f AcceleratorDivision
Introduction to RF for Particle AcceleratorsIntroduction to RF for Particle AcceleratorsPart 1: Transmission LinesPart 1: Transmission Lines
Dave McGinnis
Introduction to RF – Part 1 – Transmission Lines 2
f AcceleratorDivision MotivationMotivation
Future accelerators at Fermilab will require substantial RF expertise ILC Proton Driver SNUMI
These lectures are intended as an introduction to RF terminology and techniques
Lectures are based on the notes for the Microwave Measurement Class that is taught at the US Particle Accelerator School
Introduction to RF – Part 1 – Transmission Lines 3
f AcceleratorDivision Part 1 - Transmission LinesPart 1 - Transmission Lines
Phasors Traveling Waves Characteristic Impedance Reflection Coefficient Standing Waves Impedance and
Reflection Incident and Reflected
Power
Smith Charts Load Matching Single Stub Tuners dB and dBm Z and S parameters Lorentz Reciprocity Network Analysis Phase and Group Delay
Introduction to RF – Part 1 – Transmission Lines 4
f AcceleratorDivision Part 2 - RF CavitiesPart 2 - RF Cavities
Modes Symmetry Boundaries Degeneracy
RLC model Coupling
Inductive Capacitive Measuring
Q Unloaded Q Loaded Q Q Measurements
Impedance Measurements Bead Pulls Stretched wire
Beam Loading De-tuning Fundamental Transient
Power Amplifiers Class of operation Tetrodes Klystrons
Introduction to RF – Part 1 – Transmission Lines 5
f AcceleratorDivision Part 3 - Beam SignalsPart 3 - Beam Signals
Power Spectral Density Spectra of bunch loading patterns Betatron motion AM modulation Longitudinal motion FM and PM modulation Multipole distributions
Introduction to RF – Part 1 – Transmission Lines 6
f AcceleratorDivision Terminology and ConventionsTerminology and Conventions
tcosVtV o
tjjo
tjo eeVReeVRetV
1j
Sinusoidal Source
joeV is a complex
phasor
Introduction to RF – Part 1 – Transmission Lines 7
f AcceleratorDivision PhasorsPhasors
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
oV
cosVo
sinVo
Re
Im
sinjVcosVeV ooj
o
tje
Introduction to RF – Part 1 – Transmission Lines 8
f AcceleratorDivision TEM Transmission Line TheoryTEM Transmission Line Theory
Charge on the inner conductor:
xVCq lwhere Cl is the capacitance per unit length
Azimuthal magnetic flux:xILl
where Ll is the inductance per unit length
Introduction to RF – Part 1 – Transmission Lines 9
f AcceleratorDivision Electrical Model of a Transmission LineElectrical Model of a Transmission Line
i ii
v vv
xLl
xCl
Voltage drop along the inductor:
dt
dixLvvv l
Current flowing through the capacitor:
dt
dvxCiii l
Introduction to RF – Part 1 – Transmission Lines 10
f AcceleratorDivision Transmission Line WavesTransmission Line Waves
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
Introduction to RF – Part 1 – Transmission Lines 11
f AcceleratorDivision Phase Velocity and Characteristic ImpedancePhase Velocity and Characteristic Impedance
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
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
Introduction to RF – Part 1 – Transmission Lines 12
f AcceleratorDivision Pulses on a Transmission LinePulses on a Transmission Line
Pulse travels down the transmission line as a forward going wave only (v+). However, when the pulse reaches the load resistor:
oo
L
Zv
Zv
vvR
i
v
LRv
so a reverse wave v- and i- must be created to satisfy the boundary condition imposed by the load resistor
Introduction to RF – Part 1 – Transmission Lines 13
f AcceleratorDivision Reflection CoefficientReflection Coefficient
The reverse wave can be thought of as the incident wave reflected from the load
oL
oL
ZR
ZR
v
v Reflection coefficient
Three special cases:
RL = ∞ (open) = +1
RL = 0 (short) = -1
RL = Zo = 0A 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
Introduction to RF – Part 1 – Transmission Lines 14
f AcceleratorDivision Sinusoidal WavesSinusoidal Waves
tjxj eeVRextcosVv
vel
phase velocity
2
vel
f2wave number
By using a single frequency sine wave we can now define complex impedances such as:
LIjV dt
diLv
dt
dvCi Cj
1Zcap
LjZind
CVjI
