Transmission Lines

1
Chapter 3

• Transmission Lines

2
Contents

• Features of Transmission Lines
• Low Frequency Characters of Microstrip Line
• High Frequency Characters of Microstrip Line
• Discontinuities of Microstrip Line

3
Features of Transmission Lines
4
Microwave Integrated Circuit (MIC)

• The current trend of circuit design is toward
  miniaturization and integration.
• An MIC consists of an assembly that combines
  different circuit functions that are connected by
  transmission lines.
• The advantages of MIC compare to traditional
  circuit using printed circuit
• Higher reliability
• Reproducibility
• Better performance
• Higher Integrated
• Smaller size
• Two classes of MIC
• HMIC
• MMIC

• Planar configuration
• Easy fabrication
• Lower cost
• Lighter weight

5

• Hybrid Microwave Integrated Circuit (HMIC)

6
Photograph of one of the 25,344 hybrid integrated
T/R modules used in Raytheons Ground Based Radar
system. This X-band module contains phase
shifters, amplifiers, switches, couplers, a
ferrite circulator, and associated control and
bias circuitry.
7

• Monolithic Microwave Integrated Circuit (MMIC)

8
Photograph of a monolithic integrated X-band
power amplifier. This circuit uses eight
heterojunction bipolar transistors with power
dividers/combiners at the input and output to
produce 5 watts.
9

• Material selection is an important consideration
  for any type of MIC characteristics such as
  electrical conductivity, dielectric constant,
  loss tangent, thermal transfer, mechanical
  strength, and manufacturing compatability must be
  evaluated.
• Features of HMICs
• Alumina, quartz, and Teflon fiber are commonly
  used for substrates.
• During HMICs testing, tuning or trimming for each
  circuit is allowed to adjust components values.
• Features of MMICs
• The substrate of an MMIC must be a semiconductor
  material to accommodate the fabrication of active
  devices. Hence GaAs is the most common substrate.
  Besides, Si, sapphire, and InP are also used.
• All passive and active components are grown or
  implanted in the substrate. A single wafer can
  contain a large number of circuits.
• Circuit trimming after fabrication will be
  difficult, even impossible.

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• Conventional coaxial lines and waveguides are
  remain useful in
• High power transmission (e.g. KWMW transmitters)
• High Q component needed (e.g. low loss filter)
• Some millimetricwavelength systems (e.g. MW
  automotive radar)
• Very low loss transmission systems
• Precision instrumentation equipment
• Planar technology are already tending to overcome
  problems in areas (2) and (3), but not (1) or (4).

12
Transmission Line and Waveguide Structures
13
Transmission Line and Waveguide Comparisons
14
Planar Transmission Line Structures
15
Modifications of Planar Transmission Line
Structures
16

• Image Line

• Behavior likes a dielectric slab waveguide (thick
  strip) for use at operation frequency into
  hundreds GHz.
• Several thousand unloaded Q-factor. But fop? ?
  Q ?.
• Poor compatibility with active devices, mutual
  coupling, and radiation from discontinuities and
  bends.

• Microstrip

• The most popular MIC TL with a very simple
  geometric planar structure.
• Advantage Zero cutoff frequency , light weight,
  small size, low cost, easy fabrication and
  integration, low dispersion , and broadband
  operation (frequency range from a few GHZ, or
  even lower, up to at least many tens of GHz).
• At millimetre-wave range, problems such as loss,
  higher-order modes, and fabrication tolerances
  become exceedingly difficult to meet using HMICs.

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• Finline (E-plane circuit)

• Advantage
• 1) Low loss (typically a factor about three
  better than microstrip.
• 2) Simpler fabrication in comparison with
  inverted and trapped-inverted microstrip.
• 3) Operation frequency up to 100GHz.
• Disadvantage in biasing problem.
• Application in compatibility with solid-state
  device is fairly good, especially in the case of
  beam-lead devices, 10 bandwidth of band pass
  filters, quadrature hybrids, waveguide
  transitions, and balanced mixer circuits.

• Inverted Microstrip (IM)

• Advantages in comparison with microstrip
• 1) Wider line width for the same Z0, and this
  both reduces conductor dissipation and relaxes
  fabrication tolerances.

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• 2) Structure utilizing air between the strip and
  ground plane gives higher Q, wavelength,
  operation frequency, and avoids interference.

