2 rf network analysis - mixedsignal august 2008 © 2006 by fabian kung wai lee chapter 2 3...
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1
Chapter 2August 2008 © 2006 by Fabian Kung WaiLee 1
2 . T r a n s m i s s i o n L i n e
C i r c u i t s a n d R F / M i c r o w a v e
N e t w o r k A n a l y s i s
The information in this work has been obtained from sources believed to be reliable.The author does not guarantee the accuracy or completeness of any informationpresented herein, and shall not be responsible for any errors, omissions or damagesas a result of the use of this information.
Chapter 2
Agenda
• 1.0 – Terminated transmission line circuit.
• 2.0 – Smith Chart and its applications.
• 3.0 – Practical considerations for stripline implementation.
• 4.0 – Linear RF network analysis – 2-port network parameters.
August 2008 © 2006 by Fabian Kung Wai Lee 2
2
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 3
References
• [1] R.E. Collin, “Foundation for microwave engineering”, 2nd edition, 1992, McGraw-Hill.
• [2] T. C. Edwards, “Foundations for microstrip circuit design”, 2nd edition, 1992 John-Wiley & Sons (3rd Edition is also available).
• [3] D.M. Pozar, “Microwave engineering”, 2nd edition, 1998 John-Wiley & Sons (3rd edition, 2005 from John-Wiley & Sons is also available).
A very advanced and in-depth book on microwave
engineering. Difficult to read but the information is
very comprehensive. A classic work. Recommended.
Contains a wealth of practical microstrip design
information. A must have for every microwave
circuit design engineer.
Good coverage of EM theory with emphasis on
applications.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 4
Review of Previous Lecture
• In previous lecture we have studied how a transmission line (Tline)
structure can guide a travelling EM wave.
• We covered the various type of propagation modes for EM waves, in
particular we are interested in TEM and quasi-TEM mode operation.
• Under these two modes, the Tline can be represented by distributed
circuit model consisting of RLCG network, the E field corresponds to the
transverse voltage Vt and the H field corresponds to the axial current It,
Vt and It are also propagating waves in the Tline.
• We have also covered how to derived the RLCG parameters under low
loss condition.
• Finally, in the last section of previous chapter, we also studied the
design procedure of stripline structures on printed circuit board.
• In this chapter, we are going to study the characteristics of Tline
terminated with impedance, and how to use Tline in RF/microwave
circuits and systems.
3
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 5
1.0 Terminated Transmission Line Circuit
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 6
The Lossless Transmission Line Circuit
• A transmission line circuit consists of source, load networks and the
Tline itself.
• We will use the coordinate as shown. Some basic parameters will be
derived in the following slides.
Source Network Load Network
VLZL
ILTline
Zs Vs Vt(z)
It(z)
l
+z
z = 0z = -l
The schematic
The physical
system
4
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 7
Voltage and Current on Transmission Line Circuit
• Assumption: Tline is
lossless (α=0), and
supporting TEM mode.
• At a position z along the
Tline:
• At z=0:
( ) Loo VVVV =+= −+0
( ) Loo IIII =−= −+0
( ) zjo
zjo eVeVzV
ββ +−−+ +=
( ) zjo
zjo eIeIzI
ββ +−−+ −=(1.1a)
VL
l
+z
z = 0
Towards
source ZL
IL
Z=-l
zjo
zjo eIeV
ββ −+−+
Incident V and I waves
zjo
zjo eIeV
ββ +−+− -
Reflected waves
Incident and
reflected
waves combine
to produce load
voltage and
current
Using the definitionof Zc
( )−+ −= ooc
L VVZ
I1
−
+==
−+
−+
oo
ooc
L
LL
VV
VVZ
I
VZ (1.1b)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 8
+
−=Γ
o
oL
V
V (1.2a)
Γ−
Γ+=
L
LcL ZZ
1
1
cL
cLL
ZZ
ZZ
+
−=Γ
c
LL
L
LL
Z
ZZ
Z
Z=
+
−=Γ
1
1
(1.2b)
L
o
oI
I
IΓ−=
−=Γ
+
−
(1.2d)
(1.2c)
Reflection Coefficient (1)
• The ratio of Vo- over Vo
+ is described by a voltage reflection coefficient Γ. At the load end a subscript ‘L’ is inserted to denote that this is the ratio at load impedance. :
• Using (1.1b):
• Similarly we could also derive the current reflection coefficient:
ZL+oV
−oV
or
5
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 9
Reflection Coefficient (2)
• At a distance l from the load, the voltage reflection coefficient is given
by:
• Note that this equation is only valid when the z=0 reference is at the
load impedance, AND l is always positive.
• From now on we will deal exclusively with voltage reflection
coefficient.
(1.2e)
ZL+
oV−
oV
z = 0
z
z = -l
LΓ
l
This product is usually calledelectrical length, is given aunit of radian or degree
( )
( ) ljL
lj
o
olj
o
ljo
el
eV
V
eV
eVl
β
ββ
β
2
2
−
−+
−
+
−−
Γ=Γ⇒
==Γ
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 10
Reflection Coefficient (3)
• Reflection coefficient for various load impedance values. Make sure
you know the physical implication of these.
zz = 0
ZcZc
0=ΓL
zz = 0
ZL=0Zc 1−=ΓL
zz = 0
ZL→∞Zc
1=ΓL
zz = 0
ZL=R+jX
Zc
( ) ( ) ( )( )( ) ( )( )
( )( ) ( )( )( ) ( )ωω
ωω
ωω
ωωω
jXcZR
jXcZR
cZjXRcZjXR
L
++
+−
++
−+
=
=Γ
cL
cLL
ZZ
ZZ
+
−=Γ
6
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 11
Why Do Reflection Occur? (1)
VL
zz = 0
ZL
ILzj
ozj
o eIeVββ −+−+
coI
oVZ=
+
+
At z = 0: LZoV
LI+
=
If Zc ≠ ZL then +≠ oL II
Assuming that IL < Io+, electric
charges pile up at theTline and load intersection. Since like charges repel each other, this force the excess electric charge to flow back to the Tline. This constitutes the reflected current. From our understanding of Tline theory, if there is current, thereis also associated voltage, hencea reflected voltage and currentwave occur.
zjo
zjo eIeV
ββ +−+− -
Note that the total voltage at the Tline-loadintersection is Vo
+ + Vo- = VL. Thus the current
drawn by the load ZL will increase fromVo
+/ZL until and equilibrium is reached.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 12
Why Do Reflection Occur? (2)
• We can visualize the current flow as due to positive charges
(conventional current).
When Io+ > IL
When Io+ < IL
IL
Io+
Io+
IL
-Io-
Tline
Tline
Load impedance
Load impedance
The Tline supply more charges per unit time than the load can absorbso excess positive charges reflected back
Positive
charge
Negative
charge
-(-Io-)
At the interface positive and negative charge pairs are created. The positive charge flows into the load, while the negative charge flows back to the source, constituting the reflected currentwith negative polarity. Of course there is only free
electron in conductor. When
a region is said to contain
positive charge, it actuallyhas less free electrons as
compare to equilibrium state.
7
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 13
( ) ( )( )
−
+==−+
−+
c
ooooLLL
Z
VVVVIVP
*
21*
21 ReReLet Zc be real
2
21 += ocinc VYP
(1.3)( )
Γ−=Γ−=⇒
+++ 2
2122
2122
21 1 oZLoZLoZL VVVP
ccc
22
21
2
21
Lococr VYVYP Γ== +−
Power Delivered to Load Impedance
• Power to load:
• Thus when ΓL = 0, all incident power is absorbed by ZL. We say that the load is matched to the Tline. Otherwise there will be reflected power in the form of:
• The maximum power to load is called the incident power Pinc:
ZLPinc
Pr
PL
Pincident Preflected
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 14
LImpedance matching - Make |ΓΓΓΓL | 0
Impedance Matching
• Thus the purpose of impedance matching is to reduce reflection from
both the load and the source.
• We strive to get maximum power from the source and transport this
power (the available power) to the load.
• In other words impedance matching provides a ‘smooth’ flow of EM
wave along a system of interconnect.
8
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 15
Voltage Standing Wave Ratio (VSWR) (1)
( )
( ) ( ) ( )ljL
ljo
zjL
zjo
zjo
zjo
eeVeeVzV
eVeVzV
ββββ
ββ
22 11 −+−+
−−+
Γ+=Γ+=⇒
+=
( ) ( )ljo eVzV
βθρ 21 −+ +=
θρ jL e=Γ
( )
( ) ρ
ρ
−
+==
1
1VSWR
min
max
zV
zV
(1.5)
• At any point z along the Tline:
• Expressing ΓLin polar form
• The ratio of maximum |V| to mininum |V| is known as VSWR.
