Design of RF BAW-Based ΣΔ Modulators

Diomadson Belfort∗, Sebastian Y. C. Catunda∗ and Hassan Aboushady †

∗Automation and Computing Engineering Department

Federal University of Rio Grande do Norte†University of Pierre and Marie Curie - UPMC, Paris VI

Laboratory of Informatics of Paris 6 - LIP6

Email: diomadson@ieee.org, catunda@dca.ufrn.br, hassan.aboushady@lip6.fr

Abstract—Analysis of the Bulk Acoustic Wave (BAW) res-onator modeled as a two-port device is presented for applica-tions in Sigma Delta Modulator (ΣΔM) based Analog-to-DigitalConverter (ADC). The design of BAW-based Continuous Time(CT) ΣΔM starting from a Discrete Time (DT) model, whichis designed from scratch, is presented and explained in details.In order to convert a DT ΣΔM into CT BAW-based ΣΔM,Finite Impulse Response (FIR) filters are used in the feedbackpath increasing the degrees of freedom. We discuss and validatethe calculation of the FIR filter coefficients by using a designexample of BAW-based ΣΔMs.

I. INTRODUCTION

Software-Defined Radio (SDR) was envisioned in the late

1990’s. Since then, a lot of effort has been taken to implement

a radio transceiver where most of the analogue functions, such

as down-conversion and channel-selection, are moved to the

digital domain. With these features SDR could be programmed

and adapted to new standards.

CT ΣΔMs has been used to convert and digitize Interme-diate Frequency (IF) signals for Radio Frequency (RF) digital

receiver architectures [1]. Nonetheless, new attempts to push

the digitization of RF signals even further, allowing direct

digitization and moving most of the signal processing to the

flexible and programmable digital domain has been proposed

using RF CT ΣΔMs [2].The main choice when to implement RF modulators are LC

filters [2]. Transconductors are normally employed to couple

resonators, with the side effect of increasing the noise and

power consumption, and also limiting the linearity of the filter.

BAW resonators appear as candidates to replace that filters,

due to their passive nature, high quality factor, no need for

additional power and also for its process compatibility with

silicon technologies [1].

Fig. 1 shows the modified Butterworth Van Dyke (mBVD)

model [3], which can be used as a 2-port or as a single-

port, where the second port is grounded. Even through BAW

resonators for ΣΔM application are modeled as a single-

port device. A compensation technique that makes use of the

second port was proposed [4], [5]. It is possible to verify that,

as the second port is grounded, that compensation technique is

not valid. Moreover, once compensated, this filter behaves as

a stop-band filter and not a bandpass as previously mentioned

in the literature [4], [5].

In the present work, that misinterpretation of the single-

port is investigated. It also presents analysis, compensation

technique and design example of ΣΔM having a compensated

2-port BAW device as loop filter.

For convenience, the naming convention used in the present

work refers to the numbers of ports, being the 2-port the model

shown in Fig. 1 and the single port this model when the second

port is connected to ground.

Rs

Rp Rm

Lm

Cm

Cp

Fig. 1. modified Butterworth Van Dyke (mBVD) model.

II. CRITICAL ANALYSIS OF THE BAW MODEL USED TO

DESIGN BANDPASS ΣΔ MODULATOR

In general, in the literature, BAW for ΣΔM has been

modeled as a single-port device [4], [5]. In this section we

carry on a critical analysis of this mode of operation. It will

be shown that the proposed compensation technique for this

topology is not valid and when it is correctly compensated it

behaves a stop-band filter not a bandpass as stated in [4], [5].

A. Analysis

The BAW device, when used as a single port device, has

its characteristic impedance described by:

ZBAW (s) =

(1

sCp+Rp

) (sLm +

1

sCm+Rm

)sLm +

1

sCp+ 1

sCm+Rp +Rm

+Rs. (1)

It is easy to show, by neglecting the resistances, that the real

part of zeros and poles are placed at:

ZBAW1p−Zeros=

{1√

Cm Lm

}(2)

ZBAW1p−Poles=

{0,±

√− Cp + Cm

Cp Cm Lm

}(3)

978-1-4799-2507-0/14/$31.00 ©2014 IEEE

0.5 1 1.5 2 2.5 3

x 109

0

10

20

30

40

50

60

70

Mag

nitu

de (

dB)

Frequency (Hz)

Fig. 2. Frequency response of single port BAW device (Rm = 0.65Ω,Lm = 79.4nH , Cm = 80 fF , Cp = 1.8 pF , Rp = 0.85Ω and Rs =

1.02Ω).

