An Introduction
S-PARAMETERS
S-parameters are a useful method for representing a circuit as a “black box”
S-parameters are a useful method for representing a circuit as a “black box”
The external behaviour of this
black box can be predicted
without any regard for the contents
of the black box.
S-parameters are a useful method for representing a circuit as a “black box”
The external behaviour of this
black box can be predicted
without any regard for the contents
of the black box.
This black box could contain
anything:
a resistor,
a transmission line
or an integrated circuit.
A “black box” or network may have any number of ports.
This diagram shows a simple
network with just 2 ports.
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.
S-parameters are measured by sending a single frequency signal into the
network or “black box” and detecting what waves exit from each port.
Power, voltage and current
can be considered to be in
the form of waves travelling
in both directions.
S-parameters are measured by sending a single frequency signal into the
network or “black box” and detecting what waves exit from each port.
Power, voltage and current
can be considered to be in
the form of waves travelling
in both directions.
For a wave incident on Port 1,
some part of this signal
reflects back out of that port
and some portion of the signal
exits other ports.
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 the
signal incident at Port 1.
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 the
signal incident at Port 1.
Scattering parameter S11
is the ratio of the two
waves b1/a1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
Now lets look at S21.
S21 refers to the signal
exiting at Port 2 for the
signal incident at Port 1.
Scattering parameter S21
is the ratio of the two
waves b2/a1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
Now lets look at S21.
S21 refers to the signal
exiting at Port 2 for the
signal incident at Port 1.
Scattering parameter S21
is the ratio of the two
waves b2/a1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
Now lets look at S21.
S21 refers to the signal
exiting at Port 2 for the
signal incident at Port 1.
Scattering parameter S21
is the ratio of the two
waves b2/a1.
I have seen S-parameters described as S11, S21, etc. Can you explain?
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...)
S-parameters are complex (i.e. they have magnitude and angle)
because both the magnitude and phase of the input signal are
changed by the network.
(This is why they are sometimes referred to as complex scattering parameters).
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.
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 much
the signal has been phase shifted.
What do S-parameters depend on?
S-parameters depend upon the network
and the characteristic impedances of the
source and load used to measure it, and
the 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?
S-parameters depend upon the network
and the characteristic impedances of the
source and load used to measure it, and
the frequency measured at.
i.e.
if the network is changed, the S-parameters change.
if the frequency is changed, the S-parameters change.
In the Si9000e
if the load impedance is changed, the S-parameters
change.
S-parameters are
quoted with source and load
if the source impedance is changed, the S-parameters change.
impedances of 50 Ohms
A little math…
This is the matrix algebraic representation
of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry about
the leading diagonal.
A little math…
This is the matrix algebraic representation
of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry about
the leading diagonal.
In the case of a 2-port network, that means that S21 = S12 and interchanging
the input and output ports does not change the transmission properties.
A little math…
This is the matrix algebraic representation
of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry about
the leading diagonal.
In the case of a symmetrical 2-port network, that means that S21 = S12 and
interchanging the input and output ports does not change the transmission
properties.
A transmission line is an example of a symmetrical 2-port network.
A little math…
Parameters along the leading diagonal,
S11 & S22, of the S-matrix are referred to as
reflection coefficients because they refer to
the reflection occurring at one port only.
A little math…
Parameters along the leading diagonal,
S11 & S22, of the S-matrix are referred to as
reflection coefficients because they refer to
the 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.
Larger networks:
A Network may have any number of ports.
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.
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 the
number of ports.
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 the
number of ports.
For the S-parameter subscripts “ij”, “j” is the port that is excited (the input port)
and “i” is the output port.
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 the
number of ports.
For the S-parameter subscripts “ij”, “j” is the port that is excited (the input port)
and “i” is the output port.
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 1st!)
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:
Further reading:
Agilent papers
http://www.sss-mag.com/pdf/an-95-1.pdf
http://www.sss-mag.com/pdf/AN154.pdf
National Instruments paper
http://zone.ni.com/devzone/nidzgloss.nsf/webmain/D2C4FA88321195FE8625686B00542
EDB?OpenDocument
Other links:
http://www.sss-mag.com
http://www.microwaves101.com/index.cfm
http://www.reed-electronics.com/tmworld/article/CA187307.html
http://en.wikipedia.org/wiki/S-parameters
Online lecture OLL-140 Intro to S-parameters - Eric Bogatin
Online lecture OLL-141 S11 & Smith charts - Eric Bogatin
www.bethesignal.com
Description
TRANSMISSION LINES
Terminology and Conventions
Sinusoidal
Source
V (t ) = Vo cos (ω t + φ )
{
}
{
V (t ) = Re Vo e j(ω t + φ ) = Re Vo e jφ e jω t
j=
Vo e jφ
}
−1
is a complex phasor
36
Phasors
Im
Vo
Vo e jφ = Vo cos (φ ) + jVo sin (φ )
Vo sin (φ )
ω
φ
Vo cos (φ )
Re
• In these notes, all sources are sine waves
• Circuits are described by complex phasors
• The time varying answer is found by multiplying
phasors by e jωt and taking the real part
37
TEM Transmission Line Theory
Charge on the inner conductor:
∆ q = C l ∆ xV
where Cl is the capacitance per unit length
Azimuthal magnetic flux:
∆Φ = L l ∆ xI
where Ll is the inductance per unit length
38
Electrical Model of a Transmission Line
L l∆ x
i + ∆i
i
v
C l∆ x
v + ∆v
Voltage drop along the inductor:
di
v − (v + ∆ v ) = L l ∆ x
dt
Current flowing through the capacitor:
dv
i + ∆i = i − C l ∆x
dt
39
Transmission Line Waves
Limit as ∆x->0
∂B
∇×E = −
∂t
∂D
∇×H =
∂t
∂v
∂i
= −Ll
∂t
∂x
∂i
∂v
= −C l
∂x
∂t
Solutions are traveling waves
v (t , x ) = v
+
v+
i (t , x ) =
Zo
x 
x 
−
t −
+ v t +

