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 Raytheon’s 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
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.

11. Conventional coaxial lines and waveguides are remain useful in :
High power transmission (e.g. KW~MW transmitters)
High Q component needed (e.g. low loss filter)
Some millimetric–wavelength 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.

17. 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.

18. Trapped Inverted Microstrip (TIM)
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.

19. 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.

20. 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.

21. Summary of TL Properties
Z0 and Q-factor are criterion for circuit applications.

23. 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 =10.1~10.3), 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.

27. Properties of Some Typical Substrate Materials

29. 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.

30. 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.

32. 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; it’s 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. er er Effective Dielectric Constant e w h w h
Procedure for calculating the distributed capacitance:
Effective Dielectric Constant e er w h
For very wide lines, w / h >> 1 er w h
For very narrow lines, w / h << 1

38. (Parallel-Plate Model)
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.
It’s 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 z=0 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/h=1.5
From q-curve  q=0.66
eff = 1+q (r +1)=1+0.66(4.5-1)=3.31
2nd iteration
From Zo1 curve  w/h=1.7
From q-curve  q=0.68
eff = 1+q (r +1)=1+0.68(4.5-1)=3.38
3rd iteration
Stable result
w/h=1.88; 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 h=0.127 cm, with eff =2.20.
Solution
Guess w/h>2
Matched with guess
Then w=3.081h=0.391 (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. Planar Waveguide Model
Dispersion in Microstrip (Frequency Dependence)
Microstrip Line
Planar Waveguide Model  
air
substrate
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
Example3: Design a 50-W microstrip line on a mm thick ceramic substrate (er=9.9). Calculate the wavelength of the line at 1 and 10 GHz. Assume that G = Z0 in Getsinger’s expression.
Solution

60. 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.

62. 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 w+2d, where d accounts for the microstrip side-fringing capacitance: d=2h.
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.

71. Surface-Wave Propagation   f 3~4 , h3~4
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 3~4 , h3~4
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
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 < 0.1pF and < 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, h=0.635mm), 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 w=0
Example4: Calculate the parasitic effects for a bend on an w=0.75mm and h=0.5mm alumina substrate (er=9.9).
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 Narrow Transverse Slit
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 < 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