(1.4)
Argand
Diagram
10
( )lje
βθρ 21 −+
ρ
ρ+1ρ−1
Re
Im
ZL( )zVLΓ
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 16
Voltage Standing Wave Ratio (2)
( ) ( )ljeIzI
βθρ 21 −+ −=
• A similar expression can be obtained for I(z).
• VSWR measures how good the load ZL is matched to the Tline. For good
match, ρ = 0 and VSWR=1. For mismatch load, ρ >0 and VSWR >1.
• We can view the incident and reflected wave as interfering with each
other, causing standing wave along the Tline.
• A similar phenomenon also exist in waveguide, however it is the E and H
field standing wave the is being measured, so generally the alphabet ‘V’ is
dropped when dealing with waveguide.
• Why SWR is a popular ? - In the early days waveguides are widely used,
and a simple way to measure SWR is to use the slotted line waveguide
with diode detector.
(1.6)
9
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 17
Example 1.1
• A Tline with 2 conductors and filled with air. A sinusoidal voltage of
magnitude 2V and frequency 3.0GHz is launched into the Tline. The
characteristic impedance of the Tline is 50Ω and one end of the Tline is
terminated with load impedance ZL=100+j100 @3.0GHz. Assume
phase velocity = C, the speed of light in vacuum.
– Find the load reflection coefficient ΓL.
– Find the power delivered to the load ZL.
– Plot |V | and |I | versus l, the distance from ZL.
– Determine the VSWR of the system.
l zz = 0
ZL
Zc
Zc=502<0o
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 18
Example 1.1 Cont...
• The solution (as calculated using MathCAD):
10
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 19
Example 1.1 Cont...
• Plotting out the voltage and current phasor along the transmission line:
Magnitudes of V and Iare plotted using (1.4) and(1.6)
Note that |V | and |I | are always quarter wavelength or 90o out of phase. The values of |V | and |I | repeat themselves every half wavelength or 180o.
1 0.8 0.6 0.4 0.2 00
0.5
1
1.5
21.62
0.38
V i ∆z⋅( )
I i ∆z⋅( ) Zc⋅
01− i ∆z⋅
λl
ZL
|V(l)||I(l)|
o1802
↔λ
o904
↔λ
z
ZLZc=50
Voltage phasormagnitude
Normalized current phasor magnitude
Distance in terms
of wavelength λ
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 20
( ) ( )( ) ( )lj
olj
ocZ
ljo
ljo
ineVeV
eVeV
lI
lVlZ
ββ
ββ
−−+
−−+
−
+==
1
( )( )( )ljYY
ljYYYlY
Lc
cLcin
β
β
tan
tan
+
+=
( )( )( )ljZZ
ljZZZlZ
Lc
cLcin
β
β
tan
tan
+
+=
Use (1.2b)
(1.7a)
Input Impedance of a Terminated Tline
• At any length l from the termination impedance, we can compute the
impedance looking towards the load:
ZLZc
l
Zin(l)
(1.7b)
( )lj
L
lj
lj
L
lj
cinee
eeZlZ
ββ
ββ
−
−
Γ−
Γ+=
( ) ( )ljle lj βββ sincos +=
cL
cLL
ZZ
ZZ
+
−=Γ
11
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 21
Special Cases of Terminated Lossless Tline(1)
l zz = 0
ZL=0Zc
( )( )
( )ljZZ
ljZZlZ c
c
ccin β
βtan
0
tan0=
+
+=
• When ZL=0:
• When ZL→∞
l zz = 0
Zc
( )( )
( )ljZljZ
ZZlZ c
L
Lc
LZin ββ
cottan
−=
→
∞→
(1.8a)
(1.8b)
ZL=0
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 22
Special Cases of Terminated Lossless Tline (2)
( ) ( ) jXljZlZ cin == βtanjXLjZ == ω
Zc L
Reactance
( )( )
( ) jBljYljZ
lY cc
in ==−
= ββ
tancot
1
Zc
jBCjY == ω
Susceptance
A length of shorted Tline can be used to
synthesize an inductor or reactance, while
a length of opened Tline can be used to
synthesize a capacitor or susceptance.
βl
X
02π π
23π π2
B
βl02π π
23π π2
This area corresponds toβl < π/2, or l < λ/4
12
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 23
Special Cases of Terminated Lossless Tline (3)
( ) ( ) jXljZlZ cin == βtan
Zc
( )( )
( ) jBljYljZ
lY cc
in ==−
= ββ
tancot
1
Zc
A length of shorted Tline can be used to
approximate a parallel LC resonator, while
a length of opened Tline can be used to
approximate series LC resonator.
Cp LpZ
Cs
LsZ
βl
X
02π π
23π π2
B
βl02π π
23π π2
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 24
Example 1.2
• A lossless Tline of length l = 10 cm supports TEM propagation mode.
The per unit length L and C are given as L = 209.4 nH/m, C =
119.5pF/m. The Tline is terminated with a series RL load impedance:
• Plot the real and imaginary part of Zin from f = 1.0 GHz to 4.0 GHz.
2.5nH
120
10 cm ZL
13
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 25
Example 1.2 Cont...
1 .109
1.5 .109
2 .109
2.5 .109
3 .109
3.5 .109
4 .109
100
50
0
50
100
150
Re Zin 2 π. fi
.
Im Zin 2 π. fi
.
fi
... 999.2 ,999.1 ,9995.0
2
2
GHzGHzGHzf
nf
nl
nl
l
v
v
f
p
p
=
=⇒
=⇒
=
π
πβ
π
Note that the pattern of the waveform repeatat an interval of close to 1GHz, this is due tothe periodic nature of tan(βl) function.
n=1,2,3,4…( )9
8
105.2120
10999.11
86.41
−×+==
×==
==
ωω
β jZv
LCv
C
LZ
Lp
p
c
( )( )( )ljZZ
ljZZZlZ
Lc
cLcin
β
β
tan
tan
+
+=
frequency
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 26
+
+
+
+
==≡
11
22
1
2
oltageincident v
voltagedtransmitte
IZ
IZ
V
VT
c
c
( ) ( )0110
1
1
1
2
211
Γ+=+==⇒
=+
+
−
+
+
+−+
V
V
V
VT
VVV
z = 0
Zc1 Zc2
ZL=Zc2
Zc2zjzj
eIeVββ −+−+
11
zjzjeIeV
ββ +−+−11 -
zjzjeIeV
ββ =+−+22
( )12
220
cc
c
ZZ
ZT
+=( )
12
120cc
cc
ZZ
ZZ
+
−=ΓUsing (1.9)
At z = 0:
Matched termination
Cascading Transmission Lines -Transmission Coefficient
14
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 27
Relationship between Reflection and Transmission Coefficient
LLT Γ+=1
Tline1 Tline2 or a load Impedance
( )( ) cL
cLL
ZZ
ZZ
+
−=Γ
ω
ω ( )( ) cL
LL
ZZ
ZT
+=
ω
ω2
V1+
V1-
V2+Zc
ZL
(1.10)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 28
Power Relations
2
12
2
2
2
2
1
2
1
1
2
1
2
1
2
1
2
1
c
cinc
ctr
incc
ref
cinc
Z
ZPT
Z
VP
PZ
VP
Z
VP
==
Γ==
=
+
−
+
Zc1 Zc2
Pinc
Pref
Ptr
Pinc = Pref + Ptr
15
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 29
Return Loss and Insertion Loss
• Sometimes both voltage reflection and transmission coefficient are
expressed in dB, these are then termed return loss (RL) and
insertion loss (IL).
dB log20 Γ−=RLdB log20 TIL −=
2 – Port Zc2
Zc2Zc1
Zc1Vs
(1.11a)(1.11b)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 30
Example 1.3
• Find the return loss (RL) and insertion loss (IL) at the intersection between the Tlines. Assume TEM propagation mode at f = 1.9 GHz for both Tlines, Z1 =100, Z2 =50.
zz = 0
Z1Z2
0.05Zin
z = -0.05
Interface
z
Z1
16
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 31
Example 1.3 Cont...
542.9log20
3333.0Z
05.010
21
2105.0
=Γ−
=+
−=Γ
−=
−=
z
z ZZ
Z
499.2log20
3333.12Z
05.010
21
105.0
−=−
=+
=
−=
−=
z
z
T
ZZT
zz = 0
Z1Z1Z2
0.05Zin
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 32
( ) ( ) 60tan jljZlZ cin == β
For a transmission line terminated with short circuit:
From Example 5.1 (Chapter 1):
781.948
9
10591.1
104.22 ===×
×⋅πωβpv
m00924.0876.0tan781.9416011 =⋅=
= −
cZl
β
Exercise 1.1
• We would like to use a short length of transmission line to implement a
reactance of X = 60 at 2.4GHz. Show how this can be done using the
microstrip line of Example 5.1, Chapter 1 – Advanced Transmission
Line Theory. Hint: use equation (1.8a).
