extraordinary transmission line effects_05

7/28/2019 Extraordinary Transmission Line Effects_05

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CHAPTER 5

Extraordinary Transmission LineEffects

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.2 Frequency-Dependent Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 246

5.3 High-Frequency Properties of Microstrip Lines .. . . . . . . . . . . . . . . 253

5.4 Multimoding on Transmission Lines .. . . . . . . . . . . . . . . . . . . . . . . . . . 260

5.5 Parallel Plate Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

5.6 Microstrip Operating Frequency Limitations . . . . . . . . . . . . . . . . . . . 267

5.7 Power Losses and Parasitic Effects .. . . . . . . . . . . . . . . . . . . . . . . . . . 274

5.8 Lines on Semiconductor Substrates .. . . . . . . . . . . . . . . . . . . . . . . . . 275

5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

5.1 Introduction

In the previous chapter, the basic operation of transmission lines wasdiscussed with a concentration on designing and understanding theoperation of microstrip lines. This chapter focuses on the frequency-dependent behavior of microstrip lines and on designing the lines so thatundesirable behavior is avoided. The major limitation on the dimensions oftransmission lines is determined by considering when higher-order modes(field orientations) can exist. Different modes on a transmission line travelat different velocities. Thus the problem is that if a signal on a line is split

between two modes, then the information sent from one end of the line willreach the other end in two packets arriving at different times. A combination

of the two modes will be detected and in general the information will belost as the two modes combine in an incoherent manner. This multimodingmust be avoided at all costs. The purpose of this chapter is to help you gainan understanding of frequency-dependent behavior and multimoding ontransmission lines. You will also be able to design the dimensions of linesto avoid the excitation of higher-order modes, and will be able to debug RFand microwave circuits that fail to work correctly because of multimoding.


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246 MICROWAVE AND RF DESIGN: A SYSTEMS APPROACH

Z0

Z0

E

1 2x

Z0

t =0

t =1

t =2

z

Figure 5-1 Dispersion of a pulse along a line.

5.2 Frequency-Dependent Characteristics

All interconnects have frequency-dependent behavior. In this section theorigins of frequency-dependent behavior of the microstrip line are examined

because the microstrip line has the most significant frequency-dependenceamong interconnects of general interest. Frequency-dependent behaviorother than multimoding often results in dispersion. The effect can be seenin Figure 5-1 for a pulse traveling along a line. The pulse spreads out as thedifferent frequency components travel at different speeds. You can imagine

a long line with this problemsuccessive pulses would start merging andthe signal would become unintelligible.The most important frequency-dependent effects are

Changes of material properties (permittivity, permeability, andconductivity) with frequency (Section 5.2.1)

Current bunching (Section 5.2.3) Skin effect (Section 5.2.4) Internal conductor inductance variation (Section 5.2.4) Dielectric dispersion (Section 5.2.5)

Multimoding (Section 5.4).

While the discussion focuses on microstrip lines, the effects occur with otherplanar and nonplanar transmission lines.

5.2.1 Material Dependency

Changes of permittivity, permeability, and conductivity with frequency areproperties of the materials used. Fortunately the materials of interest in


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EXTRAORDINARY TRANSMISSION LINE EFFECTS 247

microwave technology have characteristics that are almost independentof frequency, at least up to 300 GHz or so. Skin effect, charge bunching,and internal conductor inductance variation are due to the same physicalphenomenon but have different effects on frequency dependence. Theeffects are due to the finite time it takes to transfer information from onepart of a distributed structure to another. No information can travel fasterthan the speed of light in the medium. Dielectric dispersion is also relatedto the finite time required to transfer information, and the effect resultsfrom changes in the distribution of energy in the different dielectrics of aninhomogeneous structure. So dielectric dispersion relates to the differencesof the speed of light in different media. As frequency increases it is possiblefor field distributions involving looping or variations of the fields to exist.

This is called multimoding.

5.2.2 Frequency-Dependent Charge Distribution

Skin effect, current bunching, and internal conductor inductance are all dueto the necessary delay in transferring EM information from one location toanother. This information cannot travel faster than the speed of light in themedium. In a dielectric material, the speed of light will be slower than thatin free space by a factor of

r, where r is the relative permittivity of the

material. The speed of light in the dielectric is reduced from that in free spacetypically by a factor ranging from just over 1 to 300. However, the velocity ina conductor is extremely low because of high conductivity. In brief, current

bunching is due to changes related to the finite velocity of informationtransfer through the dielectric, and skin effect and internal conductorinductance variation are due to the very slow speed of information transferinside a conductor. As frequency increases, only limited information torearrange charges can be sent before the polarity of the signal reverses andinformation is sent to reverse the charge changes. The charge distributionon the conductors depends on how fast the signal changes. What occurs is

best described by considering a sinusoidal signal. Both the skin effect andcharge bunching effect on a microstrip line are illustrated in Figure 5-2.

5.2.3 Current Bunching

Consider the charge distribution for a microstrip line shown in thecrosssectional views in Figure 5-2. The microstrip crosssections shown hereare typical of an interconnect on a printed circuit board with the top layer

being solder resist. (For a monolithically integrated circuit, the top coveris a passivation layer.) The thickness of microstrip is often a significantfraction of its width, although this is exaggerated in Figure 5-2. The chargedistribution shown in Figure 5-2(a) applies when there is a positive DCvoltage on the strip (the top conductor). In this case there are positivecharges on the top conductor arranged with a fairly uniform distribution.The individual positive charges (caused by the absence of some balancing


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0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1

5 V

0 V

DC

5 V

0 V

1 GHz

HIGHFREQUENCY EFFECT ON CHARGE DISTRIBUTION

(a) (b) (c)

SKIN EFFECT ALONE

0 V

5 V

0 V

1 GHz

0 V

5 V

0 V

10 GHz

5 V

10 GHz

5 V

0 V

1 GHz

5 V

10 GHz

CURRENT BUNCHING ALONE

Figure 5-2 Crosssectional view of the charge distribution on an interconnect at different frequencies. The

+ and indicate charge concentrations of different polarity and corresponding current densities. There isno current bunching or skin effect at DC.

electrons exposing positively charged ions) tend to repel each other, butthis has little effect on the charge distribution for practical conductors withfinite conductivity. (If the conductor had zero resistance then these chargeswould be confined to the surface of the conductor.) The bottom conductor isknown as the ground plane and there are balancing negative charges, or a

surplus of electrons, so that electric field lines begin on the positive chargesand terminate on the negative charges. The negative charges on the groundplane are uniformly distributed across the whole of the ground plane. Animportant point is that where there are unbalanced, or net, charges therecan be current flow. So the charge distribution at DC, shown in Figure 5-2(a), indicates that for the top conductor, current would flow uniformly


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(a)

(b)

(c)

(d)

(e)

Figure 5-3 Normalized magnitudes of current and charge on an alumina microstrip line at 1 GHz: (a)normalized scale; (b) longitudinal current, iz , on the strip (1026 A/m); (c) on the ground plane (0

3.2 A/m); (d) the charge on the strip (80400 nC/m2); and (e) on the ground plane (033 nC/m2). (For

alumina, = 10.0 and r = 1.)

distributed throughout the conductors and the return current in the groundplane would be distributed over the whole of the ground plane.

The charge distribution becomes less uniform as frequency increasesand eventually the signal changes so quickly that information to rearrangecharges on the ground plane is soon (half a period latter) countered byreverse instructions. Thus the charge distribution depends on how fast thesignal changes. The effects are seen in the higher-frequency views shown inFigures 5-2(b) and 5-2(c). (The concentration of charges near the surface of

the metal is a separate effect known as the skin effect.) The longitudinalimpact of current or charge bunching alone is illustrated in Figures 5-3,5-4, and 5-5. These figures present amplitudes of the current and chargephasors at various frequencies and were calculated using the SONNETEM simulator. In interpreting these figures, please take into account themagnitudes of the current and charge distributions as identified in thecaptions, as the scales are normalized. An alternative view (or time-domainview) is the instantaneous snapshot of current and charge shown in Figure5-6. This situation is not just confined to the transverse plane, and regionsfurther along the interconnect also send instructions. The net effect is

bunching of charges and hence of current on both the ground plane andthe strip.

