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Microwave Circuits 1
Microwave Engineering
1.
2.
3.
Microwave: 300MHz ~ 300 GHz, 1 m ~ 1mm.
a. Not only apply in this frequency range. The
real issue is wavelength. Historically, as early
as WWII, this is the first frequency range we
need to consider the wave effect.
b. Why microwave engineering? We all know
that the ideal of capacitor, inductor and
resistance are first defined in DC.
i. Circuit theory only apply in lower
frequency.
ii. Balance between two extreme: circuit and
fullwave.
Today’s high tech is developed long time ago.
a. Maxwell equation 1873. Theory before
Experiment.
b. Waveguide, Radar, Passive circuit, before
WWII.
Transmission Line Theory
a. Key difference between circuit theory and
transmission line is electrical size.
i. Ordinary circuit: no variation of current
and voltage.
ii. Transmission line: allow variation of
current and voltage.
Microwave Circuits 2
b.
c.
iii. Definition of Voltage and current is
ambiguous in real waveguide except for
TEM wave. Give examples.
Derive transmission line equations.
i. Purpose: to show that with only the
assumption of varying voltage and current,
we reach wave solution.
ii. Show the characteristics of lossless
transmission lines.
iii. Show the characteristics of low loss
transmission lines (p. 170)
Waveguides:
i. General solutions for TEM, TE and TM
waves
(1) Rectangular waveguide.
(2) Coaxial waveguide.
(3) Surface waveguide.
(4) Microstrip.
(5) Coplanar Waveguide.
ii. Under what condition, no TEM wave
exists.
iii. Importance of cut off frequency.
Microwave Circuits 3
Microwave Engineering
!
!
Grading policy.
"
Weekly Homework 40%
"
Midterm exam, final exam 30% each.
Office hour: 1:00 ~ 3:00 pm, Monday and 2:00 ~
6:00 pm, Wednesday.
!
Textbook: D. M. Pozar, “Microwave Engineering,
4th Ed.”
!
Contents
"
A Review of Electromagnetic Theories
"
Transmission Line Theory
"
Transmission Line and Waveguides
"
Microwave Network Analysis
"
Impedance Matching and Tuning
"
Microwave Resonators
"
Power Dividers and Directional Coupler
"
Microwave Filters
Microwave Circuits 4
A Review of Electromagnetic Theories
Maxwell Equations (1873)
: Electric field intensity
: Electric flux density
: Magnetic field intensity
: Magnetic flux density
: Electric current density
: Volume charge density
Constituent relationship
where
: Permittivity
: Permeability
Continuity relationship
Microwave Circuits 5
Divergence Operator
an
Physical meaning
V
Divergence Theorem
Curl Operator
an
S
Physical meaning
C
Stoke’s theorem:
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Time-Harmonic Fields
Time-harmonic:
: a real function in both space and time.
: a real function in space.
: a complex function in space. A
phaser.
Thus, all derivative of time becomes
.
For a partial deferential equation, all derivative of
time can be replace with , and all time
dependence of
can be removed and becomes a
partial deferential equation of space only.
Representing all field quantities as
,
then the original Maxwell’s equation becomes
Wave Equations
Microwave Circuits 7
Source Free:
Plane wave
From wave equation
where
, free space wave number or
propagation constant.
In Cartesian coordinates, considering the x
component,
.
Assume
is independent of x and y, then
.
The solutions are
In time domain,
Constant phase
Microwave Circuits 8
, intrinsic impedance or wave
impedance
Vector Potential
: vector potential
Flow of Electromagnetic Power and Poynting
Vector
Since
we have
or
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: stored energy
: energy dissipated
By conservation of energy, define Poynting vector
, the power density vector associated with an
electromagnetic field.
For time-harmonic wave
thus,
the time average power density.
Like wise
Boundary Conditions
Microwave Circuits 10
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Reciprocity Theorem
Two sets of solutions with two sets of excitation:
satisfying Maxwell’s Equations
Then
1. No sources
2. Bound by a perfect conductor
Microwave Circuits 12
3. Unbounded
Uniqueness Theorem
Let
be two sets of solutions of the same
excitation, then
The surface integral vanishes if
1.
