May_EDFA_Digital

edfas.org ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 22 NO. 2 32 odically turned “on” and “off,” and the excitation signal can be considered as a square wave function, illustrated in the top image in Fig. 4. Looking at this signal from a system-theory perspective, it is noticed that the power spectrum of such function can be expressed as a recti- fied sinc-function as shown in the bottom image of Fig. 4, from which it becomes clear that such signal con- tains energy not only at the fundamental frequency, but also at even multiples (harmonics) of it. Translated into LIT-parameters, the fundamental frequency refers to the lock-in frequency of the measurement. Hence, the signal that is employed for exciting the sample does contain energy not only at the lock-in frequency but also at even multiples of it. From Fig. 4, however, it can also be seen that the energy the signal contains at these harmonic frequencies decreases with 1/n, with n being the number of the harmonic. Regardless, because the signal does contain energy at additional frequencies, appropriate processingmay be applied to extract this information. The describedapproach enables the opportunity todetermine amplitude and phase values in a spectrumof frequencies from just a single lock-in measurement, reducing the measurement time for estimating the phase-shift-versus- frequency behavior required for 3D-localization. Recently, an approach analyzing the time-resolved temperature response (TRTR) has been introduced. [6,10] Figure 5 shows the excitation signal in green color and the time-trace of the corresponding thermal response measured at a hot-spot in red color. The thermal responses of all lock-in cycles need to be averaged on-the-fly during acquisition to gain the full benefit of lock-in measurements and to ensure reliable data-handling. In this acquisition mode, a precise synchronization between the excitation and acquisition hardware has to be ensured. Upon acquir- ing a 3D time-resolved data set of the samples’ thermal response, Fourier-analysis is performed and the ampli- tude and phase values are extracted for all frequencies of interest up to half of the camera frame rate, respecting the Nyquist criterion. The amplitude and phase values estimated by Fast- Fourier-Transformation (FFT) are equal to those obtained by classical LIT, with the only difference that classical LIT requires repetition of the measurements at individual lock-in frequencies. The approach of acquiring the entire TRTR reduces the measurement effort as it allows extrac- tion of the phase-shift-versus-frequency characteristics froma singlemeasurement. The practical applicability of this approachwill however, dependon the achievable SNR at the individual frequencies and can vary sample-depen- dent with respect to the thermal attenuation, the appli- cable electrical power, and the electrical resistance value of the defect. The power spectrum in Fig. 4 shows that the power at the individual frequencies (harmonics in this case) decreases as frequency increases, resulting in a decreas- ing SNR that may prevent analyzing the thermal signal at all frequencies. To overcome this limitation, a novel excitation approach for LIT has been recently developed alongside with the implementation and evaluation of the TRTR analysis. [7,10] In this approach, custom and applica- tion specific signal shapes are designed and employed for repetitive electrical stimulation of the sample during the lock-in measurement. According to system theory, the transfer function of a linear time-invariant system (LTI-system) can be computed by normalizing the output to the applied input signal. In the case of a LIT measure- ment, the thermal response is normalized to the electrical excitation signal in order to obtain the system's thermal transfer function. This transfer function can be expressed as the spectra of amplitude and phase. Hence, accord- ing to system theory any signal should be applicable for excitation in LIT-measurements. This hypothesis was confirmed by practical investigations, tailoring excitation signals according to a sample's characteristics, the desired spectral range, and distribution for analyzing the TRTR. [7-8] Figure 6 contains a custom-designed, arbitrarily shaped signal that consists of multiple superimposed sinusoidal signals of differing frequencies. The graph at the bottom of Fig. 6 shows the power spectrum of the time signal. It can be seen that energy is contained at distinct frequen- cies distributed across the entire spectral range available (camera frame rate was 244 Hz → f max ≈ 120 Hz). Although this approach is generally valid, some boundary condi- tions may need consideration; for example, if too many frequencies are contained in the signal, the energy at the individual frequencies will decrease. Fig. 5 Waveform of the applied voltage signal and the re- sulting temperature at the sample surface at 0.5Hz. [6]