s parameters (1)nn
TRANSCRIPT
-
8/18/2019 s Parameters (1)Nn
1/107
A
-
8/18/2019 s Parameters (1)Nn
2/107
S-parameters are a useful method for representing a circuit as a “black box”
-
8/18/2019 s Parameters (1)Nn
3/107
S-parameters are a useful method for representing a circuit as a “black box”
.
-
8/18/2019 s Parameters (1)Nn
4/107
.
,
.
S-parameters are a useful method for representing a circuit as a “black box”
-
8/18/2019 s Parameters (1)Nn
5/107
A “black box” or network may have any number of ports.
This diagram shows a simple
network with just 2 ports.
-
8/18/2019 s Parameters (1)Nn
6/107
A “black box” or network may have any number of ports.
This diagram shows a simple
network with just 2 ports.
Note :
A port is a terminal pair of lines.
-
8/18/2019 s Parameters (1)Nn
7/107
S-parameters are measured by sending a single frequency signal into thenetwork or “black box” and detecting what waves exit from each port.
Power, voltage and current
can be considered to be inthe form of waves travellingin both directions.
-
8/18/2019 s Parameters (1)Nn
8/107
Power, voltage and current
can be considered to be inthe form of waves travellingin both directions.
For a wave incident on Port 1,some part of this signal
reflects back out of that portand some portion of the signalexits other ports.
S-parameters are measured by sending a single frequency signal into thenetwork or “black box” and detecting what waves exit from each port.
-
8/18/2019 s Parameters (1)Nn
9/107
I have seen S-parameters described as S11, S21, etc. Can you explain?
First lets look at S11.
S11 refers to the signal
reflected at Port 1 for thesignal incident at Port 1.
-
8/18/2019 s Parameters (1)Nn
10/107
First lets look at S11.
S11 refers to the signal
reflected at Port 1 for thesignal incident at Port 1.
Scattering parameter S11
1/1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
-
8/18/2019 s Parameters (1)Nn
11/107
Now lets look at S21.
S21 refers to the signal
exiting at Port 2 for thesignal incident at Port 1.
Scattering parameter S21
2/1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
-
8/18/2019 s Parameters (1)Nn
12/107
Now lets look at S21.
S21 refers to the signal
exiting at Port 2 for thesignal incident at Port 1.
Scattering parameter S21
2/1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
-
8/18/2019 s Parameters (1)Nn
13/107
Now lets look at S21.
S21 refers to the signal
exiting at Port 2 for thesignal incident at Port 1.
Scattering parameter S21
2/1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
-
8/18/2019 s Parameters (1)Nn
14/107
-
8/18/2019 s Parameters (1)Nn
15/107
A linear network can be characterised by a set of simultaneous equations
describing the exiting waves from each port in terms of incident waves.
S11 = b1 / a1
S12 = b1 / a2
S21 = b2 / a1
S22 = b2 / a2
Note again how the subscript follows the parameters in the ratio (S11=b1/a1, etc...)
I have seen S-parameters described as S11, S21, etc. Can you explain?
-
8/18/2019 s Parameters (1)Nn
16/107
-
8/18/2019 s Parameters (1)Nn
17/107
S-parameters are complex (i.e. they have magnitude and angle)
because both the magnitude and phase of the input signal arechanged by the network.
(This is why they are sometimes referred to as complex scattering parameters).
-
8/18/2019 s Parameters (1)Nn
18/107
These four S-parameters actually contain eight separate numbers:
the real and imaginary parts (or the modulus and the phase angle)of each of the four complex scattering parameters.
-
8/18/2019 s Parameters (1)Nn
19/107
Quite often we refer to the magnitude only as it is of the most interest.
How much gain (or loss) you get is usually more important than how muchthe signal has been phase shifted.