9.24 mm
A number of plated throughhole to reduce the parasitic inductance of the short.
TopView
2.86mm
1.57 mmεr=4.7 µr=1.0
Zc ≅ 50
vp = 1.591x108
From Example 5.1, Chapter 1:
17
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 33
Terminated Lossy Tline (1)
• In the case of a lossy Tline we replace jβ with γ = α + jβ in equations
(1.2) to (1.9). As seen from equation (2.7) (Advanced Transmission
Line Theory), the characteristic impedance Zc becomes a complex
value too.
• Furthermore:
• The losses have the effect of reducing the standing-wave ratio SWR
towards unity as the point of observation is moved away from the load
towards the generator.
• Most of the time the losses are so small that for short length of Tline,
the neglect of α is justified. However as frequency increases beyond
3 GHz, the skin effect and dielectric loss become important for typical
PCB dielectric and conductor.
( )( )( )lljZZ
lljZZ
ej
ejlZ
Lc
cLljl
L
ljlL
inαβ
αββα
βα
++
++=
Γ−
Γ+=
−−
−−
tanh
tanh
1
1
22
22
(1.12)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 34
( ) ( )
( ) ( ) lc
lL
lc
elVYlP
eeVYVIlP
α
αα
222
21
2222
*
1
2
1Re
2
1
Γ−=⇒
Γ−==
+
−+
(1.13)
( ) ( )
( ) ( )
−Γ+−=
Γ−−
Γ−=−
−+
++
lL
lc
Lcl
cL
eeVY
VYelVYPlP
αα
α
2222
22222
112
1
12
11
2
1
Loss due to incident wave Loss due to reflected wave
Power delivered increases as weproceed towards the generator!
Terminated Lossy Tline (2)
• At some point z = -l from the load-Tline interface, the power directed
towards the load is:
• Of the power given by (1.11), the power dissipated by the load is
given by (1.3), the remainder is dissipated by the lossy line.
18
Chapter 2
Exercise 1.2
• Consider a transmission line circuit below, determine Vin and VL.
August 2008 © 2006 by Fabian Kung Wai Lee 35
( )( )[ ]
ljZZ
ljZZ
cin Lc
cLZZ β
β
tan
tan
+
+=
Thus ( )lj
L
lj
osZZ
Z
in eeVVVins
in ββ −+
+Γ+==
Solving for Vo+:
CL
cL
ZZ
ZZ
L +
−=Γ
( )( )[ ][ ] ( )[ ]lj
cZLZcZLZlj
lcjZLZ
lLjZcZ
sZcZ
s
ee
V
oV ββββ −
+
−
+
+++
+ ⋅= 1
1tan
tan
VLZL
ILTline
Zs Vin Vt(z)
It(z)
+
-Vs Zc , β
z0
Zin
l
Chapter 2
Exercise 1.2 Cont…
• Knowing Vo+ , we can find Vin and VL:
August 2008 © 2006 by Fabian Kung Wai Lee 36
( ) ( )( ) ( )LoL
lj
L
lj
oin
VzVV
eeVlzVV
Γ+===
Γ+=−==+
−+
10
ββ
19
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 37
Demo – Transmission Line Simulation Exercise
VLVin
Vs
R
RL
R=100 Ohm
MLIN
TL1
Mod=Kirschning
L=100.0 mm
W=2.9 mm
Subst="MSub1"
Tran
Tran1
MaxTimeStep=1.0 nsec
StopTime=100.0 nsec
TRANSIENT
R
Rs
R=10 OhmVtPulse
SRC1
Period=1000 nsec
Width=6 nsec
Fall=Trise psec
Rise=Trise psec
Edge=linear
Delay=0 nsec
Vhigh=1 V
Vlow=0 Vt
MSUB
MSub1
Rough=0 mil
TanD=0.02
T=1.38 mil
Hu=3.9e+034 mil
Cond=5.8E+7
Mur=1
Er=4.6
H=1.57 mm
MSub
VAR
VAR1
Trise=200.0
EqnVar
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 38
2.0 Smith Chart and Its Applications
20
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 39
Introduction (1)
• In analyzing electrical circuits, one very important parameter is the
impedance Z or admittance Y seen at a terminal/port.
• For time-harmonic circuits Z or Y is dependent on frequency and a
complex value.
• To visualize arbitrary Z or Y value, we would need an infinite 2D plane:
R
X
0
Z = R + jX Z = V/I = R + jX
Y = I/V = G + jB
ResistanceReactance
Conductance Susceptance
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 40
Introduction (2)
• In RF circuit design, we can represent an impedance Z = R+jX in terms
of its reflection coefficient with respect to a reference impedance (Zo):
• Usually we would take Zo = Zc , the characteristic impedance of a Tline
in the system.
• Γ is also a complex value, however we have learnt that its magnitude is
always < 1 for passive impedance value.
• Effectively if reflection coefficient is plotted, all possible passive Z and Y
values can be fitted into a finite 2D area.
impedance reference 1 ==−==Γ+
−
−
−
oYooYYoYY
oZZoZZ
Z
-1 0
-1
1
1
Re(Γ)
Im(Γ)
Region forpassive Z or Y
21
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 41
Introduction (3)
• To facilitate the evaluation of reflection coefficient, a graphical
procedure based on conformal mapping is developed by P.H. Smith in
1939.
• This procedure, now known as the Smith Chart permits easy and
intuitive display of reflection coefficient Γ as well as impedance Z and
admittance Y in one single graph.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 42
Introduction (4)
22
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 43
Formulation (1)
1
1
1
1
+
−=
+
−
=+
−=Γ
z
z
Z
Z
Z
Z
ZZ
ZZ
o
o
o
o impedance normalized1
1=
Γ−
Γ+=z
Let z = r + jx and Γ = U + jV:
ThenjVU
jVUjxr
−−
++=+
1
1
Equating real and imaginary part:
( )
( )2
22
2
22
111
1
1
1
xxVU
rV
r
rU
=
−+−
+=+
+−
Equation of circles for U and V:
Depends only on r (r circle)
Depends only on x (x circle)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 44
Formulation (2)
jV
U-1 1
-1
1
r circles
x circles
The complex plane
for reflection coefficient Γ:
|Γ|=1
Imaginary axis
Real axis
23
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 45
Formulation (3)
Note that:
Γ−
Γ+=
Γ+
Γ−==
π
π
j
j
e
e
zy
1
1
1
11
Let y = g + jb and Γ = U + jV:
jVU
jVUjbg
++
−−=+
1
1
Again proceeding as before we obtain:
( )
( )2
22
2
2
2
111
1
1
1
bbVU
gV
g
gU
=
+++
+=+
++Depends only on g (g circle)
Depends only on b (b circle)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 46
Formulation (4)
jV
U-1 1
-1
1
g circles
b circles
24
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 47
R=50
R=100
R=200X=10
X=30
X=-10
X=-30
R and X
circles
G circles
+ B circles
- B circles
The Complete Smith Chart with r,x,g and b circles
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 48
Smith Chart Example 1
• Zo = 50Ohm.
• Z = 50 + j0
25
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 49
Smith Chart Example 2
• Zo = 50Ohm.
• Z = 50 + j30
• Z = 50 - j30
Z50
j30R=50 circle
X=30 circle
X=-30 circle
Z50
j30
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 50
Smith Chart Example 3
• Zo = 50Ohms
• Z = 100 + j30
Z100
j30
Question: What wouldyou expect if the real partof Z is negative ?
26
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 51
Smith Chart Example 4
• Zo = 50Ohm
• Z = 50//(-j40) or
• Y = 0.020 + j0.025
Y 0.02j0.025
B G
Z 50-j40 X R
B =0.025 circle
G =0.02 circle
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 52
Smith Chart Example 5
• Zo = 50Ohm
• Z = 50//(j40) or
• Y = 0.020 - j0.025
Z 50j40
X R
Y 0.02-j0.025
B G
G =0.02 circle
B = -0.025 circle
27
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 53
Smith Chart Summary (1)
• Thus a point on a Smith Chart
can be interpreted as reflection
coefficient Γ.
• It can also be read as
impedance Z = R + jX.
• It can also be read as
admittance Y = G + jB.
Z=30 +j20Y=0.023-j0.015Γ=0.34<2.11
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 54
Smith Chart Summary (2)
• When we add a reactance in
series with Z, the point on the
Smith Chart will move in such a
way that it remains on the
constant R circle.
• When we add a susceptance in
parallel to Y, the point on the
Smith Chart will move in such a
way that it remains on the
constant G circle.
Z = 30+j20Y = 0.023-j0.015
Y = 0.023-j0.015± jB
Z = 30+j20 ± jX+X
-X +B
-B
jB Y
jX
ZZin = ZL + jX= RL + j(XL + X)
Yin = YL + jB= GL + j(BL + B)
28
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 55
Smith Chart Example 6
• In this example we would like to observe the locus of impedance Zin as l
is changed for ZL = 200 + j0 (say at certain operating frequency fo).
