symbols for decimal submultiples and multiples of
units
Prefix Symbol Factor
deca da 101
hecto h 102
kilo k 103
mega M 106
giga G 109
tera T 1012
peta P 1015
exa E 1018
zetta Z 1021
yotta Y 1024
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Rules
Special rules for prefixes:
prefixes are always used together with units
notation: prefix without any space before unit, compose a new unit
at potentiation, the exponent is also valid for the prefix
1 mm2
= 1 mm · 1 mm = 10−3
m · 10−3
m = 10−6
m2
= 1 µm2
General rules for SI units:
must be written as stipulated by law or standard
may not be modified by appending additional information such as
indices or superscripts or subscripts
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Quantities
Physical quantities:
to quantitatively describe physical phenomena
product of numerical value and unit
change of the unit ←→ change of the numerical value
U = 0.1 V = 100 mV
Notation:
half space between numerical value and unit
symbols shall have only one letter
indicate a special meaning −→ add indices to symbol
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Equations
Quantity equations:
letter symbols −→ phys. symbols or math. symbols
independent of the selected units
numerical values and units are treated as independent factors
Example
U = R · I (1)
always yields the same result for U
irrespective of the units of R and I
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Equations
Scaled quantity equations:
every quantity appears with its unit in the denominator
advantage −→ units cancel, only numerical values
still irrespective of the choice of units
recommended for representing results
Example
U
kV
= 10−3
·
R
Ω
·
I
A
(2)
derivation by expansion with the units
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Equations
Numerical value equations:
letter symbols −→ numerical values of physical quantities or
mathematical symbols
depend on the choice of units
are considered outdated and should no longer be used
Example
U[kV] = 10−3
· R[Ω] · I[A] wrong! (3)
U = 10−3
· R · I U[kV], R[Ω], I[A] wrong! (4)
U = 10−3
· R · I U in kV, R in Ω, I in A correct (5)
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Attenuation and gain figures
Definition:
logarithmic ratio of two electrical quantities
characterizes a two-port or a transmission path
arguments of the logarithm are numerical values
Units:
common (decadic) logarithm lg −→ decibel (dB)
natural logarithm ln −→ neper (Np)
dimensionless pseudounit, no SI unit
should not be modified by appending additional information
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Not to be confused with:
Figure: “Figure” of Otto von Guericke in Magdeburg
source: http://commons.wikimedia.org/wiki/File:Magdeburg_Guericke.jpg#/media/File:
Magdeburg_Guericke.jpg
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Origin of the unit name bel
Alexander Graham Bell
(1847–1922)
speech therapist, engineer
and inventor
made the telephone
commercially successful
after his death all
telephones in the US were
silenced for one minute
Figure: Alexander Graham Bell
(ca. between 1914–1919)
source:
http://commons.wikimedia.org/wiki/File:
Alexander_Graham_Bell.jpg
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Reminder of the logarithmic identities
Preconditions: x, y, b, r > 0 and b = 1
Product:
logb(x · y) = logb x + logb y (6)
Quotient:
logb
x
y
= logb x − logb y (7)
Power:
logb (xr
) = r logb x (8)
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Definition for power quantities
Example: real power
P1: input power