Experiment: Send a SINGLE frequency () sine wave into a transmission line and measure how the line responds
Introduction to RF – Part 1 – Transmission Lines 15
f AcceleratorDivision Standing WavesStanding Waves
LZoZ
0x
x
d
At x=0
oL
oL
ZZ
ZZVV
Along the transmission line:
xcosV2e1VV
eVeVVxj
xjxj
traveling wave standing wave
Introduction to RF – Part 1 – Transmission Lines 16
f AcceleratorDivision VVoltage oltage SStanding tanding WWave ave RRatio atio (VSWR)(VSWR)
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
The VSWR is always greater than 1
2
1
Large voltageLarge current
Introduction to RF – Part 1 – Transmission Lines 17
f AcceleratorDivision VVoltage oltage SStanding tanding WWave ave RRatio atio (VSWR)(VSWR)
VSWR
1
1
1V
1V
V
V
min
max
The VSWR is always greater than 1
2
1
Incident wave
Reflected wave
Standing wave
Introduction to RF – Part 1 – Transmission Lines 18
f AcceleratorDivision Reflection Coefficient Along a Transmission LineReflection Coefficient Along a Transmission Line
LZoZ
0x
x
d
oL
oLL ZZ
ZZ
Gtowards load
towards generator
xjL
xj eVeVV
d2jL)d(j
)d(j
Lgenreverse
forwardG e
eV
eV
V
V
Wave has to travel down and back
Introduction to RF – Part 1 – Transmission Lines 19
f AcceleratorDivision Impedance and ReflectionImpedance and Reflection
G
L
Re
Im
d2
There is a one-to-one correspondence between G and ZL
oG
oGG ZZ
ZZ
G
GoG 1
1ZZ
d2jL
d2jL
oGe1
e1ZZ
Introduction to RF – Part 1 – Transmission Lines 20
f AcceleratorDivision
Impedance and Reflection: Open Impedance and Reflection: Open CircuitsCircuits
For an open circuit ZL= ∞ so L = +1Impedance at the generator:
dtan
jZZ o
G
For d<<1
dCj
1
d
jZZ
l
oG
looks capacitive
For d = /2 or d=/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
Introduction to RF – Part 1 – Transmission Lines 21
f AcceleratorDivision
Impedance and Reflection: Short Impedance and Reflection: Short CircuitsCircuits
For a short circuit ZL= 0 so L = -1
Impedance at the generator:
dtanjZZ oG
For d<<1
dLjdjZZ loG looks inductive
For d = /2 or d=/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
Introduction to RF – Part 1 – Transmission Lines 22
f AcceleratorDivision Incident and Reflected PowerIncident and Reflected Power
LZoZ
0x
x
d
sP
dx
dj
oL
dj
oG
djL
djG
eZ
Ve
Z
V)d(II
eVeV)d(VV
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
forward powerreflected power
Introduction to RF – Part 1 – Transmission Lines 23
f AcceleratorDivision Incident and Reflected PowerIncident 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:
0L
which implies:
oL ZZ When ZL = Zo, the load is matched to the transmission line
Introduction to RF – Part 1 – Transmission Lines 24
f AcceleratorDivision Load MatchingLoad 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.
0
LZsP 0Z M
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.
Introduction to RF – Part 1 – Transmission Lines 25
f AcceleratorDivision Normalized ImpedanceNormalized Impedance
jxrZ
Zz
o
It will be easier if we normalize the load impedance to the characteristic impedance of the transmission line attached to the load.
1
1z
Since the impedance is a complex number, the reflection coefficient will be a complex number
jvu
22
22
vu1
vu1r
22 vu1
v2x
Introduction to RF – Part 1 – Transmission Lines 26
f AcceleratorDivision Smith ChartsSmith Charts
The impedance as a function of reflection coefficient can be re-written in the form:
22
22
vu1
vu1r
22 vu1
v2x
22
2
r1
1v
r1
ru
2
22
x
1
x
1v1u
These are equations for circles on the (u,v) plane
Introduction to RF – Part 1 – Transmission Lines 27
f AcceleratorDivision Smith Chart – Real CirclesSmith Chart – Real Circles
1 0.5 0 0.5 1
1
0.5
0.5
1
Re
Im
r=0r=1/3
r=1r=2.5
Introduction to RF – Part 1 – Transmission Lines 28
f AcceleratorDivision Smith Chart – Imaginary CirclesSmith Chart – Imaginary Circles
1 0.5 0 0.5 1
1
0.5
0.5
1
Re
Im
x=1/3
x=1 x=2.5
x=-1/3 x=-1 x=-2.5
Introduction to RF – Part 1 – Transmission Lines 29
f AcceleratorDivision Smith ChartSmith Chart
Introduction to RF – Part 1 – Transmission Lines 30
f AcceleratorDivision Smith Chart Example 1Smith Chart Example 1
Given:
50Zo
455.0L
What is ZL?