• Recall equations (1.2d) and (1.7) in this chapter:
( )( )( )ljZZ
ljZZlZ
Lc
cLin
β
β
tan
tan
+
+= ( ) lj
Lin elβ2−Γ=Γ=Γ
l zz = 0
ZL
Zc Zc=50
Vs<0o Zin , Γin
Tline with β and
λπβ 2=with where λ is the wavelength
6.0250150
5020050200 ===Γ
+−
L
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 56
Smith Chart Example 6 Cont...
Locus for Γin
(the SWR circle) Linl Γ=Γ= , 0β
84or λπβ == ll
42or λπβ == ll
83
43 or λπβ == ll
ljLin e
β2−Γ=Γ
Direction away fromload or towardsgenerator
Direction
away fromgenerator
or towards load
The locus in which|Γin| is constant is alsoknown as constantSWR circle (as the SWR depends on themagnitude of reflectioncoefficient).
29
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 57
Typical Applications of Smith Chart (1)
• Smith Chart is used in impedance transformation and impedance
matching. Although this can be performed using analytical method,
using graphical tool such as Smith Chart allows us to visualize the
effect of adding a certain element in the network. The effective
impedance of a load after adding series, shunt or transmission line
section can be read out directly from the coordinate lines of the Smith
Chart.
• Smith Chart is used in RF active circuits design, such as when
designing amplifiers. Usually certain contours in the form of circles
are plotted on the Smith Chart.
• 2-port network parameters such as s11 and s22 are best viewed in
Smith Chart.
Chapter 2
Typical Applications of Smith Chart (2)
• Classically Smith Chart can also be used to find the position along the
transmission line with maximum and minimum voltage or current.
August 2008 © 2006 by Fabian Kung Wai Lee 58
ZL( )zVLΓ
Zc , β
Zc
zz=0z=-l
( ) ( )( ) ( )( ) ( )lj
o
ljlj
o
lj
L
lj
o
eVlzV
eeVlzV
eeVlzV
βθ
βθβ
ββ
ρ
ρ
2
2
2
1
1
1
−+
−+
−+
+=−=⇒
+=−=⇒
Γ+=−=
( ) ( )ρ−=−=+
1oVlzV
( ) ( )ρ+=−=+
1oVlzV
When voltage hasmaximum magnitude
When voltage hasminimum magnitude
πβθ nl =− 2
πβθ nl 22 =−
LΓ
30
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 59
Exercise 2.1 - Impedance Transformation
• Employing the software fkSmith (http://pesona.mmu.edu.my/~wlkung/),
find a way of transforming a load impedance of ZL = 10 + j75 into Z = 50
using either lumped L, C or section of Tline. Assume an operating
frequency of 1.8GHz.
Zc
ZL=10+ j75Vs<0o 2 - port network
Z = 50
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 60
3.0 Practical Considerations for Stripline Implementation
31
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 61
Practical Transmission Line Design and Discontinuities
• Discontinuities in Tline are changes in the Tline geometry to
accommodate layout and other requirements on the printed circuit
board.
• Virtually all practical distributed circuits, whether in waveguide, coaxial
cables, microstrip line etc. must inherently contains discontinuities. A
straight uninterrupted length of waveguide or Tline would be of little
engineering use.
• The following discussion consider the effect and compensation for
discontinuities in PCB layout. This discussion is restricted to TEM or
quasi-TEM propagation modes.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 62
Ground plane gap
trace
plane
gap
trace
ground planeBend
bend
Socket-trace interconnection
socket
pin
trace via
Via
plane
cylindertrace
pad
Junction
Bend
Line to ComponentInterface
StepOpen
Practical Transmission Line Discontinuities Found in PCB (1)
Here we illustrate thediscontinuities usingmicrostripline. Similarstructures apply toother transmission lineconfiguration as well.
32
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 63
Practical Transmission Line Discontinuities Found in PCB (2)
• Further examples of microstrip and co-planar line discontinuities.
Gap Pad or Stub Coupled lines
Examples of bend and viaon co-planar Tline.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 64
Discontinuities and EM Fields (1)
• Introduction of discontinuities will distort the uniform EM fields present
in the infinite length Tline. Assuming the propagation mode is TEM or
quasi-TEM, the discontinuity will create a multitude of higher modes
(such as TM11 , TM12 , TE11 , …) in its vicinity in order to fulfill the
boundary conditions (Note - there is only one type of TEM mode !!).
• Most of these induced higher order modes are evanescent or non-
propagating as their cut-off frequencies are higher than the operating
frequency of the circuit. Thus the fields of the higher order modes are
known as local fields.
• The effect of discontinuity is usually reactive (the energy stored in the
local fields is returned back to the system) since loss is negligible.
• The effect of reactive system to the voltage and current can be
modeled using LC circuits (which are reactive elements).
• For TEM or quasi-TEM mode, we can consider the discontinuity as a
2-port network containing inductors and capacitors.
33
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 65
See Chapter 5, T.C. Edwards, “Foundation for microstrip circuit design” [4], or Chapter 3 [F. Kung]
Discontinuities and EM Fields (2)
• Modeling a discontinuity using circuit theory element such as RLCG is a
good approximation for operating frequency up to 6 – 20 GHz. This upper
limit will depends on the size of the discontinuity and dielectric thickness.
• The smaller the dimension of the discontinuity as compared to the
wavelength, the higher will be the upper usable frequency.
• As a example, the 2-port model for microstrip bend is usually accurate up
to 10GHz.
A
A’B’ B
0.25λ
0.25λ
Minimum distance
A
A’
B
B’
Two port
networks
0.25λ
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 66
Discontinuities and EM Fields (3)
• For instance for a microstrip bend, a snapshot of the EM fields at a
particular instant in time:
Non-TEM modefield here*
Quasi-TEM
field
H field
E field
*This field can be decomposed into
TEM and non-TEM components
Direction of propagation
34
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 67
Open:
Cf
The open end of a stripline contains fringing E field.This is manifested as capacitor Cf . The effect of Cf
is to slightly increase the phase of the inputimpedance.
The approximate value of Cf have been derived by Silvester and Benedek from the EM fields of an open-end structure using numerical method and curve fitted as follows:
P. Silvester and P. Benedek,”Equivalent capacitances
of microstrip open circuits”, IEEE Trans. MTT-20, No. 8
August 1972, 511-576.
pF/m log2036.2exp5
1
1
= ∑
=
−
i
i
if
h
WK
W
Cε
0.0840- 0.0147- 0.0133- 0.0073- 0.0267- 0.0540- 5
0.0240- 0.0267- 0.0087- 0.0260- 0.0133- 0.0033- 4
0.2170 0.1317 0.1308 0.1062 0.1367 0.1815 3
0.2220- 0.2233- 0.238- 0.2535- 0.2817- 0.2892- 2
2.403 1.938 1.738 1.443 1.295 1.110 1
i
51.0 16.0 9.6 4.2 2.5 1.0 rε
(3.1)
h=dielectric thicknessW=width
Microstrip Line Discontinuity Models (1)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 68
eff
fceo
CcZl
ε≅ (3.2)
Zc Cf
Zc
le
o
Zf
Zeo
Microstrip Line Discontinuity Models (2)
• Assuming the effect of Cf can be represented by a short length of Tline:
• Thus in microstrip Tline design, we need to fore-shorten the actual physical length by leo to compensate for fringing E field effect.
35
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 69
Microstrip Line Discontinuity Models (3)
Shorted via or through-hole:
+
≅ 1
4ln2.0
d
hhLs (3.3a)
Ls in nHCp in pFh in mmd and d2 in mmεr = dielectric constant of PCBN = number of GND planes
NCdd
hdrp −
≅2
056.0ε
(3.3b)
Cp/2
Ls
Cp/2
This is the capacitance between the via andinternal plane. If there are multiple internal conducting planes, then there should be one Cp corresponding to each internal plane.
Cross section of a Via
h
d = diameter of via
GND planes
d2
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 70
Microstrip Line Discontinuity Models (4)
C1
C2
C1
T1 T2
Gap:
See Chapter 5, T.C. Edwards, “Foundation
for microstrip circuit design” [4], or B. Easter, “The equivalent circuit of some microstrip
discontinuities”, IEEE Trans. Microwave Theory and Techniquesvol. MTT-23 no.8 pp 655-660,1975.
36
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 71
Microstrip Line Discontinuity Models (5)
90o Bend:
L
C
L
See Edwards [4], chapter 5
Approximate quasi-static expressions for L1, L2 and C:
( ) ( )pF/m
/
25.283.15.1214
dwwC rd
wr −−+
=εε
T2
T1
w
( ) pF/m 0.72.525.15.9 +++= rdw
rwC εε
1<dw
1>dw
nH/m 21.44100
−=
dw
dL
for
for
(3.4)
εr = dielectric constant of substrate,assume non-magnetic.d = thickness of dielectric in meter.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 72
Example 3.1 - Microstrip Line Bend
• For a 90o microstrip line bend, with w=2.8mm, d=1.57mm, εr = 4.2. Find the value of L and C.