• Slotline

• Guide mode of architecture makes it particularly
  suitable for applications where substrate is
  ferrite (components such as circulators and
  isolators).
• Disadvantages
• 1) Z0 below 60 are difficult to realize.
• 2) Q factor is significantly lower than other
  structures considered here.
• 3) Circuit structures often involve difficult
  registration problems ( especially with
  metallization on the opposite side to the slot).

• Trapped Inverted Microstrip (TIM)

• Advantages is similar to that of IM moreover, a
  slot or channel-shaped ground plane provides
  inherent suppression of some higher-order modes
• Manufacturing difficulties are particularly
  significant with HMICs.

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• Coplanar Waveguide (CPW)

• Advantages in comparison with microstrip
• 1) Easier grounding of surface-mounted ( or BGA
  mounted) component.
• 2) Lower fabrication costs.
• 3) Reduced dispersion and radiation losses.
• 4) Photolithographically defined structures
  with relatively low
• dependence on substrate thickness.

• The major problem is non-unique Z0 because
  infinite range of ratio between centre strip
  width and gap width (In micrpstrip, Z0 is unique
  decided by strip width, substrate height, and
  substrate permittivity).

• Coplanar Strip (CPS) and Differential Line

• CPS one of the conductors is ground
  Differential line neither of the conductors is
  grounded.
• Advantage of differential line
• 1) It is suitable for RFICs and high-speed
  digital ICs (but not for HMIC due to radiation
  losses and most passive components are
  single-ended).
• 2) This line is popular for use in long bus
  lines and clock distribution nets on chip as the
  signal return path.

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• The differential line has a virtual ground
  itself, which means that a real metallic ground
  is not necessary.

• Stripline

• Completely filled microstrip, i.e. a symmetrical
  structure results in TEM transmission
• Advantages
• 1) lower loss.
• 2) Fairly high Q-factor.
• 3) Waveguide modes can easily to exited at
  higher frequencies.
• Disadvantages
• 1) Insufficient space for the incorporation of
  semiconductor devices.
• 2) Mode suppression gives rise to design
  problem.
• 3) Not compatible with shunt-mounted devices.

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• Summary of TL Properties

• Z0 and Q-factor are criterion for circuit
  applications.

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Substrate Choice for HMIC

• Many factors, mechanical, thermal, electronics,
  and economic, leading to the correct choice of
  substrate deeply influence MIC design.
• The kinds of questions include
• 1) Cost
• 2) Thin-film or thick-film technology
• 3) Frequency range
• 4) Surface roughness (this will influence
  conductor losses and metal-film adhesion)
• 5) Mechanical strength, flexibility, and thermal
  conductivity
• 6) Sufficient surface area

24
Commonly used substrate materials

• Organic PCBs (Printed Circuit Boards)

• FR4
• 1) Low cost, rigid structure, and multi-layer
  capability.
• 2) Applications for operation frequency below a
  few GHz. fop? ? Loss ?

• RT/Duroid
• 1) Low loss and good for RF applications.
• 2) Board has a wide selected range for
  permittivity. e.g. RT/Duroid 5870 with ?r 2.33,
  RT/Duroid 5880 with ?r 2.2, and RT/Duroid 6010
  with ?r 10.2.
• 3) Board is soft leading to less precise
  dimensional control.

• Softboard

1) Plastic substrate with good flexibility. 2)
This board is suitable for experimental circuits
operating below a few GHz and array antennas
operating up to and beyond 20 GHz.
25

• Ceramic Substrate (Alumina)

1) Good for operation frequency up to 40
GHz. 2) Metallic patterns can be implemented on
ceramic substrate using thin-film or thick-film
technology. 3) Passive components of extremely
small volume can be implemented because the
ceramic substrate can be stacked in many tens of
layers or more, e.g. low temperature co-fired
ceramic (LTCC). 4) Good thermal
conductivity. 5) Alumina purity below 85 should
result in high conductor and dielectric losses
and poor reproducibility.

