Download Derivation of the: Improved Log Formula

Derivation of the:
Improved Log Formula
O.Berrig
Thanks to:
H.Day, V.Vaccaro, C.Vollinger
The longitudinal impedance is defined
as a serial impedance
dZ/dl=The beam impedance per length
L
The inductance per length
C
The capacity per length
The L and the C are characteristics
of the transfer line
A moving pulse
through a lossless transmission line
The movement of the pulse is described by the equation:
Where:
l = the length of the cable
𝛽 = the propagation1 factor. For a lossless cable: 𝛽 = 𝜔 ∙ 𝐿 ∙ 𝐶
1The propagation
factor is also called the wave number and then is named “k”
A moving pulse through a transmission line
with losses
The movement of a pulse through a line with losses, is described by the same equation as
moving through a line without losses (NB! |V1+|>|V2+| and |V2-|>|V1-|) :
Where:
𝑙 = The length of the cable
𝛽 = Propagation factor. The propagation factor is a complex number for a lossy line:
𝛽 =𝜔∙ 𝐶∙𝐿∙
1−
𝑖∙𝑅
𝜔∙𝐿
∙ 1−
𝑖∙𝐺
𝑤∙𝐶
See ref. [1]
[1] http://pcwww.liv.ac.uk/~awolski/Teaching/Liverpool/PHYS370/AdvancedElectromagnetism-Part6.pdf
A moving pulse through a transmission line
with losses
The general formula:
𝛽 = 𝜔 ∙ 𝐶𝐷𝑈𝑇 ∙ 𝐿𝐷𝑈𝑇 ∙
𝑖 ∙ 𝑅𝐷𝑈𝑇
𝑖 ∙ 𝐺𝐷𝑈𝑇
1−
∙ 1−
𝜔 ∙ 𝐿𝐷𝑈𝑇
𝑤 ∙ 𝐶𝐷𝑈𝑇
Can be rewritten as:
𝛽 = 𝜔 ∙ 𝐶𝑅𝐸𝐹 ∙ 𝐿𝑅𝐸𝐹 ∙
𝑑𝑍
1+
𝑑𝑙
𝑖 ∙ 𝜔 ∙ 𝐿𝑅𝐸𝐹
See ref. [2]
The last formula corresponds to an additional “beam impedance per length” :
dZ/dl=The beam impedance per length
[2] \\cern.ch\dfs\Websites\o\OEBerrig\Impedance\Verification_Erk_ImprovedLogFormula.nb
Getting to the expression
S 21  e
 i l
The scattering
parameters are
defined in the
following way:
The movement
of the pulse is
described by
the equation:
Where the
waves and
their voltages
are defined as:
Combining these three
equations give:
S 21  e
 i  l
NB! The waves must be
matched to the two-port
network
Derivation of the
Improved Log Formula
𝑑𝑍
𝑆21𝐷𝑈𝑇 = 𝑒
−𝒊∙𝜔∙ 𝐶𝑅𝐸𝐹 ∙𝐿𝑅𝐸𝐹 ∙
𝑆21𝑅𝐸𝐹 = 𝑒 −𝒊∙𝜔 ∙
The formula is proved,
but the devil is in the
details 𝑍𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 =
𝑑𝑙
1+
𝒊∙𝜔∙𝐿𝑅𝐸𝐹
∙𝑙
𝐶𝑅𝐸𝐹 ∙𝐿𝑅𝐸𝐹 ∙ 𝑙
𝑆21
𝑑𝑍
𝑑𝑙
∙ 𝑙 = −2 ∙ 𝑍0 ∙ 𝐿𝑛
𝑆21𝐷𝑈𝑇
𝑆21𝑅𝐸𝐹
∙
1+
𝒊∙𝐿𝑛 𝑆21𝐷𝑈𝑇
𝑅𝐸𝐹
2∙𝜃
where:
𝑍0 = The characteristic impedance of the REF line 𝑍0 =
𝐿𝑅𝐸𝐹
𝐶𝑅𝐸𝐹
θ = The electrical length of the REF line 𝜃 = 𝛽𝑅𝐸𝐹 ∙ 𝑙 = 𝜔 ∙
𝐶𝑅𝐸𝐹 ∙ 𝐿𝑅𝐸𝐹 ∙ 𝑙
In conclusion: when doing measurements, the VNA
cables must be matched to the device under test
Matching resistors –
prevent reflections inside the DUT
ZDUT
Z0=50 Ω
Yes, but this is
not what is
really done !!!
The real setup
Matching resistors –
Matched to vacuum pipe
ZDUT
Problem:
Reflection when the wave