-
8/18/2019 s Parameters (1)Nn
20/107
S-parameters depend upon the network
and the characteristic impedances of thesource and load used to measure it, andthe frequency measured at.
i.e.
if the network is changed, the S-parameters change.
if the frequency is changed, the S-parameters change.
if the load impedance is changed, the S-parameters change.
if the source impedance is changed, the S-parameters change.
What do S-parameters depend on?
-
8/18/2019 s Parameters (1)Nn
21/107
What do S-parameters depend on?
S-parameters depend upon the network
and the characteristic impedances of thesource and load used to measure it, andthe frequency measured at.
i.e.
if the network is changed, the S-parameters change.
if the frequency is changed, the S-parameters change.
if the load impedance is changed, the S-parameters change.
if the source impedance is changed, the S-parameters change.
In the Si9000e S-parameters arequoted with source and loadimpedances of 50 Ohms
-
8/18/2019 s Parameters (1)Nn
22/107
A little math…
This is the matrix algebraic representation
of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry aboutthe leading diagonal.
-
8/18/2019 s Parameters (1)Nn
23/107
A little math…
This is the matrix algebraic representation
of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry aboutthe leading diagonal.
In the case of a 2-port network, that means that S21 = S12 and interchangingthe input and output ports does not change the transmission properties.
-
8/18/2019 s Parameters (1)Nn
24/107
A little math…
This is the matrix algebraic representation
of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry aboutthe leading diagonal.
In the case of a symmetrical 2-port network, that means that S21 = S12 andinterchanging the input and output ports does not change the transmission
properties.
A transmission line is an example of a symmetrical 2-port network.
-
8/18/2019 s Parameters (1)Nn
25/107
A little math…
Parameters along the leading diagonal,
S11 & S22, of the S-matrix are referred to asreflection coefficients because they refer tothe reflection occurring at one port only.
-
8/18/2019 s Parameters (1)Nn
26/107
A little math…
Parameters along the leading diagonal,
S11 & S22, of the S-matrix are referred to asreflection coefficients because they refer tothe reflection occurring at one port only.
Off-diagonal S-parameters, S12, S21, are referred to as transmission coefficients because they refer to what happens from one port to another.
-
8/18/2019 s Parameters (1)Nn
27/107
Larger networks:
A Network may have any number of ports.
-
8/18/2019 s Parameters (1)Nn
28/107
Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n2 coefficients (S-parameters),each one representing a possible input-output path.
-
8/18/2019 s Parameters (1)Nn
29/107
Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n2 coefficients (S-parameters),each one representing a possible input-output path.
The number of rows and columns in an S-parameters matrix is equal to thenumber of ports.
-
8/18/2019 s Parameters (1)Nn
30/107
Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n2 coefficients (S-parameters),each one representing a possible input-output path.
The number of rows and columns in an S-parameters matrix is equal to thenumber of ports.
For the S-parameter subscripts “ij”, “j” is the port that is excited (the input port)
and “i” is the output port.
-
8/18/2019 s Parameters (1)Nn
31/107
Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n2 coefficients (S-parameters),each one representing a possible input-output path.
The number of rows and columns in an S-parameters matrix is equal to thenumber of ports.
For the S-parameter subscripts “ij”, “j” is the port that is excited (the input port)
and “i” is the output port.
-
8/18/2019 s Parameters (1)Nn
32/107
-
8/18/2019 s Parameters (1)Nn
33/107
Sum up…
• S-parameters are a powerful way to describe an electrical network
• S-parameters change with frequency / load impedance / source impedance / network• S11 is the reflection coefficient • S21 describes the forward transmission coefficient (responding port 1
st!)• S-parameters have both magnitude and phase information• Sometimes the gain (or loss) is more important than the phase shift and the phase
information may be ignored• S-parameters may describe large and complex networks
• If you would like to learn more please see next slide:
-
8/18/2019 s Parameters (1)Nn
34/107
Further reading:
//..//51.
//..//154.
//..//.//24321152500542
//..
//.101./.
//..///130.
