Microwave Engineering

Microwave Circuits 1 Microwave Engineering 1. 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. 2. Today s high tech is developed long time ago. a. Maxwell equation 1873. Theory before Experiment. b. Waveguide, Radar, Passive circuit, before WWII. 3. 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. iii. Definition of Voltage and current is

Microwave Circuits 2 ambiguous in real waveguide except for TEM wave. Give examples. b. 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) 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 wave exists. 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, 4 th 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 Physical meaning a n V Divergence Theorem Curl Operator a n Physical meaning Stoke s theorem: S C

Microwave Circuits 6 Time-Harmonic Fields Time-harmonic: phaser. : a real function in both space and time. : a real function in space. : a complex function in space. A 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 impedance Vector Potential, intrinsic impedance or wave : vector potential Flow of Electromagnetic Power and Poynting Vector 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

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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 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 constant. We have the solutions :complex propagation : 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, reflected wave 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 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. Since Y a rotation of angle

Microwave Circuits 20 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.

Example 2.2 Basic Smith Chart Operations Microwave Circuits 21

Example 2.3 Smith Chart Operations Using Admittances Microwave Circuits 22

Microwave Circuits 23 Example 2.4 Impedance Measurement with a Slotted Line

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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 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. 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:

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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 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

Microwave Circuits 37 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) 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 cutoff frequency. Solving from the Helmholtz wave equation, This equation must satisfy the boundary conditions of 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 conditions of 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 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 The power lost in the conductor due to the surface

Microwave Circuits 46 resistance ( is the conductance of the conductor). Total Loss TE 10 modes

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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 and, we have

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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

where Microwave Circuits 53

Microwave Circuits 54 Coplanar Waveguide (CPW) Benefit: 1. Lower dispersion. 2. Convenient connecting lump circuit elements.

Microwave Circuits 55 Microwave Network Analysis 1. General Properties 2. Waveguide Discontinuity 3. 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. 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 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 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 voltage at the reference plane will be, the where. Let We have

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, 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 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.

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Microwave Circuits 71, Solving for : Correction: Eq. (4.77a)

Correction: Eq. (4.77b) Microwave Circuits 72

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 fields in, we have the total tangential 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 above equations lead to (449) where Note that is the normalization constant of mode in waveguide. Rewriting the above Eq. (188) in matrix form, we have where (454)

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 where (462) (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

Likewise, let and, we have Microwave Circuits 79

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, 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 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, 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 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 (, ) 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 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 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 Matching with Lumped Elements (L Networks) jx jx Z 0 jb Z L Z 0 jb Z L (a) (b) 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) Example 5.1 a. Move along constant resistance curve until intercept with the constant admittance curve equal to 1. b. Convert to admittance plot. c. Find the required susceptance.

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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. 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.

Example 5.2 Microwave Circuits 90

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 stub 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

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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 total reflection 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 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 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 a function of, that is. By approximating with stair case functions, using small reflection formula, we have be 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 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. is equal ripple in passband. Then oscillates between 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. 3. The passband reflection coefficient cannot be zero unless. 4. As R and/or C increases, the quality of match must decrease. Thus, higher-q circuits are intrinsically harder to match than are lower Q circuits.

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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 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 be the 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 be the solution in the same cavity with shape perturbation. We have Then By divergence theorem Since Thus

Example 6.8 Microwave Circuits 119