, tangential electric field equals specified
2.
, tangential magnetic field specified.
then
That is, in a enclosed volume, if the source in the
volume and the tangential fields on the boundary
are the same, the fields are the same everywhere
inside the volume.
Microwave Circuits 13
Image Theory: an application of Uniqueness
Theorem.
Microwave Circuits 14
Transmission Line Theory
:series resistance per unit length in
.
:series inductance per unit length in
.
:shunt conductance per unit length in
.
:shunt capacitance per unit length in
.
By Kirchhoff’s voltage law:
By Kirchhoff’s current law:
As
,
For time-harmonic circuits
Microwave Circuits 15
Thus
where
:complex propagation
constant.
We have the solutions
: positive z-direction propagation wave.
: negative z-direction propagation wave.
: constants.
Also
Define characteristic impedance
Then
and
For lossless line
Microwave Circuits 16
Terminated Lossless Transmission Line
Assume incident wave
then
At
,
Define return loss:
Special case:
1.
(short):
.
2.
(open):
.
3. Half wavelength line:
, reflected wave
Microwave Circuits 17
4. Quarter wavelength line:
Two-transmission Line Junction
At
,
: transmission coefficient.
Define Insertion loss:
Conservation of energy
Incident power:
Reflected power:
Transmitted power:
Microwave Circuits 18
Voltage Standing Wave Ratio (VSWR)
Define Standing Wave Ratio
Smith Chart
Suppose a transmission line terminated by a load
impedance .
Define normalized impedance
, where
is the characteristic impedance of the
transmission line. Then,
Equating the real and imaginary parts, we have
Microwave Circuits 19
Rearranging, we have
Summary
Constant resistance circle
1. Center:
,
2. Radius:
,
3. Always passes
,
4.
decreases, radius increase,
5.
(short), unit circle,
6.
(open), point
.
Constant reactance circle
1. Center:
2. Radius:
,
,
3. Always passes:
,
4.
decreases, radius increase,
5.
(short), axis,
6.
(open), point
.
Microwave Circuits 20
Since
 a rotation of angle
clockwise.
Calculation of VSWR:
where
. Therefore,
is
the resistance value at the intersection point of the
positive and constant circle.
Admittance Smith Chart
, where
. Therefore,
Admittance Smith Chart is a rotation of 180 degree
of impedance Smith Chart.
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Example 2.2 Basic Smith Chart Operations
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Example 2.3 Smith Chart Operations Using
Admittances
Microwave Circuits 23
Example 2.4 Impedance Measurement with a
Slotted Line
Microwave Circuits 24
Microwave Circuits 25
Example 2.5 Frequency Repsonse of a Quarterwave Transformer
Microwave Circuits 26
Generator and Load Mismatches
1. Load Matched to Line
2. Generator Matched to Loaded Line
3. Conjugate Matched
Note this result means maximum power delivered
to the load under fixed . In reality, our concern is
how much portion of total power is delivered to the
load which is related to
.
Microwave Circuits 27
Lossy Transmission Lines
Low-Loss Line
Distortionless Line
Method of Evaluation Attenuation Constants
1. Perturbation
Power loss per unit length:
ex. 2.7
2. Wheeler Incremental Inductance Rule
Microwave Circuits 28
or
, where
and
are
changes due to recess of all conductor walls by an
amount of .
Ex. 2.8
Microwave Circuits 29
Plane Waves in Lossy Media
If the material is conductive
,
where
is the complex permittivity.
we have
Or if the material has dielectric loss with
where
is the attenuation constant,
is the loss tangent.
Low-Loss Dielectrics:
and
, or
the phase constant.
Microwave Circuits 30
Good Conductor:
and
Skin depth or depth of penetration:
Meaning: plane wave decay be a factor of
. At
microwave frequencies, is very small for a good conductor,
thus confined in a very thin layer of the conductor surface.
Let
be the equivalent surface conductivity defined by
Surface Resistance:
Microwave Circuits 31
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Microwave Circuits 33
Transmission Lines and Waveguides
General Solutions for TEM, TE, and TM Waves
Rectangular Waveguides
Coaxial Lines
Microstrip
Strip Lines
Coplanar Waveguides
Microwave Circuits 34
General Solutions for TEM, TE, and TM
Waves
Assuming a wave propagating in the
direction,
the electric and magnetic fields can be expressed
as
where
and
are the transverse
electric and magnetic field components.
