David A. Feld

After 60 years: A new formula for computing quality factor is warranted

A new formulation for computing the unloaded quality factor (Q) of a resonator or of any terminated electromagnetic or piezoelectric cavity from its measured S parameters is proposed. This formulation, based on the work of Dicke, and Bode, computes Q at a continuum of frequencies from below the resonant frequency (fs) to above the anti-resonant frequency (fp). Sixty years ago Dicke and Beringer derived a pair of formulations for Q in terms of the measured impedance (Z) and admittance (Y). The new formulation has two important advantages over Dickes formulations: (1) Even for resonators that are free of spurious resonances - whose S-parameters are near ideal -- Dickes Z and Y based formulations yield erroneous Q values over a range of frequencies (called ldquodead zonesrdquo) in the vicinity of the frequencies at which the functions Z and Y are singular (at fp and fs respectively). Since the S parameters of such an ideal resonator do not have any singularities, the new Q formulation provides reliable Q values over the full range of frequencies. (2) For non-ideal resonators exhibiting spurious lateral-mode resonances in certain frequency ranges, both the new and Dicke formulations report Q values that are clearly erroneous since they oscillate from positive to negative. Dicke does not explain why his formulation breaks down in such regions nor have we found such an explanation in the literature. In contrast to this, this work explains why the S parameter based Q formula is invalidated over regions of frequency space in which ln(mag(S)) fluctuates wildly. We show that in such regions that pre-conditioning of the S-parameter data with a Gaussian window average is useful in yielding a meaningful value for Q. We propose that the S parameter based Q formulation, being devoid of ldquodead zonesrdquo should be used as a standard by which piezoelectric resonators fabricated in different technologies e.g. FBAR, SAW, BAW be compared.

A wafer level encapsulated FBAR chip molded into a 2.0 mm /spl times/ 1.6 mm plastic package for use as a PCS full band Tx filter

A high performance 2.0 mm /spl times/ 1.6 mm /spl times/ 1.0 mm full band filter is demonstrated. The small filter size is achieved by: 1) encapsulating the die in a chip scale hermetic package; and by 2) molding that chip scale package into a plastic chip-on-board package. This new packaging scheme reduces the filter cost and allows for further size reduction. The filter has a minimum insertion loss of /spl sim/ 1 dB, is guaranteed to have a 3.5 dB worst-case insertion loss in the Tx band (1850-1910 MHz), and typically exhibits 35 dB of rejection in the Rx band (1930-1990 MHz). The filter comprises high-Q FBAR resonators that enable it to make a sub-10 MHz transition from the pass to stop band. Hence the required rolloff within the 20 MHz (1%) guard band can be achieved while accounting for both manufacturing variations and operating temperature drifts. The filter achieves /spl sim/ 25 dB of rejection at the second harmonic (3700-3820 MHz) and typically 15 dB of rejection at the third harmonic (5550-5730 MHz).

Employing a ground model to accurately characterize electronic devices measured with GSG probes

Traditionally, when measuring an electronic device, the nonideal (non-50 /spl Omega/) electrical behavior of the ground-signal-ground probes is removed through calibration. However, this procedure does not allow for an accurate measurement of devices that exhibit an unbalanced flow of electrical currents through the two ground fingers of the probe. We found that a simple interface circuit can be used by a circuit simulator, such as ADS, to reproduce the measurements of devices in which unbalanced ground currents flow in the return paths. A simple experimental method to determine the interface circuit is given.

Low insertion loss, high rejection handset duplexer for UMTS-1 (WCDMA) band

A manufacturable UMTS-1 band handset duplexer is developed (Tx band: 1920 - 1980 MHz, Rx band: 2110 - 2170 MHz) using thin film bulk acoustic resonator (FBAR) technology. The worst case Tx-ANT and Rx-ANT insertion loss is typically better than 1.5/2.0 dB respectively over an operating temperature range of -30C to +85C. Over this temperature range the Tx-ANT & Rx-ANT rejection are greater than: 46/51 dB respectively. The Tx/Rx isolation are greater than: 48/51 dB respectively. The second harmonic rejection for both Tx-ANT and Rx-ANT is greater than 20 dBs. Bluetooth rejection (@2.4 - 2.5 GHz) for both Tx-ANT and Rx-ANT is greater than 30 dBs. The duplexers low insertion loss and high rejection/isolation enables 3G handsets to operate with low transmit power and high receiver sensitivity in a small package size of 3.8 mm x 3.8 mm x 1.4 mm. Real estate is at a premium in a 3G handset, given the large volume of hardware that is required for analog and digital signal processing. This places the FBAR duplexer at an advantage relative to that of ceramic duplexers which are capable of delivering similar RF performance but in a substantially larger package volume. Although UMTS-1 duplexers designed using SAW technology occupy a similar volume to that of the FBAR duplexer, the high Qs of the FBAR resonators comprising the FBAR duplexer give it a favorable insertion loss to rejection (& isolation) tradeoff over that of SAWs. The spec and size constraints, as well as the topologies that were considered that led to the final design are discussed.

Unbalanced device comprised of FBAR resonators

A splitbar is comprised of two identical thin film bulk acoustic (FEAR) resonators, whose electrodes are cross connected such that the C-axes of the piezoelectric films are parallel/anti-parallel to an applied electric field stimulus. The RF response of a splitbar is near identical to that of a single resonator, but unlike the single resonator, because the resonators are cross wired, their H2 responses annihilate one another enabling the splitbar to have extremely low harmonic emissions. If, however, the series resonances of the two resonators are placed at slightly different frequencies, a narrow LC resonance, whose center frequency is in between the series resonance frequencies, is formed, and the splitbar now exhibits a large H2 response at twice the center frequency of the LC resonance. To study this resonance, we vary the frequency spacing by biasing the splitbar with a dc voltage. We explain why the H2 power emitted by the splitbar is: (1) is proportional to its stored energy, (2) increases strongly with resonator frequency separation, and (3) increases as resonator Q is increased. We also analyze the powerbar, which is the dual of the splitbar, and which exhibits analogous behavior.

Vectorial measurement of the 2 nd harmonic response of an FBAR resonator

We have measured the phase and magnitude (“vector”) response of the 2nd harmonic emissions of a thin film bulk acoustic (FBAR) resonator. The phase measurement is the difference in phase between the emitted 2nd harmonic waveform and that of the incident waveform. Although the magnitude has been measured previously by various investigators [1], [2], we are not aware of a phase measurement prior to this work. The phase measurement presents a challenge because it requires an elaborate measurement setup with a sophisticated calibration procedure. Using Agilents non-linear vector network analyzer (NVNA) we have measured the standard S11 response of a resonator as well as the vector response of its 2nd harmonic emissions. We have observed that: (1) phase vs. frequency plots of the 2nd harmonic have the same shape as the phase of S11 and (2) the magnitude vs. frequency plots of the 2nd harmonic are proportional to the group delay of S11.

A generic passive matching network for multiplexers

A systematic approach to matching an arbitrary number of bulk acoustic wave or surface acoustic wave filters is proposed. The filters can be modeled as capacitor networks at frequencies above and below their passbands. The generic matching network consists of a single shunt inductor (or a transmission line) and series capacitors in front of each filter.