Lecture 3 2017/2018
Lecture 32017/2018
2C/1L, MDCR Attendance at minimum 7 sessions (course + laboratory) Lectures- assistant professor Radu Damian Monday 16-18, P2 E – 50% final grade problems + (? 1 topic teory) + (2p atten. lect.) + (3 tests) +
(bonus activity)▪ 3p=+0.5p
all materials/equipments authorized Laboratory – assistant professor Radu Damian Monday 18-20 II.12 even weeks Thursday 8-14 odd weeks II.12 ? L – 25% final grade P – 25% final grade
RF-OPTO
http://rf-opto.etti.tuiasi.ro
David Pozar, “Microwave Engineering”, Wiley; 4th edition , 2011
1 exam problem Pozar
Photos
sent by email: [email protected]
used at lectures/laboratory
Not customized
ADS 2016 EmPro 2015 based on IP from outside
university or campus
“Engineering” Sinapses
> 2010 < 1950
Complex numbers arithmetic!!!! z = a + j · b ; j2 = -1
standard unit for angles – radians microwaves traditional unit for angles –
degrees in decimal form (55.89°)
rad 180
180
rad
0,2
,2
0,0,arctan
0,0,arctan
0,arctan
arg
anedefinit
baa
b
baa
b
aa
b
z
Attention to angle numerical values!! math software – work in standard unit: radians
▪ a conversion is necessary before and after using a trigonometric function (sin, cos, tan, atan, tanh)
scientific calculators have the built-in option of choosing the angle unit▪ always double check current working unit
rad 180
180
rad
0 dBm = 1 mW
3 dBm = 2 mW5 dBm = 3 mW10 dBm = 10 mW20 dBm = 100 mW
-3 dBm = 0.5 mW-10 dBm = 100 W-30 dBm = 1 W-60 dBm = 1 nW
0 dB = 1
+ 0.1 dB = 1.023 (+2.3%)+ 3 dB = 2+ 5 dB = 3+ 10 dB = 10
-3 dB = 0.5-10 dB = 0.1-20 dB = 0.01-30 dB = 0.001
dB = 10 • log10 (P2 / P1) dBm = 10 • log10 (P / 1 mW)
[dBm] + [dB] = [dBm]
[dBm/Hz] + [dB] = [dBm/Hz]
[x] + [dB] = [x]
Behavior (and description) of any circuit depends on his electrical length at the particular frequency of interest E≈0 Kirchhoff
E>0 wave propagation
lllE 2
2
Source matched to load ?
Ei
Zi
ZL
I
V
impedance values ? existence of
reflections ?
*iL ZZ
*iL
The source has the ability to sent to the load a certain maximum power (available power) Pa
For a particular load the power sent to the load is less than the maximum (mismatch) PL < Pa
The phenomenon is “as if” (model) some of the power is reflected Pr = Pa – PL
The power is a scalar !
Ei
ZiPa
aL
iL
PP
ZZ
*
Ei
Zi ZL
PL
Ei
Zi
ZL
Pa PL
Pr
+
TEM wave propagation, at least two conductorsI(z,t)
V(z,t)
z
I(z,t)
V(z,t)
Δz
I(z+Δz,t)
V(z+Δz,t)
L·ΔzR·Δz
G·Δz C·Δz
TEM wave propagation, at least two conductors
time domain
harmonic signals
t
tziLtziR
z
tzv
,,
,
t
tzvCtzvG
z
tzi
,,
,
zILjR
dz
zdV
zVCjG
dz
zdI
02
2
2
zVdz
zVd
02
2
2
zIdz
zId
CjGLjRj
022 EE
022 HH
j 22
zz
y eEeEE
Characteristic impedance of the line
zz eVeVzV 00
zz eIeIzI 00
zILjR
dz
zdV
zz eVeVzV 00
zz eVeVLjR
zI
00
CjG
LjRLjRZ
0
CjGLjRj
2 fv f
0
00
0
0
I
VZ
I
V
R=G=0
CLjCjGLjRj
CL ;0
C
L
CjG
LjRZ
0 Z0 is real
zjzj eVeVzV 00
zjzj eZ
Ve
Z
VzI
0
0
0
0LC
2
LCv f
1
voltage reflection coefficient
ΓL
Z0 ZL
0
0
0
0
ZZ
ZZ
V
V
L
L
l
0
0
0
I
VZL 0
00
00 ZVV
VVZL
zjzj eVeVzV 00
zjzj eZ
Ve
Z
VzI
0
0
0
0
Z0 real
voltage reflection coefficient seen at the input of the line