In a source-free region, Maxwell’s equations can
be written as
Microwave Circuits 35
With an
dependence in direction, the above
equations can be reduced to the following:
Solving the four transverse field components in
terms of
and , we have
Microwave Circuits 36
where
Case 1.
Waves)
(Transverse Electromagnetic
Property: No cutoff frequency.
where
.
Property: Voltage can be uniquely defined.
From Maxwell’s equations
Microwave Circuits 37
Property:
satisfies Laplace’s equation.
To sum up, the transverse fields of an TEM wave
have the same properties of an electrostatic field
except that it is in two dimension.
Define wave impedance
Thus,
Property: The phase constant, wave impedance
and relationship of electric field and magnetic field
are the same as an plane wave.
Property: TEM waves can exist when two or more
conductors are present.
Case 2.
(Transverse Electric Waves)
Microwave Circuits 38
where
Property: is a function of the physical structure
of the waveguide and frequency. For a fixed ,
when
,
is real and when
,
imaginary, not a propagating wave. At
the wave stop to propagate, we call this
off frequency.
Solving
is
,
the cut-
from the Helmholtz wave equation,
This equation must satisfy the boundary conditions
of the specific guide geometry.
Microwave Circuits 39
Define TE wave impedance
Case 3.
Solving
(Transverse Magnetic Waves)
from the Helmholtz wave equation,
This equation must satisfy the boundary conditions
of the specific guide geometry.
Define TM wave impedance
Microwave Circuits 40
Rectangular Waveguides
y
b
μ ε
a
x
z
TE Modes
Task: solve
Assume
Substitute to the above equation, we have
Possible solution of the above equation is
Microwave Circuits 41
Thus
From boundary conditions
That is
where
is an arbitrary constant,
except
The propagating constant is
Cut-off frequency
and
Microwave Circuits 42
Guide wavelength
Phase velocity
Group velocity
The mode with the lowest cutoff frequency is called
the dominant mode or the fundamental mode,
which is the
mode.
Microwave Circuits 43
TM Modes
Task: solve
Assume
Substitute to the above equation, we have
Possible solution of the above equation is
Thus
From boundary conditions
That is
Microwave Circuits 44
where
is an arbitrary constant,
and
.
The propagating constant, cutoff frequency, guide
wavelength, phase velocity and group velocity are
the same as TE modes.
The mode with the lowest cutoff frequency is the
mode.
Microwave Circuits 45
Loss in a Waveguide
Dielectric Loss
Let
be the loss tangent of a dielectric. The
complex propagation constant can be
expressed as
Since
we have
where
. Thus
is the attenuation constant due to
dielectric loss.
Conductor Loss
Let power flow be
Then the power loss per unit length along the line
is
Microwave Circuits 46
The power lost in the conductor due to the surface
resistance
conductor).
Total Loss
TE10 modes
( is the conductance of the
Microwave Circuits 47
Microwave Circuits 48
Coaxial Line
TEM mode
Let be the inner radius of the coaxial line and
be the outer radius of the coaxial line.
Let be the potential function of the TEM mode,
then satisfies Laplace’s equation
. In polar
coordinate
and the boundary condition
Due to symmetry,
, we have
Use the boundary condition to solve
have
and
, we
Microwave Circuits 49
Microwave Circuits 50
Microstrip Line
Formulas
,
Or
where
Microwave Circuits 51
Loss
where
Operating
frequency limits
The lower-order strong coupled TM mode:
The lowest-order transverse microstrip resonance:
Frequency Dependence
where
Microwave Circuits 52
Strip Line
Formulas
where
.
Or
where
.
Loss
Microwave Circuits 53
where
Microwave Circuits 54
Coplanar Waveguide (CPW)
Benefit:
1.
Lower dispersion.
2.
Convenient connecting lump circuit elements.
Microwave Circuits 55
Microwave Network Analysis
1.
2.
3.