5.67j5.67
35.1j35.150ZL
Introduction to RF – Part 1 – Transmission Lines 31
f AcceleratorDivision Smith Chart Example 2Smith Chart Example 2
Given:
50Zo
25j15ZL
What is L?
5.0j3.050
25j15zL
124618.0L
Introduction to RF – Part 1 – Transmission Lines 32
f AcceleratorDivision Smith Chart Example 3Smith Chart Example 3
Given:50Zo
50j50ZL
What is Zin at 50 MHz?
0.1j0.150
50j50zL
64445.0L
nS78.6
?Zin
2jL
d2jLin ee
2442 180445.0in
190.0j38.050ZL
2442
Introduction to RF – Part 1 – Transmission Lines 33
f AcceleratorDivision AdmittanceAdmittance
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
2Z1Z
21
21L ZZ
ZZZ
ZIV
For parallel loads it is better to work with admittance.
YVI2Y1Y
21L YYY 11 Z
1Y
Impedance is well suited when working with series configurations. For example:
Introduction to RF – Part 1 – Transmission Lines 34
f AcceleratorDivision Normalized AdmittanceNormalized Admittance
jbgYZY
Yy o
o
1
1y
22
22
vu1
vu1g
22 vu1
v2b
22
2
g1
1v
g1
gu
2
22
b
1
b
1v1u
These are equations for circles on the (u,v) plane
Introduction to RF – Part 1 – Transmission Lines 35
f AcceleratorDivision
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
Admittance Smith ChartAdmittance Smith Chart
Re
Im
g=1/3
b=-1 b=-1/3
g=1g=2.5 g=0
b=2.5 b=1/3
b=1
b=-2.5
Im
Re
Introduction to RF – Part 1 – Transmission Lines 36
f AcceleratorDivision
Impedance and Admittance Smith Impedance and Admittance Smith ChartsCharts
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.
Introduction to RF – Part 1 – Transmission Lines 37
f AcceleratorDivision
• Procedure:
• Plot 1+j1 on chart
• vector =
• Flip vector 180 degrees
Admittance Smith Chart Example 1Admittance Smith Chart Example 1
Given:
64445.0
What is ?
1j1y
116445.0
Plot y
Flip 180 degrees
Read
Introduction to RF – Part 1 – Transmission Lines 38
f AcceleratorDivision
• Procedure:
• Plot
• Flip vector by 180 degrees
• Read coordinate
Admittance Smith Chart Example 2Admittance Smith Chart Example 2
Given:
What is Y?
455.0 50Zo
Plot
Flip 180 degrees
Read y
36.0j38.0y
mhos10x2.7j6.7Y
36.0j38.050
1Y
3
Introduction to RF – Part 1 – Transmission Lines 39
f AcceleratorDivision Matching ExampleMatching Example
0
100sP 50Z0 M
Match 100 load to a 50 system at 100MHz
A 100 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
Introduction to RF – Part 1 – Transmission Lines 40
f AcceleratorDivision Matching ExampleMatching 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
Introduction to RF – Part 1 – Transmission Lines 41
f AcceleratorDivision Matching ExampleMatching Example
y=0.5+j0
Before we add the admittance, add a mirror of the r=1 circle as a guide.