5.2.4 Skin Effect and Internal Conductor Inductance

From the previous discussion it was seen that at low frequencies currentsare distributed uniformly throughout a conductor. Thus there are magneticfields inside the conductor and hence magnetic energy storage. As a result,there is internal conductor inductance. Transferring charge to the interior of


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(a)

(b)

(c)

(d)

(e)

Figure 5-4 Normalized magnitudes of current and charge on an alumina microstrip line at 10 GHz:

(a) normalized scale; (b) longitudinal current, iz , on the strip (1028 A/m); (c) on the ground plane (0

4.1 A/m); (d) the charge on the strip (114512 nC/m2); and (e) on the ground plane (039 nC/m2).

(a)

(b)

(c)

(d)

(e)

Figure 5-5 Normalized magnitudes of current and charge on an alumina microstrip line at 30 GHz: (a)

normalized scale; (b) longitudinal current, iz , on the strip(1031 A/m); (c) on the ground plane (06 A/m);

(d) the charge on the strip (200575 nC/m2); and (e) on the ground plane (068 nC/m2).

conductors is particularly slow, and as the frequency of the signal increasescharges are confined close to the surface of the metal. Another equallyvalid interpretation is that time-varying EM fields are not able to penetratethe conductors as much when the frequency increases. This phenomenonis known as the skin effect. With fewer internal currents, the internalconductor inductance reduces and the total inductance of the line drops,thus the redistribution of the current results in a change of the inductancewith frequency. This is principally because, for the same current, magneticenergy is stored inside as well as outside the conductors at low frequencies.As frequency increases, the magnetic field becomes confined almost entirelyto the region outside the conductors and the line inductance asymptoticallyreduces to a constant as the internal conductor inductance goes to zero. Only


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CURRENT

(b)

(a) (c)

CHARGE

+_

(e)

_+

(d) (f)

Figure 5-6 Normalized instantaneous current and charge distribution on an alumina microstrip line and

ground plane at 30 GHz. Current: (a) normalized scale; (b) magnitude of the longitudinal current, ix, on

the microstrip; and (c) on the ground plane. Arrows indicate the direction of current flow. Charge: (d)

scale; (e) microstrip; and (f) ground plane. Signs indicate the polarity of charge.

above a few gigahertz or so can the line inductance be approximated as aconstant for the transverse interconnect dimensions of a micron to severalhundred microns.

Note that the line inductance is not the inductance of the conductor, butthe inductance (per unit length) of the entire transmission line. Inductance isa measure of the energy stored in a magnetic field. Calculation of inductancerequires a volumetric integral of energy stored in a magnetic field. If there is

no magnetic field in a region then there is no inductance associated with theregion. So as frequency increases and the magnetic field inside the conductorreduces further from the conductor surface, the internal inductance of theconductor goes to zero.

The skin effect is characterized by the skin depth, , which is the distanceat which the electric field, or equivalently the charge density, reduces to 1/eof its value at the surface. By considering the attenuation constant of fieldsin a conductive medium, the skin depth is determined to be

= 1/

f 02 . (5.1)

Here f is frequency and 2 is the conductivity of the conductor. Thepermeability of metals typically used for interconnects (e.g., gold, silver,

copper, and aluminum) is that of free space,0.The skin effect is illustrated in Figure 5-2(b) at 1 GHz. (The reader should

distinguish between skin effect and current bunching.) The situation ismore extreme as the frequency continues to increase (e.g., to 10 GHz)as in Figure 5-2(c). There are several important consequences of this. Onthe top conductor the positive charges are not uniformly distributed withrespect to the depth of penetration into the conductor. Consequently, as


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frequency increases, current flow is mostly concentrated near the surface ofthe conductors and the effective crosssectional area of the conductor, as faras the current is concerned, is less. Thus the resistance of the top conductorincreases. A more dramatic situation exists for the charge distribution in theground plane. From the previous discussion of current bunching it was seenthat charge is not uniformly distributed over the whole of the ground plane,

but instead becomes more concentrated under the strip. In addition to this,charges and current are confined to the skin of the ground conductor so thefrequency-dependent relative change of the resistance of the ground planewith increasing frequency is greater than for the strip.

The skin effect, and to a lesser extent current bunching, results infrequency dependence of line resistance, R, with

R(f) = RDC + Rs(f). (5.2)

RDC is the resistance of the line at DC and Rs is the skin resistance:

Rs(f) = RDCk

f . (5.3)

Here k is a constant, and while Equation (5.3) indicates proportionality tof, this is an approximation and the actual frequency dependence may be

different, but Rs always increases more slowly than frequency.

5.2.5 Dielectric Dispersion

Dispersion is principally the result of the velocity of the various frequencycomponents of a signal being different. The electric field lines shift as a resultof the different distributions of charge, with more of the electric energy

being in the dielectric. Thus the effective permittivity of the line increaseswith increasing frequency. At high frequencies, the fundamental result ofthe field rearrangement is that the capacitance of the line increases, but thischange can be quite smalltypically less than 10% over the range of DCto 100 GHz. (This effect is described by the frequency dependence of theeffective permittivity of the transmission line.) To a lesser extent, dispersionis also the result of other parameters changing with frequency, such as aninterconnects resistance. For an IC where the interconnects can have verysmall transverse dimensions (e.g., microns) of digital interconnects, the lineresistance is the most significant source of dispersion. The qualitative effect

of dispersion is the same whether it is related to the resistance (resistance-induced dispersion) or change in the effective permittivity (dielectric-inhomogeneity-induced dispersion).

Different interconnect technologies have different dispersion characteris-tics. For example, with a microstrip line the effective permittivity changeswith frequency as the proportion of the EM energy in the air region to that inthe dielectric region changes. Dispersion is reduced if the fields are localized


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and cannot change orientation with frequency. This is the case with copla-nar interconnectsin particular, coplanar waveguide (CPW) and coplanarstrip (CPS) lines have lower dispersion characteristics than does microstrip(for small geometries). The stripline of Figure 4-8(a) also has low dispersion,as the fields are confined in one medium and the effective permittivity is

just the permittivity of the medium. Thus interconnect choices can have asignificant effect on the integrity of a signal being transmitted.

As discussed earlier, as the frequency is increased, the fields becomemore concentrated in the region beneath the stripwhere the substratepermittivity has already resulted in a relatively large electric fielddisplacement. Since the fields are forced into the dielectric substrate to anincreasing extent as the frequency rises, a frequency-dependent effective

microstrip permittivity, e(f), can be defined. This quantity clearly increaseswith frequency and the wave is progressively slowed down. The effectivemicrostrip permittivity is now

e (f) = {c/ [vp (f)]}2 . (5.4)

Fundamentally the dispersion problem then consists of solving thetransmission line fields for the phase velocity, vp(f). The limits of e(f)are readily established; at the low-frequency extreme it reduces to thestatic-TEM value e (or e(0) or e(DC)), while as frequency is increasedindefinitely e(f) approaches the substrate permittivity itself, r. This issummarized as follows:

e (f) e as f 0r as f . (5.5)Between these limits e(f) changes smoothly.

5.3 High-Frequency Properties of Microstrip Lines

Here the high-frequency properties of microstrip lines are discussed andformulas are presented for effective permittivity, characteristic impedance,and attenuation loss which incorporate frequency dependence. In theprevious chapter, frequency-dependent dispersion was not incorporated;those results (or formulas) are called the quasi-static approximation. Theeffective dielectric constant at DC (as calculated in the previous chapter)

is denoted e(0) and the characteristic impedance at DC is Z0(0). Theseare also called the quasi-static effective dielectric constant and quasi-static characteristic impedance. Detailed analysis [51] yields the followingformula for the frequency-dependent effective permittivity of a microstripline:

e(f) = r r e(0)1 + (f /fa)

m , (5.6)


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where the corner frequency

fa =fb

0.75 +

0.75 0.332 1.73r

(w/h)(5.7)

fb =47.746 106

h

r e(0)tan1

r

e(0) 1

r e(0)

(5.8)

m =

m0mc m0mc 2.32

2.32 m0mc > 2.32(5.9)

m0 = 1 +1

1 +

w/h

+ 0.321 +w/h3

(5.10)

mc =

1 +

1.4

1 + w/h

0.15 0.235e0.45f/fa , for w/h 0.7

1, for w/h > 0.7. (5.11)

In all the equations given above, SI units are used. The accuracy of theequations above is within 0.6% for 0.1 w/h 10, 1 r 128, and forany value ofh/ provided that h < /10.1

5.3.1 Frequency-Dependent Loss

The effect of loss on signal transmission is captured by the attenuationconstant . There are two primary sources of loss resulting from the

dielectric, captured by the dielectric attenuation constant, d, and from theconductor loss, captured in the conductor attenuation constant, c. Thus

= d + c. (5.12)

For dielectric loss, Equation (4.78) provides a good estimate for theattenuation when e is replaced by the frequency-dependent effectivedielectric constant, e(f).

Frequency-dependent conductor loss results from the concentration ofcurrent as frequency increases. For a wide strip of thickness three timesgreater than the skin depth,

c(f) =Rs(f)

w Z0. (5.13)

For a narrow strip, say w/h < 1, the quasi-static loss equations (Equations(4.186) to (4.189)) can be used with acceptable accuracy.