//..//
140
Online lecture OLL-141 S11 & Smith charts - Eric Bogatinwww.bethesignal.com
-
8/18/2019 s Parameters (1)Nn
35/107
D
-
8/18/2019 s Parameters (1)Nn
36/107
C
36
( ) ( )φ+ω= tcosVtV o
( ) ( ) t j jot j
o eeVReeVRetV ωφφ+ω ==
1 j −=
SinusoidalSource
φ joeV is a complex phasor
-
8/18/2019 s Parameters (1)Nn
37/107
• ,
• C •
37
oV
( )φcosVo
( )φsinVo
φωRe
Im
( ) ( )φ+φ=φ sin jVcosVeV oo j
o
t je
ω
-
8/18/2019 s Parameters (1)Nn
38/107
38
Charge on the inner conductor:
xVCq l∆=∆
where Cl is the capacitance per unit lengthAzimuthal magnetic flux:
xIL l∆=∆Φ
where Ll is the inductance per unit length
-
8/18/2019 s Parameters (1)Nn
39/107
39
i ii ∆+
v vv ∆+
xL l∆
xC l∆
Voltage drop along the inductor:
( )
dt
dixLvvv l∆=∆+−
Current flowing through the capacitor:
dt
dvxCiii l∆−=∆+
-
8/18/2019 s Parameters (1)Nn
40/107
40
Limit as ∆x->0
t
i
Lx
v
l ∂
∂
−=∂
∂
t
B
E ∂
∂
−=×∇
t
vC
x
il∂
∂−=
∂
∂
t
DH
∂
∂=×∇
Solutions are traveling waves
( )
++
−= −+
vel
xtv
vel
xtvx,tv
( )
+−
−=
−+
vel
xt
Z
v
vel
xt
Z
vx,ti
oo
v+ indicates a wave traveling in the +x directionv- indicates a wave traveling in the -x direction
-
8/18/2019 s Parameters (1)Nn
41/107
C
41
vel is the phase velocity of the wave
llCL
1
vel =
For a transverse electromagnetic wave (TEM), the phase velocity isonly a property of the material the wave travels through
µε= 1
CL
1
ll
The characteristic impedance Zo
l
lo
C
LZ =
has units of Ohms and is a function of the material AND thegeometry
-
8/18/2019 s Parameters (1)Nn
42/107
42
Pulse travels down the transmission line as a forward going wave only(v+). However, when the pulse reaches the load resistor:
oo
L
Z
v
Z
v
vvRi
v
−+
−+
−
+==
LR+v
so a reverse wave v-
and i-
must be created to satisfy the boundarycondition imposed by the load resistor
-
8/18/2019 s Parameters (1)Nn
43/107
C
43
The reverse wave can be thought of as the incident wave reflectedfrom the load
Γ =+−=+
−
oL
oLZRZR
vv Reflection coefficient
Three special cases:
RL = ∞ (open) Γ = +1
RL = 0 (short) Γ = -1
RL = Zo Γ = 0
A transmission line terminated with a resistor equal in value to the
characteristic impedance of the transmission line looks the same tothe source as an infinitely long transmission line
-
8/18/2019 s Parameters (1)Nn
44/107
44
( ) t jx j eeVRextcosVv ωβ−+++ =β−ω=
vel=β
ω
phase velocity λ
π=
π=β
2
vel
f 2wave number
By using a single frequency sine wave we can now define compleximpedances such as:
LI jV ω=dt
diLv =
dt
dvCi =
C j
1
Zcap ω=
L jZind ω=
CV jI ω=
