Measurement of the acoustic output power of transducers in burst mode and derivation of the results to the continuous wave (CW) case reduces heating problems during power measurements with radiation force balances and absorbing targets at high power levels, but requires the knowledge of an “effective duty factor,” DR_{eff}. In this work, an alternative method for determining DR_{eff} is presented that allows the determination at any input voltage amplitude as it can be calculated from the input voltage rf signal in burst mode. Thus with this method, it is not necessary to apply CW signals at all.

## I. Background

The acoustic output power is one of the most fundamental parameters for the characterization of any ultrasound emitting device. Radiation force balances with absorbing targets are the gold standard for the measurement of this value, and in many countries, this particular setup is the national primary standard for this purpose. However, the measurement of extremely high powers in continuous wave (CW) mode, as they are used in some therapeutic applications [e.g., up to 400 W sonication in CW mode for the treatment of solid tumors in different locations using high-intensity focused ultrasound (HIFU) (Ref. 1)], with a radiation force balance with an absorbing target is a challenging task—the strong heating effects within the absorbing target due to the large amount of absorbed energy may, on the one hand, significantly increase the measurement uncertainties and, on the other hand, even destroy (i.e., melt) the target.

One possible method of reducing the heating effects and the resulting problems is to measure in burst mode instead of CW mode but with the same input amplitude and to derive the values for CW mode instead: If bursts with a known “on”-time *t*_{on}, a known pulse repetition period *prp* and hence with a known nominal duty ratio *DR* = *t*_{on}/*prp*, are applied, then the output power in CW mode in a first view simply equals the ratio of the power in burst mode and the duty ratio

However, this simple formula does not take into account possible transient effects at the beginning and the end of each burst. It has been particularly shown that ignoring these transient effects by using Eq. (1) may lead to an over- or underestimation of the derived CW output power by up to 10% and that the influence of these effects depends on the particular burst settings as well as on the employed devices like the function generator, the amplifier and the transducer.^{2}

It has thus been suggested in the literature to use an effective duty ratio *DR*_{eff} instead of the nominal one that inherently accounts for the transient effects for the particular used conditions.^{3} This effective duty ratio can be obtained as the ratio of the output power for a comparably low input amplitude *U*_{in,low} (i.e., an amplitude for which the measurement of the resulting output power in CW mode is uncritical) in burst mode to the output power for the same input amplitude in CW mode,

Using the effective duty ratio obtained with the procedure mentioned in the preceding text for the derivation of CW power values for high input amplitudes, however, implicitly assumes that the effective duty ratio itself does not depend on the input amplitude. Although no significant change of *DR*_{eff} for input voltages up to 123 V (peak-to-peak) and resulting acoustic powers up to 30 W has been observed elsewhere,^{3} it might be questionable whether this remains valid for other burst settings, other devices, and especially for much higher input amplitudes and output powers.

## II. Methods

In this work, an alternative method for determining *DR*_{eff} is presented that is based on the principle that is used for hydrophone measurements: With typical hydrophones, it is not possible to measure CW signals due to possible reflections in the water tank and electromagnetic pick-up distortions. Hence, if time-averaged values for a CW signal are to be obtained from hydrophone measurements (e.g., the time-averaged intensity *I*_{TA}), burst mode signals are measured instead, and the parameters are derived from a steady-state portion of the burst where the transient effects have disappeared. Because radiation force balances are too slow for transient measurements of the acoustic power, this method cannot be used directly with acoustic power measurements. Instead a similar procedure with the transient input voltage signal is suggested here: From a measured rf input voltage signal in burst mode, *DR*_{eff} can be calculated as the ratio of the temporal integral of the squared input voltage rf signal in burst mode to the temporal integral of the squared voltage rf signal in CW mode, where the latter can be derived by considering an integral number of cycles from the steady-state portion of the burst mode signal as well,

where both *t*_{1} and *t*_{2} are in the steady-state regime of the waveform and Δ*t* = *t*_{2} − *t*_{1} is an integer multiple of the oscillation period. While this method generally works with picking one steady-state wavecycle (i.e., Δ*t* being one oscillation period), picking several steady-state wave cycles will reduce noise effects. Figure 1 illustrates this method—an rf input voltage signal in burst mode is recorded (top) and squared (middle). For the numerator in Eq. (3) (the “burst” part), this squared signal is integrated over one *prp*, while for the denominator (the “CW” part), it is integrated only over one or a few steady-state wave cycles and then scaled to one *prp* as demonstrated in the bottom diagram of Fig. 1.