( )
0.30pF 0.00288104.309C
pF/m 104.309
0.72.42.5834.125.12.45.9
=×=⇒
=
+×++×=wC
834.1=dw
[ ]18.95pHL
nH/m 120.701 21.4834.14100
=⇒
=−=dL
0.30pF
18.05pH 18.05pH
At 1GHz:Reactance of C
Reactance of L
5.5302
1 ≅=fCcX
π
119.02 ≅= fLX L π
Typically the effect of bend is notimportant for frequency below 1 GHzThis is also true for discontinuities likestep and T-junction.
37
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 73
Microstrip Line Discontinuity Models (6)
Step: L1
C
L2
( ) pF/m 17.3log6.1233.2log1.102
1
21
−−+= rw
wr
ww
C εε
T2T1
w2w1
See Edwards [4] chapter 5
Approximate quasi-static expressions for L1, L2 and C:
pF/m 44-log1301
2
21
=
w
w
ww
C
105.1 ; 101
2 ≤≤≤w
wrε
105.3 ; 6.91
2 ≤≤=w
wrε
nH/m 0.12.0750.15.402
2
1
2
1
2
1
−+−
−=
w
w
w
w
w
w
dL
LLL
LL
mm
m
21
11
+= L
LL
LL
mm
m
21
22
+=
Lm1 and Lm2 are the per unit lengthinductance of T1 and T2 respectively.
for
for
(3.5)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 74
Microstrip Line Discontinuity Models (7)
T-Junction:
T1
T3
T2
L1
C
1
L1T1 T2
T3
L3
See Edwards [4], chapter 5for alternative model and
further details.
38
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 75
Effect of Discontinuities
• Looking at the equivalent circuit models for the microstrip discontinuities, the sharp reader will immediately notice that all these networks can be interpreted as Low-Pass Filters. The inductor attenuates electrical signal at high frequency while the capacitor shunts electrical energy at high frequency.
• Thus the effect of having too many discontinuities in a high-frequency circuit reduces the overall bandwidth of the interconnection.
• Another consequence of discontinuity is attenuation due to radiation from the discontinuity.
f
|H(f)|
0
f
|H(f)|
0
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 76
Some Intuitive Concepts on Discontinuities
• Seeing the equivalent circuit models on the previous slides, one can’t
help to wonder how does one knows which model to use for which
discontinuity?
• The answer can be obtained by understanding the relationship
between electric charge, electric field, current, magnetic flux linkage
and quantities such as inductance and capacitance.
• A few observations are crucial:
– As current encounter a bend, the flow is interrupted and the current
is reduced. Moreover there will be accumulation of electric
charges at the vicinity of the bend because of the constricted flow.
– As current encounter a change in Tline width, the flow of charge
either accelerate or decelerate.
39
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 77
Intuitive Concepts (1)
• Excess charge - whenever there is constricted flow or a sudden
enlargement of Tline geometry, electric charge will accumulate. The
amount of charge greater than the charge distribution on an infinite
length of Tline is known as excess charge. The excess charge can be
negative. Associated with excess charge is a capacitance.
• Excess flux - similarly constricted flow or ease of flow, change in Tline
geometry also result in excess magnetic flux linkage. The amount of
flux linkage greater than the flux linkage on an infinite length of Tline is
known as excess flux. This excess flux is associated with an
inductance, again the excess flux can be negative although it is usually
positive (inductance is always corresponds to resistance in current
flow, recall V=L(di/dt)).
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 78
Intuitive Concepts (2)
• Both excess charge and excess flux can be computed from the higher
order mode EM fields in the vicinity of the discontinuity. For instance
by subtracting the total E field from the normal E field distribution for
infinite Tline, we would obtain the higher order modes E field (or local
E field). From the boundary condition of the local E field with the
conducting plate, the excess charge can be calculated. Similar
procedure is carried out for the excess flux.
• This argument although presented for stripline, is also valid for coaxial
line and waveguide in general.
• Usually numerical methods are employed to determine the total E and
H field at the discontinuity, and it is assumed the fields are quasi-
static.
40
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 79
Intuitive Concepts (3)
• For instance in a bend. As current approach point B, the current density
changes. We can imagine that current flow easily in the middle as
compared to near the edges. As a result the flux linkage at B for both T1
and T2 increases as compared to the flux linkage when there is no bend.
Associated with the excess flux we introduce two series inductors, L1 and
L2 . The inductance are similar if T1 and T2 are similar in geometry.
• Also at B, more positive electric charges (we think in terms of
conventional charge) accumulate as compared to the charge distribution
for infinite Tline. Thus we associate a capacitance C1 at B.
L1
C1
L2BT1
T2
harder to flow
Easier to flow
Constricted flow - LIncrease flow - CCharge accumulation - C
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 80
Exercise 3.1
• What do you expect the equivalent circuit of the following discontinuity
to be ?
41
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 81
Methods of Obtaining Equivalent Circuit Model for Discontinuities (1)
• 3 Typical approaches…
• Method 1: Analytical solution - see Chapter 4, reference [3] on Modal
Analysis for waveguide discontinuities.
• Method 2: Numerical methods such as
– Method of Moments (MOM).
– Finite Element Method (FEM).
– Finite Difference Time Domain Method (FDTD).
– And many others.
• Numerical methods are used to find the quasi-static EM fields of a 3D
model containing the discontinuity. The EM field in the vicinity of the
discontinuity is split into TEM and non-TEM fields. LC elements are
then associated with the non-TEM fields using formula similar to (3.1)
in Part 3.
Agilent’s Momentum
Ansoft’s HFSS •CST’s Microwave Studio•Sonnet
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 82
Methods of Obtaining Equivalent Circuit Model for Discontinuities (2)
• Method 3: Fitting measurement with circuit models. By proposing an
equivalent circuit model, we can try to tune the parameters of the
circuit elements in the model so that frequency/time domain response
from theoretical analysis and measurement match.
• Measurement can be done in time domain using time-domain
reflectometry (TDR) and frequency domain measurement using a
vector network analyzer (VNA) (see Chapter 3 of Ref [4] for details).
42
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 83
Radiation Loss from Discontinuities
• At higher frequency, say > 5 GHz, the assumption of lossless
discontinuity becomes flawed. This is because the higher order mode
EM fields can induce surface wave on the printed circuit board, this wave
radiates out so energy is loss.
• Furthermore the acceleration or deceleration of electric charge also
generates radiation.
• The losses due to radiation can be included in the equivalent circuit
model for the discontinuity by adding series resistance or shunt
conductance.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 84
Reducing the Effects of Discontinuity (1)
• To reduce the effect of discontinuity, we must reduce the values of the associated inductance and capacitance. This can be achieved by decreasing the abruptness of the discontinuity, so that current flow will not be disrupted and charge will not accumulate.
Chamfering of bends
W
b For 90o bend:It is seen that the optimumchamfering is b=0.57W(see Chapter 5, Edwards [4])For further examples see
Chapter 2, Bahl [7].
W 1.42W
W
W
43
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 85
Reducing the Effects of Discontinuity (2)
T2T1
Mitering of junction Mitering of step
For more details of compensation for discontinuity,please refer to chapter 5 of Edwards [4] andChapter 2 of Bahl [7].
T1
T3
T2
W2 W1
≅ 0.7W1
T1
T3
T2
W2 W1
≅ 2W2
0.5W2 or smaller
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 86
Connector Discontinuity: Coaxial -Microstrip Line Transition (1)
• Since most microstrip line invariably leads to external connection from
the printed circuit board, an interface is needed. Usually the microstrip
line is connected to a co-axial cable.
• An adapter usually used for microstrip to co-axial transistion is the SMA
to PCB adapter, also called the SMA End-launcher.
44
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 87
Connector Discontinuity: Coaxial -Microstrip Line Transition (2)
• Again the coaxial-to-microstrip transition is a form of discontinuity,
care must be taken to reduce the abruptness of the discontinuity. For
a properly designed transition such as shown in the previous slide, the
operating frequency could go as high as 6 GHz for the coaxial to
microstrip line transition and 9 GHz for the coaxial to co-planar line
transition.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 88
Exercise 3.2
• Explain qualitatively the effect of compensation on the equivalent
electrical circuits of the discontinuity in the previous slides.
45
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 89
Example 3.2
• A Zc = 50Ω microstrip Tline is used to drive a resistive termination as
shown. Sketch the equivalent electrical circuit for this system.