P2: output power
Power attenuation figure of a two-port:
AP = 10 · lg
P1
P2
dB (9)
Power gain figure of a two-port:
GP = 10 · lg
P2
P1
dB (10)
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Definition for root power quantities
Example: rms-values of alternating voltages
U1: input voltage
U2: output voltage
Voltage attenuation figure of a two-port:
AU = 10 · lg
P1
P2
dB = 10 · lg
U2
1/R
U2
2/R
dB = 20 · lg
U1
U2
dB (11)
Voltage gain figure of a two-port:
GU = 10 · lg
P2
P1
dB = 10 · lg
U2
2/R
U2
1/R
dB = 20 · lg
U2
U1
dB (12)
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Discussion of the nomenclature
So far: field quantities
misleading
power and energy density −→ field quantity and power quantity at
the same time
electric voltage and current −→ no field quantity, but integral over
a field quantity
New term: root power quantities
quantities whose square is proportional to a power quantity
introduced in ISO 80000-1, likely to be adapted in the next
versions of IEC 60027 and DIN 5493
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Some numbers to bear in mind
Table: Conversion dB – linear values
Figure Power ratio Voltage ratio
in dB approx. exact approx. exact
0 1 1 1 1
3 2 1.995 1.4 1.412
6 4 3.98 2 1.995
10 10 10 3 3.162
20 100 100 10 10
40 10 000 10 000 100 100
60 1 000 000 1 000 000 1000 1000
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Some numbers to bear in mind
Table: Conversion dB – linear values
Figure Power ratio Voltage ratio
in dB approx. exact approx. exact
0 1 1 1 1
−3 0.5 0.501 0.7 0.798
−6 0.25 0.25 0.5 0.501
−10 0.1 0.1 0.3 0.316
−20 0.01 0.01 0.1 0.1
−40 0.0001 0.0001 0.01 0.01
−60 0.000 001 0.000 001 0.001 0.001
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Level
Definition:
logarithmic ratio of two electrical quantities
denominator −→ fixed value of a reference quantity of the same
dimension as the numerator
Unit:
common (decadic) logarithm lg −→ decibel (dB)
with specification of the reference quantity
short version: reference quantity in parentheses following the dB
with a space between dB and the parentheses
if numerical value of the reference quantity equals 1 −→ omit
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Not to be confused with:
Figure: “Level” of the river Elbe in Magdeburg
source: https://commons.wikimedia.org/wiki/File:Pegelhaus_Magdeburg.jpg
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Definition of levels
Definition for power quantities:
P: power
P0: reference value
LP (re P0) = LP/P0
= 10 · lg
P
P0
dB (13)
Definition for root power quantities:
U: voltage
U0: reference value
LU (re U0) = LU/U0
= 20 · lg
U
U0
dB (14)
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Abbreviations introduced by the ITU/UIT
dB is directly followed by a letter or a sequence of characters to
identify the reference value
IEC 60027-3 recommends not to use these abbreviations
Table: Abbreviations introduced by the International Telecommunication
Union (selection)
Quantity Letter symbol Unit, short form
Reference value long short IEC ITU/UIT
Elec. power LP (re 1 mW) LP/mW dB (mW) dBm
reference to 1 mW
Elec. voltage LU (re 1 V) LU/V dB (V) dBV
reference to 1 V
Elec. ﬁeld strength LE (re 1 µV
m ) LE/µV
m
dB (µV/m) dBµ
reference to 1 µV
m
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Calculating in dB
Example P = I2 · R:
Expand with the reference values:
P ·
P0
P0
= I2
·
I2
0
I2
0
· R ·
R0
R0
(15a)
Order:
P
P0
=
I
I0
2
·
R
R0
·
I2
0R0
P0
(15b)
Logarithmize:
10 · lg
P
P0
= 10 · lg
I
I0
2
·
R
R0
·
I2
0R0
P0
(15c)
/ 49

