Chapter 5

Filter Design

5.1 Bandpass Filters

A primary goal for this thesis is to analyze the Draper resonator bar’s suitability for its

intended application in high-performance miniaturized communications filters. This

chapter presents the target filter specification goals, analyzes three potential filter

topologies, and discusses the practical concerns and tradeoffs in implementing these

designs.

5.1.1 Bandpass Filter FOMs

Figure 5.1 illustrates most of the major bandpass filter characteristics as traditionally

defined in the literature. These magnitude-based FOMs are usually referenced to the

minimum loss point within the passband, the frequency range where power is transmitted

from the source to the load. The other two regions of interest are the stopband, at which

frequencies most of the input signal is severely attenuated at the output, and the

transitionband, which resides between the passband and stopband. The filter magnitude

characteristics each describe an aspect of one of these regions.

The passband characteristics are bandwidth, center frequency, insertion loss, and

passband ripple. Bandwidth is evaluated for a given corner frequency, for example, the

3dB bandwidth is the distance in frequency between the two points at which the filter

transmission falls below 3dB relative to the minimum loss transmission. Center

frequency is self-explanatory, but there are sometimes subtleties with its exact definition.

49

Figure 5.1: Bandpass filter definitions. Plot is in units of dB magnitude vs. frequency.

(a)

(b)

Figure 5.2: Measuring insertion loss. (a) Reference test circuit. (b) Filter test circuit.

50

FilterVS VLRL

RS

VS V0RL

RS

0 dBIL

Passbandripple 3 dB

Out of bandrejection

3dB Bandwidth

fc

Passband

StopbandStopband

It can usually be defined as either the point of minimum insertion loss or the midpoint

between the two bandwidth edges, though the latter definition is usually more accurate. A

common issue that arises is that the center frequency for a given filter topology differs

from the individual resonator frequencies according to a complicated relationship, so

after an initial design is carried out, the filter must be simulated and the resonator

frequencies adjusted to achieve the desired center frequency exactly. Insertion loss is

defined as the relative loss of a filter compared to a short circuit in the test circuit [21].

Figure 5.2 demonstrates a test setup for measuring insertion loss. VL and V0 are measured

as shown, and then the loss is given by:

(5.1)

assuming the source and load impedances are matched. If they are not, the insertion loss

may be negative. In such cases it may be preferable to reference VL to the output voltage

generated by an ideal matching network. This voltage for the test circuit of Figure 5.2b is:

(5.2)

This is known as power loss or flat loss, and is never negative for a passive filter.

Insertion loss is frequently defined as the single lowest loss at any frequency given by the

filter, and thus the other filter characteristics are referenced to it. Passband ripple is the

difference in dB between the maximum transmission in the passband and the minimum,

not including the rolloff into the transitionband.

The stopband characteristics are referenced as dBs of attenuation relative to the minimum

insertion loss, also referred to as out-of-band rejection. The nature of these requirements

vary more widely with application than the passband characteristics, but frequently

include a maximum bandwidth before attaining a certain level of attenuation, and a

minimum attenuation everywhere in the stopband. Additional spurious passbands caused

by higher order modes are usually acceptable in the stopband if they are far enough away

from the carrier frequency.

51

The transitionband characteristics usually deal with the shape or steepness of the rolloff

between the passband and the stopband. A common metric used is called the shape factor,

which is defined for two levels of attenuation, and calculated as the ratio of the

frequencies at which those two levels of attenuation are achieved, such that the answer is

always greater than one. For example, to calculate the 3 dB to 60 dB shape factor for a

filter response, the frequencies at which the attenuation equals 3 dB and 60 dB are

measured (excluding those caused by passband ripple), and then w60dB/w3dB on the high

frequency side and w3dB/w60dB on the low frequency side of the passband give the

respective shape factors. They will be roughly equal for symmetric filters, though for

some applications an asymmetric response will be acceptable or even desirable. In all

cases, a value closer to unity is better, as that means a sharper transition from passband to

stopband, permitting less power from unwanted frequencies to leak through. Shape factor

usually only applies to filters that have a monotonic transition from passband to

stopband; there are many filters designed with a transmission zero in the transitionband

which results in an exceedingly sharp passband rolloff and some ambiguity in the

definition of shape factor. For these filters, simply noting the presence of the transmission

zeros is usually sufficient.

The last major frequency-domain filter characteristic of note concerns the filter’s phase

response. At times it has no impact for a given application, but when it does, it is usually

required to maintain a certain degree of linearity. A linear phase response through a given

frequency range results in a constant group delay for signals being passed at those

frequencies, which prevents the signal distortion caused by a variable group delay.

5.1.2 Filter Design Goals and Motivations

The goal of the filter design in this chapter is to choose a filter topology that could

theoretically satisfy modern communications needs, and then analyze the Draper

resonator’s suitability for implementing that design, along with predicting any potential

obstacles or limitations in doing so. A good example of a current application which could

gain much from the successful development of micromachined resonator filters is the

52

cellular phone industry. In order to be useful in this field, filters would need to be

designed that could conform to the cellular phone band allocations currently in use. An

overview of these can be found in [22]: carrier frequencies range from 800 MHz to just

over 2 GHz, and bandwidths range from 1% up to 3.1% for most bands.

These specifications will motivate the topology and manufacturing analyses that follow.

The target achievable frequency range will be 200 MHz through 1.5 GHz: this range was

chosen during the resonator’s initial design and simulation stages, and would be suitable

for many of the example wireless phone protocols as well as other RF communications

applications. Filter relative bandwidths of 1-3% are desirable. A specification not directly

attached to most consumer-level protocols, but essential to successful product fabrication

is the impedance level at resonance, which translates into insertion loss as a filter FOM.

This needs to be minimized to attain high performance and reduce the effects of noise.

Many current piezoelectric resonators have a 50 impedance at resonance (also known

as the Butterworth Van Dyke resistance), so that is a good target value.

5.2 Bandpass Filter Topologies

Three different filter topologies are presented and analyzed in this section. Each filter

transfer function is analyzed assuming no impedance at resonance (e.g., resonator Q =

∞), though the case study simulations which follow include a resistance corresponding to

Q = 104 to allow a calculation of insertion loss. It is important to note that because all the

filter transmissions were calculated directly as voltages across a load impedance due to a

driving source, the magnitudes plotted do not take into account the insertion loss

reference. This was done so that the plots match directly with the listed equations. Since

the maximum output in each case is half the input, simply shifting each magnitude plot

up by 6 dB will allow direct assessment of insertion loss. Actual devices would be

measured using a RF probe analyzer and such data is usually reported as S21 (see Chapter

4), from which insertion loss may be directly read.

53

5.2.1 Simple Ladder

Figure 5.3: Simple ladder topology. Each resonator is modeled as an L-C pair. Resonator C0 and R are ignored during the design process.

The coupled ladder filter is an old, very well-characterized design widely used to

implement filters with discrete inductors and capacitors. Crystal resonators can be

employed in this topology by modeling each resonator only as an L-C pair. The

consequences of this assumption will be examined later. This topology can only be used

to implement narrowband bandpass filters, with fractional bandwidth less than ~0.5%.

