dhbk microwaves slides 2012
TRANSCRIPT
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Microwaves
Chapter 1 ‐ Propagation on a transmission line
Đại Học Bách Khoa Hà Nội / 2012-2013
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Propagation of a wave
2
( ) ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −×=−×=
λπω z
TtEkztEE 2coscos 00
Electromagnetic field
( ) ( )zzkttkzt δδωω +−+=−
fTkt
zvP ×==== λλωδδ
fvP ×= λ
During propagation the phase remains constant
Phase velocity
Microwave electronics concerns high frequency or small wavelength signals
The size of circuits as the same order of magnitude as the wavelength
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Microwave domain
3
AM Radio 30 kHz – 30 MHz (LW, MW, SW)
FM Radio 88 – 105 MHz
DAB (Digital Audio Broadcasting) VHF III band: 174 – 240 MHzL band: 1452 – 1492 MHz
TV VHF 54 – 72 MHz / 76 – 88 MHz / 174 – 216 MHz
TV UHF 470 – 608 MHz / 614 – 890 MHz
Cellular phones GSM 900 MHz and 1.8 GHz
Bluetooth 2.4 GHz
WiFi (802.11 a/b/g) 2.4 – 2.4835 GHz / 5 GHz
Wimax 2.5 GHz / 3.5 GHz / 5.8 GHz
Hyperlan 5 GHz
Satellite5.925 – 6.425 GHz / 3.7 – 4.2 GHzIntelsat V 14 – 14.5 GHz / 10.95 – 11.2 GHz
/ 11.45 – 11.70 GHz
Radar FMCW 77 GHz
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Wavelength and frequency
4
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Examples of transmission lines
5
Circular waveguideRectangular waveguide
Twisted or untwisted pairs line Coaxial cable
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Transmission lines on a Printed Circuit Board (PCB)
6
Microstrip line Coplanar waveguide
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Materials
7
Conductors are characterized by their conductivity Copper (Cu): σ = 5.88 107 S.m‐1
Gold (Au): σ = 4.55 107 S.m‐1
Aluminium (Al): σ = 3.65 107 S.m‐1
Silver (Ag): σ = 6.21 107 S.m‐1
Insulators, between the conductors, are of dielectric materials; they are characterized by their low conductivity, the dielectric constant and the permeability (usually µr =1).
Epoxy (FR4): εr ≈ 4Polyethylene: εr ≈ 2.25 Duroid: εr ≈ 6 or 10Alumina: εr ≈ 8.4Teflon: εr ≈ 2.1; σ = 3.3 10‐14 S.m‐1
Losses are expressed by tanδ (see later)
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Hypothesis for this lecture
8
• 2 wires conductors waveguide, quasi TEM mode
voltage and intensity are defined on the circuit
• The length of the transmission lines has the same magnitude order as the
wavelength: the propagation effect has to be taken into account
• Components have a small size QSA is valid (surface mount devices)
‐ Passive elements: R, L, C; diodes
‐ Active elements: transistors (BJT, J‐FET, MOS‐FET)
‐Integrated circuits: MMIC (monolithic microwave integrated circuits)
• Small bandwidth signal (the signal is centered on the frequency carrier,
case of a modulated signal, with a small occupied channel)
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Coaxial cable
9
211
4dhRa
πσ=
( )eedhRg +
=22πσ
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
20 Ln
21
ddhL rμμ
π
( )12
0/Ln
2ddhC rεπε
=
( )12 /Ln2
ddhG iπσ
=
Resistance of the core
Resistance of the envelop
Inductance of the core placed inside the envelop
Capacitor made of the core and the envelop
Conductance of the insulator
Only in case of a coaxial cable the wave is really TEM!