1 Note that the free-space wavelength is0, thewavelength in themediumis , and theguidewavelength is g .


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Z0

0

50

100

150

200

250

300

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

h

Z0

0

50

100

150

200

250

300

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

w/

r

()

1

2

4

6

10

20

40

Figure 5-7 Dependence ofZ0 of a microstrip line at 1 GHz for various dielectric

constants and aspect (w/h) ratios.

A third source of loss is radiation loss, r, and at the frequencies at whicha transmission line is generally used, is usually smaller than dielectric andconductor losses. So, in full,

(f) = d(f) + c(f) + r(f). (5.14)

5.3.2 Field Simulations

In this section results are presented for EM simulations of microstrip lineswith a variety of parameters. These simulations were performed usingthe SONNET EM simulator. Figure 5-7 presents calculations of Z0 forvarious aspect ratios (w/h) and substrate permittivities (r) when there isno loss. The key information here is that narrow strips and low-permittivitysubstrates have high Z0. Conversely, wide strips and high-permittivitysubstrates have low Z0. The dependence of permittivity on aspect ratio isshown in Figure 5-8, where it can be seen that the effective permittivity,e, increases for wide strips. This is because more of the EM field is in thesubstrate.

When loss is incorporated, becomes complex and the imaginarycomponents indicate loss. Figures 5-9, 5-10, and 5-11 present the frequencydependence of three microstrip lines with different substrates and aspectratios. These simulations took into account finite loss in the conductors andin the dielectric. In Figure 5-9(a) it can be seen that the effective permittivity,


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0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

e

w/h

r

1

2

4

6

10

20

40

Figure 5-8 Dependence of effective relative permittivity e of a microstrip line at

1 GHz for various dielectric constants and aspect ratios (w/h).

e, increases with frequency as the fields become confined more to thesubstrate. Also, the real part of the characteristic impedance is plotted with

respect to frequency. A drop-off in Z0 is observed at the low end of thefrequencies as frequency increases. This is due to both reduction of internalconductor inductances as charges move to the skin of the conductor andalso to greater confinement (the same phenomenon as charge or current

bunching) of the EM fields in the dielectric as frequency increases. It is notlong before the characteristic impedance increases. This effect is not due tothe skin effect and current bunching that were previously described. Ratherit is due to other EM effects that are only captured in EM simulation. It is aresult of spatial variations being developed in the fields (related to the factthat not all parts of the fields are in instantaneous contact). Figure 5-9(b)shows the imaginary parts of e and Z0. These imaginary parts are a resultof loss, primarily loss in the conductors.

5.3.3 Filling Factor, q

Defining a filling factor, q, provides useful insight into the distribution ofenergy in an inhomogeneous transmission line. The effective microstrippermittivity is

e = 1 + q(r 1), (5.15)


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(a)

(b)

Figure 5-9 Frequency dependence of the real and imaginary parts of the effective permittivity, e, and

characteristic impedance Z0 of a gold microstrip line on alumina with r(DC) = 9.9, w = 70 m, h =

500 m: (a) real parts (); and (b) imaginary parts ().


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(a)

(b)

Figure 5-10 Frequency dependence of the effective permittivity, e, and characteristic impedance Z0 of a

gold microstrip line with r(DC) = 9.9, w = 240 m, h = 635 m: (a) real parts; and (b) imaginary parts.


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(a)

(b)

Figure 5-11 Frequency dependence of the effective permittivity, e, and characteristic impedance Z0 of a

gold microstrip line with r(DC) = 9.9, w = h = 635 m: (a) real parts; and (b) imaginary parts.


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q

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0.78

0.0 0.5 1.0 1.5 2.0 2

.5 3.0 3.5 4.0

w/h

r

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0.78

0.0 0.5 1.0 1.5 2.0 2

.5 3.0 3.5 4.0

r

r

4

2

6

10

20

40

Figure 5-12 Dependence of the q factor of a microstrip line at 1 GHz for various

dielectric constants and aspect (w/h) ratios. (Data obtained using SONNET.)

where qhas the bounds1

2 q 1. (5.16)

The useful aspect of q is that it is almost independent of r. A q factor

of 1 would indicate that all of the fields are in the dielectric region. Thedependence of the q of a microstrip line at 1 GHz for various dielectricconstants and aspect (w/h) ratios is shown in Figure 5-12, thus q is almostindependent of the permittivity of the line. The properties of a microstripline, and uniform transmission lines in general, can be described very well

by considering the geometric filling factor, q, and the dielectric permittivityseparately.

5.4 Multimoding on Transmission Lines

Multimoding is a phenomenon that affects the integrity of a signal as ittravels on a transmission line. For transmission lines, multimoding occurswhen there are two or more EM field configurations that can support a

propagating wave. Different field configurations travel at different speeds sothat the information traveling in the two modes will combine incoherentlyand, if the energy in the two modes is comparable, it will be impossible todiscern the intended information being sent. It is critical that transmissionline structures be designed to avoid multimoding. The most commonmode on a transmission line is when there is no, or minimum, variationof the fields in the transverse direction (perpendicular to the direction


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of propagation). There must of course be a variation in the longitudinaldirection or else the wave will not propagate. The transmission structuresof interest here, are those that have conductors to establish boundaryconditions to guide a wave along an intended path. The conductorsare electric walls that confine the fields. The lowest-order mode withminimum transverse field variations is called the transverse EM mode(TEM). Higher-order modes occur when the fields can vary. From herethe discussion necessarily invokes EM theory. If you need to do this, seeAppendix D on Page 847, where EM theory is reviewed specifically withrespect to multimoding. One of the important concepts is that electric andmagnetic walls impose boundary conditions to the fields. Electric wallsare conductors, whereas a magnetic wall is formed approximately at the

interface of two regions with different permittivity.It is the property of EM fields that spatial variations of the fields cannotoccur too quickly. This comes directly from Maxwells equations whichrelate the spatial derivative (the derivative with respect to distance) ofthe electric field to the time derivative of the magnetic field. The sameis true for spatial variation of the magnetic field and time variation ofthe electric field. How fast a field varies with time depends on frequency.How fast an EM field changes spatially, its curl, depends on wavelengthrelative to geometry and on boundary conditions. Without electric andmagnetic walls establishing boundary conditions, as in free space, a fullwavelength is required to obtain the lowest-order variation of the fields.With electric or magnetic walls, a smaller distance will be sufficient. Betweentwo electric walls one-half wavelength of distance is required. The same

is true for magnetic walls. With one electric wall and one magnetic walla quarter-wavelength separation of the walls will support a higher-ordermode. A general rule for avoiding multimoding is that critical transversegeometries must be kept to under a fraction of a wavelength (say, < /2or < /4). Identification of exactly what the critical geometries are requiressome understanding of EM fields, of Maxwells equations, and of boundaryconditions. Appendix D on Page 847 provides a review of Maxwellsequations directed at understanding multimoding and the distribution offields on transmission lines.

One type of multimoding has already been described. In the previouschapter it was seen that the signals on a regular transmission linehave two simple solutions that are interpreted as the forward-travelingand backward-traveling modes. Each mode is a possible solution of the

differential equations describing the signals. The boundary conditions in thelongitudinal direction are established by the source and load impedances,and so the variation can be any fraction of a wavelength. This sectionis concerned with other solutions to the equations describing the fieldson a transmission line structure. In general, the other solutions arisewhen the transverse dimensions, such as the distance between the two


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Table 5-1 Properties of the EM fields at electric and magnetic walls.

E field H field

Electric wall Normal ParallelMagnetic wall Parallel Normal

conductors of a two-conductor transmission line, permits a variation ofthe fields. An EM treatment cannot be avoided if spatial modes on atransmission line are to be described. The dimensions of structures can bedesigned to avoid multimoding. Also, multimoding must be understood inresolving signal integrity problems that manifest themselves when circuits

do not function correctly. The concern is that with multimoding variouscomponents of a signal travel at different velocities and generally combineat a load incoherently. Multimoding can be easily described mathematicallyfor transmission line structures with uniform geometries.

The aim in this section is to understand moding and to develop anintuitive understanding of transmission line design and of microwavecircuits in general. The boundary conditions established at electric andmagnetic walls were derived in Section D.5 on Page 858. The propertiesof the EM walls are summarized in Table 5-1. These rules provide a quickway of understanding multimoding. Circuit structures such as transmissionlines, substrate thicknesses, and related geometries are nearly always chosenso that only one solution of Maxwells equations are possible. In particular,if the crosssectional dimensions of a transmission line are much less than awavelength then it will be impossible for the fields to curl up on themselvesand so perhaps there will be only one or, in some cases, no solutions toMaxwells equations. The simplest illustration of this phenomena, whichalso happens to be particularly relevant to planar transmission lines, issignal propagation on the parallel-plate waveguide, shown in Figure 5-13.These rules are used in the next section to describe multimoding in parallel-plate waveguides.