Experiment: Send a SINGLE frequency (ω) sine wave into atransmission line and measure how the line responds
-
8/18/2019 s Parameters (1)Nn
45/107
45
LZoZ
0x =
x
d
At x=0
oL
oL
ZZ
ZZVV+−=Γ = +−
Along the transmission line:
( ) ( )xcosV2e1VV
eVeVV
x j
x jx j
βΓ +Γ −=
Γ +=
+β−+
β++β−+
traveling wave standing wave
-
8/18/2019 s Parameters (1)Nn
46/107
()
46
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5
1
0.5
0
0.5
1
1.5
Position
( )
( )
( )
( )VSWR
1
1
1V
1V
V
V
min
max
=Γ −
Γ +=
Γ −
Γ +=
+
+
The VSWR is always greater than 1
2
1=Γ
Large voltage
Large current
-
8/18/2019 s Parameters (1)Nn
47/107
()
47
( )
( )
( )( )
VSWR
1
1
1V
1V
V
V
min
max
=
Γ −
Γ +=
Γ −
Γ +=
+
+
The VSWR is always greater than 1
2
1=Γ
Incident wave
Reflected wave
Standing wave
-
8/18/2019 s Parameters (1)Nn
48/107
C A
48
LZoZ
0x =
x
d
oL
oLL
ZZZZ
+−=Γ GΓ
towards load
towards generator
x j
L
x j
eVeVV
β++β−+
Γ +=
d2 jL)d( j
)d( j
L
genreverse
forwardG e
eV
eV
V
V β−−β−+
−β++
Γ =Γ ==Γ
Wave has to traveldown and back
-
8/18/2019 s Parameters (1)Nn
49/107
49
GΓ
LΓ
{ }Γ Re
{ }Γ Im
d2β−=θ
There is a one-to-onecorrespondence between Γ G and
ZL
oG
oG
G ZZ
ZZ
+
−=Γ
G
GoG
1
1ZZ
Γ −
Γ +=
d2 jL
d2 jL
oGe1
e1ZZ
β−
β−
Γ −
Γ +=
-
8/18/2019 s Parameters (1)Nn
50/107
: C
50
For an open circuit ZL= ∞ so Γ L = +1
Impedance at the generator:
( )dtan
jZZ oG
β
−=
For βd
-
8/18/2019 s Parameters (1)Nn
51/107
: C
51
For a short circuit ZL= 0 so Γ L = -1
Impedance at the generator:
( )dtan jZZ oG β=
For βd
-
8/18/2019 s Parameters (1)Nn
52/107
52
LZoZ
0x =
x
d
sP
dx −=
d j
oL
d j
oG
d j
L
d j
G
eZ
Ve
Z
V)d(II
eVeV)d(VV
β−+
β++
β−+β++
Γ −=−=
Γ +=−=
Voltage and Current at the generator (x=-d)
The rate of energy flowing through the plane at x=-d
{ }
o
22
Lo
2
*GG
Z
V
2
1
Z
V
2
1P
IVRe2
1P
++
Γ −=
=
forward power
reflected power
-
8/18/2019 s Parameters (1)Nn
53/107
• ! .
–
• C
–
• :
53
0L =Γ
which implies:
oL ZZ =When ZL = Zo, the load is matched to the transmission line
-
8/18/2019 s Parameters (1)Nn
54/107
54
What if the load cannot be made equal to Zo for some other reasons?
Then, we need to build a matching network so that the sourceeffectively sees a match load.
0=Γ
LZsP 0Z M
Typically we only want to use lossless devices such as capacitors,inductors, transmission lines, in our matching network so that we donot dissipate any power in the network and deliver all the availablepower to the load.
-
8/18/2019 s Parameters (1)Nn
55/107
55
jxrZ
Zz
o
+==
It will be easier if we normalize the load impedance to thecharacteristic impedance of the transmission line attached to the
load.