Calculating in dB
Example P = I2 · R (continued):
Use of logb(x · y) = logb x + logb y:
10 lg
P
P0
= 10 lg
I2
I2
0
+ 10 lg
R
R0
+ 10 lg
I2
0R0
P0
(16a)
Use of logb (xr) = r logb x:
10 lg
P
P0
= 20 lg
I
I0
+ 10 lg
R
R0
+ 10 lg
I2
0R0
P0
(16b)
Rewrite as levels:
LP/P0
= LI/I0
+ LR/R0
+ 10 lg
I2
0R0
P0
(16c)
/ 49

Calculating in dB
Example P = I2 · R (continued):
Insert ﬁxed reference values, e. g. P0 = 1 W, I0 = 1 A and R0 = 1 Ω:
LP/W = LI/A + LR/Ω (17a)
Exemplary calculation with I = 10 A and R = 100 Ω:
P = (10 A)2
· 100 Ω = 10 000 W = 10 kW (17b)
Calculation in dB:
LP/W = 20 dB (A) + 20 dB (Ω) = 40 dB (W) (17c)
/ 49

Calculating in dB
Example U = R · I:
U ·
U0
U0
= R ·
R0
R0
· I ·
I0
I0
(18a)
Order:
U
U0
=
R
R0
·
I
I0
·
R0I0
U0
(18b)
Logarithmize:
20 · lg
U
U0
= 20 · lg
R
R0
·
I
I0
·
R0I0
U0
(18c)
/ 49

Calculating in dB
Example U = R · I (continued):
20 lg
U
U0
= 20 lg
R
R0
+ 20 lg
I
I0
+ 20 lg
R0I0
U0
(19a)
Rewrite as levels:
LU/U0
= LR/R0
+ LI/I0
+ 20 lg
R0I0
U0
(19b)
Insert ﬁxed reference values, e. g. U0 = 1 V, R0 = 1 Ω and I0 = 1 A:
LU/V = LR/Ω + LI/A (19c)
/ 49

Calculating in dB
Example U = R · I (continued):
Exemplary calculation R = 100 Ω and I = 10 A:
U = 100 Ω · 10 A = 1000 V = 1 kV (20a)
Calculation in dB:
LU/V = 40 dB (Ω) + 20 dB (A) = 60 dB (V) (20b)
Attention: resistance is here converted with a factor of 20
−→ Resistance is neither a power quantity nor a root power quantity
−→ “impedance conversion ﬁgure”
/ 49

Calculating in dB
Example P = U · I:
P ·
P0
P0
= U ·
U0
U0
· I ·
I0
I0
(21a)
Order:
P
P0
=
U
U0
·
I
I0
·
U0I0
P0
(21b)
Logarithmize:
10 · lg
P
P0
= 10 · lg
U
U0
·
I
I0
·
U0I0
P0
(21c)
/ 49

Calculating in dB
Example P = U · I (continued):
10 lg
P
P0
= 10 lg
U
U0
+ 10 lg
I
I0
+ 10 lg
U0I0
P0
(22a)
Rewrite as levels:
LP/P0
= LU/U0
+ LI/I0
+ 10 lg
U0I0
P0
(22b)
Insert ﬁxed reference values, e. g. P0 = 1 W, U0 = 1 V and I0 = 1 A:
LP/W = LU/V + LI/A (22c)
/ 49

Calculating in dB
Example P = U · I (continued):
Exemplary calculation with U = 100 V and I = 10 A:
P = 100 V · 10 A = 1000 W = 1 kW (23a)
Calculation in dB:
LP/W = 20 dB (V) + 10 dB (A) = 30 dB (W) (23b)
Attention: voltage and current are here converted with a factor of 10
−→ conversion factor of 20 is based on the assumption that P ∼ U2 or
P ∼ I2
/ 49

Calculating in dB
Linear scale:
power quantity:
P
1 mW
=
P
1 W
· 1000 (24)
root power quantity:
U
1 mV
=
U
1 V
· 1000 (25)
dB scale:
power quantity:
LP/mW = LP/W + 30 dB (26)
root power quantity:
LU/mV = LU/V + 60 dB (27)
/ 49

Summary of calculation rules
Sum or difference of two figures is again a figure:
20 dB + 30 dB = 50 dB (28)
Sum of figure and level gives a level:
0 dB (mW) + 50 dB = 50 dB (mW) (29)
Difference of two levels gives a figure:
50 dB (mW) − 0 dB (mW) = 50 dB (30)
Sum of two levels does not make sense:
20 dB (mW) + 30 dB (mW) wrong! (31)
/ 49