The major design variables and exact transfer function are:

(5.3) , loaded Q

(5.4)

(5.5)

K and loaded Q are mathematical constructs that can be used to conveniently express the

more directly applicable figures of merit of the filter characteristics, like fractional

bandwidth and passband ripple. They should not be confused with resonator and Q

(unloaded Q). For example, it is desirable to have infinite unloaded Q, but the loaded Q

should be within a range of finite values bounded by other design parameters to meet

desired specifications; an infinite loaded Q would result in two peaks of arbitrarily short

width and no passband.

54

Vin C12RL

RSZ=sL+1/sC Z=sL+1/sC

Vout

Figure 5.4: Maximally flat response vs. 1 dB passband ripple

The bandwidth is primarily determined by K. The KQ product affects the passband ripple

and shape factor. For a symmetric design (Q1 = Q2), KQ = 1 results in a maximally flat

or Butterworth response, where the global maximum of the transfer function is at the

center of the passband. As KQ increases, the transmission at the center of the passband

decreases such that the maxima are at the edges of the passband, introducing passband

ripple. At the same time, the shape factor improves (Figure 5.4), resulting in a Chebyshev

response. The phase linearity degrades with increasing ripple [21].

Brief Description of Operation

An ideal L-C pair has zero impedance at resonance, where the positive reactance from the

inductor and the negative reactance from the capacitor cancel out. In either direction in

frequency, the magnitude of the impedance increases linearly without limit. The resonant

peak is very sharp and narrow, so to make a bandpass filter this peak needs to be widened

and flattened somehow. This is accomplished by the coupling capacitor. Near the

55

-30

-25

-20

-15

-10

-5 Effect of KQ Product on Ladder Filter Response

Frequency

Magnitude (dB)

KQ=1.63KQ=1.00

resonant frequency, the L-C impedances will drop to low magnitudes such that they

become insignificant compared to the coupling impedance. Thus for a band of

frequencies where this inequality holds, the filter will effectively look like the source and

load resistances with just the coupling impedance. If the resistors are sized correctly

relative to the coupling impedance near resonance, the transfer function will be relatively

flat through this band. The parameter K relates the coupling impedance to the resonator

impedance and thus affects bandwidth, while loaded Q relates the source and load

resistances to the other impedances and thus affects passband ripple.

Analytical Approximation of Bandwidth

Assuming a symmetrical design (QS = QL, RS = RL) and with KQ = 1, an expression for

the fractional bandwidth can be obtained by using a narrow band approximation and

ignoring terms of lower orders of magnitude than KQ. The narrow bandwidth

approximation is:

(5.6)

Z is the impedance of the L-C pair representing each resonator - R and C0 are ignored

during design (their contributions are discussed below). The variable x is the fractional

distance from the center frequency, and w0 is the resonant frequency, defined as .

(5.7)

(5.8)

This equation may be solved for x when the transfer function equals 1/2 (the maximum

possible transmission), or by differentiating and solving for the maximum. Both methods

56

give the filter’s true center frequency at x = -1/(2Q). By transforming variables to y = x +

1/(2Q), the equation becomes:

(5.9)

The 3dB bandwidth can be found by solving this equation when equals ,

which is 3 dB less than the maximum value of one-half. This gives:

(5.10)

Thus the approximate half-power bandwidth is

(5.11)

These results were all expressed in terms of Q because 1/K was set equal to Q. In general,

for any set value for KQ, all results may be expressed in either K or Q.

Resonator R and C0

Resonator resistance traditionally has had a small effect on filter performance for

piezoelectric types. However, it is a serious issue for the Draper resonator due to its

especially small size and high impedance level. This resistance directly increases

insertion loss at all frequencies, though this is usually acceptable to some extent for most

filter applications. Of greater concern is its effect on the filter shape: as the unloaded Q

due to the resonator resistance decreases, it no longer becomes insignificant compared to

the loaded Q expression defined above. This effectively lowers the KQ product (by

lowering the effective Q) which degrades the shape factor. Practically, this limits the

minimum load and source resistance possible, which can present difficulties in

interfacing to other RF components. The resonator resistance may be decreased by using

multiple identical resonators in parallel and adjusting other values appropriately.

The static capacitance C0 of each resonator does not affect the passband characteristics

much for a narrow bandwidth, which is why it may be ignored during design. However it

57

has very significant effects on the out-of-band attenuation. An L-C pair is supposed to

have a low impedance near resonance and high impedance at all other frequencies. At

frequencies lower than resonance, the static capacitance is a large impedance in parallel

with the L-C path and may be ignored. However at higher frequencies it shorts across the

L-C pair. This may lead to a limited attenuation at high frequencies, but using capacitive

coupling avoids this since C12 shorts as well, resulting in a capacitive voltage divider

which will provide sufficient attenuation for small enough values of K. Another effect is

the parallel resonance added, which makes the resonator impedance peak at infinity at a

frequency slightly higher than the series impedance. This adds a transmission zero to one

side of the passband, which improves the shape factor on that side. However this also

means the resulting filter characteristic is asymmetric. Finally, if the circuit is analyzed

by replacing each resonator with only its static capacitance, which is a very good

approximation outside of the passband, a gradual peak with a one pole rolloff to either

side is obtained (Figure 5.5). For this topology, it is usually desirable to place the

passband far off this peak so that the attenuation immediately next to the passband is

greater. However the attenuation will decrease in the stopband towards this gradual peak,

so this must be taken into account to ensure that stopband requirements are met.

58

Figure 5.5: Simple ladder filter wideband response.

5.2.2 Dual-Resonator Ladder

Figure 5.6: Dual-resonator ladder topology. The series resonance of Zs is matched to the same frequency as the parallel resonance of Zp. Both resonators have the same capacitance values and differ only in inductance, though that is not a requirement of the design; it is assumed here to clarify the analysis. The static capacitance C0 must be included in the resonator model to analyze this topology.

(5.12)

59

102 103 104 105 106

-140

-120

-100

-80

-60

-40

-20

Ladder Filter Out-of-Band Attenuation

Frequency (MHz)

Magnitude (dB)Vin VoutRL

RS

Zp

Zs

(5.13)

Brief Description of Operation

There are three important frequencies affecting the passband. Starting from the lowest,

they are the shunt zero (series resonance of Zp), center frequency (parallel resonance of Zp

and series resonance of Zs), and the series zero (parallel resonance of Zs). In the ideal

model with no resonator resistance, both zeros result in a filter transmission of exactly

zero, while the transmission at the center frequency will be exactly one-half, the ideal

maximum. Thus this topology results in an excellent shape factor and very symmetric

response if the source and load resistors are properly sized.

Filter Characteristics

The main drawback to this topology is that the bandwidth is fixed by a physical property

of the resonators, the ratio r of the dynamic and static capacitances. This ratio does not

change with geometry for the Draper resonator. It determines the difference in frequency

between the series and parallel resonances, which is the distance between the center

frequency and each zero in this filter transfer function. The 3dB bandwidth is about two-

thirds of the bandwidth separating the two zeros according to simulation. The bandwidth

may be adjusted a small amount by moving the two resonant frequencies either closer or

further apart at the expense of passband ripple and decreased phase linearity.