Γ −
Γ +
= 1
1
z
Since the impedance is a complex number, the reflection coefficientwill be a complex number
jvu +=Γ
( )
22
22
vu1
vu1r
+−
−−=
( ) 22 vu1
v2x
+−=
-
8/18/2019 s Parameters (1)Nn
56/107
B B
56
A dB is defined as a POWER ratio. For example:
( )Γ =
Γ =
=Γ
log20
log10
P
Plog10
2
for
revdB
A dBm is defined as log unit of power referenced to 1mW:
=
mW1
Plog10PdBm
-
8/18/2019 s Parameters (1)Nn
57/107
D
-
8/18/2019 s Parameters (1)Nn
58/107
58
We have only discussed reflection so far. What about transmission?Consider a device that has two ports:
1V 2V
2I1I
[ ] [ ][ ]IZV
IZIZV
IZIZV
2221212
2121111
=
+=
+=
The device can be characterized by a 2x2 matrix:
-
8/18/2019 s Parameters (1)Nn
59/107
()
59
−+
−+
−=
+=
iiio
iii
VVIZ
VVV
Since the voltage and current at each port (i) can be broken downinto forward and reverse waves:
We can characterize the circuit with forward and reverse waves:
[ ] [ ][ ]+−++−
++−
=
+=
+=
VSV
VSVSV
VSVSV
2221212
2121111
-
8/18/2019 s Parameters (1)Nn
60/107
60
[ ] [ ] [ ]( ) [ ] [ ]( )
[ ] [ ] [ ]( ) [ ] [ ]( ) 1o
o1
o
S1S1ZZ
1ZZ1ZZS
−
−
−+=
−+=
Similar to the reflection coefficient, there is a one-to-one
correspondence between the impedance matrix and the scatteringmatrix:
-
8/18/2019 s Parameters (1)Nn
61/107
()
61
The S matrix defined previously is called the un-normalized
scattering matrix. For convenience, define normalized waves:
io
ii
io
ii
Z2
Vb
Z2
Va
−
+
=
=
Where Zoi is the characteristic impedance of the transmission lineconnecting port (i)
|ai|2 is the forward power into port (i)
|bi|2
is the reverse power from port (i)
-
8/18/2019 s Parameters (1)Nn
62/107
()
62
The normalized scattering matrix is:
Where:
[ ] [ ][ ]asbasasb
asasb
22212122121111
=
+=
+=
j,iio
jo
j,i S
Z
Zs =
If the characteristic impedance on both ports is the same then thenormalized and un-normalized S parameters are the same.
Normalized S parameters are the most commonly used.
-
8/18/2019 s Parameters (1)Nn
63/107
63
The s parameters can be drawn pictorially
s11 and s22 can be thought of as reflection coefficientss21 and s12 can be thought of as transmission coefficients
s parameters are complex numbers where the angle corresponds to aphase shift between the forward and reverse waves
s11 s22
s21
s12
a1
a2b1
b2
-
8/18/2019 s Parameters (1)Nn
64/107
64
τ
Zo1 2
[ ]
=
ωτ−
ωτ−
0e
e0s
j
j
21 [ ]
−−=
1001s
1 2
[ ]
=
0G
00s
G
Transmission Line
Short
Amplifier
-
8/18/2019 s Parameters (1)Nn
65/107
65
[ ]
=
010
001
100
s
1
2
[ ]
=
01
00s
Zo
Isolator
1 2
3
Circulator
-
8/18/2019 s Parameters (1)Nn
66/107
66
If the device is made out of linear isotropic materials (resistors,capacitors, inductors, metal, etc..) then:
[ ] [ ]ss T =
j,ii, j ss = ji ≠
or
for
This is equivalent to saying that the transmitting pattern of anantenna is the same as the receiving pattern
reciprocal devices: transmission line
shortdirectional coupler
non-reciprocal devices: amplifier
isolator
circulator
D
-
8/18/2019 s Parameters (1)Nn
67/107
D
67
The s matrix of a lossless device is unitary:
[ ] [ ] [ ]1ss T* =
∑
∑
=
=
j
2 j,i
i
2
j,i
s1
s1for all j
for all i
Lossless devices: transmission line
shortcirculator
Non-lossless devices: amplifier
isolator
A
-
8/18/2019 s Parameters (1)Nn
68/107
68
Network analyzers measure Sparameters as a function of
frequency
At a single frequency, networkanalyzers send out forward waves a1and a2 and measure the phase and
amplitude of the reflected waves b1and b2 with respect to the forward
waves.