Top view
MSUB
MSub1
Rough=0 mil
TanD=0.02
T=1.38 mil
Hu=3.9e+034 mil
Cond=5.8E+7
Mur=1
Er=4.6
H=0.8 mm
MSub
VIAHS
V1
T=1.0 mi l
H=0.0 mm
D=20.0 mil
VIAHS
V2
T=1.0 mi l
H=0.8 mm
D=20.0 mil
VIAHS
V3
T=1.0 mil
H=0.8 mm
D=20.0 mil
VIAHS
V4
T=1.0 mil
H=0.8 mm
D=20.0 mil
VIAHS
V5
T=1.0 mil
H=0.8 mm
D=20.0 mil
VIAHS
V6
T=1.0 mil
H=0.8 mm
D=20.0 mil
R
R2
R=100 Ohm
R
R1
R=100 Ohm
C
C1
C=0.47 pF
MLIN
TL3
L=15.0 mm
W=1.45 mm
Subst="MSub1"
MLIN
TL2
L=10.0 mm
W=1.45 mm
Subst="MSub1"
MLIN
TL1
L=25.0 mm
W=1.45 mm
Subst="MSub1"
MSOBND_MDS
Bend1
W=1.45 mm
Subst="MSub1"
MSOBND_MDS
Bend2
W=1.45 mm
Subst="MSub1"
L
LSMA1
R=
L=1.2 nH
C
CSMA2
C=0.33 pF
C
CSMA1
C=0.33 pF
Port
P1
Num=1
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 90
4.0 Linear RF Network Analysis – 2-Port Network
Parameters
46
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 91
Network Parameters (1)
• Many times we are only interested in the voltage (V) and current (I)
relationship at the terminals/ports of a complex circuit.
• If mathematical relations can be derived for V and I, the circuit can be
considered as a black box.
• For a linear circuit, the I-V relationship is linear and can be written in
the form of matrix equations.
• A simple example of linear 2-port circuit is shown below. Each port is
associated with 2 parameters, the V and I.
Port 1 Port 2
R
CV1
I1 I2
V2
Convention for positivepolarity current and voltage
+
-
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 92
Network Parameters (2)
• For this 2 port circuit we can easily derive the I-V relations.
• We can choose V1 and V2 as the independent variables, the I-V
relation can be expressed in matrix equations.
2 - Ports
I2
V2V1
I1
Port 1 Port 2
R
CV1
I1 I
2
V
2
=
2
1
2221
1211
2
1
V
V
yy
yy
I
I
( )
+−
−=
2
111
11
2
1
V
V
CjI
I
RR
RR
ω
Network parameters(Y-parameters)
( ) 21
11
2
221
VCjVI
CVjII
RRω
ω
++−=⇒
=+
C
I1I2
V2jωCV2
R
V1
I1
V2
( )211
1 VVIR
−=
47
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 93
Network Parameters (3)
• To determine the network parameters, the following relations can be
used:
• For example to measure y11, the following setup can be used:
021
111
==
VV
Iy
0121
12=
=V
V
Iy
021
221
==
VV
Iy
0122
22=
=V
V
Iy
This means we short circuit the port
2 - Ports
I2
V2 = 0V1
I1
=
2
1
2221
1211
2
1
V
V
yy
yy
I
I
VYI ⋅=
or
Short circuit
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 94
Network Parameters (4)
• By choosing different combination of independent variables, different
network parameters can be defined. This applies to all linear circuits
no matter how complex.
• Furthermore this concept can be generalized to more than 2 ports,
called N - port networks.
2 - Ports
I2
V2V1
I1
=
2
1
2221
1211
2
1
I
I
zz
zz
V
V
V1 V2
I1 I2
=
2
1
2221
1211
2
1
V
I
hh
hh
I
V
H-parameters
Z-parameters
Linear circuit, because allelements have linear I-V relation
48
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 95
221
221
2
2
1
1
DICVI
BIAVV
I
V
DC
BA
I
V
+=
+=⇒
=
02
1
2 =
=
IV
VA
02
1
2 =
=
VI
VB
02
1
2 =
=
VI
ID
02
1
2 =
=
IV
IC
(4.1a)
(4.1b)
2 -Ports
I2
V2V1
I1
Take note of the direction of positive current!
ABCD Parameters (1)
• Of particular interest in RF and microwave systems is ABCD
parameters. ABCD parameters are the most useful for representing
Tline and other linear microwave components in general.
Short circuit Port 2Open circuit Port 2
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 96
ABCD Parameters (2)
• The ABCD matrix is useful for characterizing the overall response of 2-
port networks that are cascaded to each other.
=
⇒
=
3
3
33
33
1
1
3
3
22
22
11
11
1
1
I
V
DC
BA
I
V
I
V
DC
BA
DC
BA
I
VI2’
V2V1
I1 I2
V3
I3
11
11
DC
BA
22
22
DC
BA
Overall ABCD matrix
49
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 97
ZY
1
0
1
=
=
=
=
D
C
ZB
A
1
0
1
=
=
=
=
D
YC
B
A
(4.2a) (4.2b)
β , Zc
l lD
lZ
jC
ljZB
lA
c
c
β
β
β
β
cos
sin1
sin
cos
=
=
=
=
(4.2c)
2 -Ports
I2
V2V1
I1
ABCD Parameters of Some Useful 2-Port Network
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 98
Example 4.1
• Derive the ABCD parameters of an ideal transformer.
1:N
=
2
21
1
1
0
0
I
V
NI
VN
1122
12
IVIV
NVV
=
=
Hints: for an ideal transformer, the followingrelations for terminal voltages and currentsapply:
Port 1 Port 2
50
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 99
Exercise 4.1
• Derive equations (4.2a), (4.2b) and (4.2c).
• Hint : For the Tline, assume the voltage and current on the Tline to be
the superposition of incident and reflected waves. And let the
terminals at port 2 corresponds to z = 0 (assuming propagation along
z axis).
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 100
Partial Solution for Exercise 4.1
Case1: For I2= 0 (open circuit port 2), ΓL= 1, thus:
( ) ( ) ( )lVeeVlzVV oljlj
o βββcos21
21
+−+ =+=−==
( ) zjoL
zjo eVeVzV
ββ ++−+ Γ+= ( ) zj
Z
VL
zj
Z
VeezI
c
o
c
o ββ +−++
Γ−=
For (4.2c)…
First we note that for terminated Tline, voltage and current along z axis are given by:
( ) ( ) ++ =+=== oo VVzVV 21102
( ) ( )
( )lj
eelzII
c
o
c
o
Z
V
ljzj
Z
V
β
ββ
sin2
1 21
+
+
=
−=−== −−
Thus ( ) ( )lAo
o
V
lV
IV
Vβ
βcos
2
cos2
022
1 ===+
+
=
( )( )ljYC c
V
lj
IV
I
o
cZoV
ββ
cos2
sin2
022
1 ===+
+
=
β , Zc
l
Vs
z=0
z
y
V1V2
51
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 101
Partial Solution for Exercise 4.1 Cont…
β , Zc
l
VsCase 2: For V2= 0, ΓL = -1.Proceeding in a similar manner, we would be able to obtain B and D terms.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 102
Z1
Y
Z2
Hint: Consider each element as a 2-port network.
Exercise 4.2
• Find the ABCD parameters of the following network:
52
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 103
S-Parameters - Why Do We Need Them?
• Usually we use Y, Z, H or ABCD parameters to describe a linear two port network.
• These parameters require us to open or short a network to find the parameters.
• At radio frequencies it is difficult to have a proper short or open circuit, there are parasitic inductance and capacitance in most instances.
• Open and short conditions lead to standing wave, which can cause oscillation and destruction of the device.
• For non-TEM propagation mode, it is not possible to measure voltage and current. We can only measure power from E and H fields.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 104
• As oppose to V and I, S-parameters relate the reflected and incident
voltage waves.
• S-parameters have the following advantages:
1. Relates to familiar measurement such as reflection coefficient,
gain, loss etc.
2. Can cascade S-parameters of multiple devices to predict system
performance (similar to ABCD parameters).
3. Can compute Z, Y or H parameters from S-parameters if needed.
S-parameters
• Hence a new set of parameters (S) is needed which
– Do not need open/short condition.
– Do not cause standing wave.
– Relates to incident and reflected power waves, instead of voltage
and current.
53
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 105
Normalized Voltage/Current Waves (1)
• Consider an n – port network:
Each port is considered to beconnected to a Tline withspecific Zc.
Linear
n - port
network
T-line or
waveguide
Port 2
Port 1
Port n
Reference planefor local z-axis(z = 0)
Zc2
Zc1
Zcn
zz=0
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 106
Normalized Voltage/Current Waves (2)
• There is a voltage and current on each port.
• This voltage (or current) can be decomposed into the incident (+) and
reflected (-) components.
