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Advanced Transmission Lines

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1. Transmission Line Theory

:series resistance per unit length in .:series inductance per unit length in .:shunt conductance per unit length in .:shunt capacitance per unit length in .

Let : complex propagation constant.

We have the solutions

• : positive z-direction propagation wave.• : negative z-direction propagation wave.• : voltage or current amplitudes of positive or

negative propagation waves.Also

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Define characteristic impedance

Then

and

Characteristic impedance is the voltage to current ratio of the wave.

For lossless line

• : attenuation constant. Power loss due to (conductor loss)and (dielectric loss).

• : propagation constant. • : phase velocity. Decreases as or increases.• : guided wavelength. Decreases as or increases.• : increases as increases or decreases.

Terminated Lossless Transmission Line

Assume incident wave , reflected wave then

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At ,

Impedance Match

For it is required that . That is, the load impedance mustmatch the characteristic impedance of the transmission line.

When to Consider Transmission Line Effect

• When the length of the transmission line is less than 1/20th ofthe wavelength of the highest frequency component of a signal,then transmission line effects can be safely ignored and thecircuit can be modeled as a simple RLC circuit.

• The actual threshold used, 1/20th, 1/10th or 1/5th, is based onexperience.

• For example, an interconnect on a silicon chip clocking at 3GHz has an appreciable frequency component at 15 GHz. Aninterconnect reaches the 1/10th threshold when it is 0.6 mmlong. This is less than the dimensions of most chips, which canbe up to 2 cm on a side. Thus it takes a finite time for thevariation of a voltage at one end of an interconnect to impactthe voltage at the other end.

When an Interconnect Should be Treated as a Transmission Line

Rise time : the time required for the signal to change from 10% to90% of its final value.

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Time of flight delay : the time required for the signal to pass

through the line where is the line length.

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Transmission Line Structures

Figure 5 Common transmission line structures suited to planarfabrication.

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Typical characteristic impedance range

Typical Q range

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Substrate Property

Property of typical substrate

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Property of organic substrate

Skin Effect

• Skin effect: the concentration of charge and current near thesurface of a conductor.

• Skin depth : the depth at which the magnitude of the currentfalls to of its value on the surface of the conductor.

• For planar metal trace, thickness should be at least to reduceresistance. Larger than that is less effective.

Figure 11 (a) an electromagneticwave from the left incidentat the interface of air and a conductor; and (b) profiles of theelectric field magnitude (top) and current magnitude (bottom) asthe wave travels into the conductor.

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•

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2. Microstrip Line

Formulas

,

Or

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where

Loss

Dielectric loss:

Conductor loss:

where

is the surface resistance where is the conductivity.

•

•

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•

Correction for Surface Roughness

where is the attenuation of the strip without roughness, is ther.m.s. surface roughness and is the skin depth.

Surface roughness increases conductor loss.

Example: consider a copper microstrip, where the skin depth at 4GHz is 1 μm. Also assume that the r.m.s. surface roughness is ofsimilar magnitude, that is, 1 μm (typical of 99% alumina). Then

.that is to say, the loss is increased by approximately 60% whensurface roughness is taken into account.

Bends in Microstrip• Right angle bends exhibit series inductance and shunt

capacitance. • Insertion loss cannot be ignored at higher frequency.

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• Example: 0.75mm line width, 0.5mm thick substrate withrelative permittivity is 9.9. Equivalent circuit capacitance andinductance are 0.135 pF and 0.031 nH, respectively. At 10 GHz,

.

Mitered (Chamfered) Microstrip Bends• A mitered bend produces as good as, or better, performance

than curved bends. This applies to a wide range of bend angles,from up to .

• At least 70% chamfer fraction ( ) is recommended foran acute-angled bend of

• Optimum right angle bends, 60% chamfer fraction.• The mitered edge of the microstrip bend acts like a reflector to

redirect the current flow.

Operating frequency limits

The lower-order strong coupled TM mode:

Figure 16 Mitered bend.

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The lowest-order transverse microstrip resonance:

Example: , , .

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3. Strip Line

Benefit: • No dispersion.

• Shielded.Disadvantage:• Difficult to attached lumped elements.• Parallel waveguide modesFormulas

where

.

Or

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where

.

Loss

where

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4. Coplanar Waveguide (CPW)

Disadvantage of Microstrip Line

• Relative inaccessibility of the ground plane. Difficultyassociated with making shunt connections between strip andground.

• Sensitivity to substrate thickness. In integrated circuittechnology lateral dimensions can be defined photo-lithographically with great accuracy, but the substrate thicknessis often not controlled at all well.

