We introduce the self-heating effect into the time-domain simulation. Since the thermal effect will change the bias condition, it can influence the time-domain behavior, too. By the algorithm proposed in section 3.1.4, the circuit as plotted in Fig. 2.3 is simulated. The parameters are listed in Table 4.2 and the thermal resistance RT Hequals 1834.20◦C/W. Besides, the multiplier fac-tor M is set to be 3. The DC biases exercised in this section are IIN= 1.0 mA and VCC = 6.5 V. Here, the input excitation is a two-tone power source as written in Eq. (2.42). At base of the HBT, the node equation (2.35) becomes different for the voltage of this node becomes an new unknown. The utiliza-tion of power source in this simulautiliza-tion is because of the clearly identificautiliza-tion for input power level without additional calculation. The given input power is
−22 dBm per tone and the time-domain simulation is continued till 68.33 ns (123 cycles at 1.8 GHz). In contrast with our simulated result in Fig. 5.7(a)-(c), the output of HSPICE simulator is demonstrated in Fig. 5.8(a)-(c).
Time (sec)
4.6e-8 4.8e-8 5.0e-8 5.2e-8 5.4e-8
IC (A)
0.0332 0 2e-8 Time (sec)4e-8 6e-8
IC (A)
4.65e-8 4.70e-8 4.75e-8 4.80e-8
IC (A)
Figure 5.7: Zoom-in plots investigation for IC simulated by the developed solver.
Time (sec)
4.6e-8 4.8e-8 5.0e-8 5.2e-8 5.4e-8
IC (A)
4.65e-8 4.70e-8 4.75e-8 4.80e-8
IC (A)
Figure 5.8: Zoom-in plots investigation of IC simulated by HSPICE.
On the one hand, the result of our method is still stable and smooth. The usage of power source dose not affect the calculation of our developed solver.
On the other hand, the result of HSPICE, especially in Fig. 5.8(c), becomes much smooth than the result shown in Fig. 5.6(c). It is because that we utilize
Frequency (Hz)
Figure 5.9: Spectrum of POUTsimulated by our solver including the self-heating effect.
time-variant current source in the HSPICE simulation and the current source is equalized in power magnitude to the power source used in our simulator. The transient analysis of HSPICE solver can handle the nonlinear circuits with current source better than those with voltage source. But, the unstable start-up
sequence caused by numerical method is still be found in Fig. 5.6(a). Further-more, although the result seems become better in Fig. 5.6, there is still some degradation in sub-figure (c). If we transfer the outputs to frequency domain, the unapparent inaccuracy leads an erroneous FFT outcome. In the
compari-Frequency (Hz)
Figure 5.10: Spectrum of POUTsimulated by HSPICE solver including the self-heating effect.
son between Fig. 5.9 and Fig. 5.10, the noise level of our result and HSPICE’s are about 10−12 Wand 10−9W. So that the IM3 products can only be recog-nized from Fig. 5.9, which is the data for the OIP3 value calculation. In order to investigate the heating effect, we perform the simulation for the circuit in Fig. 2.3 again, but ignore the corresponding thermal network. In other words,
Frequency (Hz)
Figure 5.11: Spectrum of POUTsimulated by our solver without the self-heating effect.
all the settings are the same except the RT H, which is set to be zero in this cal-culation. Figure 5.11 shows the FFT outcome of the time-domain data. The result is quite similar to Fig. 5.9. But, without thermal effect, the performance of this device is over estimated. The output power at 1.71 and 1.89 Ghz are both 1.1663 dBm. In contrast, the result with consideration of self-heating are −1.0232 and −1.0233 dBm at 1.71 and 1.89 Ghz, respectively. The self-heating effect degrades not only the output power but also the linearity. The OIP3 value is reduced from 28.5148 dBm to 26.2577 dBm.