General Properties
Waveguide Discontinuity
Excitation of Waveguide
Microwave Circuits 56
Impedance and Equivalent Voltages and
Currents
Equivalent Voltages and Currents
Let
, we have
Also
To solve
and
, choose
, or
Example 4.1
Microwave Circuits 57
Choose
Concept of Impedance
1.
Intrinsic impedance:
2.
Wave impedance:
3.
Characteristic impedance:
Example 4.2
Microwave Circuits 58
Properties of One Port
Complex power
where
: real positive. The average power dissipated.
: real positive. The stored magnetic energy.
: real positive. The stored electric energy.
Define real transverse model fields
and
such that
and
then,
Thus, the input impedance
Properties:
1.
is related to
2.
.
equals zero if lossless.
is related to
, capacitive load.
.
, inductive load.
Microwave Circuits 59
Even and Odd Properties of
and
since
. Similarly,
.
Summary
1.
Even functions:
2.
Odd functions:
3.
Even functions:
Properties of N-Port
Define impedance matrix
where
.
.
Microwave Circuits 60
and admittance matrix
where
Reciprocal Networks
Conditions:
1.
No source in the network.
2.
No ferrite or plasma.
Lossless networks:
Example 4.3
Microwave Circuits 61
The Scattering Matrix
Define impedance matrix
where
Relationship with
Let
impedance of each port.
Thus
If lossless
be the matrix formed by the characteristic
Microwave Circuits 62
Therefore,
Since
Also
Therefore,
If reciprocal
Example 4.5
, or
Microwave Circuits 63
Shift in Reference Planes
If at port n, the reference plane is shifted out by a length of
voltage at the reference plane will be
where
We have
. Let
, the
Microwave Circuits 64
Generalized Scattering Parameters
Define the scattering parameters based on the amplitude of the incident
and reflected wave normalized to power.
Let
thus
The generalized scattering matrix is defined as
where
or
If lossless,
If reciprocal,
or
Microwave Circuits 65
The Transmission (ABCD) Matrix
Define a transmission matrix of a two port network as
or in matrix form
Relationship to impedance matrix
If reciprocal,
Cascading of ABCD matrix:
Microwave Circuits 66
Two-Port Circuits
Signal Flow Graphs
Primary Components:
Microwave Circuits 67
1.
Nodes: each port
wave to port
2.
.
has two nodes
represents
and
.
represents the incident
the reflected wave from port
.
Branches: A branch is a directed path between an a-node and a b-node,
representing signal flow from node a to node b. Every branch has an
associated S parameter of reflection coefficient.
Rules:
1.
Series rule
2.
Parallel rule
3.
Self-loop rule
4.
Splitting rule
Microwave Circuits 68
Example 4.7
Thru-Reflect Line (TRL) Network Analyzer Calibration
Purpose: to de-embed the effect of the connection between the signal lines of the
network analyzer and the actual circuit.
Microwave Circuits 69
Procedure:
1.
Measure the S parameter with direct connection of the two ports of the device
under test (DUT).
2.
Measure the S parameter with the two ports terminated by loads.
3.
Measure the S parameter with the two ports connected by a section of
transmission line.
Microwave Circuits 70
Microwave Circuits 71
,
Solving for
Correction: Eq. (4.77a)
:
Microwave Circuits 72
Correction: Eq. (4.77b)
Microwave Circuits 73
Discontinuities and Modal Analysis (4.6)
Let the modes existing in a waveguide be
Assuming two waveguides
and
are connected by an aperture
. Let the remaining areas at waveguide a and b be
Assume only the first mode incident from waveguide
and
located at
respectively.
, we have the total tangential
fields in
Likewise in waveguide
At the aperture
, the fields at both sides must be the same, that is
(438)
(439)
Microwave Circuits 74
And the electric fields at
and
must equal zero.
Integrate the above electric field equation with the mode patten of mode
waveguide
over surface
in
, we have
Due to the orthogonal properties between the modes in a waveguide, the above
equations lead to
(446)
where
Note that
is the normalization constant of mode
Rewriting the above Eq. (188) in matrix form, we have
(451)
where
in waveguide
.
