The required values for this parallel feedback topology are pF for the feedback capacitor, pF for the emitter-to-ground capacitance, 3. Bypass capacitors C b and C c should be about pF. Because of the difficulty of producing capacitors above pF at these frequencies, it may be more reasonable to use several, up to 10, capacitors in parallel to achieve these values.
For a center frequency f 0 of MHz and 30 mA operation, the component values for the oscillator are L 3 of 3. As mentioned previously, because of the high values of C 1 and C 2 , their values can only be achieved using multiple parallel capacitors of about pF each.
Figure 6 shows the simulated plot of phase noise for the MHz oscillator. The "linear" calculation indicates a resonant frequency of The predictions from both CAE tools deviate less than 1 dB from measured results for the oscillator, assuming that the input SPICE type parameters for the transistor are accurate. A variety of efforts have been made to deal with large-signal conditions for oscillator design, such as the timedomain approach. Reference 10 is a first successful attempt to calculate output power with reasonable effort, notably Eq.
There are many problems associated with both the large-signal analysis and noise analysis.
From an experimental point of view, it is almost impossible to consider all possible variations. During the creation of the Ansoft Designer CAE program, for example, it was necessary for the developers to validate the accuracy of that software's large-signal noise analysis. As part of that validation, several critical circuits were used to compare CAE predictions with measured results, from crystal oscillators to voltage-controlled oscillators VCOs.
References 12 and 13 showed that the accuracy of this software's large-signal predictions is high, within 0. This evaluation involved extensive analysis of noise in oscillators using a set of equations with a minimum number of CAE tools. The equations, derived in ref. The search for low oscillator phase noise has been well documented in the technical literature. Designers have published many different recipes, such as those based on the use of certain low-noise transistors, high-Q circuits, and various other things. In all of these approaches, however, the consequences of device large-signal operation and its effects on phase noise had not been well understood.
To help gain a better grasp on the relationship between device large-signal behavior and phase noise, a complete mathematical analysis of a MHz oscillator follows. The design steps for achieving a MHz oscillator by means of the large-signal approach include:. Calculation of the output power for the selected DC operating conditions.
For this example, the same circuit as used above for the small-signal approach will be applied for a center frequency of MHz. From ref. Calculation of the large-signal transconductance for a normalized drive level can be performed by means of Eq. This assumes an ideal intrinsic transistor. To perform the transition from an intrinsic to an extrinsic transistor, parasitics package effects, lead inductance, and bond wires are added by correcting the final results for capacitances and inductances.
The transition frequency f t of the transistor used is high enough so a phase shift correction for the small-signal transconductance g m is not necessary at these frequencies VHF. Values of the feedback capacitors can be calculated in the following way. The final design step for the MHz oscillator using the largesignal approach involves finding the value of inductor L, which can be performed by knowing the relationship of the oscillator's operating frequency to the inverse of the square root of the oscillator's inductance and capacitance and selecting a value of L for optimum phase noise.
Next month, Part 2 of this threepart article series will show the details of calculating the value of L for the MHz oscillator, and how to apply the large-signal approach and commercial software to compute the oscillator's phase noise. Langford-Smith, Ed. Vilbig, Lehrbuch der Hochfrequenztechnik, vol. Kotzebue and W. The next step for the MHz oscillator is to compute the phase-noise contribution from the different noise sources for the parallel tuned Colpitts oscillator circuit at a frequency offset of 10 kHz from the oscillator carrier frequency f 0 of MHz.
This is performed by considering the circuit parameters. For example, the base resistance r b of the transistor is 6. The Q of the resonator the Q of the inductor at MHz is , the inductance of the resonator is 39 nH, and the capacitance of the resonator is 22 pF. The feedback factor n is 5. The phase noise at an offset frequency of 10 kHz for the four noise sources can be found by applying Eqs.
The sum of the four noise sources can be expressed as expressed in the relationship shown as Eq. It should be noted that the noise contribution from the resonator is the limiting factor. For low-Q cases, this can be identified as the flicker corner frequency. When evaluating closer-in phase noise, at an offset of Hz, we have the relationships shown in Eqs. The sum of the four noise sources can be expressed by Eq. It appears the collector current, base resistance noise flicker noise from the transistor, and the noise from the resonator are the limiting factors for the overall oscillator phase noise.
The oscillator circuit of Fig. The layout is quite critical even at this frequency. The layout in Fig. A standard off-the-shelf inductor was used in the oscillator. Figure 10 , figure 11 , and figure 12 show the CAE simulated phase noise plot, the measured phase noise plot, and the simulated output power for the MHz oscillator using the large-signal approach.
