S-PARAMETERS - nanoHUB

S-PARAMETERS

An Introduction

S-parameters are a useful method for representing a circuit as a “black box”


The external behaviour of thisblack box can be predicted without any regard for the contents of the black box.

The external behaviour of thisblack 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.

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.

S-parameters are measured by sending a single frequency signal into the network or “black box” and detecting what waves exit from each port.

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.

First lets look at S11.


reflected at Port 1 for the


Scattering parameter S11is the ratio of the two waves b1/a1.


Now lets look at S21.


exiting at Port 2 for the


Scattering parameter S21is the ratio of the two waves b2/a1.


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.

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?

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.

In the Si9000e S-parameters are

quoted with source and load

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…





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…





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:


The S-matrix for an n-port network contains n2 coefficients (S-parameters),

each one representing a possible input-output path.

Larger networks:




The number of rows and columns in an S-parameters matrix is equal to the

number of ports.

Larger networks:




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 papershttp://www.sss-mag.com/pdf/an-95-1.pdfhttp://www.sss-mag.com/pdf/AN154.pdfNational Instruments paperhttp://zone.ni.com/devzone/nidzgloss.nsf/webmain/D2C4FA88321195FE8625686B00542EDB?OpenDocumentOther links:http://www.sss-mag.comhttp://www.microwaves101.com/index.cfmhttp://www.reed-electronics.com/tmworld/article/CA187307.htmlhttp://en.wikipedia.org/wiki/S-parametersOnline lecture OLL-140 Intro to S-parameters - Eric BogatinOnline lecture OLL-141 S11 & Smith charts - Eric Bogatin

www.bethesignal.com

TRANSMISSION LINES

Description

Terminology and Conventions

36

( ) ( )φ+ω= tcosVtV o

( ) ( ) tjjo

tjo eeVReeVRetV ωφφ+ω ==

1j −=

Sinusoidal Source

φjoeV is a complex phasor

Phasors

• In these notes, all sources are sine waves

• Circuits are described by complex phasors

• The time varying answer is found by multiplying phasors by and taking the real part

37

oV

( )φcosVo

( )φsinVo

φω

Re

Im

( ) ( )φ+φ=φ sinjVcosVeV ooj

o

tje

ω

TEM Transmission Line Theory

38

Charge on the inner conductor:

xVCq l∆=∆

where Cl is the capacitance per unit length

Azimuthal magnetic flux:

xIL l∆=∆Φ

where Ll is the inductance per unit length

Electrical Model of a Transmission Line

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∆−=∆+

Transmission Line Waves

40

Limit as ∆x->0

t

iL

x

vl

∂

∂−=

∂

∂

t

BE

∂

∂−=×∇

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

Phase Velocity and Characteristic

Impedance

41

vel is the phase velocity of the wave

llCL

1vel =

For a transverse electromagnetic wave (TEM), the phase velocity is only 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 the geometry

Pulses on a Transmission Line

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

vvR

i

v

−+

−+

−

+==

LR+v

so a reverse wave v- and i- must be created to satisfy the boundary condition imposed by the load resistor

Reflection Coefficient

43

The reverse wave can be thought of as the incident wave reflected from the load

Γ=+

−=

+

−

oL

oL

ZR

ZR

v

v 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 to

the source as an infinitely long transmission line

Sinusoidal Waves

44

( ) tjxj eeVRextcosVv ωβ−+++ =β−ω=

vel=β

ωphase velocity

λ

π=

π=β

2

vel

f2wave number

By using a single frequency sine wave we can now define complex impedances such as:

LIjV ω=dt

diLv =

dt

dvCi =

Cj

1Zcap

ω=

LjZind ω=

CVjI ω=

Experiment: Send a SINGLE frequency (ω) sine wave into a transmission line and measure how the line responds

Standing Waves

45

LZoZ

0x =

x

d

At x=0

oL

oL

ZZ

ZZVV

+

−=Γ= +−

Along the transmission line:

( ) ( )xcosV2e1VV

eVeVV

xj

xjxj

βΓ+Γ−=

Γ+=

+β−+

β++β−+

traveling wave standing wave

Voltage Standing Wave Ratio (VSWR)

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

Voltage Standing Wave Ratio (VSWR)

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

Reflection Coefficient Along a

Transmission Line

48

LZoZ

0x =

x

d

oL

oLL

ZZ

ZZ

+

−=ΓGΓ

towards load

towards generator

xjL

xj eVeVV β++β−+ Γ+=

d2jL)d(j

)d(j

L

genreverse

forwardG e

eV

eV

V

V β−−β−+

−β++

Γ=Γ==Γ

Wave has to travel down and back

Impedance and Reflection

49

GΓ

LΓ

ΓRe

ΓIm

d2β−=θ

There is a one-to-one correspondence between ΓG and

ZL

oG

oGG

ZZ

ZZ

+

−=Γ

G

GoG

1

1ZZ

Γ−

Γ+=

d2jL

d2jL

oGe1

e1ZZ

β−

β−

Γ−

Γ+=

Impedance and Reflection: Open Circuits

50

For an open circuit ZL= ∞ so ΓL = +1

Impedance at the generator:

( )dtan

jZZ o

Gβ

−=

For βd<<1

dCj

1

d

jZZ

l

oG

ω=

β

−≈ looks capacitive

For βd = π/2 or d=λ/4

0ZG =

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

Impedance and Reflection: Short Circuits

51

For a short circuit ZL= 0 so ΓL = -1

Impedance at the generator:

( )dtanjZZ oG β=

For βd<<1

dLjdjZZ loG ω=β≈ looks inductive

For βd = π/2 or d=λ/4

∞→GZ

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

Incident and Reflected Power

52

LZoZ

0x =

x

d

sP

dx −=

dj

oL

dj

oG

djL

djG

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 powerreflected power

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:

53

0L =Γ

which implies:

oL ZZ =

When ZL = Zo, the load is matched to the transmission line

Load Matching

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.

Normalized Impedance

55

jxrZ

Zz

o

+==

It will be easier if we normalize the load impedance to thecharacteristic impedance of the transmission line attached to theload.

Γ−

Γ+=

1

1z

Since the impedance is a complex number, the reflection coefficientwill be a complex number

jvu +=Γ

( ) 22

22

vu1

vu1r

+−

−−= ( ) 22

vu1

v2x

+−=

dB and dBm

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

Z AND S PARAMETERS

Description

Two Port Z Parameters

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:

Scattering (S) Parameters

59

−+

−+

−=

+=

iiio

iii

VVIZ

VVV

Since the voltage and current at each port (i) can be broken down into forward and reverse waves:

We can characterize the circuit with forward and reverse waves:

[ ] [ ][ ]+−

++−

++−

=

+=

+=

VSV

VSVSV

VSVSV

2221212

2121111

Z and S Parameters

60

[ ] [ ] [ ]( ) [ ] [ ]( )

[ ] [ ] [ ]( ) [ ] [ ]( ) 1o

o1

o

S1S1ZZ

1ZZ1ZZS

−

−

−+=

−+=

Similar to the reflection coefficient, there is a one-to-one correspondence between the impedance matrix and the scattering

matrix:

Normalized Scattering (S) Parameters

61

The S matrix defined previously is called the un-normalizedscattering matrix. For convenience, define normalized waves:

io

ii

io

ii

Z2

Vb

Z2

Va

−

+

=

=

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)

Normalized Scattering (S) Parameters

62

The normalized scattering matrix is:

Where:

[ ] [ ][ ]asb

asasb

asasb

2221212

2121111

=

+=

+=

j,i

io

jo

j,i SZ

Zs =

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.

Normalized S Parameters

63

The s parameters can be drawn pictorially

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

s11 s22

s21

s12

a1

a2b1

b2

Examples of S parameters

64

τ

Zo1 2

[ ]

=

ωτ−

ωτ−

0e

e0s

j

j

21 [ ]

−

−=

10

01s

1 2

[ ]

=

0G

00s

G

Transmission Line

Short

Amplifier

Examples of S parameters

65

[ ]

=

010

001

100

s

1

2

[ ]

=

01

00s

Zo

Isolator

1 2

3

Circulator

Lorentz Reciprocity

66

If the device is made out of linear isotropic materials (resistors, capacitors, inductors, metal, etc..) then:

[ ] [ ]ssT =

j,ii,j ss = ji ≠

or

for

This is equivalent to saying that the transmitting pattern of an antenna is the same as the receiving pattern

reciprocal devices: transmission line

short

directional coupler

non-reciprocal devices: amplifier

isolator

circulator

Lossless Devices

67

The s matrix of a lossless device is unitary:

[ ] [ ] [ ]1ssT* =

∑

∑

=

=

j

2

j,i

i

2

j,i

s1

s1for all j

for all i

Lossless devices: transmission line

short

circulator

Non-lossless devices: amplifier

isolator

Network Analyzers

68

Network analyzers measure S parameters as a function of

frequency

At a single frequency, network analyzers 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

=

=

02a1

221

a

bs

=

=

01a2

112

a

bs

=

=

01a2

222

a

bs

=

=

Network Analyzer Calibration

69

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

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 at these reference planes

We measure the S parameters at these reference planes

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:

71

[ ]

−

−=

10

01s

[ ]

=

ωτ−

ωτ−

0e

e0s

j

j

τ

[ ]

=

01

10sThru

Short

Delay**ωτ~90degrees

Phase Delay

72

A pure sine wave can be written as:

( )ztjoeVV β−ω=

The phase shift due to a length of cable is:

ph

ph

dv

d

ωτ=

ω=

β=θ

The phase delay of a device is defined as:

( )ω

−=τ 21ph

Sarg

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

74

( )( ) ( )tcostcosm1VV o ωω∆+=

( ) ( )( ) ( )( )[ ]tcostcos2

mVtcosVV oo ω∆−ω+ω∆+ω+ω=

The modulation can be de-composed into different frequency components

Group Delay

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∆+=

Group Delay

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

Phase Delay and Group Delay

77

Phase Delay:

( )ω

−=τ 21ph

Sarg

Group Delay:

( )( )ω∂

∂−=τ 21

grSarg

SMITH CHART

Description

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 wavelengths81

Smith Charts

82

The impedance as a function of reflection coefficient can be re-written in the form:

( ) 22

22

vu1

vu1r

+−

−−=

( ) 22vu1

v2x

+−=

( )22

2

r1

1v

r1

ru

+=+

+−

( )2

22

x

1

x

1v1u =

−+−

These are equations forcircles on the (u,v) plane

Smith Chart – Real Circles

83

1 0.5 0 0.5 1

1

0.5

0.5

1

ΓRe

ΓIm

r=0r=1/3

r=1r=2.5

Smith Chart – Imaginary Circles

84

1 0.5 0 0.5 1

1

0.5

0.5

1

ΓRe

ΓIm

x=1/3 x=1 x=2.5

x=-1/3 x=-1 x=-2.5

Smith Chart

85

Smith Chart Example 1

86

Given:

Ω= 50Zo

°∠=Γ 455.0L

What is ZL?

( )Ω+Ω=

+Ω=

5.67j5.67

35.1j35.150ZL


87

Given:

Ω= 50Zo

Ω−Ω= 25j15ZL

What is ΓL?

5.0j3.0

50

25j15zL

−=

Ω

Ω−Ω=

°−∠=Γ 124618.0L


88

Given:

Ω= 50Zo

Ω+Ω= 50j50ZL

What is Zin at 50 MHz?

0.1j0.1

50

50j50zL

+=

Ω

Ω+Ω=

°∠=Γ 64445.0L

nS78.6=τ

?Zin =

ωτ−β− Γ=Γ=Γ 2jL

d2jLin ee

°=ωτ 2442

°∠=Γ 180445.0in

( ) Ω=+Ω= 190.0j38.050ZL

°=ωτ 2442

Admittance

89

A matching network is going to be a combination of elements connected in series AND parallel.

Impedance is NOT well suited when working with parallel configurations.

21L ZZZ +=

2Z1Z

2Z

1Z

21

21L

ZZ

ZZZ

+=

ZIV =

For parallel loads it is better to work with admittance.

YVI =

2Y1Y

21L YYY +=1

1Z

1Y =

Impedance is well suited when working with series configurations. For example:

Normalized Admittance

90

jbgYZY

Yy o

o

+===

Γ+

Γ−=

1

1y

( ) 22

22

vu1

vu1g

++

−−=

( ) 22vu1

v2b

++

−=

( )22

2

g1

1v

g1

gu

+=+

++

( )2

22

b

1

b

1v1u =

+++

These are equations forcircles on the (u,v) plane

Admittance Smith Chart

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

ΓRe

ΓIm

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

ΓIm

ΓRe

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

93

• Procedure:

• Plot 1+j1 on chart

• vector =

• Flip vector 180 degrees

Given:

°∠64445.0

What is Γ?

1j1y +=

°−∠=Γ 116445.0

Plot y

Flip 180degrees

Read Γ

Admittance Smith Chart Example 2

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

36.0j38.0y −=

( )

( ) mhos10x2.7j6.7Y

36.0j38.050

1Y

3−−=

−Ω

=

Smith Chart

95

Constant Imaginary

Impedance Lines

Constant Real

Impedance Circles

Impedance

Z=R+jX

=100+j50

Normalized

z=2+j for

Zo=50

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 -j18S96

97

Toward

Generator

Away From

Generator

Constant

Reflection

Coefficient Circle

Scale in

Wavelengths

Full Circle is One Half

Wavelength Since

Everything Repeats

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

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 capacitorsso we don’t dissipate any power in the matching network

Matching Example

100

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

Matching Example

101

y=0.5+j0

Before we add the admittance, add a mirror of the r=1 circle as a guide.

Admittance Chart

Matching Example

102

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

Matching Example

103

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

Admittance Chart

( )

pF16C

CMHz1002j50

5.0j

5.0jjb

=

π=Ω

=

pF16 Ω100

Matching Example

104

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

Matching Example

105

Add positive imaginary admittance to get to

z=1+j0

Impedance Chart

pF16

Ω100

( ) ( )nH80L

LMHz1002j500.1j

0.1jjx

=

π=Ω

=

nH80

Matching Example

106

This solution would have also worked

Impedance Chart

pF32

Ω100nH160

Matching Bandwidth

107

50 60 70 80 90 100 110 120 130 140 15040

35

30

25

20

15

10

5

0

Frequency (MHz)

Re

fle

ctio

n C

oe

ffic

ien

t (d

B)

50 MHz

100 MHz

Because the inductor and capacitor impedances change with frequency, the

match works over a narrow frequency range

pF16Ω100

nH80

Impedance Chart

S-PARAMETERS - nanoHUB

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