Microwave Circuits 75
(452)
Likewise, integrate the magnetic field equation (Eq. 183) with the mode pattern of
mode
of waveguide
only over aperture
, we have
which leads to
(457)
where
Microwave Circuits 76
Rewriting the above Eq. (198) in matrix form, we have
(459)
where
(460)
From Eq. (193) and Eq. (200), we have
(461)
Thus
is solved. Using Eq. (193), we have
(463)
Thus
is solved.
Microwave Circuits 77
Modal Analysis of an H-Plane Step in Rectangular Waveguide
Assume
incident only thus only
modes reflect in guide 1 and transmit and
in guide 2. Then the modes in guide 1 can be specified as
and in guide 2
Microwave Circuits 78
Excitation of Waveguides (4.7)
Assume sources
and
exist in a waveguide between
and
. The
tangential fields outside this region can be expressed as
Assume
Let
, then
and
Let
, from reciprocity theorem, we have
and
are the fields generated by
.
and
, we have
, which are
,
,
Microwave Circuits 79
Likewise, let
and
, we have
Microwave Circuits 80
Probe-Fed Rectangular Waveguide
for
Microwave Circuits 81
Electromagnetic Theorems (1.3, 1.9)
Boundary conditions
Let the fields in media 1 denoted by subscript 1 and media2 subscript 2. At the
boundary of media 1 and 2, the electromagnetic fields satisfy the following
conditions.
where
points from media 1 to 2,
electric surface current,
the magnetic surface current,
the magnetic surface charge,
the
the electric surface
charge.
Uniqueness Theorem
In a region bounded by a close surface , if two sets of electromagnetic fields
satisfy the following conditions:
1.
The sources in the region are the same.
2.
The tangential electric fields or the tangential electric fields on the boundary
are the same
Then, these two sets of electromagnetic fields are the same everywhere in the
region.
Equivalence Principle
In a region bounded by a close surface
the exterior fields are replaced with
the same if
.
, let the field on
and
and
be
and
. If
, then the interior fields will be
are placed on surface
Microwave Circuits 82
Examples: PEC boundary, PMC boundary.
Image Theory
In front of a planar PEC, the fields are the same if the PEC is removed and the
images of the sources are placed at the other side. For an electric charge, the
image is the negative of the charge. For an magnetic charge, the image is the same
charge.
For the case of PMC, the image of an electric charge is the same, while the image of
an magnetic charge is the negative.
Reciprocity Theorem
For two sets electromagnetic fields generated by sources (
) in the same space bounded by surface
, we have
,
) and (
,
Microwave Circuits 83
Impedance Matching and Tuning (5)
Smith Charts
Let
(2.4)
be characteristic impedance of a transmission line. For a load
be the normalized impedance, then the reflection coefficient
Let
and
, let
becomes
, we have
These define the constant resistance and reactance curves.
Similarly
Thus we can conclude that
the constant conductance and susceptance curves
are the same forms as the constant resistance and reactance curves.
To sum up,
1.
Smith chart is a plot of the reflection coefficient
on the complex plane with
constant resistance and reactance curves overlapped. That is the real part of
Microwave Circuits 84
is plotted as the x coordinate, the imaginary part the y coordinate.
2.
The
3.
where is the propagation constant of the transmission line.
The constant resistance and reactance curves can used for admittance
4.
except that the Smith chart becomes a plot of
.
The admittance value can be read from Smith chart by rotating 180.
Example 2.4
at a distance
from the load is a clockwise rotation of angle
,
Microwave Circuits 85
Matching with Lumped Elements (L Networks)
jX
Z0
jX
jB
(a)
ZL
Z0
jB
ZL
(b)
Analytic Solutions
(a)
(b)
Smith Chart Solutions
1.
. Use (a)
a.
b.
c.
d.
2.
Convert to admittance plot.
Move along constant conductance curve until intercept with the
constant resistance curve equal to 1.
Convert back to impedance plot.
Find the required reactance.
. Use (b)
a.
b.
c.
Move along constant resistance curve until intercept with the constant
admittance curve equal to 1.
Convert to admittance plot.
Find the required susceptance.