a1 a2
b1 b2
02a1
111
a
bs
=
=02a
1
221
a
bs
=
=
01a2
112
a
bs
=
=01a
2
222
a
bs
=
=
A C
-
8/18/2019 s Parameters (1)Nn
69/107
69
To measure the pure S parameters of a device, we need to eliminatethe effects of cables, connectors, etc. attaching the device to the
network analyzer
s11 s22
s21
s12
x11 x22
x21
x12
y11 y22
y21
y12
yx21
yx12
Connector Y Connector X
We want to know the S parameters atthese reference planes
We measure the S parameters at thesereference planes
A C
-
8/18/2019 s Parameters (1)Nn
70/107
A C
• 10
• 10 – D
– D (D)
– B
(), D .
70
A C
-
8/18/2019 s Parameters (1)Nn
71/107
A C•
, .
• :
71
[ ]
−
−=
10
01s
[ ]
=
ωτ−
ωτ−
0e
e0s
j
j
τ
[ ]
=
01
10sThru
Short
Delay**ωτ~90degrees
D
-
8/18/2019 s Parameters (1)Nn
72/107
72
A pure sine wave can be written as:( )zt j
oeVV β−ω=
The phase shift due to a length of cable is:
ph
ph
d
v
d
ωτ=
ω=
β=θ
The phase delay of a device is defined as:
( )ω
−=τ 21phSarg
D
-
8/18/2019 s Parameters (1)Nn
73/107
D
• ,
.• ,
.
•
–
–
73
D
-
8/18/2019 s Parameters (1)Nn
74/107
D• A
–
– •
74
( )( ) ( )tcostcosm1VV o ωω∆+=
( ) ( )( ) ( )( )[ ]tcostcos2
mVtcosVV oo ω∆−ω+ω∆+ω+ω=
The modulation can be de-composed into different frequencycomponents
D
-
8/18/2019 s Parameters (1)Nn
75/107
75
( )
( ) ( )( )
( ) ( )( )ztcos2
mV
ztcos2
mV
ztcosVV
o
o
o
β∆−β−ω∆−ω+
β∆+β−ω∆+ω+
β−ω=
The waves emanating from the source will look like
Which can be re-written as:
( )( ) ( )ztcosztcosm1VV o β−ωβ∆−ω∆+=
D
-
8/18/2019 s Parameters (1)Nn
76/107
76
The information travels at a velocity
ω∂
β∂⇒
ω∆
β∆= 11vgr
The group delay is defined as:
( )( )
ω∂
∂
−=
ω∂
β∂=
=τ
21
grgr
Sarg
d
v
d
D D
-
8/18/2019 s Parameters (1)Nn
77/107
D D
77
Phase Delay:
( )
ω−=τ 21
ph
Sarg
Group Delay:
( )( )ω∂∂−=τ 21gr Sarg
-
8/18/2019 s Parameters (1)Nn
78/107
D
C
-
8/18/2019 s Parameters (1)Nn
79/107
• , , , .
• , () , &
•
, ,, , . ,
79
C
-
8/18/2019 s Parameters (1)Nn
80/107
• C
•
•
80
C
-
8/18/2019 s Parameters (1)Nn
81/107
• ,
• , .
.