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Distributed elements modelling
10
C1dzG1dz
z z+dz
i(z, t)
v(z, t)
i(z+dz, t)
v(z+dz, t)
R1dz L1dz
tiLiR
zv
∂∂
+=∂∂
− 11
tvCvG
zi
∂∂
+=∂∂
− 11
Kirchhoff laws:
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Telegraphers equation
11
Telegraphers Equations
( ) vGRtvGLCR
tvCL
zv
1111112
2112
2+
∂∂
++∂
∂=
∂
∂
( ) iGRtiGLCR
tiCL
zi
1111112
2112
2+
∂∂
++∂
∂=
∂
∂
Harmonic Solution
ωjt↔
∂∂
( ) )()()()(11111111
22
2zvGRzvGLCRjzvCL
zzv
+++−=∂
∂ ωω
( ) )()()()(11111111
22
2ziGRziGLCRjziCL
zzi
+++−=∂
∂ ωω
tjezvtzv ω)(),( =tjezitzi ω)(),( =
22
2ω−↔
∂
∂
t
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General harmonic solution
12
( )( )vjCGjLRz
v ωω 11112
2++=
∂
∂
( )( )ijCGjLRz
i ωω 11112
2++=
∂
∂
( )( )ωωγ 11112 jCGjLR ++= βαγ j+= 0and0 >> βα
Incident wave Reflected wave
VzV 22
2γ=
∂
∂
Iz
I 22
2γ=
∂
∂
)()()()( ztjzztjz eeVeeVv βωαβωα +−−−+ +=
)()()()( ztjzztjz eeIeeIi βωαβωα +−−−+ +=βω
=Pv
Propagation velocity
= Phase velocity
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Characteristic impedance of a transmission line
13
tiLiR
xv
∂∂
+=∂∂
− 11
ωω
11
11jCGjLRZ c +
+=
γω11
)(
)( jLRIV +
=+
+
cZIV
+=+
+
)(
)(
γω11
)(
)( jLRIV +
−=−
−
cZIV
−=−
−
)(
)(
Case of a lossless line1
1CLRZ cc ==
11
1CL
vp =01 =R01 =G
Zc is called the characteristic
impedance
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Example for a coaxial cable (RG‐58U)
14
-117 .m 1088.5 −Ω×=Cuσ
mm406.01 =d mm418.12 =d mm25.0=e
25.2r =ε 1=rμ-1114 .m 103.3 −− Ω×=iσ
180
0011m.s10211 −×====
rrP
cCL
vεεμε
Ω=⎟⎟⎠
⎞⎜⎜⎝
⎛== 50Ln
21
0
0
1
2
1
1
rc d
dCL
Rεεμ
π
,,
,
11 pF.m100 −=C 1
1 μH.m25.0 −=L
.
This is the same value as for a plane wave in the dielectric material.
Dielectric material polyethylene is such as:
Conductors: copper, conductivity
Geometrical characteristics
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Characteristic impedance of a microstrip line
15
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡ ++⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ +
++
++
= 222
02/11
'4
11/814
'4
11/814
'41Ln
122πεεε
επη rrr
rc w
hwh
whZ
Ω==≡ 377π1200
00 ε
μη is the characteristic impedance of free space
www Δ+='
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛
++⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ +
=Δ22
1.1//1
4Ln2/11
twht
etw r
π
επ
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Reflection coefficient
16
)(
)(
−
+=
L
LL
V
VΓ
L
LL Γ
Γz−+
=11
11
+−
=L
LL z
zΓ
c
LL Z
Zz =Normalized impedance:
L
Lc
L
LL Γ
ΓZ
IV
Z−+
==11
cL
cLL ZZ
ZZΓ
+
−=
ΓL is called thereflection coefficient
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Important cases for the reflection coefficient
17
cL ZZ = 0=LΓ
=∝LZ 1+=LΓ
0=LZ 1−=LΓ
If a transmission line has a load equal to the characteristic impedance, there is no reflection.
In this case the load is said to be matched to the transmission line.
If a transmission line is terminated by an open circuit the reflection coefficient is equal to +1.
If a transmission line is terminated by a short circuit thereflection coefficient is equal to -1.