vel 
vel 


x  v− 
x 

t −
−
t +

vel  Z o 
vel 

v+ indicates a wave traveling in the +x direction
v- indicates a wave traveling in the -x direction
40
Phase Velocity and Characteristic
Impedance
vel is the phase velocity of the wave
vel =
1
L lC l
For a transverse electromagnetic wave (TEM), the phase velocity is
only a property of the material the wave travels through
1
=
L lC l
1
µε
The characteristic impedance Zo
Zo =
Ll
Cl
has units of Ohms and is a function of the material AND the
geometry
41
Pulses on a Transmission Line
v
+
RL
Pulse travels down the transmission line as a forward going wave only
(v+). However, when the pulse reaches the load resistor:
+
−
v
v +v
= RL =
i
v+ v−
−
Zo Zo
so a reverse wave v- and i- must be created to satisfy the boundary
42
condition imposed by the load resistor
Reflection Coefficient
The reverse wave can be thought of as the incident wave reflected
from the load
v− R L − Zo
Reflection coefficient
=
=Γ
v + R L + Zo
Three special cases:
RL = ∞ (open)
Γ = +1
RL = 0 (short)
Γ = -1
RL = Z o
Γ=0
A transmission line terminated with a resistor equal in value to the
characteristic impedance of the transmission line looks the same to
the source as an infinitely long transmission line
43
Sinusoidal Waves
Experiment: Send a SINGLE frequency (ω) sine wave into a
transmission line and measure how the line responds
{
v + = V + cos (ω t − β x ) = Re V + e − jβ x e jω t
ω
= vel
β
2 πf 2 π
β=
=
vel
λ
phase velocity
}
wave number
By using a single frequency sine wave we can now define complex
impedances such as:
di
dt
dv
i=C
dt
v=L
V = jω LI
Z ind = jω L
I = jω CV
Z cap =
1
jω C
44
Standing Waves
x
Zo
ZL
x=0
d
At x=0
−
ZL − Zo
V = ΓV =
ZL + Zo
+
Along the transmission line:
V = V + e − jβ x + Γ V + e + jβ x
V = V + (1 − Γ )e − jβ x + 2 V + Γ cos (β x )
traveling wave
standing wave
45
Voltage Standing Wave Ratio (VSWR)
Large voltage
Large current
1.5
1
V + (1 + Γ ) (1 + Γ )
Vmax
=
=
+
Vmin
V (1 − Γ ) (1 − Γ )
0.5
0
= VSWR
0.5
1
Γ=
1.5
0
0.1
0.2
1
2
0.3
0.4
0.5
Position
0.6
The VSWR is always greater than 1
0.7
0.8
0.9
1
46
Voltage Standing Wave Ratio (VSWR)
V + (1 + Γ ) (1 + Γ )
Vmax
=
=
+
Vmin
V (1 − Γ ) (1 − Γ )
= VSWR
Γ=
1
2
Incident wave
Reflected wave
Standing wave
The VSWR is always greater than 1
47
Reflection Coefficient Along a
Transmission Line
x
Zo
ZL
x=0
d
towards load
ΓG
towards generator
ΓL =
ZL − Zo
ZL + Zo
V = V + e − jβ x + ΓL V + e + jβ x
Wave has to travel
down and back
Vforward