Additional drawbacks to this topology arise from its strong dependence on the static

capacitance of the resonators, C0. Once again, the off-resonance transfer function is well-

approximated by replacing each resonator with its static capacitance, forming a gradual

peak as with the previous topology. However, if the bandwidth is not very narrow, the

shape of this peak affects the passband shape significantly, and the passband must be

60

placed in the middle of the peak where this function is relatively flat. Thus the attenuation

directly off-resonance is very low. Also, the source and load resistances must be fixed at

about 1/(w0C0), which is very large for micrometer-sized devices. These drawbacks are

particularly problematic for the Draper resonator. However, this filter topology is

reasonably common when using other types of resonators such as thin-film surface

acoustic wave or bulk acoustic waves types [9, 10, 23]. These devices are generally much

larger than the Draper device, which is one factor in determining a resonator’s impedance

level. The high impedance level of the small Draper resonator will make it difficult to

interface to other components without excessive losses to parasitics. Also, the Draper

resonator cannot vary its static capacitance independently of its motional capacitance,

which determines its impedance level. If the series and shunt devices have impedance

levels that are too different, the shape of the passband is severely degraded, thus if this

topology is implemented with Draper devices the static capacitances must be very

similar. Since the off-peak attenuation is determined by a capacitive divider consisting of

the static capacitances, much additional attenuation can be gained by increasing the shunt

static capacitance relative to that of the series devices, especially if multiple resonator

stages are used, as is the common practice.

In summary, the dual resonator ladder topology nearly achieves the highest bandwidth

theoretically possible with crystal-type resonators, meaning any type for which the

Butterworth Van-Dyke model is applicable, at the expense of flexibility in choosing the

bandwidth. It also has a very good shape factor due to the presence of transmission zeros

in the transitionband. These qualities have made it a common filter implementation for

other thin-film resonators to date, but the Draper resonator’s high impedance level and

fixed r make this topology impractical.

5.2.3 Lattice Filter

61

Vin VoutR

R

Zb

Za

Zb

Za

Figure 5.7: Full lattice filter topology. Two pairs of resonators comprise the filter: one pair goes straight across each edge of the lattice section, and the other connects diagonally across, forming an X (although they do not connect in the middle). The capacitances are kept equal to simplify algebra, so that the pairs differ only in their inductance and resonant frequencies. The source and load resistances are assumed to be equal and will be referred to as R in the following analysis.

(5.16)

Brief Description of Operation

The lattice filter attenuates based on the impedance match between the two different

resonators. The filter can be viewed as having two arms, each arm having one of each

resonator Za and Zb, forming a path from the input signal to ground. The arms differ in

the order of resonator placement. The load resistance connects the two arms at the

elbows. When Za(w) equals Zb(w), the two arms are identical and the voltage at their

elbows will be equal, so no signal will be transmitted to the load. If the two impedances

are not equal, the voltages at the elbows will differ so some current will flow through the

load.

62

If resonators with identical static capacitances are used, their impedances will match very

well off-resonance and thus the out-of-band attenuation for the filter will be very high.

Near resonance, the two series resonant frequencies wa and wb will outline the passband

of the filter (Figure 5.8). Approaching the passband from lower frequencies, the lower-

frequency resonator Za hits its series resonance and effectively becomes a short. While

this does short the load resistor to the source resistor and to the input signal’s ground, Zb

steals some current from the load so the transmission is not at the maximum possible. The

maximum output voltage of Vin/2 occurs at the unique point where Za(w) = -Zb(w) with

matched source and load resistors (the derivation follows in the next discussion). When

Figure 5.8: Location of individual resonator frequency peaks relative to lattice filter

passband.

the two impedances are of opposite sign, they tend to cancel each other out, and if they

are also of equal magnitude, they effectively disappear. The sign of the Butterworth Van-

Dyke impedance changes when it passes the series resonance, so the impedances will

63

Impedance of Za and Zb

Magnitude (dB)ZaZb

Full filter response

Frequency

Magnitude (dB)

wa wb

only cancel in this way between the two series resonance frequencies. This is why a

passband is not also formed between the parallel resonances.

Resistor Sizing

Assuming the source and load resistances are equal, for a maximally flat passband, the

resistors should be sized so that the output voltage reaches its maximum possible value in

the center of the passband. One way to determine this value is to use the point at which

Za(w) = -Zb(w). If we define the magnitude of this impedance to be X, then:

(5.18)

We wish to find the maximum value the magnitude of this function may yield. With no

loss of generality, R may be expressed as a ratio on X and thus X may be set to 1. We find

that the maximum possible value is one-half at the point where R = X. Thus one could

measure or simulate the point within the passband where the two resonators have the

same magnitude of impedance and set the resistance to that value to make a maximally

flat filter. As with the two-pole simple ladder, decreasing R improves the shape factor

while adding passband ripple and degrading phase linearity.

Analytical Approximation of Source and Load Resistance

A simple expression for the value of R needed for a maximally flat response may be

obtained if the fractional bandwidth is somewhat smaller than r and the approximations

(5.19)

are used, with BW defined as the bandwidth expressed as a fraction of the center

frequency. The second approximation is saying the filter center frequency is at the

geometric mean of the two series resonance frequencies, which is about the same as the

arithmetic mean if they differ by a small percentage. Finally, we define

64

(5.21)

to obtain a simple result, even though this expression is likely to underestimate the true

3dB bandwidth.

Taking the lattice transfer function and evaluating its magnitude at the center frequency:

(5.22)

(5.23)

(5.25)

Here we define a temporary variable t:

(5.27) 2

22

BWCR

t ba ww

Setting |H|² = 1/4 and substituting C = C0r yields:

65

(5.28)

This expression should be accurate to well within a factor of 2, and is useful to use both

during the design process and to discuss the size of the load and source resistances

needed when comparing this topology to others for a given set of filter specifications.

Discussion of Bandwidth and Other Filter Characteristics

A considerable advantage of the lattice filter is the wide range of bandwidths it can

achieve. The maximum possible bandwidth is limited by the first parallel resonance, but

any bandwidth less than that is attainable. For the Draper resonators, r is about 3.2%,

which places the parallel resonance about 1.6% higher than the series resonant frequency,

allowing filter responses considered wideband. Filters with passbands higher than the

series-parallel frequency difference are still possible, but the inclusion of the parallel

resonance within the passband causes asymmetry and warping of the response, and

makes the passband phase increasingly nonlinear. It should be noted that the net 3dB

bandwidth possible with this topology can actually be greater than that achieved by the

dual resonator ladder topology. In both cases, two different resonators are employed such

that the parallel resonance of one falls on the series resonance of the other, which more or

less defines the maximum possible bandwidth possible. However, due to the zeros in the

ladder transitionband, its transmission is pulled down at the edges of the passband such

that the 3dB point is reached about two-thirds of the way to the frequency of the zero

itself. The lattice function has no such zeros, and in fact has the worst shape factor of the

three topologies examined, which allows its passband to expand generously up to and

even somewhat past its limits before beginning its gentler rolloff.

Bandwidth and out-of-band attenuation are inversely proportional to each other. This is

expected since attenuation increases the better matched the two resonator impedances are,

66

and if the resonant frequencies are closer they will match better at all frequencies outside

the passband. For most applications, this consideration is unlikely to limit the bandwidth.