• Quartz

1) Production circuits for millimetric wave
applications from tens of GHz up to perhaps 300
GHz, and suitable for use in finline and image
line MIC structures. 2) Lower permittivity of
property allows larger distributed circuit
elements to be incorporated.
26

• Sapphire

• The most expensive substrate with following
  advantages
• 1) Transparent feature is useful for accurately
  registering chip devices.
• 2) Fairly high permittivity (?r 10.110.3),
  reproducible ( all pieces are essentially
  identical in dielectric properties), and thermal
  conductivity (about 30 higher than the best
  alumina).
• 3) Low power loss.
• Disadvantages
• 1) Relatively high cost.
• 2) Substrate area is limited (usually little
  more than 25 mm square).
• 3) Dielectric anisotropy poses some additional
  circuit design problems.

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• Properties of Some Typical Substrate Materials

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MIC Manufacturing Technology

• Thin-Film Module

• Circuit is accomplished by a plate-through
  technique or an etch-back technique.

• Thick-Film Module

1) Thick-film patterns are printed and fired on
the ceramic substrate. 2) Printed circuit
technique is used to etch the desired pattern in
a plastic substrate.

• Medium-Film Module

• Above technologies are suitable for HMIC
  productions.

• Monolithic Technology

• This technology is suitable for MMIC productions.

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• Properties of Various Manufacturing Technology

31
Multi-Chip Modules (MCM)

• MCM provides small, high precision interconnects
  among multiple ICs to form a cost-effectively
  single module or package.
• Four dominant types of MCM technologies
• 1) MCM-L having a laminated PCB-like structure.
• 2) MCM-C based on co-fired ceramic structures
  similar to thick-film modules.
• 3) MCM-D using deposited metals and dielectrics
  in a process very similar to that used in
  semiconductor processing.
• 4) MCM-C/D having deposited layers on the MCM-C
  base
• Advantages of an MCM over a PCB are
• 1) Higher interconnect density.
• 2) Finer geometries enables direct chip connect.
• 3) Finer interconnect geometries enables chips
  placed closer together and it results in shorter
  interconnect lengths.

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• Comparison of MCM Technologies

33
Low Frequency Characters of Microstrip Line
34
Microstrip Line

• Microstrip line is the most popular type of
  planar transmission lines, primarily because it
  can be fabricated by photolithographic processes
  and is easily integrated with other passive and
  active RF devices.
• When line length is an appreciable fraction of a
  wavelength (say 1/20th or more), the electric
  requirements is often to realize a structure that
  provides maximum signal, or power, transfer.
• Example of a transistor amplifier input network
• Microstrip components
• Transmission line
• Discontinuities
• Step
• Mitered bend
• Bondwire
• Via ground

35

• The most important dimensional parameters are the
  microstrip width w, height h (equal to the
  thickness of substrate), and the relative
  permittivity of substrate ?r.
• Useful feature of microstrip
• DC as well as AC signals may be transmitted.
• Active devices and diodes may readily be
  incorporated.
• In-circuit characterization of devices is
  straightforward to implement.
• Line wavelength is reduced considerably
  (typically 1/3) from its free space value,
  because of the substrate fields. Hence,
  distributed component dimensions are relatively
  small.
• The structure is quite rugged and can withstand
  moderately high voltages and power levels.
• Although microstrip has not a uniform dielectric
  filling, energe transmission is quite closely
  resembles TEM its usually referred to as
  quasi-TEM.

36
Electromagnetic Analysis Using Quasi-Static
Approach (Quasi-TEM Mode)

• The statically derived results are quite accurate
  where frequency is below a few GHz.
• The static results can still be used in
  conjunction with frequency-dependent functions in
  closed formula when frequency at higher frequency.

• Characteristic Impedance Z0

For air-filled microstrip lines,
For low-loss microstrip lines,
We can derive
37
Procedure for calculating the distributed
capacitance

• Effective Dielectric Constant e

For very wide lines, w / h gtgt 1
For very narrow lines, w / h ltlt 1
38
We can express eeff as
where filling factor q represents the ratio of
the EM fields inside the substrate region, and
its value is between ½ and 1. Another approximate
formula for q is
(provided by K.C. Gupta, et. al.)

• Planar Waveguide Model

(Parallel-Plate Model)
39

• Conductor Loss ac

• In most microstrip designs with high ?r,
  conductor losses in the strip and ground plane
  dominate over dielectric and radiation losses.
• Its a factors related to the metallic material
  composing the ground plane and walls, among which
  are conductivity, skin effect, and surface
  roughness.
• Relationships

• Dielectric Loss ad

• To minimize dielectric losses, high-quality
  low-loss dielectric substrate like alumina,
  quartz, and sapphire are typically used in HMICs.
• In MMICs, Si or GaAs substrates result in much
  larger dielectric losses (approximately 0.04
  dB/mm).