• A . (
)
•
81
C
-
8/18/2019 s Parameters (1)Nn
82/107
82
The impedance as a function of reflection coefficient can be re-written in the form:
( ) 22
22
vu1
vu1
r +−
−−
=
( ) 22 vu1
v2x
+−=
( )22
2
r1
1
vr1
r
u +=+
+−
( ) 2
22
x
1
x
1
v1u =
−+−
These are equations for
circles on the (u,v) plane
C C
-
8/18/2019 s Parameters (1)Nn
83/107
83
1 0.5 0 0.5 1
1
0.5
0.5
1
{ }Γ Re
{ }Γ Im
r=0r=1/3
r=1r=2.5
C C
-
8/18/2019 s Parameters (1)Nn
84/107
84
1 0.5 0 0.5 1
1
0.5
0.5
1
{ }Γ Re
{ }Γ Im
x=1/3 x=1 x=2.5
x=-1/3 x=-1 x=-2.5
C
-
8/18/2019 s Parameters (1)Nn
85/107
85
C 1
-
8/18/2019 s Parameters (1)Nn
86/107
86
Given:
Ω= 50Zo
°∠=Γ 455.0L
What is ZL?
( )Ω+Ω=
+Ω=5.67 j5.67
35.1 j35.150ZL
C 2
-
8/18/2019 s Parameters (1)Nn
87/107
87
Given:
Ω= 50Zo
Ω−Ω= 25 j15ZL
What is Γ L?
5.0 j3.050
25 j15z L
−=Ω
Ω−Ω=
°−∠=Γ 124618.0L
C 3
GiΩ+Ω= 50 j50Z
L
-
8/18/2019 s Parameters (1)Nn
88/107
88
Given:Ω= 50Zo
L
What is Zin at 50 MHz?
0.1 j0.1
50
50 j50z L
+=
Ω
Ω+Ω=
°∠=Γ 64445.0L
nS78.6=τ
?Zin
=
ωτ−β− Γ =Γ =Γ 2 jLd2 j
Lin ee
°=ωτ 2442°∠=Γ 180445.0in
( ) Ω=+Ω= 190.0 j38.050ZL
°=ωτ 2442
A
A t hi t k i i t b bi ti f l t
-
8/18/2019 s Parameters (1)Nn
89/107
89
A matching network is going to be a combination of elementsconnected in series AND parallel.
Impedance is NOT well suited when working
with parallel configurations.
21L ZZZ +=
2
Z1
Z
2Z
1Z
21
21L
ZZ
ZZZ
+=
ZIV =
For parallel loads it is better to work withadmittance.
YVI =
2Y1Y
21L YYY +=1
1Z
1Y =
Impedance is well suited when working withseries configurations. For example:
A
jbYZY
-
8/18/2019 s Parameters (1)Nn
90/107
90
jbgYZYYy o
o+===
Γ +
Γ −=
1
1y
( ) 22
22
vu1
vu1g
++
−−=
( ) 22 vu1
v2b
++
−=
( )22
2
g1
1v
g1
gu
+=+
++
( )2
22
b
1
b
1v1u =
+++
These are equations forcircles on the (u,v) plane
A C
-
8/18/2019 s Parameters (1)Nn
91/107
91
1 0.5 0 0.5 1
1
0.5
0.5
1
1 0.5 0 0.5 1
1
0.5
0.5
1
{ }Γ Re
{ }Γ Im
g=1/3
b=-1 b=-1/3
g=1g=2.5 g=0
b=2.5 b=1/3
b=1
b=-2.5
{ }Γ Im
{ }Γ Re
A
C
-
8/18/2019 s Parameters (1)Nn
92/107
C•
,
–
– C
• 180 .
– 180
. 92
A C 1
-
8/18/2019 s Parameters (1)Nn
93/107
93
• Procedure:
• Plot 1+j1 on chart
• vector =• Flip vector 180 degrees
Given:
°∠64445.0
What is Γ
?
1 j1y +=
°−∠=Γ 116445.0
Plot y
Flip 180
degreesRead Γ
A C 2
Given:
-
8/18/2019 s Parameters (1)Nn
94/107
94
• Procedure:
• Plot Γ
• Flip vector by 180 degrees
• Read coordinate
Given:
What is Y?