One important practical consideration of this full-lattice design is the output and input do

not share the same ground, which presents some complications when interfacing to

unbalanced components. Such a transformation would require a baluns or active buffer

circuitry with differential inputs [24]. The latter choice suggests that this topology is

particularly suited for the possibility of integration with prevailing IC technologies into

complete packaged devices. Differential topologies are especially advantageous for filter

elements of very small dimension due to their excellent common-mode noise rejection.

5.3 Numerical Examples

Here one example of each filter design is presented using typical Draper resonator

parameters. All resonators were simulated with the same dynamic capacitance C = 0.1132

fF and static capacitance 3.514 fF with slightly varying inductance values to realize the

different resonant frequencies required by each design. Such freedom is available when

designing the equivalent circuit parameters of these resonators, as the inductance or

capacitance may be varied independently of the resonant frequency. Each filter was

designed to have a center frequency of approximately 800 MHz. The bar dimensions and

equivalent circuit parameters are listed for each filter example, along with the other filter

element parameters as defined for each topology in Section 5.2. The procedure used was

to first choose appropriate bar dimensions to generate the desired BVD circuit parameters

using the relationships found in Chapter 2. These electrical parameters were then input

into MATLAB as the transfer functions in equations 5.5, 5.15, 5.17 and simulated with

the bode function.

Bar Parameter

Value BVD Parameter

Value Filter Element

Value

l 6.04 m L 350.0 H RS, RL 1758 w 3.22 m C 0.1132 fF Loaded Q 1000

67

2a 0.5 m R 175.8 C12 113.2 fFw0 799.58 MHz K 10-3

Table 5.1: Simple ladder filter simulation parameters.

Figure 5.9: Bode plot of simple ladder filter transfer function.

Filter FOM

Value Shape Factor

Left Right Out of Band Attenuation

Value

fc 799.94 MHz 3dB-20dB 1.0019 1.0013 At w = 0 ∞BW3dB 0.1334% 3dB-40dB 1.0121 1.0042 At w = ∞ ∞

IL 0.9202 dB 3dB-60dB 2.1785 1.0085 Minimum 29.51 dB @ 26.6 GHz

Table 5.2: Simple ladder filter FOMs.

Bar Parameter

Zs ZpBVD

ParameterZs Zp

l 6.04 m 6.14 m L 350.0 H 361.3 Hw 3.22 m 3.17 m C 0.1132 fF 0.1132 fF2a 0.5 m 0.5 m R 175.8 178.7

68

Simple Ladder Filter Response

-35

-30

-25

-20

-15

-10

-5

Mag

nitu

de (

dB)

-270

-180

-90

0

90

Pha

se (d

egre

es)

-35

-30

-25

-20

-15

-10

-5

Mag

nitu

de (

dB)

-270

-180

-90

0

90

Pha

se (d

egre

es)

798 799 800 801 802Frequency (MHz)

798 799 800 801 802Frequency (MHz)

w0 799.58 MHz 786.98 MHz

Table 5.3: Dual resonator ladder filter simulation parameters. RL = 56649 .

Figure 5.10: Bode plot of dual resonator ladder filter transfer function.

Filter FOM

Value Shape Factor

Left Right Out of Band Attenuation

Value

fc 799.83 MHz 3dB-20dB N/A N/A At w = 0 ∞BW3dB 2.1397% 3dB-40dB N/A N/A At w = ∞ ∞

IL 0.285 dB 3dB-60dB N/A N/A Minimum 3.56 dB nearpassband

Table 5.4: Dual resonator ladder filter FOMs.

Bar Parameter

Za ZbBVD

ParameterZa Zb

l 6.09 m 6.01 m L 355.6 H 346.0 Hw 3.195 m 3.24 m C 0.1132 fF 0.1133 fF2a 0.5 m 0.5 m R 177.2 174.8

69

-35

-30

-25

-20

-15

-10

-5

Mag

nitu

de (

dB)

-35

-30

-25

-20

-15

-10

-5

Mag

nitu

de (

dB)

Dual Resonator Ladder Filter Response

740 760 780 800 820 840 860Frequency (MHz)

740 760 780 800 820 840 860Frequency (MHz)

-90

0

90

Pha

se (

degr

ees)

-90

0

90

Pha

se (

degr

ees)

w0 793.26 MHz 803.84 MHz

Table 5.5: Lattice filter simulation parameters. RL = 28 k.

Figure 5.11: Bode plot of lattice filter transfer function.

Filter FOM

Value Shape Factor

Left Right Out of Band Attenuation

Value

fc 801.13 MHz 3dB-20dB 1.0198 1.0203 At w = 0 ∞BW3dB 1.8795% 3dB-40dB 1.0822 1.0877 At w = ∞ ∞

IL 0.0676 dB 3dB-60dB 1.2514 1.2874 Minimum none

Table 5.6: Lattice filter FOMs.

Discussion of Simulation Results

Shape Factor: In order from best to worst, the shape factors of the three filter designs are

dual ladder, simple ladder, and lattice. The simple ladder function is asymmetrical due to

70

Lattice Filter Response

-30

-25

-20

-15

-10

-5

Mag

nitu

de (d

B)

-30

-25

-20

-15

-10

-5

Mag

nitu

de (d

B)

-270

-180

-90

0

90

Pha

se (d

egre

es)

-270

-180

-90

0

90

Pha

se (d

egre

es)

775 780 785 790 795 800 805 810 815 820 825 830Frequency (MHz)

775 780 785 790 795 800 805 810 815 820 825 830Frequency (MHz)

the parallel resonance caused by the static capacitance C0 off the high frequency side of

the passband. Thus the shape factor is excellent to arbitrary levels of attenuation on the

high frequency side, while it drops off much more slowly on the low frequency side,

asymptotically approaching a 20 dB/decade rolloff. However, near the resonance peak

the shape factor is very steep on both sides due to the high loaded Q of this topology

allowed by its very narrow bandwidth. To take advantage of the resonant peak’s

steepness to greater levels of attenuation, another identical filter section may be added in

series. When shape factor is measured to an attenuation level beyond this initial steep

section, very large values such as the left value for 3 dB to 60 dB shape factor in Table

5.2 will result. The dual resonator ladder filter has the best shape factor due to the

transmission zeros on both sides of its passband. The lattice filter has no transmission

zeros and thus has the most gradual transitionband slope. As with the simple ladder

however, shape factor could be improved if needed by adding additional filter stages in

series – the fact that this design had twice as many resonators as the other two already

helped it compare better with them than it otherwise would have.

Out of Band Attenuation: The lattice filter has the best attenuation characteristic, as it

increases monotonically in both directions infinitely, with a slope no less than 60

dB/decade everywhere. A close second is the simple ladder filter, which has very similar

behavior except for the large gradual minimum in attenuation which occurs far above the

passband. This peak can be moved further away if necessary, or the minimum attenuation

increased by adding additional stages; in any case, it is unlikely to be as large a problem

as the additional modes of resonance exhibited by all resonators which cause spurious

passbands throughout the stopband. Finally, the worst attenuation characteristic is that of

the dual resonator ladder filter. An attenuation of 3-4 dB extending for many filter

bandwidths on either side of the passband is almost certainly insufficient for most filter

applications. This limitation is difficult and costly for the Draper resonator to overcome

due to its fixed r.