From Eqs. (2 and 3) it is possible to identify the pole

at origin, the resonance and antiresonance frequencies. The

magnitude frequency response for this system is presented in

Fig. 2. In [4], [5] this model was used to design a 2nd order

band-pass ΣΔM. Nonetheless, as the antiresonance frequencydegrades the Signal-to-Noise Ratio (SNR) and the pole at

origin degrades the quality of inherent anti-aliasing filter, a

compensation technique to mitigate those was proposed in

[5]. In this compensation an inverted signal is applied to a

compensation capacitor, Cc, and added to the signal from the

BAW. However, knowing that a single port device has its input

and output on the same node, one can conclude that the device

was used as a 2-port, meanwhile the equations for the single

port were maintained. That misinterpretation of the models,

invalidates such a compensation technique.

B. Antiresonance Compensation

The antiresonance, due to the Cp capacitance, provides a

peak in the impedance response and changes the phase from

−90◦ to +90◦, which can bring instability issues to the ΣΔM.Moreover, with the existence of antiresonance, it is difficult to

obtain the desired transfer function for the continuous-time

ΣΔM [6].

For the single port BAW the antiresonance frequency can

be canceled by putting a “negative capacitor” in parallel to

it. A negative capacitance can be emulated by using the

circuit shown in Fig. 3. If one considers that the Operational

Amplifier (OPAMP) has a very high gain, it is easily shown

that:

V+(s) = Vout(s)R2

R1 +R2

(4)

and that the input current, Iin(s), is given by:

Iin(s) =−Vout(s) R1

R1+R2

ZCF (s)(5)

where, ZCF (s), is the impedance of the capacitor CF and

given by:

ZCF (s) =1

sCF

(6)

−

+

Negative Capacitor

CF

R1

R2

B1I1

Zin

Vout

Fig. 3. Antiresonance cancellation of a single port BAW using a negativecapacitance in parallel.

From Eqs. (4) and (5), the input impedance of the negative

capacitor can be found as follows:

Zin(s) =V+(s)

Iin(s)= −ZCF (s)

R2

R1

(7)

The frequency response of the compensated single port BAW

is shown in Fig. 4. One can see that the pole at origin wasn’t

compensated. Actually the only way to do it is adding a low-

pass filter. It can also be observed that the filter is a stop-band

not a band-pass. It is rather difficult to implement a band-pass

ΣΔM using a stop-band filter, concluding that this topology

is not well adapted to design band-pass ΣΔM.

0.5 1 1.5 2 2.5 3

x 109

0

10

20

30

40

50

60

70

80

Mag

nitu

de (

dB)

Frequency (Hz)

Fig. 4. Frequency response of compensated single port BAW device (Cf =

Cp = 1.8 pF ). Notice that the pole at origin was not compensated and itbehaves as a stop-band filter, not a band-pass.

III. BAW MODELED AS A TWO-PORT DEVICE

In this section we carry on the modeling of the BAW as

a two-port device. It will be shown that in such a mode the

compensated BAW behaves as the desired bandpass filter.

A. Analysis

In order to analyze the BAW in the 2-port mode, a non-zero

load is connected to the second port, as illustrated in Fig. 5.

One can see that this circuit is nothing more than a voltage

divider, being the output voltage, VOUT , defined as:

VOUT (s) = V1(s)RL

RL + ZBAW (s). (8)

From Eq. (8) one can conclude that, as ZBAW is on the

denominator, the zeros from Eq. (1) will become the poles

and vice-versa, but in different position, depending on RL.

The BAW when used as a 2-port device has no pole at origin,

instead it has a zero close to it. One can realize that use

a load resistor degrades the quality factor of the filter. In a

ΣΔM loop, that degradation raises the noise floor, limiting

the maximum achievable SNR. In order to circumvent that

issue, a Transimpedance Amplifier (TIA) can be used sensing

directly the current that flows through the BAW device, which

is given by:

I1(s) =V1(s)

ZBAW (s). (9)

B1

V1

Rs Rm Cm Lm

Rp Cp

RLI1

VOUT

Fig. 5. Circuit used to analyze the BAW in a 2-port mode, where a loadresistor was connected to the second port.

B. Antiresonance Compensation

For the two-port a “negative capacitor”, as that one used for

the single-port, can be used to compensate the antiresonance

frequency. However, the effect of Cp can be generated in

an additional branch and subsequently removed from the

resonator output. That can by done by using a differential

signal that flows trough a capacitor, as shown in Fig. 6. A

TIA is used for the summation and as it has ideally a zero

input impedance, it does not degrades the quality factor of the

filter. This topology seems to be more attractive in terms of

simplicity and probably with lower power consumption. The

transfer function of that topology is defined as:

T (s) = −RF

ZBAW (s)− Zcc(s)

ZBAW (s)Zcc(s)(10)

where, Zcc(s), is the impedance of the capacitor Cc and given

by:

Zcc(s) =1

sCc

. (11)

−

+

V1

V2

B1

CC

RF

VOUTA

-A

Fig. 6. Antiresonance cancellation of a 2-port BAW using a capacitance anda TIA.