Microwave Circuits 86
Example 5.1
Microwave Circuits 87
Microwave Circuits 88
Single-Stub Tuning (5.2)
Analytic Solutions
1. Shunt Stubs
Open stub:
Short stub:
Where
2. Series Stubs
Microwave Circuits 89
Open stub:
Short stub:
where
Smith Chart Solutions
Shunt (Series) Stubs
1.
Use admittance (impedance) plot.
2.
3.
Rotate clockwise along constant
curve until intercept with the constant
conductance (resistance) curve of value 1.
Compensate the remaining susceptance (reactance) by a suitable length of
open or short stub.
Microwave Circuits 90
Example 5.2
Microwave Circuits 91
Double-Stub Tuning (5.3)
Analytical Solution
Requirement:
Smith Chart Solutions
1.
Use admittance plot.
2.
Rotate the constant conductance circle of value 1 counterclockwise by a
distance d.
3.
Move
along the constant conductance curve until intercepting the rotated
circle in 2. The difference of the susceptance determines the length of the
Microwave Circuits 92
4.
stub 2.
Rotate the intercepting point back to constant conductance circle of value 1.
The susceptance value determine the length of stub 1.
Example 5.4
Microwave Circuits 93
Microwave Circuits 94
Transformers (5.4 – 5.9)
Quarter-Wave Transformer
Match a real load
impedance
to
by a section of transmission line with characteristic
and length .
The reflection coefficient becomes
for a given
, solve for
, we have
Microwave Circuits 95
Assume TEM mode,
The bandwidth becomes
Example 5.5
Microwave Circuits 96
Theory of Small Reflections
A multisection transformer consists of N equal-length sections of transmission lines.
Let
Assume that the reflection coefficients at each junction is very small, the total
reflection coefficient can be approximated by
Microwave Circuits 97
If
is symmetrical, that is,
,
If
is even, the previous equation becomes
If
is odd
,
, etc. Then,
Binomial Multisection Matching Transformers
Let
and the length of each section equals the quarter wavelength at the center
frequency. That is
.
We have
Thus
Property: flat near the center frequency
Proof:
For
Microwave Circuits 98
Thus, at
,
When frequency approaching zero, the electrical length of each section also
approaching zero. We have
The above result is not rigorous, since the limit only holds when multiple reflections
are considered.
Since
is known, every
can be computed. Also all the required
can
computed from
or
.
Bandwidth: Let
be the maximum value of reflection coefficient that can be
tolerated over the passband. Let
edge. That is
Thus
To sum up,
. We have
be the corresponding
value at the lower
Microwave Circuits 99
1.
From
,
and
2.
From
3.
If the bandwidth is not satisfied, increase
4.
Find
and the given
, find
by using Eq. 5.49.
find the bandwidth by using Eq. 5.55.
and repeat 1 and 2.
by Table 5.1 or Eq. 5.53, or the relationship
Microwave Circuits 100
Example 5.6
Chebyshev Multisection Matching Transformer
Chebyshev Polynomials
Characteristics:
1.
.
2.
3.
For
,
increases faster with
4.
Suppose the passband is
. Let
as
increases.
Microwave Circuits 101
Since
in the passband
,
in this range
and
.
Similar to previous section
Combine the previous two equations, we have
To sum up
1.
From the given
2.
Determine the bandwidth by using Eq. 5.64.
3.
If the bandwidth is not satisfied, increase
4.
From Eq. 5.62, decide
determined from
Example 5.7
,
,
, and
, find
by using Eq. 5.63..
and repeat 1 and 2.
. By Eq. 5.61, all the
or by looking up Table 5.2.
can be found.
can be
Microwave Circuits 102
Tapered Lines
Let the characteristic impedance of a section of transmission line with length
a function of , that is
. By approximating
using small reflection formula, we have
In the limit as
Exponential Taper
Let
Then
, we have the exact differential
be
with stair case functions,
Microwave Circuits 103
Note: peaks in
than
decrease with increasing length. The length should be greater
to minimize the mismatch at low frequencies.
Triangular Taper
Let
Microwave Circuits 104
Note: for
, the peaks is larger than the corresponding peaks of the
exponential case. The first null occurs at
Klopfenstein Taper
Reflection coefficient is minimum over the passband, or the length of the matching
section is shortest for a maximum reflection coefficient specified over the passband.