40

• Radiation Loss ar

• Radiation loss is major problem for open
  microstrip lines with low ?. Lower ? (?5) is used
  when cost reduction is a priority, but it lead to
  radiation loss increased.
• The use of top cover and side walls can reduce
  radiation losses. Higher ? substrate can also
  reduce the radiation losses, and has a benefit in
  that the package size decreases by approximately
  the square root of ?. This benefit is an
  advantage at low frequency, but may be a problem
  at higher frequencies due to tolerances.

41

• Formulations of Attenuation Constant a

However, the dielectric loss should occur in the
substrate region only, not the whole region.
Therefore, ad should be modified as
42
How to evaluate attenuation constant ?

• Method 1 in Chapter 2.14 ? is calculated
  from RLCG values of material.
• Method 2 Perturbation method

where Pl is power loss per unit length of line,
P0is the power on line at z0 plane.

• Method 3? is calculated from material
  parameters.

where ac is attenuation due to conductor loss
ad is attenuation due to dielectric loss
ar is attenuation due to radiation loss

• Combined Loss Effect linearly combined quality
  factors (Q)

43

• Recommendations

• Use a specific dimension ratio to achieve the
  desired characteristic impedance. Following that,
  the strip width should be minimized to decrease
  the overall dimension, as well as to suppress
  higher-order modes. However, a smaller strip
  width leads to higher losses.
• Power-handling capability in microstrip line is
  relatively low. To increase peak power, the
  thickness of the substrate should be maximized,
  and the edges of strip should be rounded ( EM
  fields concentrate at the sharp edges of the
  strip).
• The positive effects of decreasing substrate
  thickness are
• Compact circuit
• Ease of integration
• Less tendency to launch higher-order modes or
  radiation
• The via holes drilled through dielectric
  substrate contributing smaller parasitic
  inductances
• However, thin substrate while maintaining a
  constant Z0 must narrow the conductor width w,
  and it consequently lead to higher conductor
  losses, lower Q-factor and the problem of
  fabrication tolerances.

44

• Using higher ? substrate can decrease microstrip
  circuit dimensions, but increase losses due to
  higher loss tangent. Besides, narrowing conductor
  line have higher ohmic losses. Therefore, it is a
  conflict between the requirements of small
  dimensions and low loss. For many applications,
  lower dielectric constant is preferred since
  losses are reduced, conductor geometries are
  larger ( more producible), and the cutoff
  frequency of the circuit increases.
• For microwave device applications, microstrip
  generally offers the smallest sizes and the
  easiest fabrication, but not offer the highest
  electrical performance.

45

• Design a microstrip line by the method of
• Approximate Graphically-Based Synthesis

46
Example1 Design a 50? microstrip line on a FR4
substrate( ?r 4.5).
Solution

• Assume ?eff ?r 4.5
• From Zo1 curve ? w/h1.5
• From q-curve ? q0.66
• ?eff 1q (?r 1)10.66(4.5-1)3.31
• 2nd iteration
• From Zo1 curve ? w/h1.7
• From q-curve ? q0.68
• ?eff 1q (?r 1)10.68(4.5-1)3.38
• 3rd iteration
• Stable result
• w/h1.88 ?eff 3.39

47

• Formulas for Quasi-TEM Design Calculations

• Analysis procedure Give w / h to find eeff and
  Z0.

(provided by I.J. Bahl, et. al.)

• Synthesis procedure Give Z0 to find w / h.

48
Example2 Calculate the width and length of a
microstrip line for a 50 ? Characteristic
impedance and a 90 phase shift at 2.5 GHz. The
substrate thickness is h0.127 cm, with ?eff
2.20.
Solution
Guess w/hgt2
Matched with guess
Then w3.081h0.391 (cm)
The line length, l, for a 90 phase shift is
found as
49
Microstrip on an Dielectrically Anisotropic
Substrate
Empirical formula
50
Curve ? ?i 10.6 Curve ? used ?req formula
51
Effects of Finite Strip Thickness

• At larger value of t/w the significance of the
  thickness increase.