5.3 Summary of This Chapter
Comparison between our solver and HSPICE in solving time-domain circuit problem shows the advantages of the proposed numerical algorithms. With the MI method, our solver can prevent error propagation even after tremendous time-step estimations. In this chapter we demonstrated:
• the time-domain simulated results of both our and HSPICE simulator,
• the frequency-domain data after FFT with respect to previous time-domain results, and
• the time- and frequency-domain outcomes including the consideration of self-heating effect.
From the simulation and comparison described above, we execute further anal-yses, for example, the distortion analysis in the following chapter.
Intermodulation Distortion and Power Characteristics
B
y the discrete Fourier transform, the information in frequency domain is extracted from our time-domain calculated data. These discrete in-tensity or power at sampled frequencies can be converted to useful engineer data by proper calculation. In the section 3.3, the intermodulation distortion analysis is briefly described. For narrow band communication, in order to pre-vent channels from cross talk, the two-tone intermodulation nonlinearity is an important benchmark to verify the quality of the devices. In fact, all practical devices and systems are either strongly or weakly nonlinear. With a limited range of bias and operating frequencies, some lumped elements can be treated as linear components and the circuit can be divided into linear and nonlinear parts. The developed simulation kernel has its superiority on solving strongly107
nonlinear circuit. This advantage had been demonstrated in previous chapters.
By utilization of this kernel, we can perform correct intermodulation distor-tion and power analysis. In this chapter, the estimadistor-tion of OIP3 values for an HBT without thermal effects is described first. Then, the characterization of an HBT with the self-heating effect and the multi-finger devices with thermal coupling effect are discussed sequentially in the next sections. After all, a brief summary of this chapter is given.
6.1 Characteristics without Thermal Effects
Input power (dBm)
-20 -10 0 10 20
Output power (dBm)
-80 -60 -40 -20 0 20 40
Slope = 0.99726
Slope = 3.00606
Square belongs to f1 and LO-IM3 product Triangle belongs to f2 and HI-IM3 product
Figure 6.1: Output power, simulated by our solver, at the fundamental frequencies (black-filled symbol) and the IM3 products (white-filled symbol) versus input power.
As the illustration in Fig. 3.11, the third order intermodulation (IM3) prod-ucts at 2f1−f2and 2f2−f1are denoted as LO-IM3 and HI-IM3, respectively.
In our simulation, f1 and f2 are set to be 1.71 and 1.89 GHz. Therefore, the frequencies of LO-IM3 and HI-IM3 become 1.53 and 2.04 GHz. The plots of extracted output powers at the fundamental frequencies and IM3 products versus input powers are demonstrated in Fig. 6.1 and Fig. 6.2. The slopes of
our calculated data in dBm scale are 0.99726 and 3.00606. These values are almost identical to the theoretical values (1 and 3) discussed in section 3.3.
From the cross-point of the extrapolated lines, we can get the value of output third-order intercept point (OIP3), which equals 36.9 dBm. We noted that the HI-IM3 and LO-IM3 are close enough and, hence, they have almost the same OIP3 values. Unfortunately, as shown in Fig. 6.2, the slopes of the funda-mental frequencies and IM3 products from the simulated time-domain data by HSPICE solver are equal to 0.99964 and −0.28781, respectively. It leads to a
Input power (dBm)
4 6 8 10 12 14 16 18 20 22
Output power (dBm)
-30 -20 -10 0 10 20
Slope = 0.99964
Slope =-0.28781
Square belongs to f1 and LO-IM3 product Triangle belongs to f2 and HI-IM3 product
Figure 6.2: Output power, simulated by HSPICE solver, at the fundamental frequencies (black-filled symbol) and the IM3 products (white-filled symbol) versus input power.
nonpredictable OIP3 value and is caused by the errors of numerical method in time-domain.