Reconvert into linear values
Use the deﬁnition of a level, divide by 10 dB:
LP/mW = 30 dB = 10 · lg
P
1 mW
dB (32a)
30 dB
10 dB
= lg
P
1 mW
(32b)
Delogarithmize, transpose to P:
10
30 dB
10 dB =
P
1 mW
(32c)
P = 1 mW · 10
30 dB
10 dB = 103
mW = 1 W (32d)
/ 49

Advantages of the calculation in dB
very small and very large values can be given in terms of handy
numerical values
multiplication on linear scale −→ addition in dB
division on linear scale −→ subtraction in dB
simple calculation of signal chains
corresponds to the human reception of light intensity, loudness,
pressure and taste (but not temperature) −→ Weber Fechner law
/ 49

Advantages of a double-logarithmic scale
0 20 40 60 80 100
0
0.5
1
Normalized frequency f/f0
Amplituderesponse
LP of 1. order
LP of 2. order
(a) Linear scaling
10−2
10−1
100
101
102
10−3
10−2
10−1
100
Normalized frequency f/f0Amplituderesponse
LP of 1. order
LP of 2. order
(b) Logarithmic scaling
Figure: Amplitude response of low-pass ﬁlters of different order.
/ 49

Disadvantage of the calculation in dB
partially conversion necessary
necessary distinction between power quantities, root power
quantities and impedance conversion ﬁgures
conversion factors (10 vs. 20) partially not clear
unusual computation of the units
no direct addition or subtraction of values possible
no representation of the value zero in dB
no representation of negative or complex values in dB
/ 49

Recommendations
avoid a calculation in dB, if possible
always do a control calculation with linear values
clear statement of the reference values and conversion factors
clear statement of the units and reference units
no statistics with dB values
/ 49

Further reading
Products: Signal generators, spectrum analyzers, test receivers, network analyzers, power meters, audio analyzers
dB or not dB?
Everything you ever wanted to know
about decibels but were afraid to ask…
Application Note 1MA98
True or false: 30 dBm + 30 dBm = 60 dBm? Why does 1% work out to be -40 dB one time but
then 0.1 dB or 0.05 dB the next time? These questions sometimes leave even experienced
engineers scratching their heads. Decibels are found everywhere, including power levels,
voltages, reflection coefficients, noise figures, field strengths and more. What is a decibel and
how should we use it in our calculations? This Application Note is intended as a refresher on
the subject of decibels.
/ 49

Notation
Italics:
letter symbols for physical quantities, e. g. m (mass), U (electric
voltage)
letter symbols for variables, e. g. x, n
symbols for functions and operators with user-deﬁnable meaning,
e. g. f(x)
−→ recommendation of a serif font
/ 49

Notation
Roman type:
units and their preﬁxes, e. g. m, kg, s, pF, V, dB
numerals, e. g. 4.5; 67; 8-fold; 1⁄2
symbols for functions and operators with ﬁxed meaning, e. g. sin,
lg, max
indices with abbreviations, e. g. Uq, Etan, Pout
chemical elements and compounds, e. g. Cu; H2O
−→ recommendation of using a sans serif font
/ 49

Labeling of table headers
Table: Labeling of table headers and coordinate systems.
Correct Wronga
U U/V U in V E/V
m
E in V
m U [V] U U in [V]
[V]
0.1 V 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.2 V 0.2 0.2 0.2 0.2 0.2 0.2 0.2
. . . . . . . . . . . . . . . . . . . . . . . .
a
Do not put units in brackets.
/ 49

Labeling of table headers
Table: Labeling of table headers and coordinate systems for large value
ranges.
Correct Wronga
P P/W P/W P/W
1 W 1 1 1
1 mW 1 × 10−3 10−3 1 m
1 µW 1 × 10−6 10−6 1 µ
1 nW 1 × 10−9 10−9 1 n
. . . . . . . . . . . .
a
Do not use preﬁxes alone.
/ 49

Calculation in Decibels in the Scope of Electromagnetic Compatibility

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