5.5 Parallel-Plate Waveguide

The parallel-plate waveguide shown in Figure 5-13 is the closest regularstructure to planar transmission lines such as microstrip. While the profile isnot the same as a microstrip line, it is as close a structure as is available that

still has a reasonably straightforward field solution that can be developedanalytically. The aim here is to develop an understanding of the origins ofmultimoding and develop design guidelines that will enable transmissionline structures to be designed to avoid multimoding. Intuition is essential indebugging circuits that are not working properly because of multimoding.

The development begins with Maxwells equations (Equations (D.1)


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xz

y

DIRECTION OF PROPAGATIONFORWARDTRAVELING WAVE

DIRECTION OF PROPAGATIONREVERSETRAVELING WAVE

d ,

(a)

d

00 x

y

(b)

Figure 5-13 Parallel-plate waveguide: (a) three-dimensional view; and (b)

crosssectional (transverse) view.

(D.4) on Page 848). A further simplification to the equations is to assumea linear, isotropic, and homogeneous medium, a uniform dielectric, so that and are independent of signal level and are independent of the fielddirection and of position. Thus

E = Bt

(5.17)

H = J+ Dt

(5.18)

D = V (5.19) B = 0, (5.20)

where V is the charge density and J is the current density. J and V will bezero except at an electric wall. The above equations do not include magneticcharge or magnetic current density. They do not actually exist and so a

modified form of Maxwells equations incorporating these is not necessaryto solve the fields on a structure. They are invoked in Appendix D to drawan analogy between an electric wall at a conductor and a magnetic wall atthe interface between two regions of different permittivity.

In the lateral direction, the parallel-plate waveguide in Figure 5-13extends indefinitely, and this is indicated by the broken continuationof the top and bottom metal planes in Figure 5-13(a). For this regular


20/42


y

x

E

H

A

B

C

E

H

(a) (b)

Figure 5-14 Parallel-plate waveguide showing electric, E, and magnetic, H, field

lines: (a) the TEM mode; and (b) the TE mode. The electric (E) and magnetic (H)

field lines are shown with the line thickness indicating field strength. The shading of

the Efield lines indicates polarity (+ or ) and the arrows indicate direction (andalso polarity). The zdirection is into the page.

structure, and with a few assumptions, the form of Maxwells equationswith multidimensional spatial derivatives can be simplified. One approachto solving differential equations, one that works very well, is to assumea form of the solution and then test to see if it is a valid solution. Thefirst solution to be considered is called the TEM mode and corresponds tothe minimum possible variation of the fields and will be the only possiblesolution when the interconnect transverse dimensions are small (relative toa wavelength). Also, it is assumed that the variation in the z direction isdescribed by the traveling-wave equations. So the only fields of interest hereare E and H in the transverse plane; all that is seen in Figure 5-13(b). If allthe fields are in the xy plane, then it is sufficient to apply just the boundaryconditions that come from the top and bottom ground planes. At first itappears that there are many possible solutions to the differential equations.

This is simplified by assuming certain variational properties of theEx,

Ey ,Hx, and Hy fields and then seeing if these solutions can be supported. The

simplest solution is when there is no variation in the fields and then the onlypossible outcome is that Ex = 0 = Hy . This is the TEM mode indicated inFigure 5-14(a). At the boundaries, the top and bottom metal planes, there isa divergence of the electric field, as immediately inside the (ideal) conductorthere is no electric field and immediately outside there is. This divergence issupported by the surface charge on the ground planes (see Equation (D.3)).

In Figure 5-14(a), the thickness of the lines indicates relative field strengthand there is no variation in field strength in either the electric or magneticfields shown. The coefficients of the field components are determined by the

boundary conditions. The trial solution used here has Ex = 0 = Hy and Ez= 0 = Hz ; that is, the trial solution has the electric field in the y direction

and the magnetic field is in the x direction. Maxwells equations (Equations(5.17)(5.20)) become

Eyy = Bxxt

(5.21)

Hxx = Dyxt

+ J (5.22)


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Dyy = (5.23) Bxx = 0. (5.24)

Expanding the curl, , and div, , operators using Equations (D.15) and(D.16) these become (with D = Eand B = H)

xEyz

=Bxx

t=

Hxx

t(5.25)

yHx

z=

Dy y

t+ J =

Eyy

t+ J (5.26)

Dyy

=Ey

y= (5.27)

Bxx =

Hxx = 0. (5.28)

These equations describe what happens at each point in space. Equation(5.25) indicates that if there is a time-varying component of the x-directed Bfield then there must be a z-varying component of the Ey field component.This is just part of the wave equation describing a field propagating in thez direction. Equation (5.26) indicates the same thing, but now the roles ofthe electric and magnetic fields are reversed. Equation (5.25) shows thatthe y component of the electric field can be constant between the plates,

but at the plates there must be charge on the surface of the conductors(see Equation (5.27)),to terminate the electric field (as indicated by Equation(5.27)), as there is no electric field inside the conductors (assuming that theyare perfect conductors for the moment). Since magnetic charges do not exist,

captured by the zero on the right-hand side of Equation (5.28), Equation(5.28) indicates that the x component of the magnetic field cannot vary inthe x direction (i.e., Bx and Hx are constant). Thus the assumption behindthe trial solution about the form of this mode is correct. The electric andmagnetic field are uniform in the transverse plane, the xy plane, and theonly variation is in the direction of propagation, the z direction. Thus thetest solution satisfies Maxwells equations. This is the TEM mode, as all fieldcomponents are in the x and y directions and none are in the z direction.The TEM mode can be supported at all frequencies in the parallel-platewaveguide.

Putting Equations (5.25)(5.28)in phasor form, and considering the sourcefree region between the plates (so J = 0 and = 0), these become

Eyz

= Hx (5.29)

Hxz

= Ey (5.30)

Eyy

= 0 (5.31)


22/42


Hxx

= 0 (5.32)

and the solution becomes

2Eyz2

= 2Ey (5.33)

2Hxz2

= 2Hx , (5.34)

where =

and there is no variation of the Ey component of theelectric field in the y direction and no variation of the Hx component of themagnetic field in the x direction.

Another possible set of modes occurs when the electric field is only inthe y direction, but then there must be a variation of the field strength, asshown in Figure 5-14(b). The full EM development of the fields is given inAppendix E and here we summarize the results qualitatively. The simplestvariation is when there is a half-sinusoidal spatial variation of the Ey fieldcomponent. Applying the methodology described above, it is found thatthere must be a component of the magnetic field in the z direction tosupport this mode. Hence these modes are called Transverse Electric (TE)modes. (Interchanging the roles of the electric and magnetic fields yields theTransverse Magnetic (TM) modes where there is an electric field componentin the z direction.) The half-sinusoidal variation still enables the charge tosupport the existence of an Ey electric field. A key result from our previousdiscussion is that there must be enough distance for the field to curl by the

half-sinusoidal spatial variation and this is related to wavelength, . Thistransverse electric mode can only exist when h /2. When h is smallerthan one-half wavelength, this mode cannot be supported, and is said to becut off. Only the TEM mode can be supported all the way down to DC, somodes other than TEM have a cutoff frequency, fc, and a cutoff wavelength,c. The concept of wavenumber k(= 2/ =

) is also used, and for the

lowest TE mode, kc = 2/c with c = 2h. The cutoff wavelength, c, andthe cutoff wavenumber, kc, are both related to the dimension below in whicha mode cannot curl sufficiently to be self-supporting. In general, for TEmodes, there can be n variations of the electric field, and so we talk about thenth TE mode, denoted as TEn, for which kc,n = n/h and c = 2h/n. Alsonote that kc = 2/c. The propagation constant of any mode in a uniformlossless medium (not just in a parallel-plate waveguide mode) is

=

k2 k2c . (5.35)

For the TEM mode we have kc = 0. High-order modes are described by theircutoff wavenumber, kc.

Transverse magnetic (TM) modes are similarly described, and again kc =n/h for the TMn mode. A mode can be supported at any frequency above


23/42


the cutoff frequency for the mode. It just cannot be supported at frequenciesbelow the cutoff frequency, as it is not possible for the fields to vary (or curl)below cutoff.

5.5.1 Multimoding and Electric and Magnetic Walls

In the above discussion, the parallel-plate waveguide had two electricwallsthe top and bottom metal walls. Without going into much detail,results will be presented when magnetic walls are introduced. A magneticwall can only be approximated, as ideal magnetic conductors do not ex-ist (since magnetic charges do not exist). Whereas an electric wall appearsas a short circuit, a magnetic wall is an open circuit, so an open-circuitedcoaxial line appears to have a transverse magnetic wall at the open circuit.