Increasing thickness t
E-fields
,where we is effective width of strip
52
Effects of Metallic Enclosure (Housing)

• The purpose of metallic enclosure provide
  hermetic sealing, mechanical strength, EM
  shielding, connector mounting, and module
  handling.
• The conducting top and side walls lower both eeff
  and Z0, which is due to increase proportion of
  electric flux in air.

53
Effects of Propagation Delay

• One of the most significant properties of
  microstrip for applications in high speed digital
  or time-domain applications ( e.g. computer
  logic, digit communication, sampler for
  oscilloscope, counter) to carry signal pulses is
  propagation delay.
• Crosstalk between adjacent circuits is a serious
  problem in pulse systems.

For example, a 50? microstrip line on high-purity
alumina eeff 6.7

• High-speed gates typically have around 50 ps
  delay per gate, it means that 5-10 mm of
  microstrip is needed to realize such a gate. For
  instance, such length of line is not feasible to
  implement in chips.

54
Recommendations to The Static-TEM Approaches

• The Static-TEM formulas will exhibit significant
  errors once operation frequency beyond a few GHz.
• Always start with a slightly lower impedance than
  the actually desired, i.e. larger w/h, if
  trimming (etch or laser-trim) is contemplated.
• The physical lengths of line should slightly
  longer than required for adjusting operation
  frequency. In general, 1 reduction in length can
  be expected approximately a 1 increase in
  frequency.
• The length of a top-cover shield might be
  adjusted to trim the performance of MICs.

55
High Frequency Characters of Microstrip Line
56
Dispersion in Microstrip (Frequency Dependence)
High loss Low dispersion
Microstrip Line
Medium loss High dispersion
Low loss Low dispersion
Good for Applications

• As frequency goes higher, EM fields tend to
  distribute in the substrate region in a higher
  ratio.

57
Frequency-Dependent Effective Dielectric Constant
eeff (f ) for Microstrip Line

• The reason of dispersion generated
• 1) Higher TE and TM modes
• (hybrid mode) generated
• 2) Surface wave couples with
• dominate mode

58
Example3 Design a 50-W microstrip line on a
0.635 mm thick ceramic substrate (er9.9).
Calculate the wavelength of the line at 1 and 10
GHz. Assume that G 0.6 0.009 Z0 in
Getsingers expression.
Solution
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• Other accurate formulas of eeff (f )

• Edwards and Owens expression applicable for
  alumina and sapphire substrate under the range
  10? ?r ?12 (alumina type) and f?18 GHz.

• Yamashita expression suitable for
  millimetre-wave design (up to 100GHz) but not
  accuracy for frequency below 18 GHZ.

• Advantage of these formulas are calculated-based
  design and inexpensively integrated into CAD
  tools. However, these approximate approaches
  based on some limited applications are their
  drawback.

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Frequency-Dependent of Microstrip Characteristic
Impedance (Z0)

• The problem of characteristic impedance as a
  function of frequency is difficult to settle.
  Because there are several definitions of Z0 used
  different assumptions to derive results.
• Planar waveguide model

• For a 50? line the increase is about 10 over
  0-16GHz range

63

• Dispersion of lossy gold microstrip on a 635?m
  thick alumina substrate (?r 9.8, w 635?m, Z0
  50?)

• Dispersion of lossy copper microstrip on a 650?m
  thick high resistivity silicon substrate (?r
  11.9,
• w 70?m, Z0 83?)

64

• Variation of effective permittivity and
  characteristic impedance for a lossy gold
  microstrip on a 635?m thick alumina substrate (?r
  9.8)

65
Operation frequency Limitation

• Two possible spurious effects restrict the
  desirable operating frequency
• 1) The lowest-mode TM mode the most significant
  modal limitation in microstrip are associated
  with strong coupling between the dominant
  quasi-TEM mode and the lowest-order TM mode.
• 2) The lowest-order transverse microstrip
  resonance.
• TM mode it is identified when the associated two
  phase velocities are close.

Effective mode
air

• The maximum restriction on usable substrate
  thickness

fTEM1
TM0
substrate
Quasi-TEM

• hM? ? fTEM1?
• fTEM1 can be regarded as the upper limitation of
  operating frequency.

66

• fTEM1 as a function of substrate thickness h and
  relative permittivity ?r .