°+∠=Γ 455.0 Ω= 50Zo
Plot Γ
Flip 180degrees
Read y
36.0 j38.0y −=
( )
( ) mhos10x2.7 j6.7Y
36.0 j38.050
1Y
3−−=
−Ω=
C
Constant ImaginaryImpedance Lines
-
8/18/2019 s Parameters (1)Nn
95/107
95
Constant RealImpedance Circles
Impedance
Z=R+jX
=100+j50
Normalized
z=2+j for
Zo=50
C• (50)
– 1 = 100 + 50
-
8/18/2019 s Parameters (1)Nn
96/107
1 = 100 + 50 – 2 = 75 100 – 3 = 200 – 4 = 150 – 5 = ( )
– 6 = 0 ( ) – 7 = 50 – 8 = 184 900
•, . :
– 1 = 2 + – 2 = 1.5 2
– 3 = 4
– 4 = 3 – 5 = – 6 = 0 – 7 = 1 – 8 = 3.68 18
96
Toward
Generator ConstantReflection
-
8/18/2019 s Parameters (1)Nn
97/107
97
Generator
Away FromGenerator
ReflectionCoefficient Circle
Scale inWavelengths
Full Circle is One HalfWavelength Since
Everything Repeats
C
-
8/18/2019 s Parameters (1)Nn
98/107
• , = 50 + 100
•
• ( )
• A . ,
.
98
-
8/18/2019 s Parameters (1)Nn
99/107
99
0=Γ
Ω100sP Ω= 50Z0 M
Match 100Ω load to a 50Ω system at 100MHz
A 100Ω resistor in parallel would do the trick but ½ of thepower would be dissipated in the matching network. We wantto use only lossless elements such as inductors and capacitors
so we don’t dissipate any power in the matching network
We need to go from
-
8/18/2019 s Parameters (1)Nn
100/107
100
We need to go fromz=2+j0 to z=1+j0 on
the Smith chart
We won’t get anycloser by adding seriesimpedance so we willneed to add somethingin parallel.
We need to flip overto the admittance
chart
ImpedanceChart
y=0.5+j0
-
8/18/2019 s Parameters (1)Nn
101/107
101
y 0.5 j0
Before we add theadmittance, add a
mirror of the r=1circle as a guide.
AdmittanceChart
-
8/18/2019 s Parameters (1)Nn
102/107
102
y=0.5+j0
Before we add the
admittance, add amirror of the r=1circle as a guide
Now add positive
imaginary admittance.
AdmittanceChart
y=0.5+j0
-
8/18/2019 s Parameters (1)Nn
103/107
103
y j Before we add the
admittance, add amirror of the r=1
circle as a guide Now add positive
imaginary admittance jb = j0.5
AdmittanceChart
( )
pF16C
CMHz1002 j50
5.0 j
5.0 j jb
=
π=Ω
=
pF16 Ω100
-
8/18/2019 s Parameters (1)Nn
104/107
104
We will now add seriesimpedance
Flip to the impedanceSmith Chart
We land at on the r=1circle at x=-1
ImpedanceChart
Add positive imaginaryd itt t t t
-
8/18/2019 s Parameters (1)Nn
105/107
105
admittance to get toz=1+j0
ImpedanceChart
pF16
Ω100
( ) ( )
nH80L
LMHz1002 j500.1 j0.1 j jx
=
π=Ω=
nH 80
This solution wouldh ve ls rked
-
8/18/2019 s Parameters (1)Nn
106/107
106
have also worked
ImpedanceChart
pF32
Ω100nH160
B
5
0
nH80
-
8/18/2019 s Parameters (1)Nn
107/107
107
50 60 70 80 90 100 110 120 130 140 15040
35
30
25
20
15
10
Frequency (MHz)
R e f l e c t i o n C o e f f i c i e n t ( d B )
50 MHz
100 MHz
Because the inductor and capacitorimpedances change with frequency, thematch works over a narrow frequency range
pF16Ω100
ImpedanceChart