V + e + jβ ( − d )
ΓG =
= ΓL
= ΓL e − j2β d
Vreverse gen
V + e − jβ ( − d )
48
Impedance and Reflection
Im {Γ}
There is a one-to-one
correspondence between ΓG and
ZL
ΓL
θ = − 2β d
ΓG
Re {Γ }
ZG − Zo
ΓG =
ZG + Zo
1 + ΓG
ZG = Zo
1 − ΓG
ZG = Zo
1 + ΓL e − j2β d
1 − ΓL e − j2β d
49
Impedance and Reflection: Open Circuits
For an open circuit ZL=
∞
so ΓL = +1
Impedance at the generator:
− jZ o
ZG =
tan (β d )
For βd<<1
ZG ≈
− jZ o
1
=
βd
jω C l d
looks capacitive
For βd = π/2 or d=λ/4
ZG = 0
An open circuit at the load looks like a short circuit at the generator
if the generator is a quarter wavelength away from the load
50
Impedance and Reflection: Short Circuits
For a short circuit ZL=
0
so ΓL = -1
Impedance at the generator:
Z G = jZ o tan (β d )
For βd<<1
Z G ≈ jZ o β d = jω L l d
looks inductive
For βd = π/2 or d=λ/4
ZG → ∞
A short circuit at the load looks like an open circuit at the generator
if the generator is a quarter wavelength away from the load
51
Incident and Reflected Power
x
Zo
Ps
ZL
d
x = −d
x = 0
Voltage and Current at the generator (x=-d)
VG = V ( − d ) = V + e + jβ d + ΓL V + e − jβ d
V + + jβ d
V + − jβ d
I G = I( − d ) =
e
− ΓL
e
Zo
Zo
The rate of energy flowing through the plane at x=-d
1
P = Re VG I G *
2
{
+2
forward power
P=
}
+2
1V
1
2 V
− ΓL
2 Zo
2
Zo
reflected power
52
Incident and Reflected Power
•
•
Power does not flow! Energy flows.
– The forward and reflected traveling waves are power orthogonal
• Cross terms cancel
– The net rate of energy transfer is equal to the difference in power of the
individual waves
To maximize the power transferred to the load we want:
ΓL = 0
which implies:
ZL = Zo
When ZL = Zo, the load is matched to the transmission line
53
Load Matching
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 source
effectively sees a match load.
Ps
M
Z0
ZL
Γ=0
Typically we only want to use lossless devices such as capacitors,
inductors, transmission lines, in our matching network so that we do
not dissipate any power in the network and deliver all the available
power to the load.
54
Normalized Impedance
It will be easier if we normalize the load impedance to the
characteristic impedance of the transmission line attached to the
load.
Z
z=
Zo
z=
= r + jx
1+ Γ
1− Γ
Since the impedance is a complex number, the reflection coefficient
will be a complex number
Γ = u + jv
r=
1− u 2 − v2
2
(1 − u )
+v
2
x=
2v
(1 − u )2 + v 2
55
dB and dBm
A dB is defined as a POWER ratio. For example:
 Prev 