Let
where
Microwave Circuits 105
and
is the modified Bessel function. Then,
where
Define the passband as when
oscillates between
.
is equal ripple in passband. Then
for
Example 5-8
The Klopfenstein taper is seen
to give the desired response of
for
, which
is lower than either the
triangular or exponential taper
responses.
Microwave Circuits 106
The Bode-Fano Criterion
1.
Bode-Fano criterion
gives for certain
canonical types of load
impedances a
theoretical limit on the
minimum reflection
coefficient magnitude
that can be obtained
with an arbitrary
matching network.
2.
For a given load, a
broader bandwidth can
be achieved only at the
expense of a higher
reflection coefficient in
the passband.
cannot be zero unless
.
3.
The passband reflection coefficient
4.
As R and/or C increases, the quality of match must decrease. Thus, higher-Q
Microwave Circuits 107
circuits are intrinsically harder to match than are lower Q circuits.
Microwave Circuits 108
Microwave Resonators (6)
What is resonance?
1.
The natural modes of a system.
a.
Metallic cavity.
b.
A long beam.
c.
Musical instrument.
d.
LC circuit.
2.
Self-sustained if lossless.
3.
Energy grows to infinity if fed by a source which has a spectrum containing
the resonant frequency if lossless.
Quality Factor
Series and Parallel Resonant Circuits(6.1)
Series Resonant Circuit
Power Loss:
Average Stored Magnetic Energy:
Microwave Circuits 109
Average Stored Electric Energy:
Resonant Frequency:
Quality Factor:
At
,
Near resonance, let
Let complex resonant frequency be
Treat the circuit as lossless, and use the complex resonant frequency to account for
the loss
Microwave Circuits 110
Half-power bandwidth
Parallel Resonant Circuit
Similarly,
Microwave Circuits 111
Loaded and Unloaded Q
Define the Q of an external load
as
The loaded Q can be expressed as
Transmission Line Resonators (6.2)
Short-Circuited
Line
Assume small loss,
Assume a TEM line,
Since
and
Thus
at
,
, then
Microwave Circuits 112
Similar to a series RLC circuit,
Short-Circuited
Line
Similar to a parallel RLC circuit,
Open-Circuited
Line
Similar to a parallel RLC circuit,
Microwave Circuits 113
Rectangular Waveguide Cavities (6.3)
Cufoff wavenumber
Resonant Frequcney of mode
or
Circular Waveguide Cavities (6.4)
Dielectric Resonators (6.5)
Fabry-Perot Resonators (6.6)
Excitation of Resonators (6.7)
Critical Coupling  matching, maximum power
Microwave Circuits 114
Define coefficient coupling,
: undercoupled to the feedline
: critically coupled to the feedline
: overcoupled to the feedline
A Gap-Coupled Microstrip Resonator
where
.
Condition of resonance:
which is a function of
.
Note: assume ideal transmission line such that
Characteristic near resonance
By Taylor’s expansion near resonant frequency
Microwave Circuits 115
First,
So,
Compare to series RLC circuit
Using complex frequency
where
is approximated by the
,
to include the effect of loss, we have
of the open-circuit
the gap capacitance is very small. For critical coupling
Example 6.6
An Aperture-Coupled Cavity
transmission line since
Microwave Circuits 116
where
, similarly to previous section,
For a rectangular waveguide,
Thus
Use complex frequency,
This is similar to a parallel RLC circuit with
At critical coupling,
Microwave Circuits 117
Cavity Perturbations (6.8)
Material Perturbations
Let
be the solution in a metallic cavity with material
the solution in the same cavity with material
. Let
. We have
Then
By divergence theorem
To sum up, as
or
increases, the resonant frequency decreases.
be
Microwave Circuits 118
Example 6.7
Shape Perturbations
Let
be the solution in a metallic
cavity with material
. Let
be
the solution in the same cavity with
shape perturbation
. We have
Then
By divergence theorem
Since
Thus
Example 6.8
Microwave Circuits 119
Homework 6.8, 22