Admittance Chart
Introduction to RF – Part 1 – Transmission Lines 42
f AcceleratorDivision Matching ExampleMatching 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
Introduction to RF – Part 1 – Transmission Lines 43
f AcceleratorDivision Matching ExampleMatching 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
Admittance Chart
pF16C
CMHz1002j50
5.0j
5.0jjb
pF16 100
Introduction to RF – Part 1 – Transmission Lines 44
f AcceleratorDivision Matching ExampleMatching 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
Introduction to RF – Part 1 – Transmission Lines 45
f AcceleratorDivision Matching ExampleMatching Example
Add positive imaginary admittance to get to z=1+j0
Impedance Chart
pF16100
nH80L
LMHz1002j500.1j
0.1jjx
nH80
Introduction to RF – Part 1 – Transmission Lines 46
f AcceleratorDivision Matching ExampleMatching Example
This solution would have also worked
Impedance Chart
pF32
100nH160
Introduction to RF – Part 1 – Transmission Lines 47
f AcceleratorDivision
50 60 70 80 90 100 110 120 130 140 15040
35
30
25
20
15
10
5
0
Frequency (MHz)
Re
flect
ion
Co
effi
cie
nt (
dB
)
Matching BandwidthMatching Bandwidth
1
1
vi
Im gammai
11 ui Re gammai
50 MHz
100 MHz
Because the inductor and capacitor impedances change with frequency, the match works over a narrow frequency range
pF16100
nH80
Impedance Chart
Introduction to RF – Part 1 – Transmission Lines 48
f AcceleratorDivision dB and dBmdB and dBm
A dB is defined as a POWERPOWER ratio. For example:
log20
log10
P
Plog10
2
for
revdB
A dBm is defined as log unit of power referenced to 1mW:
mW1
Plog10PdBm
Introduction to RF – Part 1 – Transmission Lines 49
f AcceleratorDivision Single Stub TunerSingle Stub Tuner
Match 100 load to a 50 system at 100MHz using two transmission lines connected in parallel
0
1001
2
Introduction to RF – Part 1 – Transmission Lines 50
f AcceleratorDivision Single Stub TunerSingle Stub Tuner
Adding length to Cable 1 rotates the reflection coefficient clockwise.
Enough cable is added so that the reflection coefficient reaches the mirror image circle
nS49.3
MHz1003602251
1
1
Impedance Chart
2511
Introduction to RF – Part 1 – Transmission Lines 51
f AcceleratorDivision Single Stub TunerSingle Stub Tuner
The stub is going to be added in parallel so flip to the admittance chart.
The stub has to add a normalized admittance of 0.7 to bring the trajectory to the center of the Smith Chart
Admittance Chart
Introduction to RF – Part 1 – Transmission Lines 52
f AcceleratorDivision Single Stub TunerSingle Stub Tuner
Admittance Chart
An open stub of zero length has an admittance=j0.0
By adding enough cable to the open stub, the admittance of the stub will increase.
70 degrees will give the open stub an admittance of j0.7
702
nS97.0
MHz100360270
2
2
Introduction to RF – Part 1 – Transmission Lines 53
f AcceleratorDivision Single Stub TunerSingle Stub Tuner
Admittance Chart
01003.5nS
0.97nS
Introduction to RF – Part 1 – Transmission Lines 54
f AcceleratorDivision Single Stub TunerSingle Stub Tuner
Admittance Chart
01001.5nS
1.4nS
This solution would have worked as well.
Introduction to RF – Part 1 – Transmission Lines 55
f AcceleratorDivision
50 60 70 80 90 100 110 120 130 140 15040
35
30
25
20
15
10
5
0
Frequency (MHz)
Re
flect
ion
Co
effi
cie
nt (
dB
)
0 1003.5nS
0.97nS
Single Stub Tuner Matching BandwidthSingle Stub Tuner Matching Bandwidth
50 MHz
100 MHz
Because the cable phase changes linearly with frequency, the match works over a narrow frequency range
Impedance Chart
Introduction to RF – Part 1 – Transmission Lines 56
f AcceleratorDivision Two Port Z ParametersTwo Port Z Parameters
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:
Introduction to RF – Part 1 – Transmission Lines 57
f AcceleratorDivision Scattering (S) ParametersScattering (S) Parameters
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
Introduction to RF – Part 1 – Transmission Lines 58
f AcceleratorDivision Z and S ParametersZ and S Parameters
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:
Introduction to RF – Part 1 – Transmission Lines 59
f AcceleratorDivision Normalized Scattering (S) ParametersNormalized Scattering (S) Parameters
The S matrix defined previously is called the un-normalized scattering matrix. For convenience, define normalized waves:
io
ii
io
ii
Z2
Vb
Z2
Va
Where Zoi is the characteristic impedance of the transmission line connecting port (i)
|ai|2 is the forward power into port (i)
|bi|2 is the reverse power from port (i)
Introduction to RF – Part 1 – Transmission Lines 60
f AcceleratorDivision Normalized Scattering (S) ParametersNormalized Scattering (S) Parameters
The normalized scattering matrix is:
Where:
asb
asasb
asasb
2221212
2121111
j,iio
joj,i S
Z
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.