• Increased radiation when thick substrates are used.

CPW Benefit:• Easier grounding of surface-mounted (or ball grid array (BGA)-

mounted) components• Lower fabrication costs. No via needed.• Reduced dispersion• Decreased radiation losses• Photo-lithographically defined structures with relatively low

dependence on substrate thickness.

CPW Disadvantage:• Difficult in routing.• Can produce unwanted common mode if asymmetric.• Air bridge needed to tie the two ground to the same potential.

. Insensitive to substrate thickness.

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Coupled Transmission Lines

1. Structure: symmetrical, two signal wire. Supportdifferential (odd) and common (even) modes withdifferent characteristic impedances and phasevelocities.

2. Benefit: reduce interference.3. Disadvantage: difficult to layout. When symmetry

is broken due to bend or branch, common modewill be generated, contaminating differentialsignals.

4. Extra consideration needed to match bothdifferential and common modes.

Example: microstrip parallel coupled line

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Example:

Coupling: 0.05 for a length of 1/4 wavelength.t

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Moisture Effect on Dielectric

• PCB constructed with a dielectric that tends to absorb moisture(such as FR4) is exposed to a humid environment for asignificant amount of time, both the loss tangent and thedielectric permittivity will increase.

• Fig. 25 shows an example of the insertion losses measured in adry environment such as an Arizona winter (15% RH and 60NF)and a humid environment such as Malaysia (95% RH and95NF). The dramatic increase in insertion losses corresponds toan increase in dissipated power.

• Fig. 26 depicts measured values of the same dielectric materialfor both low- and high-humidity environments. At about 7.5GHz, the increase in the loss tangent is greater than 50%.

• Fig. 27 shows a relatively small increase of less than 5% in thedielectric permittivity.

• Fig. 28 shows the measured change in insertion loss (S21) at 10GHz for a microstrip transmission line built on FR4 that wasdry at beginning and then exposed to humid conditions for 55days. In this example, the dielectric transmission-line losses at10 GHz were 6.3 dB under dry conditions at t = 0. Thedielectric became almost fully saturated within the first 7 days,increasing the loss to 8.9 dB. Finally, the transmission-lineloss stabilized to a value of 9.3 dB of loss at t = 48 days.

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Figure 25

Figure 26

Figure 27

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Figure 28

Differential Signaling

• Single-ended signaling generally work well up to approximately 1to 2 Gb/s.

• As data rates increase, it becomes increasingly difficult tomaintain adequate signal integrity because of system noise due toswitching, crosstalk and nonideal current return paths (Fig. 1).

• To reduce the effect of the system noise dramatically is to dedicatea pair of transmission lines for each bit on the bus. The twotransmission lines are driven in the odd mode, and the differencesbetween the voltages are used to recover the signal at the receiverusing a differential amplifier (Fig. 2 and Fig. 3).

Figure 1 How system noise can severely degrade signal integrity onsingle-ended buses. The ideal versus noisy receiver voltages comparedto the reference voltage.

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Benefits of Differential Signaling1. Reduce common mode noise.2. Due to the cancellation effect of the positive and negative signal

pair, crosstalk of differential line is lower than single-end line forthe same separation.

3. For the same reason, radiation loss is also smaller, reducing EMI.

Figure 2 Differential signaling where each bit istransmitted from the driver to a receiver using a pair oftransmission lines driven in the odd mode. The signal isrecovered at the receiver with a differential amplifier.

Figure 3 Example of how common-mode noise is eliminated withdifferential signaling: (a) single-ended waveforms at each leg of adifferential receiver showing common mode noise; (b) differentialwaveform.

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Note: crosstalk of single-end signal line might not be higher if the samearea is used (Fig. 4 and 5).

Figure 4 Cross sections used to compare differentialversus single-ended crosstalk: (a) two single-ended signals;(b) two differential pairs with the same pair spacing as thatof the single-ended signals in (a); (c) two widely spacedsingle-ended signals that occupy the same board area as thedifferential pairs (dimensions in mils).

Figure 5 Crosstalk for the cross sections inFig. 4 demonstrating that differential signalingdoes not reduce crosstalk compared tosingle-ended signaling if the same board area isused. However, if the spacing betweendifferential pairs is similar to the spacingbetween single-ended signals, crosstalk inreduced (5-in. microstrip).

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4. Virtual Ground. Helpful for signal integrity when not perfectground exists.

