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Microwave Circuits 1
Microwave Engineering1. Microwave: 300MHz ~ 300 GHz, 1 m ~ 1mm.
a. Not only apply in this frequency range. Thereal issue is wavelength. Historically, as earlyas WWII, this is the first frequency range weneed to consider the wave effect.
b. Why microwave engineering? We all knowthat the ideal of capacitor, inductor andresistance are first defined in DC.i. Circuit theory only apply in lower
frequency.ii. Balance between two extreme: circuit and
fullwave.2. Today’s high tech is developed long time ago.
a. Maxwell equation 1873. Theory beforeExperiment.
b. Waveguide, Radar, Passive circuit, before WWII.
3. Transmission Line Theorya. 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.iii. Definition of Voltage and current is
Microwave Circuits 2
ambiguous in real waveguide except forTEM wave. Give examples.
b. Derive transmission line equations.i. Purpose: to show that with only the
assumption of varying voltage andcurrent, we reach wave solution.
ii. Show the characteristics of losslesstransmission lines.
iii. Show the characteristics of low losstransmission lines (p. 170)
c. 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 waveexists.
iii. Importance of cut off frequency.
Microwave Circuits 3
Microwave Engineering
! Grading policy.
" 2 midterm exams, 30% each.
" Final exam, 40%.
! Office hour: 3:00 ~ 6:00 pm, Monday and 3:00 ~
6:00 pm, Thursday.
! 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
V
an
S
an
C
Divergence Operator
Physical meaning
Divergence Theorem
Curl Operator
Physical meaning
Stoke’s theorem:
Microwave Circuits 6
Time-Harmonic Fields
Time-harmonic:
: a real function in both space and time.: a real function in space.
: a complex function in space. Aphaser.
Thus, all derivative of time becomes.
For a partial deferential equation, all derivative oftime can be replace with , and all timedependence of can be removed and becomes apartial 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 waveFrom wave equation
where , free space wave number or
propagation constant.In Cartesian coordinates, considering the xcomponent,
.
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 PoyntingVector
Since
we have
or
Microwave Circuits 9
: 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
Microwave Circuits 11
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
3. Unbounded
Microwave Circuits 12
Uniqueness Theorem
Let be two sets of solutions of the sameexcitation, then
The surface integral vanishes if1. , tangential electric field equals specified2. , tangential magnetic field specified.then
That is, in a enclosed volume, if the source in thevolume and the tangential fields on the boundaryare the same, the fields are the same everywhereinside the volume.
Microwave Circuits 13
Image Theory: an application of UniquenessTheorem.
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 , reflected wave then
At ,
Define return loss:
Special case:1. (short): .2. (open): .3. Half wavelength line:
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 loadimpedance .
Define normalized impedance , where
is the characteristic impedance of thetransmission line. Then,
Equating the real and imaginary parts, we have
Microwave Circuits 19
Rearranging, we have
SummaryConstant 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 .
Since Y a rotation of angle
Microwave Circuits 20
clockwise.
Calculation of VSWR:
where . Therefore, is
the resistance value at the intersection point of thepositive and constant circle.
Admittance Smith Chart
, where . Therefore,
Admittance Smith Chart is a rotation of 180 degreeof impedance Smith Chart.
Microwave Circuits 21
Example 2.2 Basic Smith Chart Operations
Microwave Circuits 22
Example 2.3 Smith Chart Operations UsingAdmittances
Microwave Circuits 23
Example 2.4 Impedance Measurement with aSlotted Line
Microwave Circuits 24
Microwave Circuits 25
Example 2.5 Frequency Repsonse of a Quarter-wave 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 deliveredto the load under fixed . In reality, our concern ishow 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. 2.72. Wheeler Incremental Inductance Rule
or , where and are
changes due to recess of all conductor walls by an
Microwave Circuits 28
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, the phase constant.
is the loss tangent.
Low-Loss Dielectrics: , or
and
Microwave Circuits 30
Good Conductor:
and
Skin depth or depth of penetration:
Meaning: plane wave decay be a factor of . Atmicrowave 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
Microwave Circuits 32
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 TMWavesAssuming a wave propagating in the direction,the electric and magnetic fields can be expressedas
where and are the transverse electric and magnetic field components.
In a source-free region, Maxwell’s equations canbe written as
Microwave Circuits 35
With an dependence in direction, the aboveequations can be reduced to the following:
Solving the four transverse field components interms of and , we have
Microwave Circuits 36
where
Case 1. (Transverse Electromagnetic
Waves)
Property: No cutoff frequency.
where .
Property: Voltage can be uniquely defined.
From Maxwell’s equations
Property: satisfies Laplace’s equation.
To sum up, the transverse fields of an TEM wavehave the same properties of an electrostatic fieldexcept that it is in two dimension.
Define wave impedance
Microwave Circuits 37
Thus,
Property: The phase constant, wave impedanceand relationship of electric field and magnetic fieldare the same as an plane wave.
Property: TEM waves can exist when two or moreconductors are present.
Case 2. (Transverse Electric Waves)
where
Property: is a function of the physical structure
of the waveguide and frequency. For a fixed ,
Microwave Circuits 38
when , is real and when , is
imaginary, not a propagating wave. At ,
the wave stop to propagate, we call this the cut-
off frequency.
Solving from the Helmholtz wave equation,
This equation must satisfy the boundary conditionsof the specific guide geometry.
Define TE wave impedance
Case 3. (Transverse Magnetic Waves)
Microwave Circuits 39
Solving from the Helmholtz wave equation,
This equation must satisfy the boundary conditionsof the specific guide geometry.
Define TM wave impedance
Rectangular Waveguides
Microwave Circuits 40
TE Modes
Task: solve
Assume
Substitute to the above equation, we have
Possible solution of the above equation is
Thus
Microwave Circuits 41
From boundary conditions
That is
where is an arbitrary constant, and
except
The propagating constant is
Cut-off frequency
Guide wavelength
Phase velocity
Microwave Circuits 42
Group velocity
The mode with the lowest cutoff frequency iscalled the dominant mode or the fundamentalmode, 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, guidewavelength, phase velocity and group velocity arethe 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. Thecomplex propagation constant can beexpressed 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 lineis
The power lost in the conductor due to the surface
Microwave Circuits 46
resistance ( is the conductance of the
conductor).
Total Loss
TE10 modes
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 polarcoordinate
and the boundary condition
Due to symmetry, , we have
Use the boundary condition to solve and , wehave
Microwave Circuits 49
Microwave Circuits 50
Microstrip Line
Formulas
,
Or
where
Microwave Circuits 51
Loss
where
Operatingfrequency 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. General Properties2. Waveguide Discontinuity3. Excitation of Waveguide
Microwave Circuits 56
Impedance and Equivalent Voltages andCurrents
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 . equals zero if lossless.
2. is related to . , inductive load.
, capacitive load.
Even and Odd Properties of and
Microwave Circuits 59
since . Similarly,
.
Summary
1. Even functions: .
2. Odd functions:
3. Even functions: .
Properties of N-Port
Define impedance matrix
where
and admittance matrix
Microwave Circuits 60
where
Reciprocal NetworksConditions:1. No source in the network.2. No ferrite or plasma.
Lossless networks:
Example 4.3
Microwave Circuits 61
The Scattering MatrixDefine impedance matrix
where
Relationship with
Let be the matrix formed by the
characteristic impedance of each port.
Thus
Like wise,
Microwave Circuits 62
If lossless
Therefore,
Since
Also
Therefore, , or
If reciprocal
Example 4.5
Microwave Circuits 63
Shift in Reference Planes
If at port n, the reference plane is shifted out by a length of , the
voltage at the reference plane will be
where . Let
We have
Microwave Circuits 64
Generalized Scattering Parameters
Define the scattering parameters based on the amplitude of the incidentand reflected wave normalized to power.
Let
thus
The generalized scattering matrix is defined as
where
or
If lossless, or
If reciprocal,
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:
Two-Port Circuits
Microwave Circuits 66
Signal Flow Graphs
Primary Components:
1. Nodes: each port has two nodes and . represents the incident
Microwave Circuits 67
wave to port . represents the reflected wave from port .
2. 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 anassociated S parameter of reflection coefficient.
Rules:1. Series rule2. Parallel rule3. Self-loop rule4. 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 thenetwork 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 located at
. Let the remaining areas at waveguide a and b be and respectively.
Assume only the first mode incident from waveguide , we have the total tangential
fields in
Likewise in waveguide
At the aperture , the fields at both sides must be the same, that is
(441)
(442)
Microwave Circuits 74
And the electric fields at and must equal zero.
Integrate the above electric field equation with the mode patten of mode in
waveguide over surface , we have
Due to the orthogonal properties between the modes in a waveguide, the aboveequations lead to
(449)
where
Note that is the normalization constant of mode in waveguide .
Rewriting the above Eq. (188) in matrix form, we have
(454)
where
Microwave Circuits 75
(455)
Likewise, integrate the magnetic field equation (Eq. 183) with the mode pattern of
mode of waveguide only over aperture , we have
which leads to
(460)
where
Microwave Circuits 76
Rewriting the above Eq. (198) in matrix form, we have
(462)
where
(463)
From Eq. (193) and Eq. (200), we have
(464)
Thus is solved. Using Eq. (193), we have
(466)
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 , from reciprocity theorem, we have
Let , then and are the fields generated by , which are , ,
and .
Let and , we have
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 theboundary of media 1 and 2, the electromagnetic fields satisfy the followingconditions.
where points from media 1 to 2, the magnetic surface current, the
electric surface current, the magnetic surface charge, 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 sameThen, these two sets of electromagnetic fields are the same everywhere in theregion.
Equivalence Principle
In a region bounded by a close surface , let the field on be and . If
the exterior fields are replaced with and , then the interior fields will be the
same if and are placed on surface .
Examples: PEC boundary, PMC boundary.
Microwave Circuits 82
Image TheoryIn front of a planar PEC, the fields are the same if the PEC is removed and theimages of the sources are placed at the other side. For an electric charge, the imageis the negative of the charge. For an magnetic charge, the image is the samecharge.
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 ( , ) and ( ,
) in the same space bounded by surface , we have
Microwave Circuits 83
Impedance Matching and Tuning (5)
Smith Charts (2.4)
Let be characteristic impedance of a transmission line. For a load , let
be the normalized impedance, then the reflection coefficient becomes
Let and , we have
These define the constant resistance and reactance curves.
Similarly
Thus we can conclude that the constant conductance and susceptance curvesare 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
is plotted as the x coordinate, the imaginary part the y coordinate.
Microwave Circuits 84
2. The at a distance from the load is a clockwise rotation of angle ,
where is the propagation constant of the transmission line.
3. The constant resistance and reactance curves can used for admittance
except that the Smith chart becomes a plot of .
4. The admittance value can be read from Smith chart by rotating 180E.
Example 2.4
Microwave Circuits 85
jX
ZLjBZ0
(a)
Z0 jB
jX
ZL
(b)
Matching with Lumped Elements (L Networks)
Analytic Solutions
(a)
(b)
Smith Chart Solutions
1. . Use (a)
a. Convert to admittance plot.b. Move along constant conductance curve until intercept with the
constant resistance curve equal to 1.c. Convert back to impedance plot.d. Find the required reactance.
2. . Use (b)
a. Move along constant resistance curve until intercept with the constantadmittance curve equal to 1.
b. Convert to admittance plot.c. Find the required susceptance.
Example 5.1
Microwave Circuits 86
Microwave Circuits 87
Microwave Circuits 88
Single-Stub Tuning (5.2)
Analytic Solutions1. Shunt Stubs
Open stub:
Short stub:
Where
2. Series Stubs
Microwave Circuits 89
Open stub:
Short stub:
where
Smith Chart Solutions
Shunt (Series) Stubs1. Use admittance (impedance) plot.
2. Rotate clockwise along constant curve until intercept with the constant
conductance (resistance) curve of value 1. 3. 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 Solutions1. 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 thestub 2.
Microwave Circuits 92
4. 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 to by a section of transmission line with characteristic
impedance and length R.
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 totalreflection coefficient can be approximated by
Microwave Circuits 97
If is symmetrical, that is, , , , etc. Then,
If is even, the previous equation becomes
If is odd
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 alsoapproaching zero. We have
The above result is not rigorous, since the limit only holds when multiple reflectionsare 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 be the corresponding value at the lower
edge. That is . We have
Thus
To sum up,
Microwave Circuits 99
1. From , and , find by using Eq. 5.49.
2. From and the given find the bandwidth by using Eq. 5.55.
3. If the bandwidth is not satisfied, increase and repeat 1 and 2.
4. Find 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 as increases.
4.
Suppose the passband is . Let
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 , , , and , find by using Eq. 5.63..
2. Determine the bandwidth by using Eq. 5.64.
3. If the bandwidth is not satisfied, increase and repeat 1 and 2.
4. From Eq. 5.62, decide . By Eq. 5.61, all the can be found. can be
determined from or by looking up Table 5.2.
Example 5.7
Microwave Circuits 102
Tapered Lines
Let the characteristic impedance of a section of transmission line with length be
a function of , that is . By approximating with stair case functions,
using small reflection formula, we have
In the limit as , we have the exact differential
Exponential Taper
Let
Then
Microwave Circuits 103
Note: peaks in decrease with increasing length. The length should be greater than
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 exponentialcase. The first null occurs at
Klopfenstein Taper
Reflection coefficient is minimum over the passband, or the length of the matchingsection 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 . is equal ripple in passband. Then
oscillates between for
Example 5-8
The Klopfenstein taper is seento give the desired response of
for , which
is lower than either thetriangular or exponential taperresponses.
Microwave Circuits 106
The Bode-Fano Criterion
1. Bode-Fano criteriongives for certaincanonical types of loadimpedances atheoretical limit on theminimum reflectioncoefficient magnitudethat can be obtainedwith an arbitrarymatching network.
2. For a given load, abroader bandwidth canbe achieved only at theexpense of a higherreflection coefficient inthe passband.
3. The passband reflection coefficient cannot be zero unless .
4. As R and/or C increases, the quality of match must decrease. Thus, higher-Qcircuits are intrinsically harder to match than are lower Q circuits.
Microwave Circuits 107
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 forthe loss
Half-power bandwidth
Microwave Circuits 110
Parallel Resonant Circuit
Similarly,
Loaded and Unloaded Q
Microwave Circuits 111
Define the Q of an external load as , then
The loaded Q can be expressed as
Transmission Line Resonators (6.2)
Short-Circuited Line
Assume small loss, Assume a TEM line,
Since at ,
and
Thus
Similar to a series RLC circuit,
Microwave Circuits 112
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
Define coefficient coupling,
: undercoupled to the feedline
: critically coupled to the feedline
: overcoupled to the feedline
A Gap-Coupled Microstrip Resonator
Microwave Circuits 114
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
First,
Microwave Circuits 115
So,
Compare to series RLC circuit ,
Using complex frequency to include the effect of loss, we have
where is approximated by the of the open-circuit transmission line since
the gap capacitance is very small. For critical coupling
Example 6.6
An Aperture-Coupled Cavity
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 . Let bethe solution in the same cavity with material
. We have
Then
By divergence theorem
To sum up, as or increases, the resonant frequency decreases.
Microwave Circuits 118
Example 6.7
Shape Perturbations
Let be the solution in a metallic
cavity with material . Let bethe solution in the same cavity with
shape perturbation . We have
Then
By divergence theorem
Since
Thus
Microwave Circuits 119
Example 6.8