1=Lz
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Reflection coefficient at a distance d from a load
18
djdLdL eeΓΓ βα 22
,−−=
For a lossless transmission line: djLdL eΓΓ β2
,−=
LΓ
LZ
dLΓ ,
0
d
zd−
Moving from the load toward generator = clockwise rotation on a constant reflection coefficient modulus circle
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Standing wave ratio
19
( )dLddd ΓVVVdV ,)()()( 1)( +=+= +−+
L
LΓΓ
VVSWR
−+
==11
min
max
-1 0
ΓL,d
1+ΓL,d
dLΓVdV ,)( 1)( += +
LΓ
LZ
dLΓ ,
0
d
zd−
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Standing wave ratio
20
Standing waves in case of a reflection coefficient equal to 0,6 (λ = 1 cm)SWR = 4
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Smith Chart
21
jqpΓ +=
ϕjeΓΓ =
jxrz +=11
+−
=zzΓ
jxrjxrjqp
+++−
=+11
( )22
2
11
1 rq
rrp
+=+⎟
⎠⎞
⎜⎝⎛
+− ⎟
⎠⎞
⎜⎝⎛
+= 0,
1rrC
r+=
11R
( ) 2
22 111
xxqp =⎟
⎠⎞
⎜⎝⎛ −+− ⎟
⎠⎞
⎜⎝⎛=
x1,1C x
1=R
The Smith chart is a representation of the reflection coefficient in polar coordinates
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Smith
chart
22
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Microwaves
Chapter 2 – Impedance matching
Đại Học Bách Khoa Hà Nội / 2012-2013
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Power received by a load
24
*21
LLL IV=PZLVL
IL
( )LLP PRe=
( ) ⎟⎠⎞⎜
⎝⎛ −==
+2
2)(
121Re L
cLL Γ
R
VP P
v
i
v(t)i(t)p(t) =
)cos(21)( ϕLLIVtp >=<
( )( ) ( ) ( )**)()(*)()()()(* 1121
21
21
LLLLL ΓIΓVIIVVIV −+=++== ++−+−+P
)()( ++ = IRV c)()( −− −= IRV c ( )( )**)()( 11
21
LLL ΓΓIV −+= ++P
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Incident and reflected power
25
⎟⎠⎞
⎜⎝⎛= +++ *)()()(
21Re IVP 2)(
2)()(
21
21 +
++ == IR
R
VP c
c
⎟⎠⎞
⎜⎝⎛= −−− *)()()(
21Re IVP
2)(
2)()(
21
21 −
−− −=−= IR
R
VP c
c
ccL R
V
R
VP
2)(2)(
21
21
−+
−= )()( −+ −= PPPL
)()( ++ = IRV c The incident wave propagates seeing an impedance Rc
The incident wave propagates seeing an impedance ‐Rc)()( −− −= IRV c
incident power
reflected power
The power transmitted to the load is the sum of the incident power and of the reflected power
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Normalized waves
26
21 )(+
=V
Ra
c 21 )(−
=V
Rb
c
abΓ L =
⎟⎠⎞⎜
⎝⎛ −=−= 2222 1 LL ΓabaP
ZLV (+) V (-)
ΓL
ZL
b
a
ΓL
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Unit of power
27
⎟⎟⎠
⎞⎜⎜⎝
⎛=
W1)W(log10dB
PP
⎟⎟⎠
⎞⎜⎜⎝
⎛=
mW1mW)(log10dBm
PP
30dBdBm += PP
in
outPP
G =
⎟⎟⎠
⎞⎜⎜⎝
⎛=
in
outPP
G log10dB
Decibel
Gain
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Power transfer from a source to a load
28
ZL
ΓL
a
b
ΓS
ZS
eS
*SL ΓΓ = 2
20
1
1
SSav
ΓaP
−=PL is maximum if: Available
power
aΓΓabΓaa LSSSS +=+= 00
LS
SΓΓ
aa−
=1
0
Sc
cS
cS ZR
RE
Ra
+=
21
0
2
22
022
1
1
LS
LSL
ΓΓ
ΓabaP
−
−=−=
)cos(21
1
1
122
22
02)(
22
0LSLSLS
LS
jLS
LSL
ΓΓΓΓ
Γa
eΓΓ
ΓaP
LS ϕϕϕϕ +−+
−=
−
−=
+
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Impedance matching
29
ZL
ΓLΓin
(C)
a1
b1 a2
b2
ΓoutΓS
ZS
eS
Matching network (matching to the source and to the load)
*Lout ΓΓ =
*Sin ΓΓ =
For a given source and a given load:matching is obtained by a matching cell placed between source and load