ΓL
Z0 ZL
-l 0
Zin
ΓIN
zjzj eVeVzV 00
ljlj eVeVlV 00
0
00V
VL 000 VVV
zV
zVz
0
0
lj
lj
lj
IN eeV
eVl
2
0
0 0
00 2 ljel
ljel 20
time-average Power flow along the line
zjzj eeVzV 0 zjzj ee
Z
VzI
0
0
Total power delivered to the load = Incidentpower – “Reflected” power
Return “Loss” [dB]
the input impedance seen looking toward the load
ΓL
Z0 ZL
-l0
lI
lVZin
lj
lj
ine
eZZ
2
2
01
1
ljlj eVeVlV 00
ljlj eZ
Ve
Z
VlI
0
0
0
0
Zin
lj
Llj
L
ljL
ljL
ineZZeZZ
eZZeZZZZ
00
000
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
input impedance of a length l of transmission line with characteristic impedance Z0 , loaded with an arbitrary impedance ZL
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
ΓL
Z0 ZL
-l 0
Zin
input impedance is frequency dependent through
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
ΓL
Z0 ZL
-l 0
Zin
2fv f
l
fv
ll
v
fll
ff
222
frequency dependence is periodical, imposed by the tan trigonometric function
l = k·λ/2 l = λ/4 + k·λ/2
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
ΓL
Z0 ZL
-l 0
Zin
kll
20ZZin 0tan l
ltan
L
inZ
ZZ
2
0
quarter-wave transformer
purely imaginary for any length l
+/- depending on l value
lZjZin tan0
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
purely imaginary for any length l
+/- depending on l value
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
lZjZin cot0
SWR is defined as the ratio between maximum and minimum (Voltage) Standing Wave Ratio
real number 1 ≤ VSWR < a measure of the mismatch (SWR = 1 means a matched line)
zjzj eeVzV 0
zjzj eeVzV 20 1
je
zjeVzV 20 1
12 zje maximum magnitude value for 10max VV
12 zje 10min VV
1
1
min
max
V
VVSWR
minimum magnitude value for
lZjZ
lZjZZZ
L
Lin
tan
tan
0
00
ΓL
Z0 ZL
-l 0
Zin
zz eVeVzV 00
zz eIeIzI 00
ljel 20
Impedance Matching with Impedance Transformers (Lab 1)
Feed line – input line with characteristic impedance Z0
Real load impedance RL
We desire matching the load to the fider with a second line with the length λ/4 and characteristic impedance Z1
)tan(
)tan(
1
11
ljRZ
ljZRZZ
L
Lin
lj
lj
ine
eZZ
2
2
11
1
1
1
0
0
ZR
ZR
V
V
L
LO
In the feed line (Z0) we have only progressive wave
In the quarter-wave line (Z1) we have standing waves
24
2
l
0
0
ZZ
ZZ
in
inin
L
inR
ZZ
2
1
0in LRZZ 01 L
Lin
RZZ
RZZ
0
2
1
0
2
1
1T
The Multiple-Reflection Viewpoint
The Multiple-Reflection Viewpoint
00 0
2
1 LRZZ
(only) at f0
24
2 0
0
0
l
4
0l
)tan(
)tan(
1
11
lZjZ
lZjZZZ
L
Lin
)tan( lt
not
tZjZ
tZjZZZ
L
Lin
1
11
LZZZ 01
lnot
matching quality power reflection coefficient
222 1tan1sec
cos
1sec
t
we assume that the operating frequency is near the design frequency (narrow bandwidth)
0ff 1tan1sec 22 4
0l2
cos2 0
0
L
L
ZZ
ZZ
we set a maximum value Гm for an acceptable reflection coefficient magnitude then the bandwidth of the matching transformer, θm
for TEM lines
00
0
24
12
4 f
f
f
v
v
fl
f
f
02 ff m
m
0
0
2
1
0
0
0
2
1cos
42
42
2
ZZ
ZZ
f
ff
f
f
L
L
m
mmm
When non-TEM lines (such as waveguides) are used, the propagation constant is no longer a linear function of frequency, and the wave impedance will be frequency dependent, but in practice the bandwidth of the transformer is often small enough that these complications do not substantially affect the result