Introduction to RF – Part 1 – Transmission Lines 61
f AcceleratorDivision Normalized S ParametersNormalized S Parameters
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
Introduction to RF – Part 1 – Transmission Lines 62
f AcceleratorDivision Examples of S parametersExamples of S parameters
Zo1 2
0e
e0s
j
j
21
10
01s
1 2
0G
00s
G
Transmission Line
Short
Amplifier
Introduction to RF – Part 1 – Transmission Lines 63
f AcceleratorDivision Examples of S parametersExamples of S parameters
010
001
100
s
1
2
01
00s
Zo
Isolator
1 2
3
Circulator
Introduction to RF – Part 1 – Transmission Lines 64
f AcceleratorDivision Lorentz ReciprocityLorentz Reciprocity
If the device is made out of linear isotropic materials (resistors, capacitors, inductors, metal, etc..) then:
ss T
j,ii,j ss jior
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 couplernon-reciprocal devices: amplifier
isolator
circulator
Introduction to RF – Part 1 – Transmission Lines 65
f AcceleratorDivision Lossless DevicesLossless Devices
The s matrix of a lossless device is unitary:
1ssT*
j
2j,i
i
2j,i
s1
s1for all j
for all i
Lossless devices: transmission line
short
circulator
Non-lossless devices: amplifier
isolator
Introduction to RF – Part 1 – Transmission Lines 66
f AcceleratorDivision Network AnalyzersNetwork 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.
a1 a2
b1 b2
02a1
111 a
bs
02a1
221 a
bs
01a2
112 a
bs
01a2
222 a
bs
Introduction to RF – Part 1 – Transmission Lines 67
f AcceleratorDivision Network Analyzer CalibrationNetwork 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
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
Introduction to RF – Part 1 – Transmission Lines 68
f AcceleratorDivision Network Analyzer CalibrationNetwork 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.
Introduction to RF – Part 1 – Transmission Lines 69
f AcceleratorDivision Network Analyzer CalibrationNetwork Analyzer Calibration
Since we measure four S parameters for each calibration standard, we need at least three independent standards.
One possible set is:
10
01s
0e
e0s
j
j
01
10sThru
Short
Delay**~90degrees
Introduction to RF – Part 1 – Transmission Lines 70
f AcceleratorDivision Phase DelayPhase Delay
A pure sine wave can be written as:
ztjoeVV
The phase shift due to a length of cable is:
ph
phd
v
d
The phase delay of a device is defined as:
21ph
Sarg
Introduction to RF – Part 1 – Transmission Lines 71
f AcceleratorDivision Phase DelayPhase 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
Introduction to RF – Part 1 – Transmission Lines 72
f AcceleratorDivision Group DelayGroup 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
tcostcosm1VV o
tcostcos2
mVtcosVV oo
The modulation can be de-composed into different frequency components
Introduction to RF – Part 1 – Transmission Lines 73
f AcceleratorDivision Group DelayGroup Delay
ztcos2
mV
ztcos2
mV
ztcosVV
o
o
o
The waves emanating from the source will look like
Which can be re-written as:
ztcosztcosm1VV o
Introduction to RF – Part 1 – Transmission Lines 74
f AcceleratorDivision Group DelayGroup Delay
The information travels at a velocity
11vgr
The group delay is defined as:
21
grgr
Sarg
d
v
d
Introduction to RF – Part 1 – Transmission Lines 75
f AcceleratorDivision Phase Delay and Group DelayPhase Delay and Group Delay
Phase Delay:
21ph
Sarg
Group Delay:
21
grSarg
Introduction to RF – Part 1 – Transmission Lines 76
f AcceleratorDivision Transmission Line TopicsTransmission Line Topics
Phasors Traveling Waves Characteristic Impedance Reflection Coefficient Standing Waves Impedance and
Reflection Incident and Reflected
Power
Smith Charts Load Matching Single Stub Tuners dB and dBm Z and S parameters Lorentz Reciprocity Network Analysis Phase and Group Delay