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Matching with discrete reactive elements
30
ZL
ΓLΓS
jX
jBpRc
(a)
ZL
ΓLΓS
jXs
jBRc
(a)
)(1
)/(11
LLpLLpc jBGjB
jXjXRjB
jXR++
+=++
+=)(11
LsLc XXjRjB
Z +++=
Thanks to the Smith Chart it is possible to avoid calculation
Serial association on the Smith chart: the point is moving on a constant real part circle of the impedance
Parallel association on the Smith chart: the point is moving on a constant real part circle of the admittance
)(' sLL xxjrz ++=
)(' pLL bbjgy ++=
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Matching with a single stub placed in parallel
31
ZL
d
l
Rc
)tan()(1)tan(
)tan(11
djbgjdjjbg
lj LL
LLββ
β ++++
+=
)tan(1)tan(1
djydjy
zy
L
L
LdLd β
β++
==
Ldstub yy +=1
Two solutions for d then two solutions for the length l
)tan()(1)tan(
)tan(11
djbgjdjjbg
lj LL
LLββ
β ++++
=−
Thanks to the Smith Chart it is possible to avoid calculation
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Matching with two stubs
32
ZL
d1
l1
Rc
l2
d2
Two distances d1 and d2 are fixed, the length l1 and l2 have to be determined
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Matching with a quarter wavelength line
33
RL
d = λ/4
RxRc
Lcx RRR =
11
+
−= x
L
xLx
LrrΓ
x
LxL R
Rr =11
11
114,
+
−
=+
−=−=== −−
xL
xL
xL
xLx
Ljx
Ljx
Lx
dL
r
rrrΓeΓeΓΓ π
λβ
xx
dLdL RrR ,, = xL
xcdL R
RR
RR ==,
This method concerns only real loads
L
xxL
xdL R
R
rr ==
1,
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Microwaves
Chapter 3 – The S parameters
Đại Học Bách Khoa Hà Nội / 2012-2013
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System with n connections
35
"Connection" equal to "port" (input or output)
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Scattering parameters
36
For a linear system with n ports (1, 2, … i, …j, …n)The normalized waves at the output can be expressed as a linear combination
of the normalized waves at the input.
∑=j
jiji aSb ( ) [ ]( )aSb =
Reflection coefficient at port "i" when all the other ports are matched to their corresponding transmission line.
Transmission coefficient from port "j" to port "i" when ports "k≠j" are matched to their corresponding transmission line.
0=≠
=ijai
iii a
bS
0=≠
=
jkaj
iij a
bS
[ ]S Smatrix
ijS Scattering or S parameters
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Case of a quadripole
37
2221212
2121111aSaSb
aSaSb+=+=
01
111
2=⎟⎟⎠
⎞⎜⎜⎝
⎛=
aabS
01
221
2=⎟⎟⎠
⎞⎜⎜⎝
⎛=
aabS
02
222
1=⎟⎟⎠
⎞⎜⎜⎝
⎛=
aabS
02
112
1=⎟⎟⎠
⎞⎜⎜⎝
⎛=
aabS
Q(1) (2)
a2a1
b1 b2
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S Parameters: 2 ports devices
38
Attenuator: ⎟⎟⎠
⎞⎜⎜⎝
⎛=
0'0
ββ
SR3
R2
R1
R3
R2
R1
Amplifier: ⎟⎟⎠
⎞⎜⎜⎝
⎛=
000
21SSPort-1 Port-2
Filter: low-pass, band-pass, high-pass or band-reject ⎟⎟⎠
⎞⎜⎜⎝
⎛=
0)()(0
21
12fS
fSS
0=iiSIdeal devices are matched at the different ports:
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Three ports devices
39
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
333231
232221
131211
SSSSSSSSS
S
jiij SS =
0=iiS
A lossless reciprocal three ports device cannot have all ports simultaneously matched.