Maxwells equations impose the following boundary conditions:

Electric wall Perpendicular electric fieldTangential magnetic field

Magnetic wall Perpendicular magnetic fieldTangential electric field

The lowest-order modes that can be supported by combinations of electricand magnetic walls are shown in Figure 5-15. With two electric or twomagnetic walls, a TEM mode (having no field variations in the transverseplane) can be supported. Of course, there will be variations in the field

components in the direction of propagation. The modes with the simplestgeometric variations in the plane transverse to the direction of propagationestablish the critical wavelength. In Figure 5-15 the distance between thewalls is d. For the case of two like walls (Figures 5-15(a) and 5-15(c)), c = 2h,as one-half sinusoidal variation is required. With unlike walls (see Figure 5-15(b)), the varying modes are supported with just one-quarter sinusoidalvariation, and so c = 4h.

5.6 Microstrip Operating Frequency Limitations

Different types of higher-order modes can exist with microstrip and the twomaximum operating frequencies of microstrip lines are the (a) the lowest-order TM mode, and (b) the lowest-order transverse microstrip resonance

mode. In practice, multimoding is a problem when two conditions are met.First it must be possible for higher-order field variations to exist, and second,that energy can be effectively coupled into the higher-order mode. Generallythis requires significant discontinuity on the line or that the phase velocitiesof two modes coincide. An early discussion said that the phase velocities oftwo modes would be different and this would be true when the dielectricis uniform. However, with a nonhomogeneous line like microstrip there


24/42


ELECTRIC

y

x

MODES WITHSIMPLESTVARIATION

ExyE

xH yH

TE modeTM mode

MODE WITHNO

VARIATION

yEyE xE

xH

zHyH

y

z z

z

y

y

Ez zx

E H H E

x

z

y

TE modeTM mode

yE

Ex EzyH zno H

xH

y

zE zH

y

x

y

x

OR

h

0

ELECTRIC

ELECTRIC

E HE H H E

(a)

0 hy0 h y

TEM mode

OR

h

0MAGNETIC

E

y0 h 0 h

H

NONE(b)

H E

y

0 h

E

y0 h

H

E

OR

h

0

MAGNETIC

MAGNETIC

0 h

yE

0 h

Ex

H

y0 h

xE

y0 h

H

E

(c)

y0 h

y

0 h y

E

no HEyH

TEM mode

H

z H

H

y0 h

y0 h y0 h 0 h

Figure 5-15 Lowest-order modes supported with combinations of electric and

magnetic walls.


25/42


can be frequencies where the phase velocities of two modes can coincide.Since discontinuities are inevitable it is always a good idea to use thefirst consideration. As was noted previously, the modal analysis that waspossible with the parallel-plate waveguide cannot be repeated easily for themicrostrip line because of the irregular crosssection, but the phenomena isvery similar. Instead of a TEM mode, there is a quasi-TEM mode and thereare TE and TM modes.

5.6.1 Microstrip Dielectric Mode

A dielectric on a ground plane with an air region (of a wavelength or moreabove it) can support a TM mode, generally called a microstrip dielectricmode, substrate mode, or slab mode. The microstrip dielectric mode is a

problem for narrowmicrostrip lines. However, only at high frequencies doesit become important. Whether this mode exists in a microstrip environmentdepends on whether energy can be coupled from the quasi-TEM mode(which is always generated) of the microstrip line into the TM dielectricmode. The critical frequency at which the TM mode becomes important iswhen there is significant coupling. Coupling is a problem with a microstripline having a narrow strip, as the field orientations of the quasi-TEMmode and the dielectric mode align. Also, coupling occurs when the phasevelocities of the two modes coincide. A detailed analysis reported inEdwards and Steer [51, page 143], and Vendelin [74], shows that this occursat the first critical frequency,

fc1 =c tan1 (r)2hr 1 . (5.36)

So at fc1, in Equation (5.36), the dielectric mode will be generated even ifthere is not a discontinuity. If there is a discontinuity, say a split of onemicrostrip line into two microstrip lines, multimoding will occur when thedielectric mode can exist. From Figure 5-15(b), the dielectric slab mode can

be supported when h > g/4, where g is the wavelength in the dielectric.Now g = 0/

r = c/(f

r), so the second critical frequency is

fc2 =c

4h

r. (5.37)

The development here assumes that the interface between the dielectric andair forms a good magnetic wall. With a dielectric having a permittivity of 10,typical for microwave circuits, the effective value of h would be increased

by up to 10%. However, it is difficult to place an exact value on this.In summary, fc2 is the lowest frequency at which the dielectric mode will

exist if there is a discontinuity, and fc1 is the lowest frequency at which thedielectric mode will exist if there is not a discontinuity.


26/42


EXAMPLE 5. 1 Dielectric Mode

The strip of a microstrip has a width of 1 mm and is fabricated on a lossless substrate that is2.5 mm thick and has a relative permittivity of 9. At what frequency does the substrate (orslab) mode first occur?

=9

+0.4h

h

w

w

r

Solution:There are two frequencies that must be considered. One that comes from the dimensions ofthe dielectric slab and the other from considerations of matching phase velocities. Fromphase velocity consideration, the development behind Equation (5.36), the first criticalfrequency is

fc1 =c tan1 (r)2h

r 1

= 13.9 GHz. (5.38)

The first slab mode occurs when a variation of the magnetic or electric field can besupported between the ground plane and the approximate magnetic wall supported by thedielectric/free-space interface; that is, when h =

4= 0/(4

9) = 2.5 mm 0 = 3 cm.

Thus the second slab mode critical frequency is

fc2 = 10 GHz. (5.39)

Since discontinuities cannot be avoided (you cannot build an interesting circuit with just atransmission line), fc2 is the critical frequency to use.

5.6.2 Higher-Order Microstrip Mode

If the crosssectional dimensions of a microstrip line are smaller than afraction of a wavelength, then the electric and magnetic field lines will beas shown in Figure 4-4 on Page 169. These field lines have the minimumpossible spatial variation and the fields are almost entirely confinedto the transverse plane; this mode is called the quasi-TEM microstripmode. However, as the frequency of the signal on the line increases itis possible for these fields to have one-quarter or one-half sinusoidal

variation. Determining the frequency at which this higher-order modecan be supported requires more involved intuition than is appropriate todevelop here.

The following is a summary of a more complete discussion on OperatingFrequency Limitations in Edwards and Steer [51]. Some variations of thefields or modes do not look anything like the field orientations shown in


27/42


Figure 4-4. However, the variations that are closest to the quasi-TEM modeare called higher-order microstrip modes and the one that occurs at thelowest frequency corresponds to a half-sinusoidal variation of the electricfield between the edge of the strip and the ground plane. This path isa little longer than the path directly from the strip to the ground plane.However, for a wide strip, most of the EM energy is between the stripand the ground plane (both of which are electric walls) with approximatemagnetic walls on the side of the strip. The modes are then similar to theparallel-plate waveguide modes described in Appendix D on Page 847. Thenext highest microstrip mode (or parallel-plate TE mode) occurs when therecan be a half-sinusoidal variation of the electric field between the stripand the ground plane. This corresponds to Figure 5-15(a). However, for

finite-width strips the first higher-order microstrip mode occurs at a lowerfrequency than implied by the parallel-plate waveguide model. In reality,multimoding occurs at a slightly lower frequency than would be implied

by this model. This is because the microstrip fields are not solely confinedto the dielectric region, and in fact the electric field lines do not follow theshortest distance between the strip and the ground plane. Thus the fieldsalong the longer paths to the sides of the strip can vary at a lower frequencythan if we considered only the direct path. With detailed EM modeling andwith experimental support it has been established that the first higher-ordermicrostrip mode can exist at frequencies greater than [51, page 143]:

fHigherMicrostrip =c

4h

r 1. (5.40)

This is only an approximate guide, and it is best to use the thinnest substratepossible. Most EM software programs used to model microstrip and otherplanar transmission lines report when higher-order moding can occur.

EXAMPLE 5. 2 Higher-Order Microstrip Mode

The strip of a microstrip has a width of 1 mm and is fabricated on a lossless substrate thatis 2.5 mm thick and has a relative permittivity of 9. At what frequency does the first highermicrostrip mode first propagate?