Load Impedance: The smallest load impedance is required by the simple ladder topology,

followed by the lattice, and the largest matching impedance is required by the dual

71

resonator ladder design. The simple ladder requires a small load impedance because its

narrow bandwidth limitation allows its loaded Q to be very high. However the impedance

varies directly with the bandwidth, and can only be lowered independently of it to some

degree by allowing passband ripple. The lower limit for RL with this topology is set by

the impedance at resonance of the resonators: as RL approaches this value the shape of the

passband degrades considerably. The load impedance for the dual ladder is the

impedance of C0 at resonance, which is very high for the small Draper resonators. If C0 =

3 fF (in the middle of the range for these devices), the range for RL given the target

frequency range of 200 MHz – 1.5 GHz would be 35 k – 265 k. The impedance

required by the lattice design is about half that for the dual ladder at the same bandwidth,

but as with the simple ladder, the load impedance required is lowered with a smaller

bandwidth.

Bandwidth: The three filter topologies have very different bandwidth characteristics. The

dual ladder bandwidth is essentially fixed by the physical properties of the resonator

used, though as it is the maximum bandwidth generally possible for that resonator this

topology is useful for many applications. The simple ladder can only be used for very

narrow bandwidths while maintaining a good passband shape, less than about 0.5% of the

center frequency. The lattice topology has the most flexible bandwidth options by far, as

it essentially allows all bandwidths up to that achieved by the dual ladder.

Insertion Loss: The insertion loss of each filter essentially depends on how the

resonator’s BVD resistance compares with RS and RL. Thus it is subject to vary

depending on the parameters chosen for each filter. In general, the simple ladder topology

will always exhibit more insertion loss than the dual ladder because the dual ladder load

resistance will always be a good deal larger, which helps make the effects of a small

additional resistance less significant. The lattice topology is particularly insensitive to this

parasitic resistance in terms of insertion loss because unlike the other two topologies, the

center frequency of the lattice does not coincide with the series resonance of any of its

resonators. Since by design the resonators are not at zero reactance at this point, their

72

existing reactance helps reduce the impact of the nonideal resistance, which will be a

very small percentage of the reactance if the resonator Q is high.

Topology Achievable Bandwidths

Shape Factor

Out of Band Attenuation

Load Impedance

Insertion Loss

Simple ladder < 0.5% of fc Good Good wcL/QL GoodDual ladder 2.2% of fc Excellent Poor 1/wcC0 Very GoodFull lattice < 2.2% of fc Fair Excellent %BW/wcC Excellent

Table 5.7: Filter FOM comparison for three filter topologies.

5.4 Manufacturing Limits and Tolerances

5.4.1 Resonator Bar Dimensional and FOM Limits

As shown in Eqns. (2.22)-(2.25), the equivalent circuit parameters of the resonator

depend on the bar dimensions and material parameters. Resonator FOMs such as the

resonant frequency w0 and the impedance level at resonance may also be expressed

directly in terms of the bar dimensions. Therefore, manufacturing and design limits on

the bar determine the filter characteristics this resonator may attain. “Manufacturing”

limits are imposed by what is physically or practically realizable, for example, very thick

films can be prohibitively time-consuming and expensive, and it is hard to control their

stress. “Design” limits help optimize the final resonator characteristics, for example,

setting a lower bound on the length/width ratio helps maintain good modal isolation. All

the limits in the following discussion are not permanent and are subject to further

optimization and improvement as technology advances.

Thickness

There is a limit on the minimum total thickness (2a) that can be sputtered by a given

facility, and it becomes increasingly costly to make it very thick. An preliminary limit of

73

0.2–1.5 m will be employed throughout this section, though it should be noted that

commercial fabrication technologies could likely achieve a range of at least 0.05–3.0 m.

Length and Resonant Frequency

The minimum length depends on the width of the tethers holding the bar. Since they were

ignored in the original analysis, their contribution to bar flexing should be minimized,

which can be accomplished by restricting their width relative to the bar’s length. This

suggests a design limit of at least five times the tether width. Current technology can

make the tethers as little as 0.5 m wide, therefore the minimum bar length is 2.5 m.

The maximum length will be restricted by the bar’s likelihood of hitting the substrate

upon a sharp acceleration. As the bar length increases, it becomes easier to bend and twist

up and down. If it should touch the substrate 1 m below, it may stick or break. Applying

the force-deflection relation for a simple cantilever beam:

(5.31)

where F is the applied force, E is the Young’s modulus of AlN, d is the deflection, and l,

w, and t are the beam dimensions (Figure 5.12), the approximate deflection for a given

force may be calculated. This calculation ignores the electrical contacts on the bar and

displacement due to twisting of the tethers, and assumes the deflection is relatively small.

As an example impact force, 1000 g times the mass of the bar will be used for F. With

thickness 2a and length 0.5l:

(5.32)

(5.33)

74

l

dt

F

Figure 5.12: Cantilever beam deflection under an applied force. Under no deflection the beam is a rectangular prism. The width w goes into the page and is not shown.

Figure 5.13: Deflection vs. bar length under a force of 1000mg for various thicknesses.

2a (m) lmax (m)

0.2 198

0.5 313

1.0 442

Table 5.8: Total bar length allowing a maximum 1 m deflection under a 1000mg force.

Another means by which the bar may hit the substrate is through twisting of the tethers.

Modeling this mechanism is somewhat complicated. To obtain a simple estimate for the

limiting lengths in this case, the tethers will be modeled as elastic cylinders with diameter

halfway between the width and thickness of the tethers. The torque-twist relation for

cylinders is:

75

2a=0.2 m

2a=1.0 m

2a=0.5 m

0 100 200 300 400 5000

1

2

3

4

5

6

7

89

10

Bar Length (m)

Deflection (m)

(5.34)

where G is the shear modulus of AlN and dt is the diameter of the cylinder. The other

variables are defined by two clamp points along the cylinder, for example, if one end is

attached to a fixed surface while the other is attached to a lever applying torque. The

applied torque T and twist θ are measured between these two points, and h is the distance

between them (Figure 5.14). Assuming no deflection, w is two-thirds l, tether width is 0.5

m, tether length is 30 m (a conservative maximum), and a 1000 mg force is applied at

the beam’s midpoint:

(5.35)

(5.36)

Clearly the maximum bar length is constrained most by twisting of the tethers, although

this estimate is extremely conservative in using a tether length of 30 m. A shorter tether

Figure 5.14: Elastic linear torsion. The twisting cylinder has diameter dt and length h which goes into the page. The near end is assumed fixed while the far end is attached to one end of a beam with length 0.5l. A force F on the beam creates a torque on the cylinder, and displacement d of the beam’s end.

76

d

0.5l

θdt

F

4

5

6

2a=0.2 m2a=0.5 m

Figure 5.15: Deflection vs. bar length under a torque of 250mgl for various thicknesses.