For the power measurements in the next section, the German national primary standard radiation force balance at Physikalisch-Technische Bundesanstalt (PTB) was used and is described elsewhere,^{4} and for the electrical measurements, a 100:1 voltage probe (model TT-HV 150, Testec, Frankfurt, Germany) was attached to an oscilloscope (model DPO 7104, Tektronix, Beaverton, OR). HIFU signals were created using an arbitrary function generator (model AFG 3101, Tektronix), an rf amplifier (model 1040L, E&I, Rochester, NY), and a HIFU transducer (*f*_{0} = 1.06 MHz, model H-101, Sonic Concepts, Bothell, WA).

## III. Results

The first question to be investigated here is whether the effective duty ratio obtained from the rf input voltage signal (the “electrically determined effective duty ratio,” abbreviated *DR*_{eff,el} in the following) is equivalent to the one obtained from small amplitude acoustic power measurements as described elsewhere^{3} (the “acoustically determined effective duty ratio,” abbreviated *DR*_{eff,ac} in the following). For this purpose, both values have been measured for a series of burst signals with a fixed *prp* of 1 ms and an increasing duty ratio. The results, shown in Fig. 2, clearly show that *DR*_{eff,el} and *DR*_{eff,ac} agree within 1% (except for one point), which is on the order of the typical random variations for this type of measurement. Besides, the results in Fig. 2 show that the deviation of the effective duty ratios from the nominal ones (i.e., the deviation from “1” for the ratios *DR*_{eff}/*DR*_{nom}) increases with decreasing nominal duty ratio. This agrees well with what can be expected because the influence of the transient effects should increase with decreasing number of steady-state wavecycles.

Another question concerns a possible dependence of the effective duty ratios on the input amplitude. For this purpose, another series of measurements was performed, where *DR*_{eff,el} and *DR*_{eff,ac} were obtained for increasing input voltages and for a constant nominal duty ratio of *DR*_{nom} = 0.0283. The results, given in Fig. 3 show a clear tendency for both values to decrease with increasing voltage. The obviously slightly different slopes for this decrease for the two values might require further investigation as this might already be due to heating effects in the measurements in CW mode that are indispensable for the determination of *DR*_{eff,ac}. However, the difference between the two values again is on the order of 1% only.

## IV. Discussion

Determination of an effective duty ratio from the rf input signals in burst mode seems to be an appropriate method for the investigated conditions. The values obtained with this method are in good agreement with those obtained with a method suggested in the literature. However, this agreement is not universally valid of course—using an electrically determined effective duty ratio to upscale the acoustic power implicitly assumes that the input voltage and the output intensity show similar burst characteristics, especially with respect to the transient effects at the beginning and the end of each burst. This is of course dependent on the electroacoustic characteristics, which might be different from transducer to transducer. This issue needs further investigation. Nevertheless, the method presented here might be helpful for cases where it is not possible to apply CW signals for any reason because it does not require application of CW signals at all. Furthermore this method can be used to reduce uncertainty contributions due to heating effects in high output power measurements, for instance, of HIFU transducers, using radiation force balances with absorbing targets.

Additionally, it was found for the investigated conditions that the values of the effective duty ratios obtained from both methods (acoustically and electrically) seem to depend on the input voltage. From this result, it follows that the method described in the literature to use an acoustically determined effective duty for high input voltages, which was obtained at lower input voltages, is debatable. However, this effect is also rather small (∼1% per 100 V) and needs further investigation.