We ignored also the effect of reactancesassociated with discontinuities when there is a step change in the dimensions of a transmission line (Z0 -> Z1). This can often be compensated by making a small adjustment in the length of the matching section
Bandwidth depends on the initial mismatch
increased bandwidth for smaller load mismatches
A quarter-wave matching transformer to match a 10Ω load to a 50 Ω transmission line at f0=3GHz
Determine the percent bandwidth for SWR<1.5
ADS Simulation
GHzf 88.0
51033 GHz
2933.03
88.0
0
f
f
The quarter-wave transformer can match any real load to any feed line impedance
If a greater bandwidth for the match is required we must use multiple sections of transmission lines transformers:
binomial
Chebyshev
jjj eTTeTTeTT 622
332112
42
232112
2321121
12
121
ZZ
ZZ
12
2
23
ZZ
ZZ
L
L
21
2121
21
ZZ
ZT
21
1212
21
ZZ
ZT
0
223
2321121
n
jnnnj eeTT
If the discontinuities between the impedances Z1 Z2 and Z2 ZL are small we can approximate
0
223
2321121
n
jnnnj eeTT
11
1
0
xx
xn
n
j
j
e
e2
31
231
1
je 231
We also assume that all impedances increase or decrease monotonically across the transformer
This implies that all reflection coefficients will be real and of the same sign
Previously, 1 section
01
011
ZZ
ZZ
nn
nnn
ZZ
ZZ
1
1
NL
NLN
ZZ
ZZ
1,1 Nn
je 231
jNN
jj eee 242
210
assume that the transformer can be made symmetrical
Note that this does not imply that the impedances are symmetrical
22110 ,, NNN
jNN
jj eee 242
210
442
2210
NjNjNjNjjNjNjN eeeeeee
nNNNe njN 2cos2coscos2 10
last item: even2
12/ NN
oddcos2/1 NN
Input reflection coefficient
we can choose the coefficients so we obtain a desired behavior (of the polynomial)
jNN
jj eee 242
210
xe j 2
NN xaxaxaaxf 2
210
The response is as flat as possible near the design frequency, also known as maximally flat
For N sections the first N-1 derivatives of the |Γ(θ)| functions are annuled
0;02 2
n
n
d
d
NxAxf 1
NjeA 21
NNN
jjN
j AeeeA cos2
1,1 Nn24
ll
A, θ 0 , 0 length sections, the sections disappear
Binomial expansion
Reflection coefficient:
NNN
nnNNN
NxCxCxCCxxf 101
!!
!
nnN
NC n
N
0
0
0
0 220ZZ
ZZA
ZZ
ZZA
L
LN
L
LN
jNN
jj eee 242
210
nNn CA
NjeA 21
nNn CA
n
n
nn
nnn
Z
Z
ZZ
ZZ 1
1
1 ln2
1
1
1
12ln
xx
xx
00
01 ln22222lnZ
ZC
ZZ
ZZCA
Z
Z LnN
N
L
LNnNn
n
n
0
1 ln2lnlnZ
ZCZZ Ln
NN
nn
Manual design procedure
0
02ZZ
ZZA
L
LN
Bandwidth, Γm maximum acceptable value
N
mN
mm A cos2
Nm
mA
1
1
2
1cos
N
mmm
Af
ff
f
f
1
1
0
0
0 2
1cos
42
42
2
Design a three-section binomial transformer to match a 30Ω load to a 100 Ω line at f0=3GHz, Γm=0.1
N = 3
30LZ 1000Z
0.07525ln2
12
01
0
0
Z
Z
ZZ
ZZA L
NL
LN
1!0!3
!30
3
C 3!1!2
!31
3
C 3!2!1
!32
3
C
0n
4.455100
30ln12100lnln2lnln 3
0
0301
Z
ZCZZ LN
03.861Z
1n
77.542Z
2n
552.3100
30ln3277.54lnln2lnln 3
0
2323
Z
ZCZZ LN
87.343Z
4.003100
30ln3203.86lnln2lnln 3
0
1312
Z
ZCZZ LN
74.00.07525
1.0
2
1arccos
42
2
1arccos
42
311
0
N
m
Af
f
GHzf 22.2
Similarly Lab. 1
GHzf 169.2
6105.33 GHz