Reciprocal device, the S matrix is symmetric:
All ports matched:
Lossless device, conservation of energy:
∑∑==
=3
1
23
1
2
ii
ii ab
![Page 40: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/40.jpg)
Power splitters (dividers)
40
cR2
cR2
cR
cR
cR
Port 1
Port 2
cR2
Port 3
4/λ
4/λ
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
011101110
21SPort 1
Port 2
Port 3
3/cR3/cR
3/cR
inout PP41
2 =
inout PP41
3 =
Resistive divider:
Wilkinson divider:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
=001001110
2jS
![Page 41: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/41.jpg)
Circulators
41
Circulator: Source 1 23
Matched load⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=0000
00
θ
θ
θ
j
j
j
ee
eS
31 aeb jθ= 12 aeb jθ= 23 aeb jθ=
03 =a 01 =b
The power returned at Port‐2 is totally absorbed by the matched load at Port‐3.
The source placed at Port‐1 is protected against any kind of change at Port‐2.
![Page 42: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/42.jpg)
Directionnel couplers
42
(3)
(2)
(4)
(1)in
isolated coupled
trough
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
000000
00
αβαββα
βα
ϕ
θ
ϕ
θ
j
j
j
j
ee
ee
S
αlog20log20log10 212
1 −=−== SPPIL
βlog20log20log10 313
1 −=−== SPPC
4131
41
4
3 log20log20log10SS
SPPD β
=−==
414
1 log20log10 SPPI −==
Insertion loss:
Coupling factor:
Directivity:
Isolation:
(1) (2)
(3)
(4)
dBdBdB CDI +=
Port‐4 internally matched
![Page 43: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/43.jpg)
Some examples of circuits: filters
43
![Page 44: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/44.jpg)
Bandpass filters
44
![Page 45: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/45.jpg)
Couplers
45
![Page 46: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/46.jpg)
Measurement planes
46
a1a’1 a’2a2
b’1 b1 b2 b’2
d1 d2
Q
(S)1 2
P’1 P1 P2 P’2
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= −
−
−
−
2
1
2
1
00
00' dj
dj
dj
dj
eeS
eeS β
β
β
β
11 1' beb dγ−=
11 1' aea dγ=
22 2' beb dγ−=
22 2' aea dγ=
![Page 47: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/47.jpg)
Measurement of the S parameters
47
RBSij = R
AS jj =
Att
DUTAtt (i)(j)
S
A B
R
Rc
Phas
The measurement instrument is called a network analyzer
The inputs at A, B and R are matched
![Page 48: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/48.jpg)
Network analyzer: basic structure
48
(1)(2)
(3)
(1)(2)
(3)
DUT
(4)
(4)
PowerSplitter
RFSource
Att. Phas
R B
A
Rc
Rc
ForwardPort-2
Port-1
Att.
