Content of the presentation

1
Content of the presentation

• Characterisation of the SOI substrate
• S parameters
• Why S-parameters?
• Mason rule
• S-parameters measurements
• Calibration
• Measurement
• Evaluation of the transmission line parameters
  on SOI
• Integrated passive inductances

2
S-parameters

• S parameters
• Why S-parameters?
• Mason rule

3
Why S-parameters ?

• At low frequencies, a 2-port is characterised by
  currents and voltages.
• Z parameters are used to fully describe the
  circuit
• V1 Z11 I1 Z12 I2
• V2 Z21 I1 Z22 I2

4
Why S-parameters ?

• Z11 is recovered by measuring V1/ I1 with I2 0
  (output port open circuited)
• Z12 is recovered by measuring V1/ I2 with I1 0
  (input port open circuited)
• Z22 and Z21 are recovered similarly
• Other matrices are also used for the
  characterisation of 2-ports Y-matrix,
  H-matrix,
• They are all using current/voltage representation

5
Why S-parameters ?

• In the microwave frequency range, the length of
  the elements are often of the same order of
  magnitude as the wavelength transmission
  line theory is applicable
• current and voltage waves are used
• V1 V10 cos (?t - ?z) ReV10 exp j(?t - ?z)
• it is often difficult
• to measure current and/or voltage
• to open circuit and/or short circuit an element

• Generalisation by using waves that are square
  roots of watts

6
Why S-parameters ?

• Definition of generalised waves
• ai (Vi Zci Ii )/2 Rci where Rci
  ReZci
• bi (Vi - Zci Ii )/2 Rci
• where ai is the incident wave at port i
• bi is the reflected wave at port i
• Zci is the reference impedance at port i
• The reference impedances are arbitrarily chosen
  but could be taken as the characteristic
  impedances of the transmission lines at the ports
• The reflected wave bi is equal to zero for
  conjugate matching

7
Why S-parameters ?

• Assuming that Rci gt 0, the exchange power from
  the source at port i is ai 2
• The power delivered to port i is
• ( ai 2 - bi 2 )
• The scattering matrix is related to reflexion
  and transmission coefficients. The relationship
  is straightforward if we assume that the
  characteristic impedances are real
• Zci Zci
• S11 ?11 and S22 ?22
• S12 Rc1/Rc2 T12 and S21 Rc2/Rc1
  T21

8
Why S-parameters ?

• S parameters fully describe a 2-port
• b1 S11 a1 S12 a2
• b2 S21 a1 S22 a2
• or
• b S a S21
• a1 b2
• S parameters are measured as S22
• S11 b1/a1 a20 S12 b1/a2 a10
  S11
• b1 a2
• S21 b2/a1 a20 S22 b2/a2 a10
  S12
• For real characteristic impedance conjugate
  matching line matching

9
Why S-parameters ?

• In which condition is the characteristic
  impedance real ?
• for lossless TEM transmission lines
• How are S parameters related to other matrices ?
• S F (Z-G) (Z G)-1 F-1
• where G is a diagonal matrix diag(Zci)
• F is a diagonal matrix diag(2 Rci)-1

10
MASON rule

• Calculates transfer functions from graphs

b1 N __ ____ a1 D
S12
with D 1 - ? T ? T - ? T T
represents all the loop transmittances T
represents the product 2 by 2 of all the
transmittances of loops that do not touch
each other T represents the product 3 by
3 of all N ? Tab Dp Dp same as
D, Tab being suppressed from the graph
11
MASON rule
S12
b1 N S11(1- ?L S22) S21
?L S12 ??in???__ ____ ___________________
___________ a1 D
1 - ?L S22
12
S parameters measurement

• Vector network analyser
• Measurement principle
• calibration

13
S parameters measurement

• Basic principle for reflexion measurement
• REFLECTOMETRY

b3
a3
DUT
Source
Matched load

• Calibration by known loads such as
• short, open, transmission (thru or line)

14
S parameters measurement

• Two port measurement calibration methods
• standard method (12 terms)
• port 1 and 2 calibration (1-port procedure)
• isolation (2 broadband loads)
• thru connexion
• self calibration procedure
• TRL thru-reflect-line
• LRL line-reflect-line
• TRM thru-reflect-match
• ...

15
S parameters measurement

• One port error modeling

Measured data (?m)
?a
Imperfect reflectometer
Source
Measured data (?m)
a0
b0
a0
b1
Error box
?a
a1
b0
Perfect reflectometer
Source
4 complex error terms
16
S parameters measurement
e10
b1
a0
e00
?a
e11
?m
b0
a1
e01
b0 N e00 (1- ?a e11)
e10 ?a e01 (e10 e01) ?a
??m???__ ____ __________________________
____ e00 ____________ a0 D
1 - ?a e11 1 - ?a e11
?m is the measured reflexion coefficient
17
S parameters measurement

• Two port error modeling

Z0
Error box
Source
b2
18
S parameters measurement - standard

• 12 terms calibration

e11
e10
e01
e00
e30
e32
b2
e22
19
S parameters measurement - standard
Forward
e30
e32
b1
a2
b3
e10
a0
S12A
S11A
e00
e11
S22A
e22
S21A
b0
b2
a1
e01
Reverse
S21A
e01
a1
b2
b0
e11
e22
S22A
S11A
e00
b3
a0
e10
S12A
a2
e32
b1
e30
20

• self calibration procedure

Error box
b2
21
S parameters measurement - self cal
8 error terms
a2
b1
e32
S12A
e10
a0
b3
e00
e11
S11A
e22
e33
S22A
b0
a3
e23
S21A
e01
a1
b2
Thru and delay ?l, e00, e33, (e10 e01)/ e11,
(e23 e32)/ e22, e10 e32 , e23 e01 Reflect
termination e11, e22