V1+ V1
-
Linear
n - port
Network
Port 2
Port 1
Port n
z = 0
V1
I1
+z
Port 1
−+ += 111 VVV
====V1
V1+
+
- V1-
+
( )−+
−+
−=
−=
111
1
111
VV
III
cZ
( )
( ) −+
−−+
+==
+=
111
11
0 VVVV
eVeVzVzjzj ββ
( )
( ) −+
−−+
−==
−=
111
11
0 IIII
eIeIzI zjzj ββ
54
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 107
Normalized Voltage/Current Waves (3)
• The port voltage and current can be normalized with respect to the impedance connected to it.
• It is customary to define normalized voltage waves at each port as:
ciii
ci
ii
ZIa
Z
Va
+
+
=
=(4.3a)
Normalizedincident waves
Normalizedreflected waves
ciii
ci
ii
ZIb
Z
Vb
−
−
=
=
(4.3b)
i = 1, 2, 3 … n
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 108
Normalized Voltage/Current Waves (4)
• Thus in general:
Vi+ and Vi
- are propagatingvoltage waves, which canbe the actual voltage for TEM
modes or the equivalent voltages for non-TEM modes.(for non-TEM, V is definedproportional to transverse Efield while I is defined propor-
tional to transverse H field, see[1] for details).
a2
b2
a1 b1
an
bn
Linear
n - port
Network
T-line or
waveguide
Port 2
Port 1
Port nZc1
Zc2
Zcn
55
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 109
Scattering Parameters (1)
• If the n – port network is linear (make sure you know what this means!),
then there is a linear relationship between the normalized waves.
• For instance if we energize port 2:
a2
b1
bn
Port 2
Port 1
Port nZc1
Zc2
Zcn
b2
Linear
n - port
Network
2121 asb =
2222 asb =
22asb nn =
Constant thatdepends on the network construction
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 110
Scattering Parameters (2)
• Considering that we can send energy into all ports, this can be
generalized to:
• Or written in Matrix equation:
• Where sij is known as the generalized Scattering (S) parameter, or just
S-parameters for short. From (4.3), each port i can have different
characteristic impedance Zci.
nn asasasasb 13132121111 ++++= L
nn asasasasb 23232221212 ++++= L
nnnnnnn asasasasb ++++= L332211
=
nnnnn
n
n
n a
a
a
sss
sss
sss
b
b
b
:
...
:::
...
...
:
2
1
21
22221
11211
2
1
O
(4.4a)
(4.4b)aSb = or
56
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 111
Linear Relation Between ai and bi
• That ai and bi are related by linear relationship can be proved using Green’s Function Theory for partial differential equations.
• For a hint on proof of this, you can refer to the advanced text by R.E. Collins, “Field theory of guided waves”, IEEE Press, 1991, or you can see F. Kung’s detailed mathematical proof.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 112
=
=
2
1
2
1
2221
1211
2
1
a
aS
a
a
ss
ss
b
b
012
112
012
222
021
221
021
111
====
====
aaaaa
bs
a
bs
a
bs
a
bs
(4.5a)
(4.5b)
S-parameters for 2-port Networks
• For 2-port networks, (4.4) reduces to:
• Note that ai = 0 implies that we terminate i th port with its characteristic impedance.
• Thus zero reflection eliminates standing wave.
• Good termination can be established reliably for RF and Microwave frequencies in a number of transmission system. This will be illustrated in the following slides.
57
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 113
Measurement of S-parameter for 2-port Networks
2 – Port Zc2
Zc2Zc1
Zc1Vs
a1
01
221
01
111
22 ==
==
aaa
bs
a
bs
b1
b2
b1
2 – PortZc1
Zc2Zc1
Zc2Vs
b2
a2
02
112
02
222
11
==
==
aaa
bs
a
bs
Measurement of s11 and s21:
Measurement of s22 and s12:
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 114
Example of Terminations
DIY 50Ω Termination for grounded
co-planar Tline
2 × 100Ω SMT resistor(0603 package, 1% tolerance)
Another exampleof broadband 50Ω co-axialtermination
-20dB
0dB
Measured s11 from 10 MHz to 3 GHzusing Agilent’s 8753ES VNA
OSL calibration
performed on Port 1
58
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 115
An example of VNA byAgilent Technologies. Other manufacturersof VNA are Advantek, Wiltron, Anritsu, Rhode &Schwartz etc.
Device under test(DUT)
Port 1
Port 2
Practical Measurement of S-parameters
• Vector Network Analyzer (VNA) - an instrument that can measure the
magnitude and phase of S11, S12, S21, S22.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 116
General Vector Network Analyzer Block Diagram
Measuring s11and s21:
DUT
Receiver /
Detector
Processor and Display
Incident (R)
Reflected (A)
Transmitted (B)
Signal
Separation
Directional coupler
LO1
Reference
Oscillator
Sampler
+
ADC
Port 1 Port 2
59
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 117
Relationship Between Port Voltage/Current and Normalized Waves
• From the relations:
• One can easily obtain a and b from the port voltages and currents (for
instance a 2-port network):
• This shows that S-parameters can be computed if we know the port’s
voltage and current (take note these are phasors).
• Most RF circuit simulator software uses this approach to derive the S-
parameters.
( )
( )1111
1
1111
1
2
1
2
1
IZVZ
b
IZVZ
a
cc
cc
−=
+=(4.6a)
2-ports network
V1 V2
I1 I2
a2a1
b2b1
Zc1 Zc2
−+ += 111 VVV ( )−+−+ −=−= 111
111 VVIIIcZ
( )
( )2222
2
2222
2
2
1
2
1
IZVZ
b
IZVZ
a
cc
cc
−=
+=
Usually Zc1 = Zc2 = Zc
(4.6b)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 118
Example 4.2 - Measuring Normalized Waves
• This setup shows how one can measure a1 and b1 with oscilloscope.
• It can be done for frequency < 1 MHz where parasitic inductance and
capacitance are not a concern.
2-ports network
V1V2
I1
Zc = 50
Zc = 50
Rtest = 1
Channel 2
Channel 1
To oscilloscope
Signal generator
Assume Zc1 = Zc2 = 50
Rtest shouldbe as small
as possible
60
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 119
Example 4.3 - S11 Measurement Example in Time-Domain (1)
• This experiment is done using a signal generator and Agilent
Technologies Infiniium Digital Sampling Oscilloscope.
Test resistor, as smallas possible, (V2-V1)/Rtest
gives the current flowinginto the network
Sample RLC network
Signal generator model
We would like to find what is s11 at 10 kHz
V2 V1
R
R2
R=180 Ohm
C
C1
C=50.0 nF
R
Rtest
R=10 Ohm
R
Rs
R=50 OhmVtSine
SRC1
Phase=0
Damping=0
Delay=0 nsec
Freq=10 kHz
Amplitude=0.5 V
Vdc=0 V
Tran
Tran1
MaxTimeStep=1.0 usec
StopTime=5 msec
TRANSIENT
L
L1
R=
L=47.0 uH
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 120
Example 4.3 - S11 Measurement Example in Time-Domain (2)
• Experiment setup.
From signal
generator
61
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 121
Example 4.3 - S11 Measurement Example in Time-Domain (3)
V2 – V1
(16x averaging10mV perdivision)
V2 – V1 peak-to-peakamplitude ≅ 40.0mV Frequency ≅ 10 kHz V2 peak-to-peak
amplitude ≅ 668mV
V2 waveform(200mV perdivision)
To = 1/10000 = 100usec
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 122
Example 4.3 - S11 Measurement Example in Time-Domain (4)
• Zooming in to measure the phase angle.
Phase = positiveif voltage leadscurrent, zero if both voltage andcurrent samephase, and negativeotherwise.
Phase calculation:
radian 503.02 =⋅= ∆ πθoTt
∆t≅ 8us
Voltage leads current
V2
V2 – V1
10mV/division
62
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 123
Example 4.3 - S11 Measurement Example in Time-Domain (5)
Measured results
Converts into current phasore.g. I1 = Iamp ejθ , where θ = phase
V1 is used as the reference forphase measurement, hence V1 = Vampe
j0
where Vamp = Vpeak
Voltage and ‘current’magnitude
VSWR 3.896=VSWRs11 1+
1 s11−:=
s11 0.564 0.179i−=s11b1
a1:=
b1V1 I1 Zc⋅−
2 Zc⋅:=a1
V1 I1 Zc⋅+
2 Zc⋅:=
V1 V2amp:=
I1 1.753 103−
× 9.635i 104−
×+=
I1 I1amp exp i I1phase⋅( )⋅:=
I1amp 2 103−
×=
I1phase 0.503=I1ampV21amp
Rtest:=I1phase
∆t
To2⋅ π⋅:=
∆t 8 106−
⋅:=To 1 104−
×=To1
fo:=
fo 10000:=
V21amp0.04
2:=V2amp
0.668
2:=Rtest 10:=Zc 50:=
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 124
Example 4.3 - S11 Measurement Example in Time-Domain (6)
Compare this withmeasured results
V2 V1
R
R21
R=180 Ohm
R
Rtest
R=10 Ohm
C
C1
C=50.0 nF
VSWR
VSWR1
VSWR1=vswr(S11)
VSWR
S_Param
SP1
Step=1.0 kHz
Stop=10.0 kHz
Start=10.0 kHz
S-PARAMETERS
Term
Term1
Z=50 Ohm
Num=1
L
L1
R=
L=47.0 uH
freq
10.00kHz
S(1,1)
0.554 - j0.168
VSWR1
3.755
63
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 125
( )
( )iici
iii
iiciiii
baZ
III
baZVVV
−=−=
+=+=
−+
−+
1
( ) ( )22
2
1*Re
2
1iiiii baIVP −==
Power Waves
• Consider port i, since (assuming z = 0):
• Power along a waveguide or transmission line on Port i :
• Because of this, ai and bi are sometimes refer to as incident and
reflected power waves. S-parameters relate the incident and reflected
power of a port.