2a (m) lmax (m)

0.2 35

0.5 40

1.0 50

Table 5.9: Total bar length allowing a maximum 1 m deflection under a 250mgl torque.

length can almost double the maximum length, since to first order l4 h-1, but these

conservative estimates will serve as acceptable limits for this discussion. Bounding the

values for bar length allows calculation of the resonant frequencies achievable by this bar

design, since w0 depends only on l:

(5.37)

Now the full range of possible resonator frequencies can be calculated and compared to

the target range of 200 MHz to 1.5 GHz.

2a (m) lmax (m) wmin (MHz)

0.2 35 138

0.5 40 121

1.0 50 96.7

Table 5.10: Minimum resonant frequency achievable at three thicknesses, 30m tethers

Tether width (m) lmin (m) wmax (GHz)

0.2 1.0 4.83

77

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

78910

Bar Length (m)

Deflection (m)

2a=1.0 m

0.5 2.5 1.93

1.0 5.0 0.97

Table 5.11: Maximum resonant frequency achievable at various tether widths.

Width and Impedance Level

The final bar geometric parameter is width. It is essentially only limited on the lower end

by lateral manufacturing capabilities to a few tenths of a micron. As it turns out, it will

usually be desirable to keep width as large as possible to minimize impedance, so the

lower limit is not too significant. As width increases relative to the length, the frequency

of the longitudinal width mode grows closer to that of the length mode. To maintain good

modal isolation and to make sure the desirable longitudinal length mode is the lowest in

frequency, a design limit for maximum width is set to two-thirds the length. Another

approach is to make the width much greater than the length, resulting in additional modes

below the length resonant frequency. The advantage to this approach is it reduces the

impedance level of the resonator, and may be explored later for this application, but it

will be ignored in the present discussion.

It is useful to express these geometric limits in terms of the resonator FOMs used during

filter design. If the resonator bar is modeled as an L-C pair, two relevant parameters are

the resonant frequency w0 and the “impedance level”, defined as and

respectively. The impedance level is the magnitude of the impedance of either the

inductor or capacitor at resonance (they are equal), and is important for scaling the other

elements of the filter appropriately. In terms of material parameters and the bar

dimensions l, w, and a:

(5.38)

As it turns out, the BVD parameter R is equal to the impedance level divided by Q. This

resistance is also known as the “impedance level at resonance”. In general this term is a

78

pure parasitic and it is desirable to reduce it as much as possible. Since a common RF

matching impedance is 50 , that is a good order-of-magnitude target for R.

(5.39)

The impedance level at resonance is inversely proportional to both Q and w. Q is

determined by quality of the manufacturing process and higher-order loss mechanisms, so

it will be treated as a given parameter for now, and estimated at the target minimum value

of 10000. Now given a bar thickness and resonant frequency, the maximum width and the

minimum R can be calculated as shown in Tables 5.12 and 5.13.

2a (m) lmax (m) w0 (MHz) w (m) R ()

0.2 35 138 ≤ 23 ≥ 9.87

0.5 40 121 ≤ 27 ≥ 21.0

1.0 50 96.7 ≤ 33 ≥ 34.4

Table 5.12: Impedance level at resonance achievable for minimum frequencies.

Tether width (m) lmin (m) w0 (GHz) w (m) R ()

0.2 1.0 4.83 ≤ 0.667 ≥ 851

0.5 2.5 1.93 ≤ 1.67 ≥ 340

1.0 5.0 0.97 ≤ 3.33 ≥ 170

Table 5.13: Impedance level at resonance achievable for maximum frequencies. Bar thickness 2a was set to 0.5 m.

The target impedance at resonance of 50 is difficult to achieve without going to lower

frequencies. Advanced commercial technologies can produce smaller thicknesses and

impedances than those discussed here, but real-life deviations from these ideal analyses

must also be expected to reduce such gains. One way to address this issue is to increase

the impedance level of components interfacing to the filter to 200 – 400 , as has already

been demonstrated in some integrated designs. Another possible solution is to fabricate

two identical resonators in parallel with each other. The resulting pair has an equivalent

impedance of a single resonator with half the resistances and inductances and twice the

79

capacitances, using the BVD model. Thus the resonant frequency remains the same while

both the impedance level and the impedance at resonance are halved.

5.4.2 Manufacturing Tolerances and FOM Sensitivities

It is important to consider the effects of the imperfect machining capabilities of whatever

equipment is used to fabricate a device. Each physical parameter is subject to finite

manufacturing tolerances, representing the precision with which the machinery can

achieve a target value for that parameter. In the case of the Draper resonator, it was a

specific design goal to have the various resonator properties defined lithographically,

over a wide range. Thus any error in producing the correct bar dimensions will translate

strongly to error in meeting the desired resonator characteristics. The dependence of a

design parameter q on a fabrication parameter f maybe be expressed as the sensitivity S:

(5.40)

which compares the relative change in q due to a relative change in p. Table 7 lists the

sensitivities of the BVD circuit parameters to bar geometry.

L C R C0

l 1 1 0 1

w -1 1 -1 1

a 1 -1 1 -1

Table 5.14: Sensitivities of equivalent circuit parameters to bar parameters l, w, and a.A sensitivity of one indicates that a 1% change in the independent parameter causes a 1%

change in the dependent parameter. Since the geometric parameters were originally

intended to define the resonator characteristics, the dependence is strong; thus filter

designers should be aware when using this resonator that they are limited by the

tolerances of the technology used.

Typical tolerances with current technology are 1% for thickness (a) and about 2% for

small (~3-5 m) lateral dimensions across the wafer (l, w). These tolerances translate into

80

a 2% difference in resonant frequency w0 between different devices, and a 3% difference

in impedance level . Various tuning methods are under investigation, such as

trimming some mass from the bar corners after fabrication, applying a DC voltage across

the resonator, and coupling multiple resonators mechanically to force them to lock on to

the same resonant frequency even if they are slightly different in length.

5.5 Filter Design

The preceding sections have outlined the capabilities and limitations of the Draper

resonator, and given some examples of how it may be used in a filter. This section will

summarize the process of designing a filter around this resonator, from an application-

specific point of view. In this case a set of filter specifications is the starting point, such

as a center frequency, minimum bandwidth, and load impedance, specified by a particular

application. In addition, a set of constraints is provided by the capabilities of the Draper

resonator and the technology being used to fabricate it. In general, both sets of

specifications will allow some leeway in the exact resonator characteristics, but some

back-and-forth iteration will be required to find a solution that meets all the requirements.

5.5.1 Filter-level Resonator Requirements

When designing the filter, a topology is chosen and element values are selected for the

electrical components. At this level the designer uses an electrical model of the resonator

and must take into account whatever restrictions the chosen resonator places on the

possible values for each component. For the Draper resonator, it is important to note that

there are fundamentally only two degrees of freedom in its BVD equivalent circuit. The

ratio of the motional and static capacitances is fixed at 1/31.04 (according to simulation),

and choosing L and C determines R through Q: . Thus a good way to define

the resonator at this level is with its resonant frequency and impedance level, which

together completely determine its geometry and other characteristics. Usually the

resonant frequency is established by the application, which leaves only the impedance

81

level free for adjustment. The other common filter-level properties of interest, Q and r,

have already been set by the process and technology, so the designer should confirm that

these values would allow the filter specifications to be met. Otherwise this resonator is

not suitable for the application.