For an input power, we can also estimate the OIP3 value with output spec-trum. If the slopes of the fundamental frequencies and IM3 products are 1.0 and 3.0, the OIP3 values is directly given by
OIP3 Value = PFFO + 1
2(PFFO − PIM3O ), (6.1) where PFFO and PIM3O represent the output powers of the fundamental frequen-cies and IM3 products, respectively. Fig. 6.3 shows the OIP3 values with respect to different frequency spacing (∆f = f1 − f2) between fundamental frequencies. The OIP3 values of IM3 and LO-IM3 are denoted by HI-OIP3 and LO-HI-OIP3. Besides, the central frequency fc of each OIP3 calcula-tion is identical and equals to 1.8 GHz. As shown in Fig. 6.3, there are only slight deviations of OIP3 values as ∆f increases. Variation of ∆f from 360 to 20 MHzproduces |36.5294−36.5014| = 0.0180 dBm difference in LO-OIP3.
In addition, the difference between LO-OIP3 and HI-OIP3 are 0.0719 and 0.0034782 dBmwhen ∆f varies from 360 to 20 MHz. With this observation, our approach enables us to efficiently calculate the two-tone intermodulation distortion with a larger ∆f. On the one hand, for ∆f = 20 MHz, we have to preform the computation with over 180 cycles for good FFT resolution in frequency domain (see also Sec. 3.3). On the other hand, for ∆f = 360 MHz, there are only ten cycles required. From our numerical experience, we would like to point out that our method can compute the simulation condition with a
Frequency spacing (MHz)
0 50 100 150 200 250 300 350 400
OIP3 Value (dBm)
35.8 36.0 36.2 36.4 36.6 36.8 37.0 37.2
LO-OIP3 HI-OIP3
Figure 6.3: A deviation plot of OIP3 value versus frequency spacing ∆f.
narrow tone spacing of 10 MHz. In our practical implementation, this method provides a more efficient computing alternative and may overcome one of the weakness of the conventional time-domain approaches, such as the enormous requirement of computational resources.
JC (kA/cm^2)
0 2 4 6 8 10 12 14
OIP3 value (dBm)
25 30 35 40
LO-OIP3 (our result) HI-OIP3 (our result) Measured OIP3
Figure 6.4: Plots of OIP3 value versus JC, where symbols are measured data and lines are simulated results of our proposed method.
By utilization of on-wafer device testing with harmonic load-pull system mentioned in Sec. 3.4, we compare the results of simulation with the data from measurement. Fig. 6.4 shows the OIP3 values versus collector current density JC. Here, we note that unit of JC is cmkA2. Our results of HI-OIP3 (dotted line) and LO-OIP3 (solid line) are almost coincident in the scale of this figure. Comparing with the the measured data (square symbols), our results indicate their accuracy for different bias conditions. Fig. 6.5 plots the results
JC (kA/cm^2)
0 2 4 6 8 10 12 14
OIP3 value (dBm)
25 30 35 40
LO-OIP3 (Agilent ADS result) HI-OIP3 (Agilent ADS result) Measured OIP3
Figure 6.5: Plots of OIP3 value versus JC, where symbols are measured data and lines are simulated results of harmonic balance method.
of Agilent Advanced Design System (ADS), which is a well-known harmonic balance based RF circuit solver [40]-[63], and measured data. We can also observe a quite fitting results between simulation and measurement. But, there is over a 1 dBm difference between the values of HI-OIP3 and LO-OIP3.
Our measurement is performed on an HBT device with 2.8 × 12 µm2 emitter area size × 104 fingers. In this case, we simply set the multiplier factor M (see also Sec. 2.1) to be 104 in both the our simulator and ADS. The 104 fingers are assumed to be 104 parallel connected one-finger devices. In our
simulation with ADS, we find that the differences between HI-OIP3s and LO-OIP3s always exits and it is difficult to let the measured data, simulated HI-OIP3 and LO-HI-OIP3 fit well at the same time. We should note that, for the curve fitting in Fig. 6.4 and Fig. 6.5, the parameters of device are adjusted to different values when utilize different simulators.