67
Lowest-Order Transverse Microstrip Resonance

• Transverse microstrip resonance For a
  sufficiently wide microstrip the resonant mode
  can also couple strongly to quasi TEM mode.
• To suppress transverse resonance, slot can
  introduce into metal strip but sometimes it might
  excite resonance. A practice method is a change
  in circuit configuration to avoid wide microstip
  lines close adjacent.
• At the cutoff frequency of transverse resonant
  mode, line has a length equivalent to w2d, where
  d accounts for the microstrip side-fringing
  capacitance d2h.
• The cutoff frequency

68

• Parameters governing the choice of substrate for
  any microstrip application.

69
Power Losses and Parasitic Coupling

• Four separate mechanisms can be identified for
  power losses and parasitic coupling
• 1) conductor losses
• 2) dissipation in the dielectric of substrate
• 3) radiation loss
• 4) surface-wave propagation
• The dissipative losses may be interpreted in
  terms of Q factor or can be lumped together as
  the attenuation coefficent ?.

Dissipative effects
Parasitic phenomena

• Conductor Loss

? ? f -1/2 , h-1

• In practice the loss is approximately 60
  increased when surface roughness is taken into
  account.

70

• Dielectric Loss

? Independent f , h

• In general conductor loss greatly exceed
  dielectric loss for most microstrip lines on
  alumina or sapphire substrates, but opposite
  condition to have larger dielectric loss for Si
  or GaAs substrates.

• f ? ? ? ? ? Q ?
• However Q factor will be limited by parasitic
  effects at high frequencies.

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• Radiation

? ? f 2 , h 2

• Microstrip is an asymmetric TL structure and is
  often used in unshielded or poorly shielded
  circuits where any radiations is either free to
  propagate away or to induce currents in the
  shielding. Further power loss is the net result.
• Discontinuities of microstrip form essential
  features of a MIC and are the major sources of
  radiations unavoidably.
• Various techniques may be adopted to reduce
  radiation
• 1) Metallic shielding or screening.
• 2)A lossy (absorbent) material near any
  radiation discontinuity.
• 3) Possibly shape the discontinuity in some way
  to reduce the radiation efficiency.

• Surface-Wave Propagation

? ? f 34 , h34

• Surface wave trapped just beneath the surface of
  substrate dielectric, will be propagated away
  from microstrip discontinuities in the form of a
  range of TE or TM modes.
• This effect can be reduced by above methods 1 and
  2 , or by cutting slots into the substrate
  surface just in front of an open-circuit.

72

• Power losses versus frequency for open-end
  discontinuity (?r 10.2, w 24 mil, h 25 mil)

73

• Parasitic Coupling

• If shielding cannot be adopted due to space
  limitation as to use the absorbent material, the
  method will reduces the Q-factor .
• High degree of isolation can suppress the
  parasitic coupling.
• Various methods for increasing isolation
• 1) Use relatively high permittivity substrate.
• 2) Use fairly thin substrate.
• 3) Employ high impedance stubs, wherever this is
  feasible.

Conclusion Attenuation is mainly due to
conductor and dielectric losses.
Radiation and surface-wave losses are negligible.
This face can be observed from the relative
degree that these losses dependent to
frequency.
74
Recommendations for Higher frequency
Considerations

• Select the substrate such that the TM mode effect
  is avoided. fTEM1 , hM
• Check that the first-order transverse resonance
  cannot be exited at the highest frequency. If a
  resonance is occur, above mentioned solutions can
  be adopted to suppress. fCT
• Calculate the total losses and Q-factor to check
  if they satisfy the design requirement. A
  reappraisal of design philosophy may be necessary
  when Q-factor is too low.
• Evaluate the frequency-dependent effective
  microstrip parameters to account for
  high-frequency effects. e.g. ?eff (f ), Z0(f )

75
Discontinuities of Microstrip Line
76
The Main Discontinuities

• All practical distributed circuits must
  inherently contain discontinuities. Such
  discontinuities give rise to small capacitances
  and inductances ( often lt 0.1pF and lt 0.1nH) and
  these reactances become significant at high
  frequencies.
• Several form of discontinuities
• Open-end circuit (Stub)
• Series coupling gaps
• Short-circuit through to the ground plane (Via)
• Right-angled corner (Bend)
• Step width change
• Transverse slit
• T-junction
• Cross-junction

77

• A HMIC microwave amplifier using a GaAs MESFET,
  showing several discontinuities in the microstrip
  lines.