Impedance Level

Given a resonator with a certain impedance level, the elements of a filter designed to use

it will have parameter values scaled according to that impedance level. If the resonator’s

impedance level were changed, the other filter components could be scaled by the same

factor to maintain the same filter characteristic. Thus an obvious way to choose an

impedance level is to meet a desired source or load impedance requirement.

Unfortunately, there are several other factors that change with impedance level, like area

and cost, and modal isolation.

The decision is usually simplified by consideration of parasitics, especially in a high-

frequency integrated system. In such systems an overriding concern is the effect of

parasitic capacitances. At high frequencies, these capacitances have small impedances

and steal signal current away by appearing as shorts. To counteract this effect, RFIC

systems usually try to minimize their operational impedance levels, with the goal of being

very small compared to the parasitic impedances. Using typical numbers, this usually

means for the Draper resonator that the smallest impedance level possible before modal

overlap becomes significant is chosen. In other words, there is a minimum load

impedance possible for each filter topology using this resonator, and it is usually

somewhat high so the minimum possible value is chosen. In summary, though this can

vary with application and technology, the filter designer generally does not have much

flexibility with this resonator, and it becomes a decision of whether this resonator will

satisfy the requirements as is.

5.5.2 Resonator Fabrication Tradeoffs

82

The existence of only two independent electrical parameters for the Draper resonator can

be explained mathematically by inspecting the equations relating bar dimensions to those

parameters. Width and thickness always appear in the form w/a, which means their only

contribution to determining the electrical parameters is through their ratio. Since the

resonant frequency is uniquely determined by the length, once again there is only one

degree of freedom for a given frequency. However since there are two parameters that go

into this ratio, the choices are not as limited as during the filter design stage.

The same constraints do apply here, such as the desire to minimize cost and area, and

reduce the impedance level. Both of these can be accomplished by maximizing w/a. The

width can only be increased until it reaches the limit of two-thirds the length to maintain

the chosen modal isolation. So the only other choice is to decrease the thickness. There is

an overall minimum set by the process technology on thickness, as well as the

consideration for multiresonator systems that all the thicknesses must be the same. In

practice this means that the thickness will likely be set wafer-wide as small as the

technology will allow. The relative magnitude of the width of each resonator in any

coupled topology can be calculated based on considerations of impedance-matching the

various stages, and the largest relative width will be set at the design maximum, which

minimizes impedance throughout the circuit while maintaining modal isolation. Thus,

once again for a given application and technology the design is likely to be constrained at

one of its limits.

5.5.3 Simple Ladder Filter Design Process

As an example of the filter design procedure, a step-by-step outline using the simple

ladder topology follows. First, the filter requirements must be checked to make sure they

are realizable with the Draper resonator: a center frequency between about 200 MHz and

2 GHz, and a bandwidth less than about 3% of the center frequency. An additional

requirement of this particular topology is that the percentage bandwidth be 1% at the

highest, and preferably less than 0.5%. Next, the filter requirements should be converted

to filter K and loaded Q values using a published table or analytic equation. One final

83

check before component selection can begin is to make sure the loaded Q is sufficiently

less than the resonator Q. The bigger the difference the better, with a minimum resonator

Q of three times the loaded Q to ensure an acceptable filter characteristic.

Equations to generate normalized k and q values for any number of filter stages can be

found in the Appendix. These parameters relate the impedance values of elements of

neighboring coupled filter stages, as defined in [25]. There is a k between each pair of

stages, so k1,2 relates the first and second stage, k2,3 relates the second and third stage, etc.

while there are usually only two q parameters, for the source and load stages. The

lowercase signifies values normalized to a fractional bandwidth of one, whereas

unnormalized values for specific bandwidths are usually written in uppercase. The

difference is just a multiplication or division by the fractional bandwidth as shown in the

Appendix. Throughout this thesis, the symbols K and Q are used to represent these values

for the two stage, symmetrical topology of Sec. 5.2.1.

When calculating the coupled filter parameters by hand, k and q are most easily obtained

by calculating an intermediate set of parameters called g-codes, used to design low-pass

filter prototypes (i.e. a low pass filter with the same bandwidth and rolloff). There are n +

1 g-codes for a n-order coupled filter.

To illustrate the process, here are two examples for a two-resonator filter, a Butterworth

response (zero passband ripple) and a 1 dB ripple Chebychev response. With n = 2 there

are 4 g-codes. For the Butterworth response, g0 and g3 are both 1, and g1 and g2 are:

(5.41)

(5.42)

With 1 dB ripple, g0 is still 1, and substituting values for the other codes gives:

(5.43)

Obtaining normalized q values from the g-codes is straightforward [25]:

(5.44)

84

(5.45)

For symmetric designs, qSOURCE = qLOAD. The Butterworth q is , and the 1 dB ripple q is

1.8219. Calculating k in general is slightly more complicated because the number of

coefficients increases with the order of the filter. Specifically, there are n – 1 coefficients,

each corresponding to a coupling impedance between two neighboring resonators, which

are numbered from 1 to n going from source to load. For a two-resonator design there is

only one k, and it is labeled k1,2 because it is between the first and second resonators.

(5.46)

This results in for Butterworth and 1.1171 with ripple. K and Q are obtained by

multiplying k by the fractional bandwidth, and dividing q by the same. Note that the

product kq is constant for all bandwidths given n and the passband ripple.

Continuing with the design process, the motional inductance and capacitance of the

resonator are defined as L and C, the impedance level as , and the loaded Q as

QL. All the other filter elements may be expressed in terms of w0 and Z0:

(5.41)

(5.42)

using the topology shown in Figure 3.3. Since w0 is usually given as a filter specification,

this leaves the choice of Z0 to be made. The possible values are bounded by maximum bar

width and minimum thickness on the lower end, proceeding up to very high values for

smaller widths. The choice within this range is determined by the desired interface

impedance, and possibly also to limit the size of C12. Once the value for Z0 is set, bar

dimensions can be calculated from w0 and Z0 and the design is finished.

Some further comments about filter K and QL: they are uniquely determined for a given

number of resonators combined with bandwidth and passband ripple or shape factor

specifications. One thing this means is that if bandwidth, passband ripple, and shape

85

factor are simultaneously specified, they may be impossible to realize without adding

additional resonators. Second, the equations generating K and Q ignore the effects of the

resonator impedance at resonance and its static capacitance. Thus the filter should be

overdesigned to exceed specifications to account for this. Finally, filter specifications are

usually given as a bounded value, for example, a minimum bandwidth, a maximum

passband ripple, or a minimum shape factor. If a filter can be designed that exceeds these

bounds, some flexibility in K and Q is present. For example, with some leeway in

bandwidth, Q and K may be changed to hit a more favorable value for load impedance

without falling out of specification.

5.6 Filter Bank Considerations

One of the most promising applications for an integrated, high-Q resonator is to make an

integrated filter bank. While it is premature at this point to carry out a full design, some

analysis can be made on the eventual feasibility of such a device, and some potential

difficulties may be enumerated.