The frequency response of that topology is shown in Fig.

7. It behaves as a band-pass filter in the zone of interest,

making it well suitable to be used as loop filter of a band-

pass continuous-time ΣΔM.

0.5 1 1.5 2 2.5 3

x 109

−20

−10

0

10

20

30

40

50

60

Mag

nitu

de (

dB)

Frequency (Hz)

Fig. 7. Frequency response of compensated single port BAW device. Noticethat there is no pole at origin and it behaves as a band-pass.

IV. DESIGN OF ΣΔ MODULATORS USING BAW

This section describes how the compensated BAW device

can be used as loop filter of a continuous-time ΣΔM. The goalis to start from a discrete-time model and obtain a continuous-

time equivalent model, having a compensated BAW device as

loop filter, conceptually illustrated in Fig. 8.

...

...

X(s) Y (z)

NRZ

FIRMe−tdsFIRC

BAW Filter

u(z)

v(s)

α0α1α2αn

z−1z−1z−1

Fig. 8. Model of a ΣΔM using a compensated BAW device as loop filter.Notice that the feedback paths has FIR filters.

A. Band-pass ΣΔ Modulator

The BAW device, when used in the 2-port configuration and

having its antiresonance frequency compensated, behaves as a

band-pass filter. That makes it proper to be used as the loop

filter of a band-pass ΣΔM.The design of the band-pass ΣΔM starts with the definition

of the low-pass discrete-time Noise Transfer Function (NTF),

the loop gain can then be obtained by using Eq. (12) and

finally the transformation z → −z2 is applied. Using sucha transformation into Eq. (13), one can obtain the band-pass

loop gain as follows:

HLG(z) =1−NTF (z)

NTF (z)(12)

HLGBP(z) =

−z−2

1 + z−2. (13)

which has its poles placed around 1/4 of the sampling

frequency and a Non-Return-to-Zero (NRZ) Digital-to-Analog

Converter (DAC). Taking into account the loop delay of

0.4Ts, the discrete-time equivalent for the continuous-timeloop gain using a 2-port BAW (Rm = 0.65Ω, Lm = 79.4nH ,Cm = 80 fF , Cp = 1.8 pF , Rp = 0.85Ω and Rs = 1.02Ω)obtained with c2d function, which is a Matlab function thatconverts a system from continuous to discrete time, and it

gives the following FIR coefficients:

FIRm = 2.20− 13.91 z−1 (14)

FIRc = −0.09− 0.41 z−1. (15)

One can observe that two coefficients are needed per branch,

0 0.5 1 1.5 2 2.5 3 3.5 4

x 109

−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency (Hz)

PS

D (

dBv)

Discrete−TimeContinuous−Time

Fig. 9. Output spectrum of the discrete-time ΣΔM and its bandpass BAW-based continuous-time counterpart, with and without the lossy components.

this number is manually increased when then solution of

the linear system has a relatively high Mean Squared Error

(MSE). The output spectrum of the discrete-time ΣΔM and

its continuous-time band-pass BAW-based ΣΔM counterpart,

with and without losses, is shown in Fig. 9. A good agreement

between the discrete-time model its continuous-time coun-

terpart was obtained, as shown in Fig. 10. The maximum

SNR obtained was about 54 dB for all models for a sampling

frequency of 8GHz and an Oversampling Ratio (OSR) of 64,which means a Bandwidth (BW) of about 62MHz.

V. CONCLUSION

The BAW resonators are attractive due their passive nature

and compatibility with Complementary Metal Oxide Semi-

conductor (CMOS) process. The analysis of the BAW device

using a single or two port was presented. This analysis was

motivated by misinterpretation of the mBVD model found in

the literature and, could lead to wrong design and consequently

waste of time. All compensation techniques were validated by

using Simulation Program with Integrated Circuit Emphasis

(SPICE) simulator. Finally, the design of BAW-based 2nd order

ΣΔM that makes use of the two-port and compensated by a

single capacitor driven by a differential signal was presented.

The maximum SNR obtained was equivalent to that of its

discrete-time counterpart.

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0−10

0

10

20

30

40

50

60

Input (dBv)

SN

R (

dB)

Continuous−TimeDiscrete−Time

Fig. 10. Input voltage versus SNR of the designed band-pass BAW-BasedΣΔM and its discrete-time counterpart

ACKNOWLEDGMENT

The authors would like to thank the CNPq (Brazilian Na-

tional Council for Scientific and Technological Development)

for its support to this work.

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