=9

+0.4h

h

w

w

r

Solution:

The higher-order microstrip mode occurs when a half-wavelength variation of the electricfield between the strip and the ground plane can be supported. When h = /2 = 0/(32) =2.5 mm; that is, the mode will occur when 0 = 15 mm. So


28/42


(a)

ELECTRIC WALL

WALLMAGNETIC

WALL

MAGNETIC

(b)

Figure 5-16 Approximation of a microstrip line as a waveguide: (a) crosssection of microstrip; and (b)

model with magnetic and electric walls.

fHigherMicrostrip = 20 GHz. (5.41)

A better estimate of the frequency where the higher-order microstrip mode becomes aproblem is given by Equation (5.40):

fHigherMicrostrip = c/(4hr 1) = 10.6 GHz. (5.42)

So two estimates have been calculated for the frequency at which the first higher-ordermicrostrip mode can first exist. The estimate in Equation (5.41) is approximate and is basedon a half-wavelength variation of the electric field confined to the direct path between thestrip and the ground plane. Equation (5.42) is more accurate as it considers that on the edgeof the strip the fields follow a longer path to the ground plane. It is the half-wavelengthvariation on this longer path that determines if the higher-order microstrip mode will exist.Thus the more precise determination yields a lower critical frequency.

=9

Longer half wavelength pathLower frequency estimate

Shorter half wavelength pathHigher frequency estimate

h

w

r

5.6.3 Transverse Microstrip Resonance

For a wide microstrip line, a transverse resonance mode can exist. This isthe mode that occurs when EM energy bounces between the edges of thestrip with the discontinuity at the strip edges forming a weak boundary.This is illustrated in Figure 5-16, where the microstrip shown in crosssectionin Figure 5-16(a) is approximated as a rectangular waveguide in Figure 5-

16(b) with magnetic walls on the sides and an extended electrical wall onthe sides of the strip. The transverse resonance mode corresponds to thelowest-order H field variation between the magnetic walls. At the cutofffrequency for this transverse-resonant mode the equivalent circuit is aresonant transmission line of length w + 2d, as shown in Figure 5-17, whered = 0.2h accounts for the microstrip side fringing. A half-wavelength must


29/42


Ey

+2w d

0

wd

h

x

r

Figure 5-17 Transverse resonance: standing wave (|Ey|) and equivalent transmis-sion line of length w + 2dwhere d = 0.2h.

be supported by the length w + 2d. Therefore the cutoff half-wavelength is

c2

= w + 2d = w + 0.4h (5.43)

orc

2fcr= w + 0.4h. (5.44)

Hencefc =

cr (2w + 0.8h)

. (5.45)

EXAMPLE 5. 3 Transverse Resonance Mode

The strip of a microstrip has a width of 1 mm and is fabricated on a lossless substrate that is2.5 mm thick and has a relative permittivity of 9.

=9

+0.4h

h

w

w

r

(a) At what frequency does the transverse resonance first occur?

(b) What is the operating frequency range of the microstrip line?


30/42


Solution: h = 2.5 mm, w = 1 mm , = 0/r = 0/3

(a) The magnetic waveguide model of Figure5-16 can be used in estimating the frequencyat which this occurs. The frequency at which the first transverse resonance modeoccurs is when there is a full half-wavelength variation of the magnetic field betweenthe magnetic walls, that is, the first transverse resonance when +0.4h = /2 = 2 mm:

03 2 = 2 mm 0 = 12 mm,

and sofTRAN = 25 GHz . (5.46)

(b) All of the critical multimoding frequencies must be considered here and the minimumtaken. The slab mode critical frequencies are fc1 (Equation (5.38)) and fc2 (Equation(5.39)); the higher-order mode critical frequency is fHighMicrostrip (Equation (5.42));and the transverse resonance frequency (Equation (5.46)). So the operating frequencyrange is DC to 10 GHz.

5.6.4 Summary of Multimoding on Microstrip TransmissionLines

There are four principal higher-order modes that need to be considered withmicrostrip transmission lines.

Mode Critical

frequencyDielectric (or substrate) mode with no discontinuity Equation (5.36)Dielectric (or substrate) mode with discontinuity Equation (5.39)Higher Order Microstrip Mode Equation (5.42)Transverse Resonance Mode Equation (5.45)

5.7 Power Losses and Parasitic Effects

Four separate mechanisms lead to power losses in microstrip lines:

(a) Conductor losses

(b) Dissipation in the dielectric of the substrate

(c) Radiation losses

(d) Surface-wave propagation.

The first two items are dissipative effects, whereas radiation losses andsurface-wave propagation are essentially parasitic phenomena. The readeris directed to Edwards and Steer [51] for an extensive treatment. Here,summary results are presented.


31/42


Conductor losses greatly exceed dielectric losses for most microstriplines fabricated on low-loss substrates. Lines fabricated on low-resistivitysilicon wafers, however, can have high dielectric loss. These wafers aremost commonly used for digital circuits and the interconnect transversedimensions are generally very small so that line resistance is very high, andagain, resistive losses dominate.

Radiation from a microstrip line results from asymmetric structures. Inparticular, discontinuities such as abruptly open-circuited microstrip (i.e.,open ends), steps, and bends will all radiate to a certain extent. Suchdiscontinuities form essential features of a microwave IC and thereforeradiation cannot be avoided. Efforts must be made to reduce such radiationand its undesirable effects. In circuits such as filters, amplifiers, etc., this

radiation is an acknowledged nuisance. In most cases, radiation can berepresented as shunt admittances.Surface waves, trapped just beneath the surface of the substrate dielectric,

will propagate away from microstrip discontinuities as TE and TM modes.The effect of surface waves can also be treated as shunt conductance.

Various techniques can be used to suppress radiation and surface waves:

(a) Metallic shielding or screening.

(b) The introduction of lossy (i.e., absorbent) material near any radiativediscontinuity.

(c) The utilization of compact, planar, inherently enclosed circuits such asinverted microstrip and stripline.

(d) Reducing the current densities flowing in the outer edges of any metalsections and concentrate currents toward the center and in the middleof the microstrip.

(e) Possibly shape the discontinuity in some way to reduce the radiativeefficiency.

5.8 Lines on Semiconductor Substrates

Propagation on transmission line structures fabricated on semiconductorsubstrates can have peculiar behavior. The interest in such lines is in relationto Monolithic Microwave Integrated Circuits (MMICs) using both silicon

(Si) and compound semiconductor technologies (especially GaAs). Morespecifically, interconnects on Metal-Oxide-Semiconductor(MOS) or Metal-Insulator-Semiconductor (MIS) systems, where an insulating layer exists

between the conductor and the semiconductor wafer (see Figure 5-18(a)),are of particular interest due to their ability to support a slow-wave.These structures find major use in distributed elements on chip at RF. Inparticular, a silicon substrate can have a significant impact on microstrip


32/42


Metallized back

Strip

SiO2

Si

Schottkycontact

Depletion

Bulk

layer

Metallized back

Strip

(a) (b)

Figure 5-18 Transmission lines on silicon semiconductor: (a) silicon-silicon dioxide

sandwich; and (b) bulk view.

, ,2 2 0

, ,011d1

d2d

y

x Axis of wavepropagation

z

Figure 5-19 Parallel-plate transmission line structure.

propagation that derives from the charge layer formed at the silicon-silicondioxide interface. The slow-wave effect is utilized in delay lines, couplers,

and filters. With Schottky contact lines, the effect is used in variable-phaseshifters, voltage-tunable filters, and various other applications.Now, an intuitive explanation of the propagation characteristics of

microstrip lines of this type can be based on the parallel-plate structureshown in Figure 5-19. For an exact EM analysis of the slow-wave effect withsilicon substrates see Hasegawa et al. [75]. As well as developing a veryuseful approximation for the important situation of transmission lines on asemiconductor, the treatment below indicates the type of approach that can

be used to analyze unusual structures. In this structure, assume a quasi-TEMmode of propagation. In other words, the wave propagation parameters, , and Z0 can be deduced from electrostatic and magnetostatic solutionsfor the per unit length parameters C, G, L, and R.

The analysis begins with a treatment of the classic Maxwell-Wagner

capacitor. Figure 5-20(a) shows the structure of such a capacitor wherethere are two different materials between the parallel plates of the capacitorwith different dielectric constants and conductivity. The equivalent circuit isshown in Figure 5-20(b) with the elements given by

C1 = 1A

d1, R1 =

1

1

d1A

(5.47)


33/42


,1 1 d1

d2 ,2d 2

R1

C1

R2

C2

Y( )

(a) (b)

Figure 5-20 Structure of the Maxwell-Wagner capacitor: (a) structure; and (b) its

equivalent circuit.