78
Open-End

• Three phenomena associated with the open-end
• Fringing fields. Cf
• Surface waves.
• Radiation.
• Terms 2 and 3 equivalent to a shunt conductance
  (G), but minimization can be carried out to
  suppress the effects.
• Curve-fitting formula (by Silvester and Benedek)

Coefficients for k?
79
Equivalent End-Effect Length

• The microstrip line is longer than it actually is
  to account for the end-effect.

• More general formula
• (by Hammerstad and Bekkadal)

Over a wide range of materials and w/h, the
expression gives error of 5. Where such error
is accepted.

• Cf equivalent and fringing capacitance
• Leo equivalent extra TL of length

• Upper limit to end-effect length (by Cohn)

80

• Normalized end-effect length (Leo /h ) as a
  function of shape ratio w /h.

81
The Series Gap

• The gap end-effect line extension may be written

• More general formula by Garg and Bahl

82
Via-Ground

• The via hole provides a fairly good short-circuit
  to ground at lower frequency range, but the
  parasitic effects increase at high frequencies.
• Optimum via-hole dimension for minimum reactance
  ( by Owens)

• For a 50? line on alumina substrate
• (?r 10.1, h0.635mm), the hole diameter
• needs 0.26mm for a good broadband
• short-circuit. To accurately and repeatably
• locate these holes or shunt posts,
• Computer-controlled laser drilling can provide
• Precision realization.

83
Right-Angle Bend or Corner

• The bend usually pass through an angle of 90 and
  the line does not change width.
• The capacitance arises through additional charge
  accumulation at the corners particularly around
  the outer part of bend where electric fields
  concentrate.
• The inductance arise because of current flow
  interruption.
• Reactance formula ( by Gupta)

84
Example4 Calculate the parasitic effects for a
bend on an w0.75mm and h0.5mm alumina substrate
(er9.9).
Solution

• The 2?/120 ? reactances in
• series/parallel connection with 50 ? line
• will have a pronounced influence
• on circuit response.

85
Mitred or Matched Bend

• A mitred bend can greatly reduce the effects of
  reactance and hence improving circuit
  performance.
• An equivalent line-length lc occurs and increase
  with enhanced mitred.
• The champing function should be restricted to
  around

• A bend acts like a reflector.

86

• Magnitude of the current densities on
• (a) a right-angled bend, and (b) an optimally
  mitred bend.

87
The Symmetrical Step

• Like the bend, the shunt capacitance is the
  dominant factor.
• Curve-fitting formulas

88
The Asymmetrical Step

• The values of reactances are about half of the
  values obtained for the symmetrical step.

The Narrow Transverse Slit

• A narrow slit yields a series inductance effect,
  and it may be used to compensate for excess
  capacitance at discontinuities or to fine-tune
  lengths of microstrip such as stubs.

• A narrow slit width causes parasitic capacitance
  to parallel connection with L. While wide slit
  forms the asymmetrical steps. Therefore b lt h.

89
T-Junction

• The junction necessarily occurs in a wide variety
  of microstrip circuits such as matching elements,
  stub filters, branch-line couplers, and antenna
  element feeds.
• Garg et. al. and Hammerstad et. al. have provided
  formulas for extracting the elements of
  equivalent circuit. However, some limitations to
  the accuracy of formulas should be noticed.

90

• Parameter trends for the T-junction.

91
Compensated T-Junction

• Dydyk have modified the microstrip in the
  vicinity of junction in order to compensate for
  reference plane shifts, at least over a specified
  range of frequencies.
• The treatment of the junction can exclude
  radiation loss with little error in circuit
  performance results, at least up to a frequency
  of 17 GHz.

92
Cross-Junction

• A cross-junction may be symmetrical or
  asymmetrical, where the lines forming the cross
  do not all have the same widths.
• Theoretical and experimental agreement is not
  good, especially for some inductance parameter.
• The coupling effects that occur with
  cross-junctions illustrates the origin of
  cross-talk in complicated interconnection
  networks.
• One kind of applications is that used two stubs
  placed on each side of microstrip to instead of
  single one. The method can prevent wider stub
  from sustaining transverse resonance modes at
  higher operating frequency.

93
Frequency-Dependence of Discontinuity Effects

• Open-Circuit

Edward Figure 7.27
Edward Figure 7.25 7.26
94

• Open-Circuit

95

• Open-Circuit

96

• Series Gap

97

• Cross-Junction

98

• Bend