ΓdB = 10 log 
 Pfor 
2
= 10 log  Γ 


= 20 log ( Γ )
A dBm is defined as log unit of power referenced to 1mW:
 P 
PdBm = 10 log 

 1mW 
56
Description
Z AND S PARAMETERS
Two Port Z Parameters
We have only discussed reflection so far. What about transmission?
Consider a device that has two ports:
I2
I1
V1
V2
The device can be characterized by a 2x2 matrix:
V1 = Z11 I1 + Z12 I 2
V2 = Z 21 I1 + Z 22 I 2
[V ] = [Z ][I ]
58
Scattering (S) Parameters
Since the voltage and current at each port (i) can be broken down
into forward and reverse waves:
Vi = Vi+ + Vi−
Z o I i = Vi+ − Vi−
We can characterize the circuit with forward and reverse waves:
V1− = S11 V1+ + S12 V2+
V2− = S 21 V1+ + S 22 V2+
[V ]= [S][V ]
−
+
59
Z and S Parameters
Similar to the reflection coefficient, there is a one-to-one
correspondence between the impedance matrix and the scattering
matrix:
[S] = ([Z ] + Z o [1])−1 ([Z ] − Z o [1])
[Z ] = Z o ([1] + [S])([1] − [S])−1
60
Normalized Scattering (S) Parameters
The S matrix defined previously is called the un-normalized
scattering matrix. For convenience, define normalized waves:
ai =
Vi+
2Zoi
bi =
Vi−
2Zoi
Where Zoi is the characteristic impedance of the transmission line
connecting port (i)
|ai|2 is the forward power into port (i)
|bi|2 is the reverse power from port (i)
61
Normalized Scattering (S) Parameters
The normalized scattering matrix is:
b1 = s11a 1 + s12 a 2
b 2 = s 21a 1 + s 22 a 2
[b ] = [s ][a ]
Where:
s i, j =
Zo j
Zoi
Si, j
If the characteristic impedance on both ports is the same then the
normalized and un-normalized S parameters are the same.
Normalized S parameters are the most commonly used.
62
Normalized S Parameters
The s parameters can be drawn pictorially
a1
s21
s11
b1
b2
s22
s12
a2
s11 and s22 can be thought of as reflection coefficients
s21 and s12 can be thought of as transmission coefficients
s parameters are complex numbers where the angle corresponds to a
phase shift between the forward and reverse waves
63
Examples of S parameters
τ
Zo
1
 0
[s ] =  − jωτ
 e
2
e − jωτ 

0 
Transmission Line
1
2
− 1 0 
[s ] = 

0
−
1


Short
1
G
Amplifier
2
0
[s ] = 
G
0
0 
64
Examples of S parameters
1
0
[s ] = 1
 0
2
0
1
1
0 
0 
0
[s ] = 
1
0
0 
0
3
Circulator
1
Zo
Isolator
2
65
Lorentz Reciprocity
If the device is made out of linear isotropic materials (resistors,
capacitors, inductors, metal, etc..) then:
[s ]T = [s ]
or
s j, i = s i , j
for
i≠ j
This is equivalent to saying that the transmitting pattern of an
antenna is the same as the receiving pattern
reciprocal devices:
non-reciprocal devices:
transmission line
short
directional coupler
amplifier
isolator
circulator
66
Lossless Devices
The s matrix of a lossless device is unitary:
[s ] [s] = [1]
*T
1 = ∑ s i, j
2
for all j
i
1 = ∑ s i, j
2
for all i
j
Lossless devices:
Non-lossless devices:
transmission line
short
circulator
amplifier
isolator
67
Network Analyzers
Network analyzers measure S
parameters as a function of
frequency
At a single frequency, network
analyzers send out forward waves a1
and a2 and measure the phase and
amplitude of the reflected waves b1
and b2 with respect to the forward
waves.
b
s11 = 1
a1 a = 0
2
b
s12 = 1
a 2 a =0
1
b
s 21 = 2
a1 a = 0
2
b
s 22 = 2
a 2 a =0
1
a1
b1
a2
b2
68
Network Analyzer Calibration
To measure the pure S parameters of a device, we need to eliminate
the effects of cables, connectors, etc. attaching the device to the
network analyzer
Connector Y
y21
y11
s21
y22
y12
Connector X
yx21
s11
x21
s22
x11
s12
x22
x12
yx12
We want to know the S parameters at
these reference planes
We measure the S parameters at these
reference planes
69
Network Analyzer Calibration
• There are 10 unknowns in the connectors
• We need 10 independent measurements to eliminate
these unknowns
– Develop calibration standards
– Place the standards in place of the Device Under Test (DUT)
and measure the S- parameters of the standards and the
connectors
– Because the S parameters of the calibration standards are
known (theoretically), the S parameters of the connectors
can be determined and can be mathematically eliminated
once the DUT is placed back in the measuring fixtures.
70
Network Analyzer Calibration
• Since we measure four S parameters for each
calibration standard, we need at least three
independent standards.
• One possible set is:
Thru
0
[s ] = 
1
Short
− 1 0 
[s ] = 