(4.7)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 126
More on S Parameters
• S parameters are very useful at microwave frequency. Most of the
performance parameters of microwave components such as
attenuators, microwave FET/Transistors, coupler, isolator etc. are
specified with S parameters.
• In fact, theories on the realizability of 3 ports and 4 ports network such
as power divider, directional coupler are derived using the S matrix.
• In the subject “RF Transistor Circuits Design or RF Active Circuit
Design”, we will use S parameters exclusively to design various small
signal amplifiers and oscillators.
• At present, the small signal performance of many microwave
semiconductor devices is specified using S parameters.
64
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 127
Extra Knowledge 1 - Sample Datasheet of RF BJT
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 128
=
+
+
+
−
−
−
nnnnn
n
n
n V
V
V
SSS
SSS
SSS
V
V
V
:
...
:::
...
...
:
2
1
''2
'1
'2
'22
'21
'1
'12
'11
2
1
O
( ) 11
=⋅=
+
++
+
cii
iici
ci
i
ZI
VIZ
Z
V
Extra Knowledge 2
• We can actually use S matrix to relate reflected voltage waves to incident voltage waves. Call this the S’ matrix to distinguish from the S matrix which relate the generalized voltage waves.
• The reason generalized voltage and current are used more often is the ratio of generalized voltage to current at a port n is always 1. This is useful in deriving some properties of the S matrix.
• Of course when the characteristic impedance of all Tlines in the system are similar, then S’ = S.
65
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 129
ZZc2Zc1
Example 4.4 – S-parameters for Series Impedance
• Find the S matrix of the 2 port network below.
See extra notes for solution.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 130
β , Zc
lcZcZ
1a
1b 2b
2a
=
−
−
2
1
2
1
0
0
a
a
e
e
b
b
lj
lj
β
β
Example 4.5 – S-parameters for Lossless Tline Section
• Derive the 2-port S-matrix for a Tline as shown below.
Case 1: Terminated Port 2
01
11
1
1
021
111 ==Γ===
+
−
+
−
= cZZcZZ
cZ
V
cZ
V
aa
bs
Since+− =
12 VeVljβ
2102
1
2
1
2
1
2 sea
a
blj
cZ
V
cZ
V
V
V====⇒
=
−+
−
+
−β
Case 2: from symmetry, s11 = s22 = 0, s12=s21=ejβ:l
β , Zc
l
Zc
a1
b1
b2
Zin = Zc
+z
z=0
z
66
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 131
IIIVVV =−=+−+−+
1−
+
−= EZZEZZS cc
1−
+
+= SEESZZ c
=
100
010
001
L
MOMM
L
L
E
Exercise 4.3 – Conversion Between Impedance and S Matrix
• Show how we can convert from Z matrix to S matrix and vice versa.
hint: use equations (4.4) and (4.6) and the fact that:
• For a system with Zc1= Zc2 = … = Zcn = Zc:
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 132
S Matrix for Reciprocal Network –Symmetry (1)
• A linear N-port network is made up of materials which are isotropic and
linear, the E and H fields in the network observe Lorentz reciprocity
theorem (See Section 2.12, ref [1]).
• A special condition arises when the linear N-port network does not contain
any sources (electric and magnetic current densities, J and M), the
network is called reciprocal.
• Reciprocal network cannot contain active devices, ferrites or plasmas.
• Active devices such as transistor contains equivalent current and source in
the model, while in ferro-magnetic material, the bound current due to
magnetization* constitutes current source. Similarly the ions moving in a
plasma also constitute current source.
Linear
n - port
NetworkPort 2
Port 1Port n
0
0
=
=
M
Jr
r
* See for instance the book by D.J. Griffith, “Introductory electrodynamics”, Prentice Hall,or any EM book for further discussion on
magnetization and µ (permeability)
67
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 133
S Matrix for Reciprocal Network –Symmetry (2)
• Under reciprocal condition, we can show that the S Matrix is symmetry,
i.e. sij = sji .
• For example for a 3-port reciprocal network:
• Many types of RF components fulfill reciprocal and linear conditions, for
example passive filters, impedance matching networks, power
splitter/combiner etc.
• You can refer to Section 4.2 and 4.3 of Ref. [3] or extra notes from F.
Kung for the derivation.
tSS = (4.8)
tS
sss
sss
sss
sss
sss
sss
S =
=
=
332313
322212
312111
333231
232221
131211
This is achieved by using Reciprocity Theorem
in Electromagnetism, and using the definition
of V and I from EM fields, one can show that
the Z matrix is symmetrical. Then using the
relationship between S and Z matrices, we
can show that S matrix is also symmetry.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 134
1*
* −=
=
SSS
tt
USSSStt
=
=
=
1..00
0::
0..10
0..01**
O
S Matrix for Lossless Network - Unitary
• When the network is lossless, then no real power can be delivered to the network. By considering the voltage and current at each port, and equating total incident power to total reflected power, we can show that:
• Again you can refer to Section 4.3 of Ref. [3] or extra note from F.Kung for the derivation.
• Matrix S of this form is known as Unitary.
222
21
222
21 nn bbbaaa +++=+++ KK
(4.9)
68
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 135
USSSS
t
=⋅=
⋅
**
Reciprocal and Lossless Network
• Thus when the network is both reciprocal and lossless, symmetry and
unitary of the S matrix are fulfilled.
• This is the case for many microwave circuits, for instance those
constructed using stripline technology.
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 136
Conversion Between ABCD and S-Parameters
( )( )
( )( )
( )( )
( )( )
21
21122211
21
21122211
21
21122211
21
21122211
2
11
2
111
2
11
2
11
S
SSSSD
S
SSSS
ZC
S
SSSSZB
S
SSSSA
o
o
−+−=
−−−=
−++=
+−+=
( )
DCZZBA
DCZZBAS
DCZZBAS
DCZZBA
BCADS
DCZZBA
DCZZBAS
oo
oo
oo
oo
oo
oo
+++
+−+−=
+++=
+++
−=
+++
−−+=
/
/
/
2
/
2
/
/
22
21
12
11
• For 2-port networks, with Zc1 = Zc2 = Zo:
(4.10a) (4.10b)
69
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 137
Shift in Reference Plane (1)
V’n+
V’n-
=
⋅⋅⋅=
−
−
−
nnlj
lj
lj
e
e
e
A
aASAb
β
β
β
...00
:::
0...0
0...0
''
22
11
O
V2+
V2-
V1+ V1
-
Vn+
Vn-
Linear
N - port
Network0
00
z=-l2
z=-l1 z=-ln
V’2+
V’2-
V’1+ V’1
-
+ve z direction
Note: For a wave propagating in
+ve z direction, Vi+ e-jβl
ci
ii
ci
ii
Z
Vb
Z
Va
−+
=='
' ''
Let:
i = 1,2,…n
(4.11)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 138
Shift in Reference Plane (2)
( )
( )
==
=
=
−+−
+−−
−
−
222211
221111
22
11
22221
122
11
2
1
2
1
'
0
0 ,
'
''
'
'
ljllj
lljlj
lj
lj
eses
esesASAS
e
eA
a
aS
b
b
βββ
βββ
β
β
• For 2-port network:
(4.12)
70
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 139
Cascading 2-port Networks
a1A a1B
a2Ba2A
b2A
b1A b1B
b2B
A B
=
=
A
AA
A
A
B
BB
B
B
a
aS
b
b
a
aS
b
b
2
1
2
1
2
1
2
1
−
−
−=
=
BABAB
BAABA
AB
B
A
B
A
DSSSS
SSDSS
SSS
a
aS
b
b
22222121
12121111
2211
2
1
2
1
1
1
New S matrix:
Let:
a1A a2B
b1Ab2B
(4.13)
Chapter 2August 2008 © 2006 by Fabian Kung Wai Lee 140
THE END