Drawback of Differential Signaling1. Occupy larger area.2. Due line routing is more difficult than single line.3. AC common-mode conversion (ACCM conversion): if the phase

difference of the two signals on a differential line is not , common mode signal will be generated. Possible causes (Fig. 7):a. Length differences between the line pair.b. Asymmetric lines.c. Coupling.d. Bends.e. Termination differences

Figure 6 For an odd-mode (or differential) signal, the fields orient sothat an ideal virtual reference plane existed between the conductors.When the physical plane is interrupted, the virtual plane provides acontinuous reference for a differential signal and helps preserve signalintegrity.

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Figure 7 Examples of asymmetry in a differential pair that can causedifferential to common-mode conversion: (a) routing-lengthdifferences; (b) impedance differences due to etching variation; (c)crosstalk differences; (d) length differences due to bends.

Figure 8 When asymmetry exists in the differential pair, part ofthe signal gets converted to common mode at the receiver.

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Example (Fig. 9): Calculate the common mode voltage of the followingcircuit.

Fiber-Weave Effect: for example, FR4 is made from a matrix of wovenbundles of fiberglass ( ) embedded in an epoxy resin ( ) asshown in Fig. 11. The glass pitch shown is typical in the printed circuitboard industry. It is possible for one leg of the differential signal to berouted between glass bundles and the other to be routed over a glassbundle that effectively gives each trace a unique propagation velocitycausing an imbalance in the differential pair that causes ACCMconversion. To avoid, signal traces should align with fiber at an angle.

Figure 9 Example circuit for computingACCM conversion.

Figure 10 Differential common-modeconversion plotted for Fig. 9 showing that adifferential signal launched at the driverbecomes common mode at the receiver whenthe frequency is about 12.5 GHz.

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Example (Fig. 12): calculate the ACCM conversion of a typical case offiber-weave effect.

Figure 11 A typical FR4substrate.

Figure 12 A type fiber-weave effect of FR4 substrate onACCM converion.

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Effects of ACCM Conversion of Differential LineBends and Compensation Techniques

1. Comparison of (a) right angle bend, (b) mitered bend, (c) roundcorner bend and (d) bend. Length 25 mm, signal voltage 1 Vand rise time 100 ps (Fig. 14 and Fig. 15).

• The 45-degree-angle bend structure has theminimum common-mode noise andreflected differential-mode noise. 60%lower than the right-angle bend.

• For common mode noise, round cornerbend is a little bit better than right anglebend and mitered bend.

• For reflected differential-mode noise,mitered bend is the worse while roundcorner bend and right angle bend is similar.

• The faster the rise-time, the higher both thecommon-mode noise and reflecteddifferential-mode noise.

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2. Dual Back-to-Back Bends• Can reduce common mode

noise from 10s mV to 1smV.

• The longer separation of thedual bends, the worse theperformance. Saturate above100 mm (Fig. 17).

Figure 14 Figure 15

Figure 16 Dual back to back bends

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3. Compensation Capacitance for Common Mode Noise Reduction(Fig. 18) • The bend of the outer line can be treated as an extra

capacitor to ground.• A capacitor can be added on the inner line to compensate.• The 1.17 pF capacitance can reduce the maximum

common-mode noise by 53.48% against the uncompensatedright-angle bend.

4. Compensation Patch for Common Mode Noise Reduction (Fig.19)• Due to space limitation and cost, difficult to add lumped

Figure 17 Remnant common-mode noise of the dual back-to-back coupled bendsobserved at the probing point g for various length as depicted in Fig. 16.

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capacitors to compensate.• Instead, use patch metal on inner bend to achieve similar

capacitive effect.• For the right-angle bends, a square patch of the side lengths

5.8 mm achieves 58% improvement.• For 45-degree bends with a fan-shaped patch, a of 4 mm

achieves 46% improvement.

Figure 18 Transient analyses at the receiver forright-angle bends with shunt variouscompensation capacitances.

Figure 19 Common-mode noise of coupledbends with the parallel patch capacitance. (a)Right-angle bends with a square patch. (b)45-degree-angle bends with a fan-shapedpatch.

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5. Slow Wave Structure• Use slow wave structure on the inner line to compensate the

shorter path.

6. Parasitic Trace

Figure 21 87.5% reductionin common mode noise.

Figure 22 About 5-dBreduction in common modenoise.

Figure 23 15 dB reduction of commonmode noise.

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Bandwidth of Digital Signal

By Fourier transform, a time-domain waveform canbe thought of equivalently as being composed of the sum of infinitesinusoidal frequency components .

,

Example: 1-GHz clock signal waveform (period 1 ns) having rise andfall times ( ) of 100 ps, a 50% duty cycle, and an amplitude of 5V.The spectrum components are

Note that because of the 50% duty cycle, the even harmonics are zero.The DC component of the waveform, is the average value of thewaveform which is 2.5 V.