0
−
1


Delay*
 0
[s ] =  − jωτ
 e
*ωτ~90degrees
τ
1
0 
e − jωτ 

0 
71
Phase Delay
A pure sine wave can be written as:
V = Vo e j(ω t − β z )
The phase shift due to a length of cable is:
θ = βd
ω
=
d
v ph
= ωτ ph
The phase delay of a device is defined as:
arg (S 21 )
τ ph = −
ω
72
Phase Delay
• For a non-dispersive cable, the phase delay is the
same for all frequencies.
• In general, the phase delay will be a function of
frequency.
• It is possible for the phase velocity to take on any
value - even greater than the velocity of light
– Waveguides
– Waves hitting the shore at an angle
73
Group Delay
• A pure sine wave has no information content
– There is nothing changing in a pure sine wave
– Information is equivalent to something changing
• To send information there must be some
modulation of the sine wave at the source
The modulation can be de-composed into different frequency
components
m
V = Vo cos (ω t ) + Vo [cos ((ω + ∆ ω )t ) + cos ((ω − ∆ ω )t )]
2
V = Vo (1 + m cos (∆ ω t )) cos (ω t )
74
Group Delay
The waves emanating from the source will look like
V = Vo cos (ω t − β z )
m
+ Vo cos ((ω + ∆ ω )t − (β + ∆ β )z )
2
m
+ Vo cos ((ω − ∆ ω )t − (β − ∆ β )z )
2
Which can be re-written as:
V = Vo (1 + m cos (∆ ω t − ∆ β z )) cos (ω t − β z )
75
Group Delay
The information travels at a velocity
v gr =
1
∆β
∆ω
⇒
1
∂β
∂ω
The group delay is defined as:
d
τ gr =
v gr
∂β
d
∂ω
∂ (arg (S 21 ))
=−
∂ω
=
76
Phase Delay and Group Delay
Phase Delay:
arg (S 21 )
τ ph = −
ω
Group Delay:
∂ (arg (S 21 ))
τ gr = −
∂ω
77
Description
SMITH CHART
Smith Chart
• Impedances, voltages, currents, etc. all repeat
every half wavelength
• The magnitude of the reflection coefficient, the
standing wave ratio (SWR) do not change, so they
characterize the voltage & current patterns on the
line
• If the load impedance is normalized by the
characteristic impedance of the line, the voltages,
currents, impedances, etc. all still have the same
properties, but the results can be generalized to
any line with the same normalized impedances
79
Smith Chart
• The Smith Chart is a clever tool for analyzing
transmission lines
• The outside of the chart shows location on the line
in wavelengths
• The combination of intersecting circles inside the
chart allow us to locate the normalized impedance
and then to find the impedance anywhere on the
line
80
Smith Chart
• Thus, the first step in analyzing a transmission line is to
locate the normalized load impedance on the chart
• Next, a circle is drawn that represents the reflection
coefficient or SWR. The center of the circle is the center
of the chart. The circle passes through the normalized
load impedance
• Any point on the line is found on this circle. Rotate
clockwise to move toward the generator (away from the
load)
• The distance moved on the line is indicated on the
outside of the chart in wavelengths
81
Smith Charts
The impedance as a function of reflection coefficient can be rewritten in the form:
2
r=
1− u − v
2
(1 − u )2 + v 2
x=
2v
(1 − u )2 + v 2
2
r 
1