The calculated, simulated, and measured results all agreed within 1 dB. For designers not having access to expensive CAE tools with the proper oscillator noise-calculation capabilities, this approach with its capabilities of calculating phase noise can be quite useful and cost-effective. This demonstrates that choosing a high-output transistor can aid in achieving a low-phase-noise oscillator design.
A second, higher-frequency, MHz oscillator example may help to further demonstrate the usefulness of the large-signal oscillator design approach. The phase noise and the carrier frequency are related in a quadratic fashion, so that a three times increase in carrier frequency will result in a 9-dB degradation in phase noise as shown in Fig. The answer lies in the fact that that even in a grounded-base condition, the large signal Re [Y22] loads the parallel tuned circuit significantly, resulting in a lower dynamic operating Q.
This is a limitation that must be overcome. Because phase noise and Q are related in a quadratic function, a doubling of Q results in a dB improvement in phase noise. Since 26 dB phase noise was lost in the switch to the higher-frequency oscillator design, the dynamic loaded operating Q must be improved by about 20 times the initial value. Since this cannot be done, it may be possible to find effects other than the deterioration in Q that effect the phase noise.
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An inspection of the [Y22] parameters for the oscillator transistor at MHz and 30 mA will reveal that loading of the tank circuit decreases the operating Q significantly. The way around this is to apply a center-tapped inductor. As the coupling at these frequencies from winding to winding is not extremely high, two separate identical inductors can be used for this purpose.
Figure 13 shows the schematic diagram of the MHz grounded base oscillator using the tapped inductor, a modification of the oscillator circuit used previously. In the case of a VCO, it would be advantageous to use a different outputcoupling scheme since the loading would vary with frequency. This can easily be achieved by adding inductive coupling, such as a printed resonator, to the oscillator circuit. Figure 14 shows the layout of the MHz oscillator circuit using a buried printed coupled-line resonator network with a stripline resonator in the middle layer of the circuit board.
The actual resonator would not be visible if performing a visual inspection of the oscillator. Figure 15 offers a plot of simulated phase noise. It shows the expected noise degradation of 9 dB, since the frequency is approximately three times higher than the earlier example MHz. This is due to internal package parasitics, which could not be compensated for externally.
Second harmonics are suppressed by 38 dB due to the higher operating Q. The results obtained so far were based on mathematical calculations, some difficult to obtain. However, by inspecting the resulting circuits, there are certain relationship between the values of the capacitance of the tuned circuit and the two feedback capacitors, the collector emitter capacitor and the emitter to ground capacitor.
The following shows the set of recommended steps for easy design of such oscillator. The accuracy of this simple approach can be evaluated by applying it to the MHz grounded-base oscillator from Fig. These results are comparable with the results above and the calculation is frequency scalable with minor corrections possibly, if necessary. Competing other alternative short formulae published in the literature may not deliver the same high performance.
Many modern applications require high-performance, low-cost oscillators and design time is critical for realizing these components. The approach shown here meets these requirements and gives detailed guidelines for better-performing oscillators. Several examples have been presented for the approach, but it can be applied to a wide range of different frequencies and oscillators.
This is related to the fact that sustained oscillations score far higher in this metric than damped oscillations, and sustained oscillations only occur outside the sparse parabolic region that spans the majority of the volume. Values below 20 dark blue in the 5-dimensional plot correspond to damped oscillations in all cases. The distribution is inhomogeneous when the amplitude is plotted against the cell density. It is more homogeneous when plotted against the IPTG level, and even more so when plotted against the aTc level. The more homogeneous the plot, the less effect changes in the variable have on the amplitude.
It might seem counter-intuitive that increasing levels of IPTG do not correspond to an increasing range of cell densities in which oscillations occur, as increased IPTG results in decreased repression of AHL production. We propose that the predicted dynamics are due to the difference in the time it takes for the enzymes to become active [ 14 ].
Since the maturation of the AHL-degradase takes longer than that of the AHL-synthase, AHL can accumulate and reach a saturation threshold which will effectively produce a steady state. When IPTG exceeds aTc, the system requires a higher cell density to oscillate than in the reverse case.
Top: 2-dimensional scatterplots separately comparing the normalized mean amplitudes to the 3 input variables. The amplitude metric varies the most as a function of cell density, and the least as a function of aTc. The 5-dimensional scatterplots show that the non-tunable system can only oscillate at cell densities above 0.
The maximum amplitude is around 60 for the tunable, and around 30 for the non-tunable circuit. Frequency variability is visualized by histograms, which show that the tunable system can produce a waveform with any number of peaks between 0 and 16, whereas the non-tunable system is comparatively limited. These simulations indicate that while damped oscillations dark blue still make up the vast majority of waveforms produced by the tunable circuit, this system is capable of a significantly more diverse behavior than the original non-tunable design.