5.6.1 Filter Bank Layout

One conceptualization of a filter bank is a set of individual filters connected in parallel

between active buffer stages, selected via electrical or possibly MEMS switches (Figure

5.16). Only one switch would be closed at a time, actuated by frequency-selection

circuitry. The maximum number of filter branches between two individual buffers would

be determined by the effect of parasitic loading of the additional paths and parasitics due

to increasing lengths and areas from laying out many multiresonator filters and switches

between the buffer pair.

86

Figure 5.16: One possible switched filter bank implementation.

5.6.2 Impedance Matching and Parasitic Loading

In such a topology, it is vital that each filter be designed for the same source and load

impedance, unless passive matching elements can be placed in series with each filter,

though it is desirable to eliminate matching elements if possible to save on die area and

cost. An advantage of fully integrated filters is the impedance of the active circuits

interfacing with the filters may be adjusted to best accommodate the filter design, for

example at load values of 1000 or more, where a higher impedance might be desirable

to minimize relative losses through parasitic resistances. The limit to this flexibility arises

from the parasitic capacitances loading each stage. Each switch is not perfect, and

presents some finite capacitance while open. Additionally there is some capacitance

linking adjacent resonators. These issues are present with single filter circuits, but are

multipled in severity when several filters are connected in parallel in a filter bank. To

prevent too much loss through these parasitic capacitances, the impedance level of each

stage should be kept low to compete favorably for the signal current. Since the parasitic

capacitances in general will be much higher within a filter bank, higher frequencies will

also be harder to achieve compared to a single filter circuit since the frequency at which

these capacitances annihilate the signal will be that much lower.

5.6.3 Harmonic Mode Cancellation

87

Another problem that is present for single filters, but particularly troublesome in a filter

bank is the existence of spurious resonant modes. Such modes would pass signals that

may be important to attenuate for a given application. One example is the width

longitudinal mode, which approximately is governed by the same equations as the length

mode, except length and width are switched. A possible solution to this additional

passband is to use multiple resonators with the same length mode frequency but differing

width mode frequencies. Wherever one resonator is used in a filter, another is added in

series. In theory, the length resonance of the first resonator will overlap fully the

resonance of the second resonator and pass signal at that frequency as originally intended,

while the two dissimilar width resonances should mostly cancel. In a coupled

multiresonator filter topology such as the simple ladder, this cancellation can be designed

into the two existing resonators to achieve this effect.

Figure 5.17: The Butterworth Van-Dyke model with an additional resonance.

Simulation

A numerical simulation to test this solution was performed using the simple ladder filter

topology described in Section 5.2.1. The resonators were modeled as shown in Figure

5.17. By adding another R-L-C branch in parallel with the BVD model, an additional

resonance/anti-resonance pair (corresponding to the width mode) is added to the

frequency response. One resonator was designed with branches with resonant frequencies

at 780 MHz (length = 6.19 m) and 860 MHz (width = 5.62 m), while the second had

frequencies of 780 MHz and 944 MHz (width = 5.12 m), which corresponds to a width

half a micron less than in the first resonator. A reference filter using the wider resonator

88

twice was also simulated for comparison. The other filter parameters and element values

are listed in Table 5.15.

Figures 5.18 and 5.19 show the wideband response of the two filters. In the reference

filter, the minimum loss at both peaks is about 6.5 dB. On the other hand, the mode

canceling filter maintains a primary insertion loss of about 6.5 dB while attenuating the

spurious peaks by about 19 dB and 21 dB. Doubling the total number of resonators

should also approximately double this attenuation while not increasing the primary

insertion loss by too much, although precise matching would become more difficult.

Overall this method of cancellation appears to be a worthwhile solution to unavoidable

spurious resonances, especially if multiple resonators were already called for to meet

attenuation or shape factor requirements.

Filter Parameter w0 RS, RL QL K C12

Value 780 MHz 2016.2 500 0.002 101.1 fFTable 5.15: Simple ladder filter simulation parameters.

89

-270

-150

-100

-50

0

Filter Transmission (dB)

700 750 800 850 950 1000-180

-90

0

90

Phase (degrees)

Figure 5.18: Filter transmission characteristic with spurious resonance

Figure 5.19: Filter transmission characteristic with cancelled spurious resonance.

Chapter 6

Summary and Conclusions

6.1 Summary of Thesis Work

Draper Resonator

90

900Frequency (MHz)

-120-100-80

-60

-40-20

0

-270

Filter Transmission (dB)

-180

-90

0

90

Phase (degrees)

700 750 800 850 900 950 1000Frequency (MHz)

The design goals behind the Draper resonator were presented. It was intended to satisfy a

strong need in the field of RF communications for a high-performance, GHz frequency

resonator compatible with integrated silicon processes. After a brief review of competing

technologies, it is clear that if the optimized resonator can meet performance

specifications, it can fulfill this purpose. The approximate first-order behavior was

derived and found to be equivalent to the Butterworth Van-dyke circuit model about

resonance. The approximate range of resonant frequencies achievable with current

technologies are 200 MHz to 2 GHz, with a range of impedances at resonance of

50~1000 possible if the target Q of 10,000 is reached.

Measurement and Modeling

A procedure for measuring the frequency response of resonators at RF frequencies and

extracting equivalent circuit parameters from these measurements was devised and tested

on optimized TFR resonators. The models and the fitting routine produced an acceptable

fit to the data. However, measurement of the first Draper devices was hindered by a large

parasitic and data could only collected on a small number of extremely large resonators.

In addition, due to this parasitic and the unoptimized experimental process, fitting was

not possible with the developed methods due to significant additional parasitics which

those methods did not account for. Hand extraction of resonant parameters estimated an

achieved Q of 542 at a resonant frequency of 148 MHz.

Filter Design

Finally, implementation of the Draper resonator in three filter designs was explored. The

different filter topologies were first analyzed generally and their tradeoffs and capabilities

discussed, and then numerical examples with typical parameters expected of the Draper

resonator calculated and presented. Some of the issues concerning single filter design

were extended to a filter bank in concept and the ramifications discussed.

6.2 Future Investigations

91

Optimization and refinement of the capabilities of the resonator will proceed, hopefully

resulting in high Q, temperature stability, and methods of fine-tuning the resonant

frequency. Additional process investigations will focus on precisely matching multiple

resonators on the same wafer to each other. One method to accomplish this may be direct

mechanical coupling of several resonator bars while keeping the various electrical

connections distinct. This will obviously be very important in the design of

multiresonator filters and filter banks.

Another important goal in the next stage of resonator improvements should be lowering

its impedance level. As it is currently, integration at high frequencies with active devices

will be difficult due to parasitic capacitances competing with the high impedances levels.

A promising avenue of research to resolve this is to make the width much wider than the

frequency-determining length. This would reduce impedance by the same factor that is

used to increase the width at the cost of moving the spurious width resonance modes to

lower frequencies. However, higher-order modes of these longitudal resonances are

attenuated proportionately to the mode number, so by placing the larger modes far

enough below the functional resonance, they may be eliminated through associated

circuitry.

Finally, once single resonator characteristics have been optimized, the issue of acoustic

isolation must also be addressed to allow multiple resonators to function on the same

substrate without interfering with each other.

92

93