C2 = 2 Ad2

, R2 = 12

d2A

(5.48)

1 = R1C1 =11

(5.49)

2 = R2C2 =22

. (5.50)

The admittance of the entire structure at radian frequency is

Y() =1

R1 + R2

(1 +1)(1 +2)

1 +, (5.51)

where

=R12 + R21

R1 + R2. (5.52)

Introducing

Y() =

e

A

d

, (5.53)

the effective complex permittivity, e =

e e , can be defined in terms ofEquation (5.51). Using Equations (5.51) and (5.53) yields

e =1 + 2 + 122(R1 + R2) [1 + ()2]

d

A. (5.54)

Consider now the case when R1 goes to infinity (i.e., layer 1 in Figure 5-20is an insulator). For this case, Equation (5.54) becomes

e =1 +

1 + d2d1

12

22

2

1 +

1 + d2d112

2 22

2

1d

d1

. (5.55)

It is clear from Equations (5.54) and (5.55) that the effective complexpermittivity has a frequency-dependent component. Consider how this


34/42


varies with a few cases of. For = 0 (static value),

e,0 = 1d

d1. (5.56)

For the case where goes to infinity (the optical value), the real part of theeffective permittivity is

e, =12 (d1 + d2)

2d1 + 1d2. (5.57)

Note also that the value of e, can be approximately achieved for a largevalue of2/2 (i.e., low-conductivity substrates can be used to ensure thatthe displacement currents dominate). In a similar way, it is clear that e,0can be achieved by having a small value of2/2. Also note that e,0 can

be made very large by making d1 much smaller than d.

EXAMPLE 5. 4 Two-Layer Substrate

Consider the structure in Figure 5-19. Determine the guide wavelength, g, and thewavelength in the insulator, 1, at a frequency of 1 GHz. SiO2 and Si are the dielectrics,with permittivities 1 = 40 and 2 = 130 (the conductivities are zero). The depths d2 andd1 of the two dielectrics are d2 = 250 m and d1 = 0.1 m.

Solution:

1 =3108109

4

= 0.15 m = 15 cm,

e =12(d1 + d2)

1d2 + 2d1= 12.990 g =

3108

10912.99= 0.0832 m = 8.32 cm (5.58)

5.8.1 Modes on the MIS (MOS) Line

The previous description of the properties of a Maxwell-Wagner capacitorleads to a discussion of the possible modes on the MIS (MOS) line. To makethe problem tractable, the transmission line shown in Figure 5-18(a) will beapproximated as having the crosssection shown in Figure 5-21(a).

Dielectric Quasi-TEM Mode

The first possible mode is the dielectric quasi-TEM mode, for which thesectional equivalent circuit model of Figure 5-21(b) is applicable. In this

mode 2 2. This implies from our earlier discussion that

e =

e, ande = 0. Thus the per unit length parameters are

L = 0d1W

(5.59)

C1 = 1W

d1(5.60)


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W

,1 0

2 2d

d1

d2 , ,

0

(a)

G2C2

C1

L

x

C1

L L21 R2

x

C

2G

L1 R

x

1

(f)

(b) (c) (d)

Figure 5-21 Modes on an MOS transmission line: (a) equivalent structure of the MOS line of Figure 5-18;

(b) quasi-TEM mode; (c) skin effect mode; and (d) slow-wave mode.

C2 = 2W

d2(5.61)

G2 = 2W

d2

. (5.62)

In SI units these have the units: H/m for L, F/m for C1, F/m for C2, andS/m for G2.

Skin-Effect Mode

The second possible mode is the skin-effect mode, for which the sectionalequivalent circuit model of Figure 5-21(c) is applicable. Here, 2 2 issuch that the skin depth = 1/

f 02 in the semiconductor is much

smaller than d2 and

e =

e,0 ,

e =0d

d1 +

2

,

ThusL1 = 0

d1W

, C1 = 1W

d1(5.63)

L2 = 01

W

2

, R2 = 2f 0

1

W

2

. (5.64)

These have the SI units: H/m for L1 and L2, F/m for C1, and /m for R2.


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Slow-Wave Mode

The third possible mode of propagation is the slow-wave mode [76, 77], forwhich the sectional equivalent circuit model of Figure 5-21(d) is applicable.This mode occurs when f is not so large and the resistivity is moderate sothat the skin depth, , is larger than (or on the order of) d2. Thus e =

e,0,but e = 0. Therefore

vp =1

e

e

=1

00

11

d1d

(5.65)

(with SI units of m/s) and g = 1

d1/d, where 1 is the wavelength in the

insulator.

5.9 Summary

In this chapter, extraordinary transmission line effects were considered.Multimoding can occur in many forms, but whenever crosssectionaldimensions are large enough compared to a wavelength multimodingoccurs. This sets the upper bound on the frequency of operation of mosttransmission line structures. There are also significant frequency-dependenteffects with all transmission lines and arise from a number of sources.However, most of these effects in transmission lines of two or moreconductors can be understood as resulting from skin effect and charge

bunching. Different transmission line structures have varying degrees offrequency-dependent properties.

5.10 Exercises

1. A medium has a relative permittivity of 13and supports a 5.6 GHz EM signal. By default,if not specified otherwise, a medium is loss-less and will have a relative permeability of 1.[Based on Appendix D.]

(a) Calculate the characteristic impedance ofan EM plane wave.

(b) Calculate the propagation constant of themedium.

2. A medium has a dielectric constant of 20.

What is the index of refractionof the medium?[Based on Appendix D.]

3. A plane wave in free space is normally inci-dent on a lossless medium occupying a halfspace with a dielectric constant of 12. [Basedon Appendix D. Parallels examples on Page866.]

(a) Calculate the electric field reflection coef-ficient referred to the interface medium.

(b) What is the magnetic field reflection coef-ficient?

4. Water, or more specifically tap water or seawater, has a complex dielectric constant re-sulting from two effects: conductivity result-ingfromdissolved ions in the water leading tocharge carriers that can conduct current underthe influence of an electric field, and dielectricloss resulting from rotation or bending of thewater molecules themselves under the influ-

ence of an electric field. The rotation or bend-ing of the water molecules results in motionof the water molecules and thus heat. The rel-ative permittivity is

wr =

wr wr (5.66)and the real and imaginary components are


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two frequencies is characterized by the rel-ative permittivity (/0) of the real {/0}and imaginary parts {/0} as follows:

Measured relative permittivity:

Frequency Real Imaginarypart part

1 GHz 3.8 0.0510 GHz 4.0 0.03

Measured relative permeability:

Frequency Real ImaginaryPart Part

1 GHz 0.999

0.001

10 GHz 0.998 0.001Since there is an imaginary part of the di-electric constant there could be either dielec-tric damping or material conductivity, or both.[Based on Appendix D. Parallels example onPage 862.]

(a) Determine the dielectric loss tangent at10 GHz.

(b) Determine the relative dielectric damp-ing factor at 10 GHz (the part of the per-mittivity due to dielectric damping).

(c) What is the conductivity of the dielectric

at 10 GHz?7. Consider a material with a relative permittiv-

ity of 72, a relative permeability of 1, and astatic electric field (E) of 1 kV/m. How muchenergy is stored in the Efield in a 10 cm3 vol-ume of the material? [Based on Appendix D.Parallels example on Page 853.]

8. A time-varying electric field in the x directionhas a strength of 1 kV/m and a frequency of1 GHz. The medium has a relative permittiv-ity of 70. What is the polarization vector? Ex-press this vector in the time domain? [Basedon Appendix D. Parallels example on Page853.]

9. A 4 GHz time-varying EM field is traveling inthe +z direction in Region 1 and is incidenton another material in Region 2, as shownin the Figure D-8. The permittivity of Region1 is 1 = 0 and that of Region 2 is 2 =(4 0.04)0. For both regions1 = 2 = 0.

[Based on Appendix D. Parallels example onPage 867.]

(a) What is the characteristic impedance (orwave impedance) in Region 2?

(b) What is the propagation constant in Re-gion 1?

10. A 4 GHz time-varying EM field is travelingin the +z direction in Medium 1 and is nor-mally incident on another material in Region2, as shown in Figure D-8. The boundary be-tween the two regions is in the z = 0 plane.The permittivity of Region 1, 1 = 0, and

that of Region 2 is 2 = 40. For both regions,1 = 2 = 0. The phasor of the forward-traveling electric field (i.e., the incident field)is E+ = 100 y V/m and the phase is normal-ized with respect to z= 0. Q0 = 0. [Based onAppendix D. Parallels example on Page 867.]

(a) What is the wave impedance of Region 1?

(b) What is the wave impedance of Region 2?

(c) What is the electric field reflection coeffi-cient at the boundary?

(d) What is the magnetic field reflection coef-ficient at the boundary?

(e) What is the electric field transmission co-

efficient at the boundary?(f) What is the power transmitted into Re-

gion 2?

(g) What is the power reflected from theboundary back into Region 1?

11. What is the skin depth on a copper microstripline at 10 GHz? Assume that the conductiv-ity of the deposited copper forming the stripis half that of bulk single crystal copper. Usethe data in the table on Page 844.

12. What is the skin depth on a silver microstripline at 1 GHz? Assume that the conductivityof the fabricated silver conductor is 75% thatof bulk single crystal silver. Use the data in thetable on Page 844.