2
u −
 +v =
1+ r 

(1 + r )2
2
1
1
2 
(u − 1) +  v −  = 2
x

x
These are equations for
circles on the (u,v) plane
82
Smith Chart – Real Circles
1
Im {Γ}
0.5
r=0
r=1/3
r=1
1
0.5
0
r=2.5
0.5
1
Re {Γ}
0.5
1
83
Smith Chart – Imaginary Circles
Im {Γ}
1
x=1/3
x=1
x=2.5
0.5
1
0.5
0
x=-1/3
0.5
1
0.5
x=-1
1
Re {Γ}
x=-2.5
84
Smith Chart
85
Smith Chart Example 1
Given:
ΓL = 0 .5∠ 45 °
Z o = 50 Ω
What is ZL?
Z L = 50 Ω (1 .35 + j1 .35 )
= 67 .5Ω + j67 .5Ω
86
Smith Chart Example 2
Given:
Z L = 15 Ω − j25 Ω
Z o = 50 Ω
What is ΓL?
15 Ω − j25 Ω
50 Ω
= 0 .3 − j0 .5
zL =
ΓL = 0 .618 ∠ − 124 °
87
Smith Chart Example 3
Given:
Z L = 50 Ω + j50 Ω
Z o = 50 Ω
τ = 6 .78 nS
Z in = ?
What is Zin at 50 MHz?
50 Ω + j50 Ω
50 Ω
= 1 .0 + j1 .0
zL =
ΓL = 0 .445 ∠ 64 °
Γin = ΓL e
− j2 β d
= ΓL e
− j2 ωτ
2 ωτ = 244 °
2 ωτ = 244 °
Γin = 0 .445 ∠180 °
Z L = 50 Ω (0 .38 + j0 .0 ) = 19 Ω
88
Admittance
A matching network is going to be a combination of elements
connected in series AND parallel.
Z1
Impedance is well suited when working with
series configurations. For example:
V = ZI
Z L = Z1 + Z 2
Impedance is NOT well suited when working
with parallel configurations.
ZL =
1
Y1 =
Z1
Z1
Z2
Y1
Y2
Z1Z 2
Z1 + Z 2
For parallel loads it is better to work with
admittance.
I = YV
Z2
YL = Y1 + Y2
89
Normalized Admittance
Y
y=
= YZ o = g + jb
Yo
1− Γ
y=
1+ Γ
2
g=
1− u − v
2
(1 + u )2 + v 2
b=
− 2v
(1 + u )2 + v 2
2