The response of this multisection impedance transformer is equal-ripple in passband
optimizes (increases) bandwidth at the expense of passband ripple
We match the Γ(θ) function with an desired Chebyshev polynomial
equal-ripple
xxT 1
12 2
2 xxT
xxxT 34 3
3
188 24
4 xxxT
xTxxTxT nnn 212
111 xTx n
We can show that:
nNNNe njN 2cos2coscos2 10
jNN
jj eee 242
210
xe j 2
NN xaxaxaaxf 2
210
cosx
xnxTn arccoscos )(coshcosh)( 1 xnxTn1x 1x
nTn coscos
1x
last item:even
2
12/ NN
oddcos2/1 NN
variable change so we map:
cos
1sec
1 xm
1 xm
m
x
cos
cos
cossec mx
We search coefficients of Γ(θ) function to obtain a Chebyshevpolynomial
cosseccossec1 mmT
12cos1seccossec 2
2 mmT
cossec3cos33cosseccossec 3
3 mmmT
112cossec432cos44cosseccossec 24
4 mmmT
nNNNe njN 2cos2coscos2 10
cossec mNjN TeA
last item:even
2
12/ NN
oddcos2/1 NN
A, θ 0 , 0 length sections, the sections disappear
mN
L
L TAZZ
ZZsec0
0
0
mNL
L
TZZ
ZZA
sec
1
0
0
00
0 ln2
11sec
Z
Z
ZZ
ZZT L
mL
L
m
mN
)(coshcosh)( 1 xnxTn
m
L
L
L
m
m
ZZ
NZZ
ZZ
N 2
lncosh
1cosh
1cosh
1coshsec 01
0
01
mm
f
ff
f
f 42
2
0
0
0
Am
Design a three-section Chebyshevtransformer to match a 30Ω load to a 100 Ωline at f0=3GHz, Γm=0.1
N = 3 30LZ 1000Z
cosseccos3cos2 33
103
mjj TAee
1.0 mA
362.1
1.02
10030lncosh
3
1cosh
2
lncosh
1coshsec 101
m
Lm
ZZ
N
76.42746.0
sec
1arccos rad
m
m
mNL
L
TZZ
ZZA
sec
1
0
0
00 AZZL 1.0A
cossec3cos33cosseccos3cos2 310 mm AA
mA 30 sec2 1263.00
cos mmA secsec32 31 1747.01
3cos
simetrie: 1203 ;
0n
4.3531263.02100ln2lnln 001 ZZ
68.771Z
1n
77.542Z
2n
62.383Z
1263.00
1747.01
4.0031747.0268.77ln2lnln 112 ZZ
654.31747.0277.54ln2lnln 223 ZZ
GHzf 15.3
045.1
180
76.4242
42
2
0
0
0
mm
f
ff
f
f
Similarly Lab. 1
GHzf 096.3
51017.43 GHz
09925.0282.2 GHz
G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks,and Coupling Structures, ArtechHouse Books, Dedham, Mass. 1980
Impedance Matching
Feed line – input line with characteristic impedance Z0
Real load impedance RL
We desire matching the load to the fider with a second line with the length λ/4 and characteristic impedance Z1
)tan(
)tan(
1
11
ljRZ
ljZRZZ
L
Lin
lj
lj
ine
eZZ
2
2
11
1
1
1
0
0
ZR
ZR
V
V
L
LO
In the feed line (Z0) we have only progressive wave
In the quarter-wave line (Z1) we have standing waves
24
2
l
0
0
ZZ
ZZ
in
inin
L
inR
ZZ
2
1
0in LRZZ 01 L
Lin
RZZ
RZZ
0
2
1
0
2
1
Bandwidth depends on the initial mismatch
increased bandwidth for smaller load mismatches
ADS Simulation
GHzf 88.0
51033 GHz
2933.03
88.0
0
f
f
The quarter-wave transformer can match any real load to any feed line impedance
If a greater bandwidth for the match is required we must use multiple sections of transmission lines transformers:
binomial
Chebyshev
We also assume that all impedances increase or decrease monotonically across the transformer
This implies that all reflection coefficients will be real and of the same sign
Previously, 1 section
01
011
ZZ
ZZ
nn
nnn
ZZ
ZZ
1
1
NL
NLN
ZZ
ZZ
1,1 Nn
je 231
jNN
jj eee 242
210
Similarly Lab. 1
GHzf 169.2
6105.33 GHz
Similarly Lab. 1
GHzf 096.3
51017.43 GHz
09925.0282.2 GHz
Microwave and Optoelectronics Laboratory http://rf-opto.etti.tuiasi.ro [email protected]