Observe that the ninth harmonic of 9 GHz has a wavelength of3.33 cm. To use lumped-circuit approximation to analyze a circuitdriven by this frequency would require that the largest dimension of thecircuit be less that 3.33mm. Similarly, to analyze a circuit that is drivenby the fundamental frequency of 1 GHz, whose wavelength is 30 cmwould restrict the maximum circuit dimensions to being less than 3 cm.This shows clearly that use of lumped-circuit analysis methods to

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analyze a circuit having a physical dimension of, say, 3 cm that isdriven by this clockwaveform would result in erroneous results for allbut perhaps the fundamental frequency of the waveform.

Approximate the trapezoidal pulse by finite number of harmonics

It shows that the result is pretty good when up to the ninth harmonic isapplied. Therefore, the required bandwidth is at least 9 GHz which is

close to . Therefore,

Figure 25 Approximating the clock waveform with firstand third harmonics.

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Figure 26 Approximating the clock waveform withfirst, third and fifth harmonics.

Figure 27 Approximating the clock waveform with upto the ninth harmonics

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Meshed-ground Plane

Benefit:1. Improve flexibility of FPC.2. Improve adhesion between different metallization layers3. Reduce weight.

Short coming:1. Lower operation frequency due to stop band of EBG.2. Generate radiation due to perturbation on ground.3. Deteriorate isolation due to holes on ground plane.

Effect:1. Reduce capacitance per unit length due to the removal of metal

under transmission lines. The larger the aperture, the smaller thecapacitance.

2. Increase inductance per unit length due to the perturbation ofcurrent path on ground plane. The larger the aperture, the largerthe inductance.

3. Increase characteristic impedance .4. The periodic cells create stop band.5. Transmission line characteristics are more frequency dependent.

Figure 1 Rectangular Cell

Figure 2 Rhombus Cell

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Benefit of rhombus cell over rectangular cell1. Characteristic impedance is less sensitive to alignment.2. Higher operation frequency due to the less perturbation of ground

current reducing the EBG effect.

Case Study: Rectangular Cell

1. Impact of Trace-to-Aperture Intersecting Angle (Fig. 4)a. When β was small, the response changed fast.b. curve was flattened dramatically from to .

c. from to .d. Inductance and Capacitance.Solid Ground: ,

. .

Figure 3

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2. Impact of Trace-to-Baseline Offset (Fig. 5)a. For , change of the offset causes large change of the

capacitance, thus the impedance changes a lot. .

b. For , almost no change of impedance.c. For , small change of impedance. .d. Inductance and Capacitance. Solid Ground: ,

.

3. Impact of Aperture Dimension (Fig. 6)a. increases when the aperture dimension increased.b. When τ approaches 1/2, the trace moved closer to the top of

a row of apertures. A slight change of aperture size will makethe return current distribution drastically different, leading to

Figure 4

Figure 5

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significant change of . c. Forsmaller τ values, the per-unit-length parameters changed

minimally with aperture dimension changes.d. did not change much as the trace shifted if aperture size

was small, e.g. .e. changed more dramatically for larger aperture sizes.f. is independent of , if the aperture size to trace width

ratio is smaller than 1/2.g. Inductance and Capacitance. Solid Ground: ,

.

Figure 6 Figure 7

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4. Impact of aperture-to-aperture separation (Fig. 7)

a. decreased as b increased for different . b. As b increased, approaching solid reference conductor case.c. changed slightly faster for the no trace offset situation

compared to the trace offset cases.d. Inductance and Capacitance. Solid Ground: ,

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Case Study: Rhombus CellLine width: 0.36 mm.Substrate thickness: 0.2 mm.Substrate dielectric constant: 3.8.

1. Line offset effect (Fig. 8)a. The middle case provides the shortest return current path,

thus the best transmitting performanceb. The center and the edge cases have the longest return current

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path, thus have poor characteristics.c. Cutoff at about 6 GHz due to EBG effect except the middle

case. The cutoff of the middle case is about twice of the othercase because the effective cell size is half.

2. Cell Size Effect (Fig. 9)a. As cell size increases, cutoff frequency decreases.

3. Improving performance by adding a trace on ground plan (Fig. 10)

Figure 8Figure 9

Figure 10

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a. A straight current path is generated on ground plane toreduce EBG effect.

b. 10, 15 and 20 TG cases maintain -3-dB insertion loss overthe 10 GHz, which show almost ideal characteristics of thegeneral microstrip line.

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