g 
1
2
 u +
 + v =
1+ g 
(1 + g )2

2
1
1
2 
(u + 1) +  v +  = 2
b

b
These are equations for
circles on the (u,v) plane
90
Admittance Smith Chart
1
1
Im {Γ}
Im {Γ }
b=-1/3
b=-1
b=-2.5
0.5
g=2.5
1
g=1/3
g=1
0.5
0
0.5
g=0
0.5
Re {Γ}1
1
0.5
0
Re {Γ}
1
b=1/3
b=2.5
0.5
0.5
0.5
b=1
1
1
91
Impedance and Admittance Smith
Charts
• For a matching network that contains elements
connected in series and parallel, we will need two
types of Smith charts
– impedance Smith chart
– admittance Smith Chart
• The admittance Smith chart is the impedance Smith
chart rotated 180 degrees.
– We could use one Smith chart and flip the
reflection coefficient vector 180 degrees when
switching between a series configuration to a
parallel configuration.
92
Admittance Smith Chart Example 1
Given:
y = 1 + j1
What is Γ?
• Procedure:
Plot y
• Plot 1+j1 on chart
• vector =
0 .445 ∠64 °
• Flip vector 180 degrees
Read Γ
Flip 180
degrees
Γ = 0 .445 ∠ − 116 °
93
Admittance Smith Chart Example 2
Given:
Γ = 0 .5∠ + 45 ° Z o = 50 Ω
What is Y?
• Procedure:
Plot Γ
• Plot Γ
• Flip vector by 180 degrees
• Read coordinate
Read y
Flip 180
degrees
y = 0 .38 − j0 .36
Y=
1
(0.38 − j0.36 )
50 Ω
Y = (7 .6 − j7 .2 )x10 − 3 mhos
94
Smith Chart
Constant Imaginary
Impedance Lines
Impedance
Z=R+jX
=100+j50
Normalized
z=2+j for
Zo=50
Constant Real
Impedance Circles
95
Smith Chart
•Impedance divided by line impedance (50
Ohms)
– Z1 = 100 + j50
– Z2 = 75 -j100
– Z3 = j200
– Z4 = 150
– Z5 = infinity (an open circuit)
– Z6 = 0 (a short circuit)
– Z7 = 50
– Z8 = 184 -j900
•Then, normalize and plot. The points are
plotted as follows:
– z1 = 2 + j
– z2 = 1.5 -j2
– z3 = j4
– z4 = 3
– z5 = infinity
– z6 = 0
– z7 = 1
– z8 = 3.68 -j18S
96
Toward
Generator
Away From
Generator
Constant
Reflection
Coefficient Circle
Scale in
Wavelengths
Full Circle is One Half
Wavelength Since
Everything Repeats
97
Smith Chart Example
• First, locate the normalized impedance on the chart for ZL
= 50 + j100
• Then draw the circle through the point
• The circle gives us the reflection coefficient (the radius of
the circle) which can be read from the scale at the bottom
of most charts
• Also note that exactly opposite to the normalized load is
its admittance. Thus, the chart can also be used to find
the admittance. We use this fact in stub matching
98
Matching Example
Ps
M
Z 0 = 50 Ω
100 Ω
Γ=0
Match 100Ω load to a 50Ω system at 100MHz
A 100Ω resistor in parallel would do the trick but ½ of the
power would be dissipated in the matching network. We want
to use only lossless elements such as inductors and capacitors
so we don’t dissipate any power in the matching network
99
Matching Example
We need to go from
z=2+j0 to z=1+j0 on
the Smith chart
We won’t get any
closer by adding series
impedance so we will
need to add something
in parallel.
We need to flip over
to the admittance
chart
Impedance
Chart
100
Matching Example
y=0.5+j0
Before we add the
admittance, add a
mirror of the r=1
circle as a guide.
Admittance
Chart
101
Matching Example
y=0.5+j0
Before we add the
admittance, add a
mirror of the r=1
circle as a guide
Now add positive
imaginary admittance.
Admittance
Chart
102
Matching Example
y=0.5+j0
Before we add the
admittance, add a
mirror of the r=1
circle as a guide
Now add positive
imaginary admittance
jb = j0.5
jb = j0 .5
j0 .5
= j2 π (100 MHz )C
50 Ω
C = 16 pF
16 pF
100 Ω
Admittance
Chart
103
Matching Example
We will now add series
impedance
Flip to the impedance
Smith Chart
We land at on the r=1
circle at x=-1
Impedance
Chart
104
Matching Example
Add positive imaginary
admittance to get to
z=1+j0
jx = j1 .0
( j1.0 )50 Ω = j2 π (100 MHz )L
L = 80 nH
80 nH
16 pF
100 Ω
Impedance
Chart
105
Matching Example
This solution would
have also worked
32 pF
160 nH
100 Ω
Impedance
Chart
106
Matching Bandwidth
0
80 nH
5
16 pF
100 Ω
Reflection Coefficient (dB)
10
100 MHz
15
20
25
30
35
40
50 MHz
50
60
70
80
90
100
110
Frequency (MHz)
120
130
140
150
Because the inductor and capacitor
impedances change with frequency, the
match works over a narrow frequency range
Impedance
Chart
107