It is clear that the changes made to the original circuit have a substantial effect on the resulting protein expression dynamics. The introduction of mathematical expressions representing chemical inducer molecules and corresponding orthogonal transcription factors revealed an unexpected range of non-obvious relationships between the system components. Comparison between the simulated circuit with and without repressors.
Simulated dynamics of the tunable and non-tunable circuits demonstrates that the introduction of tuners greatly enhances the range of cell densities under which sustained oscillations can occur. Maximum achievable amplitude is also substantially higher, as is the range of frequencies that can be generated. The aim of this study was to test the functionality of a refactored synchronized transcriptional oscillator and to investigate whether its reliability and utility could be enhanced by the introduction of chemically inducible repressors. The functionality of the basic circuit, assembled from BioBrick parts, was verified experimentally using a custom experimental platform.
These experiments revealed synchronization at an unexpected scale between spatially separated but chemically linked populations of bacteria. Computational simulations of the tunable circuit design revealed a rich landscape of non-linear relationships between the oscillatory behavior of the circuit and the control variables. The simulations suggested that, while cell density is the primary determinant of gene expression dynamics in this system, the ability to tune transcriptional feedback kinetics via inducer molecules substantially broadens the range of waveforms that this circuit can generate.
Assuming that the model upon which the simulations were based capture the actual dynamics, the tunable oscillator design described here should be highly versatile. These results offer a cursory glance at the type of methods that could be employed to study nonlinear transcriptional regulatory dynamics using this circuit.
Future work on this system should aim to validate the model before exploring more rigorous analytical methods.
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Due to its efficient single-plasmid design it also lends itself to the investigation of expression dynamics as a function of varying copy numbers using different plasmid backbones, or the effect of genomic integration. Such approaches could very well yield reliable, quantitative data if combined with advanced experimental platforms, such as fluorescence microscopy combined with microfluidics, fluorescence-based cell-sorting methods, or milliliter-scale continuous stirred-tank bioreactors.
It is our hope that in the future, this circuit may be used by others as a tool for developing, and possibly benchmarking increasingly refined modeling approaches that shed light on the intricate and elusive properties of complex genetic circuits. Parts were assembled hierarchically, two at a time using the BioBrick standard assembly method [ 18 , 22 ].
After digestion the fragments were separated via gel electrophoresis and subsequently isolated with a Qiagen Gel Extraction kit. The purified fragments were then ligated using T4 Ligase and used to transform chemically competent E. The resulting composite BioBrick part was then isolated from these liquid cultures using a Qiagen miniprep kit.
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The cultures were spun down and resuspended in 0. Since LB-amp medium was supplied from below the microdish, the growth of bacteria was restricted to the wells there nutrients could be obtained via diffusion through the porous material at the base. Data analysis and processing were done with ImageJ 1. The genetic circuit described above can be represented as a system of delay differential equations, which was adapted from Danino et al.
The hybrid promoters regulating luxI and aiiA were assumed to have the same response kinetics to LuxR-AHL as the natural lux promoter in the absence of the repressor proteins. The term after that in equations 1 and 2 is the actual tuner term and represents the influence of the repressors on the system, which in turn is dependent on the presence of either IPTG for LacI or aTc for tetR, respectively. Equations 3 and 4 contain an additional term proportional to D, which shows the diffusion of AHL throughout the cells.
The model is applicable to both the basic circuit and the tunable one, as the model for the latter can be reduced to represent the former simply by setting the concentration of the repressors to 0. Deterministic simulations were performed using the MATLAB dde23 solver in order to elucidate the relationship between inducer molecule concentrations and their effect on GFP expression relative to cell density.
The input values were chosen to cover the entirety of the controllable input space, ranging from full repression IPTG and aTc set to 0 to full induction both IPTG and aTc set to 1 in steps of 0.
Analysis and design of low phase noise CMOS oscillator circuit topologies
The cell density was also iterated from 0 to 0. However, it is treated as such for the purposes of this study. Equations 1, 2, 3, 4, 5. The main difference is that the maximum expression levels of AiiA and LuxI are limited by Hill functions that take the concentrations of their respective repressors and corresponding inducer molecules into account.
De Jong H: Modeling and simulation of genetic regulatory systems: a literature review. J Comput Biol. Curr Biol.
Foundations of Oscillator Circuit Design
Endy D: Foundations for engineering biology. Nat Rev Genet. ACS Synthetic Biol. Chaos: Interdiscipl J Nonlinear Sci. J Cell Sci. Math Biosci. J Roy Soc Interface. Sayut DJ, Sun L: Slow activator degradation reduces the robustness of a coupled feedback loop oscillator. Mol BioSystems. J Biol Eng. PloS One. Proc Nat Acad Sci.