13. An inhomogeneous transmission line is fabri-cated using a medium with a relative permit-tivity of 10 and has an effective permittivity of7. What is the fill factor q?


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14. Using Figure 5-9 on Page 257, determine thecomplex characteristic impedance and com-plex effective permittivity of a microstrip lineat 12 GHz. The line is fabricated on aluminawith r(DC) = 9.9, w = 70 m, h = 500 m.

15. A magnetic wall and an electric wall are 2 cmapart and are separeted by a lossless materialhaving an effective permittivity of 10 and aneffective permeability of 23. What is the cut-off frequency of the lowest-order mode in thissystem.

16. The strip ofa microstriphas a width of 600 mand is fabricated on a lossless substrate that is1 mm thick and has a relative permittivity of10.

(a) Draw the magnetic waveguide model ofthe microstrip line. Put dimensions onyour drawing.

(b) Sketch the electric field distribution of thefirst transverse resonance mode and cal-culate the frequency at which the trans-verse resonance mode occurs.

(c) Sketch the electric field distribution of thefirst higher-order microstrip mode andcalculate the frequency at which it occurs.

(d) Sketch the electric field distribution of theslab mode and calculate the frequency atwhich it occurs.

17. A microstrip line has a width of 352 m and isconstructed on a substrate that is 500 m thickwith a relative dielectric constant of 5.6.

(a) Determine the frequency at which trans-verse resonance would first occur.

(b) When the dielectric is slightly less thanone quarter wavelength in thickness thedielectric slab mode can be supported.Some of the fields will appear in the airregion as well as in the dielectric, extend-

ing the effective thickness of the dielec-tric. Ignoring the fields in the air (so thatwe are considering a quarter-wavelengthcriterion), at what frequency will the di-electric slab mode first occur?

18. The strip ofa microstriphas a width of 600 mand is fabricated on a lossless substrate that is

635 m thick and has a relative permittivity of4.1.

(a) At what frequency will the first trans-verse resonance occur?

(b) At what frequency will the first higher-order microstrip mode occur?

(c) At what frequency will the slab mode oc-cur?

(d) Identify the useful operating frequencyrange of the microstrip.

19. The electrical design space for transmissionlines in microstrip designs is determined by

the designproject leaderto be 20 to 100 . Thisis based on experience with the types of loadsand sources that need to be matched. The pre-ferred substrate has a thickness of 500 mand a relative permittivity of 8. What is themaximum operating frequency range of de-signs supported by this technology choice.Ignore frequency dispersion effects (i.e., fre-quency dependence of effective permittivityand losses). [Hint: First determine the dimen-sions of the stripthat will provide the requiredimpedance range and then determine the fre-quency at which lowest-order multimodingwill occur.]

20. Consider the design of transmission lines inmicrostrip technology using a lossless sub-strate with relative permittivity of 10 andthickness of 400 m. You will want to use theformulas in Section 4.10.2 on Page 223.

(a) What is the maximum characteristicimpedance that can be achieved for atransmission line fabricated in this tech-nology?

(b) Plot the characteristic impedance versusstrip width.

(c) From manufacturing tolerance consider-ations, the minimum strip width that can

be manufactured is 20 m. What is themaximum characteristic impedance thatcan be achieved in practice?

(d) If the operating frequency range is 1 to10 GHz, determine the maximum widthof the strip from higher-order mode con-siderations. You must consider the trans-


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verse resonance mode as well as higher-order microstrip modes.

(e) Identify the electrical design space (i.e.,the achievable characteristic impedancerange).

(f) Identify the physical design space (i.e.,the range of acceptable strip widths).

(g) If the electrical design space requiresthat transmission line impedances beachieved within 2 . What tolerancemust be achieved in the manufacturingprocess if the substrate thickness can beachieved exactly? [Hint: First identify the

critical physical process corner and thusthe critical strip width that is most sus-ceptible to width variations. Then deter-mine the tolerance on the strip widthto achieve the allowable characteristicimpedance variation. That is, characteris-tic impedance is a function of strip widthand height. If the substrate is perfect (noheight variation), then how much can thestrip width vary to keep the impedancewithin2 of the desiredvalue? You cansolve this graphically using a plot ofZ0versus width or you can iteratively arriveat the answer by recalculating Z0.]

(h) If the electrical design space requiresthat transmission line impedances beachieved within 2 . What tolerancemust be achieved in the manufacturingprocess if the substrate thickness toler-ance is 2 m? (Assume that the per-fectly symmetrical stripline property willbe achieved accurately.) If the substrateis not perfect (the height variation is2 m), then how much can the stripwidth vary to keep the impedance within2 of the desired value? This problemis directly applicable to real-world pro-cess/design trade-offs.

21. The strip of a microstriphas a width of500mand is fabricated on a lossless substrate that is635 m thick and has a relative permittivityof 12. [Parallels Examples 5.1, 5.2, and 5.3 onPages 270, 271 and 273. ]

(a) At what frequency does the transverseresonance first occur?

(b) At what frequency does the first higher-order microstrip mode first propagate?

(c) At what frequency does the substrate (orslab) mode first occur?

22. The strip ofa microstriphas a width of 250 mand is fabricated on a lossless substrate that is300 m thick and has a relative permittivity of15.

(a) At what frequency does the transverseresonance first occur?

(b) At what frequency does the first higher-order microstrip mode first propagate?

(c) At what frequency does the substrate (orslab) mode first occur?

23. A microstrip line has a strip width of 100 mis fabricated on a lossless substrate that is150 m thick and has a relative permittivityof 9.

(a) Draw the microstrip waveguide modeland indicate and calculate the dimen-sions of the model.

(b) Based only on the microstrip waveguidemodel, determine the frequency at whichthe first transverse resonance occurs?

(c) Based only on the microstrip waveguidemodel, determine the frequency at whichthe firsthigher-ordermicrostrip mode oc-curs?

(d) At what frequency will the slab modeoccur? For this you cannot use the mi-crostrip waveguide model.

24. A microstrip line has a strip width of 100 mand is fabricated on a lossless substrate that is150 m thick and has a relative permittivity of9.

(a) Define the properties of a magnetic wall.

(b) Identify two situations where a magnetic

wall can be used in the analysis of amicrostrip line; that is, give two loca-tions where a magnetic wall approxima-tion can be used.

(c) Draw the microstrip waveguide modeland indicate and calculate the dimen-sions of the model.


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25. Consider the design of transmission lines inmicrostrip technology using a lossless sub-strate with relative permittivity of 20 andthickness of 200 m.

(a) Using qualitative arguments, show thatthe maximum characteristic impedancethat can be achieved for a transmissionline fabricated in this technology is 116. The maximum is not actually 116 ,but a simple argument will bring you tothis conclusion. Hint: This will occur atthe minimum width possible.

(b) Plot the characteristic impedance versus

substrate width.

(c) From manufacturing tolerance consider-ations, the minimum strip width that canbe manufactured is 20 m. What is themaximum characteristic impedance thatcan be achieved in practice?

(d) If the operating frequency range is 2 to18 GHz, determine the maximum widthof the strip from higher-order mode con-siderations. You must consider the trans-verse resonance mode, the higher-orderstripline mode, and the slab mode.

(e) Identify the electrical design space (i.e.,

the achievable characteristic impedancerange).

(f) Identify the physical design space (i.e.,the range of acceptable strip widths).

(g) If the electrical design space requiresthat transmission line impedances beachieved within 2 . What tolerance

must be achieved in the manufacturingprocess if the substrate thickness can beachieved exactly? [Hint: First identify thecritical physical process corner and thusthe critical strip width that is most sus-ceptible to width variations. Then deter-mine the tolerance on the strip widthto achieve the allowable characteristicimpedance variation.]

(h) If the electrical design space requiresthat transmission line impedances beachieved within 2 . What tolerancemust be achieved in the manufacturing

process if the substrate thickness toler-ance is 2 m. (Assume that the per-fectly symmetrical stripline property willbe achieved accurately.)

26. Consider the structure in Figure 5-19. Deter-mine the guide wavelength, g , and the wave-length in the top insulator, 1, at a frequencyof 20 GHz. The dielectric permittivities are1 = 3.90 and 2 = 130. The depths of thetwo dielectrics are d2 = 100 m and d1 =1.0 m. [Parallels Examples 5.4, on Page 278.]

27. Consider a metal-oxide-semiconductor trans-mission medium as examined in Section 5.8.The structure in the form of the Maxwell-Wagner capacitor is shown in Figure 5-20(a)with d1 = 100 m, d2 = 500 m, and rel-ative permittivities r1 = 3.9 and r2 = 13.Ignore the finite conductivity. What is the ca-pacitance model of this structure (see Figure5-20(b)) and what are the values of the capaci-tance?


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extraordinary transmission line effects_05

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