goes from vacuum pipe to
DUT. The DUT is embedded in
the vacuum pipe, and we need
to modify the “improved log
formula” to take account of
this.
The real setup
Derivation from Vaccaro
From Vaccaro:
The real setup
In order to solve the measurement problem, use Vaccaro’s formula [4] :
Where X is R (for reference) or D (for device under test)
and SC is the measurement of the spacers:
Transformations of the improved log-formula
1. The improved log-formula gives the impedance for a specific length of wire.
However, in reality the impedance is specific for a given position on the wire.
The improved log-formula therefore gives an average for the specific length.
2. The improved log-formula is exact:
𝑍𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 =
𝑑𝑍
𝑑𝑙
∙ 𝑙 = −2 ∙ 𝑍0 ∙ 𝐿𝑛
𝑆21𝐷𝑈𝑇
𝑆21𝑅𝐸𝐹
∙
1+
𝒊∙𝐿𝑛
It transforms to the log-formula for 𝑍𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 << 𝑍0 : 𝑍𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 = −2 ∙ 𝑍0 ∙ 𝐿𝑛
3. The improved log-formula is transformed into Vaccaro’s formula:
𝑍𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 = 𝑍0 ∙ 𝐿𝑛
𝑆21𝑅𝐸𝐹
𝑆21𝐷𝑈𝑇
∙ 1 + 𝐿𝑛
𝑆21𝐷𝑈𝑇
𝑆21𝑅𝐸𝐹
by the equality: 𝐿𝑛[𝑆12𝑅𝐸𝐹 ] = −𝒊 ∙ θ
4) For wavelengths smaller than the length of the “device under test” (DUT);
the lumped impedance formula is exact. It can be shown that the improved log
formula can be transformed into the lumped formula, see [3]:
𝑍𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 = 2 ∙ 𝑍0 ∙
𝑆21𝑅𝐸𝐹
𝑆21𝐷𝑈𝑇
−1
𝑆21𝐷𝑈𝑇
𝑆21𝑅𝐸𝐹
2∙𝜃
𝑆21𝐷𝑈𝑇
𝑆21𝑅𝐸𝐹
The Lumped formula, can be derived in the following way (freely adapted from
F.Caspers). The first derivation of the lumped formula was made in ”On coaxial
wire measurement of the longitudinal coupling. H.Hahn and F.Pedersen”:
http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?197810003 :
ZL
a1
b1
I
V1
V2 b2
V1  Z L  I  V2
We can then calculate the transmission coefficient:
S 21, DUT 
S 21, REF 




V  Z 0  I   V2  Z 0  I   Z 0  I  Z 0  I   2  Z 0
b2 V2  Z 0  I  / 2  Z 0

 2
V1  Z 0  I  V2  Z L  I  Z 0  I  Z 0  I  Z L  I  Z 0  I  2  Z 0  Z L
a1 V1  Z 0  I  / 2  Z 0
V2  Z 0  I  / 2 
V1  Z 0  I  / 2 
Z0
Z0
 1

Because ZL is zero
And from S21, we get the lumped impedance:
S 21, DUT
S 21, REF

2  Z0
2  Z0  Z L

S
ZL
 2  21, REF  2
Z0
S 21, DUT
[3] http://cdsweb.cern.ch/record/960162/files/cer-002626446.pdf E.Jensen
[4]\\cern.ch\dfs\Websites\o\OEBerrig\Impedance\COUPLING_IMPEDANCE_MEASUREMENTS_AN_IMPROVED
_WIRE_SCANNER_METHOD_Vaccaro.pdf
[5] \\cern.ch\dfs\Websites\o\OEBerrig\Impedance\Validity_84.pdf H.Hahn
Thank you !
Extra Material
Lossy transmission line:
No losses