Calibration process = measurement with standard loadsopen circuit, short circuit and characteristic impedance load
Reverse
![Page 49: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/49.jpg)
Two ports measurement
49
Network analyzer
![Page 50: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/50.jpg)
Measuremen
t of the
4 S param
eters
50
![Page 51: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/51.jpg)
Network analyzer structure with local oscillator
51
Heterodyne technique
![Page 52: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/52.jpg)
Examples of network analyzers
52Rhode & Schwarz (Germany)
Anritsu (Japan)Agilent Technologies (USA)
![Page 53: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/53.jpg)
Microwaves
Chapter 4 – Amplification
Đại Học Bách Khoa Hà Nội / 2012-2013
![Page 54: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/54.jpg)
Amplification basics
54
LO
IFA
Γ = ‐1Γ’ =1
λ/4
Γ = ‐1
λ/4
Γ’ =1
(DC)(DC & AC)
Decoupling for biasing the amplifier
Amplifier (LNA) in a receiver stage
Operating point
![Page 55: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/55.jpg)
Amplification is based on transistors
55
RD
λ/4
λ/4
In HF
-VGS
Out HF
+VDD
C1
C2
C1
C2
RC
λ/4
λ/4
In HF
Out HF
+VCC
C1
C2
C1
C2
R1
+VCC
R1
T1
T2
J‐FET transistors BJT transistors
![Page 56: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/56.jpg)
Reflection coefficients on a quadripole
56
22211211 1 SΓ
ΓSSSΓL
Lin −
+=11
122122 1 SΓΓSSSΓ
S
Sout −
+=
a1 b2
ZL
ΓLΓin
(S)b1
a2Q
ΓoutΓS
ZS
eS
![Page 57: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/57.jpg)
Insertion gain
57
av
Li P
PG ==sourcetheatpoweravailable
loadthetopower
( )( ) 221122211
222
2111
11
SSΓΓSΓSΓ
ΓΓSG
LSLS
LSi
−−−
⎟⎠⎞⎜
⎝⎛ −⎟⎠⎞⎜
⎝⎛ −
=
22
01
1
SSav
ΓaP
−=
⎟⎠⎞⎜
⎝⎛ −=−= 22
22
22
2 1 LL ΓbabP
SinS
L ΓΓa
SΓSb
−−=
11
1 022
212
22211211 1 SΓ
ΓSSSΓL
Lin −
+=
![Page 58: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/58.jpg)
Conjugate matching
58
ZL
ΓmL
Q
Γms
ZS
eS C1 C2
ΓLΓs Γin Γout
The problem is to find matching networks (C1) and (C2) to be placedsimultaneously at the input and at the output and leading to a maximumvalue of the insertion gain, remembering that the output has an influence onthe input and the input has an influence on the output.
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Unilateral quadripole
59
012 =SProperty of an unilateral quadripole
222
211
2221
2
11
11
SΓSΓ
ΓSΓGG
LS
LSiui
−−
⎟⎠⎞⎜
⎝⎛ −⎟
⎠⎞⎜
⎝⎛ −
==
*22SΓΓ mLL == *
11SΓΓ mSS ==
Conjugate matching is easy:
⎟⎠⎞⎜
⎝⎛ −⎟
⎠⎞⎜
⎝⎛ −
=2
22
2212
11max,
1
1
1
1
SS
SGiu
There is no feedback between the output (Port‐2) to theinput (Port‐1)
![Page 60: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/60.jpg)
Unilateral practical condition
60
⎟⎠⎞⎜
⎝⎛ −⎟⎠⎞⎜
⎝⎛ −
=2
222
11
22122111
11 SS
SSSSuUnilateral figure of merit
( ) ( )22 11 u
GG
u
G iui
iu
−≤≤
+
1<<u ( ) ( )uGGuG iuiiu 2121 +≤≤−
iui GG ≈12 <<u
Examples
‐ Transistor 2N3970 at f = 750 MHz
‐ Transistor FPD4000AF at f = 3 GHz
151.011 =S 38.621 =S 061.012 =S 63.022 =S 2106.122 −=u
8.011 =S 926.121 =S 062.012 =S 488.022 =S 210342 −=u
![Page 61: DHBK Microwaves Slides 2012](https://reader034.vdocuments.site/reader034/viewer/2022050811/577cca021a28aba711a51fa7/html5/thumbnails/61.jpg)
Designing amplifiers
61
Stability
Noise figure
Gain
Bandwidth
Trade‐off: