Microwave Engineering - myweb.ntut.edu.twjuiching/microwave.pdf · Maxwell equation 1873. Theory...

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


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 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


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


V

an

S

an

C

Divergence Operator

Physical meaning

Divergence Theorem

Curl Operator

Physical meaning

Stoke’s theorem:


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


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


, intrinsic impedance or wave

impedance

Vector Potential

: vector potential

Flow of Electromagnetic Power and PoyntingVector

Since

we have

or


: 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


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


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.


Image Theory: an application of UniquenessTheorem.


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


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


Terminated Lossless Transmission Line

Assume incident wave , reflected wave then

At ,

Define return loss:

Special case:1. (short): .2. (open): .3. Half wavelength line:


4. Quarter wavelength line:

Two-transmission Line Junction

At ,

: transmission coefficient.

Define Insertion loss: Conservation of energy

Incident power:

Reflected power:

Transmitted power:


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


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


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.


Example 2.2 Basic Smith Chart Operations


Example 2.3 Smith Chart Operations UsingAdmittances


Example 2.4 Impedance Measurement with aSlotted Line


Example 2.5 Frequency Repsonse of a Quarter-wave Transformer


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 .


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


amount of .

Ex. 2.8


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


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:


Transmission Lines and Waveguides

General Solutions for TEM, TE, and TM Waves

Rectangular Waveguides

Coaxial Lines

Microstrip

Strip Lines

Coplanar Waveguides


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


With an dependence in direction, the aboveequations can be reduced to the following:

Solving the four transverse field components interms of and , we have


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


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 ,


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)


Solving from the Helmholtz wave equation,

This equation must satisfy the boundary conditionsof the specific guide geometry.

Define TM wave impedance

Rectangular Waveguides


TE Modes

Task: solve

Assume

Substitute to the above equation, we have

Possible solution of the above equation is

Thus


From boundary conditions

That is

where is an arbitrary constant, and

except

The propagating constant is

Cut-off frequency

Guide wavelength

Phase velocity


Group velocity

The mode with the lowest cutoff frequency iscalled the dominant mode or the fundamentalmode, which is the mode.


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


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.


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


resistance ( is the conductance of the

conductor).

Total Loss

TE10 modes


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


Microstrip Line

Formulas

,

Or

where


Loss

where

Operatingfrequency limits

The lower-order strong coupled TM mode:

The lowest-order transverse microstrip resonance:

Frequency Dependence

where


Strip Line

Formulas

where

.

Or

where

.

Loss


where


Coplanar Waveguide (CPW)

Benefit:1. Lower dispersion.2. Convenient connecting lump circuit elements.


Microwave Network Analysis

1. General Properties2. Waveguide Discontinuity3. Excitation of Waveguide


Impedance and Equivalent Voltages andCurrents

Equivalent Voltages and Currents

Let , we have

Also

To solve and , choose

, or

Example 4.1


Choose

Concept of Impedance

1. Intrinsic impedance:

2. Wave impedance:

3. Characteristic impedance:

Example 4.2


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


since . Similarly,

.

Summary

1. Even functions: .

2. Odd functions:

3. Even functions: .

Properties of N-Port

Define impedance matrix

where

and admittance matrix


where

Reciprocal NetworksConditions:1. No source in the network.2. No ferrite or plasma.

Lossless networks:

Example 4.3


The Scattering MatrixDefine impedance matrix

where

Relationship with

Let be the matrix formed by the

characteristic impedance of each port.

Thus

Like wise,


If lossless

Therefore,

Since

Also

Therefore, , or

If reciprocal

Example 4.5


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


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,


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


Signal Flow Graphs

Primary Components:

1. Nodes: each port has two nodes and . represents the incident


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


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.


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.


,

Solving for :

Correction: Eq. (4.77a)


Correction: Eq. (4.77b)


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)


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


(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


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.


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


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


Probe-Fed Rectangular Waveguide

for


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.


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


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.


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


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


Single-Stub Tuning (5.2)

Analytic Solutions1. Shunt Stubs

Open stub:

Short stub:

Where

2. Series Stubs


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.


Example 5.2


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.


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


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


Assume TEM mode,

The bandwidth becomes

Example 5.5


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


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


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,


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


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


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


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


Note: peaks in decrease with increasing length. The length should be greater than

to minimize the mismatch at low frequencies.

Triangular Taper

Let


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


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.


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 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:


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


Parallel Resonant Circuit

Similarly,

Loaded and Unloaded Q


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,


Short-Circuited Line

Similar to a parallel RLC circuit,

Open-Circuited Line

Similar to a parallel RLC circuit,


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


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,


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


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,


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.


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


Example 6.8

Microwave Engineering - myweb.ntut.edu.twjuiching/microwave.pdf · Maxwell equation 1873. Theory...

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Transcript of Microwave Engineering - myweb.ntut.edu.twjuiching/microwave.pdf · Maxwell equation 1873. Theory...