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. Considering a Ohm load, the RF output power can be calculated by means of.

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. First, the values of capacitors C 1 and C 2 can be found from the following approach.

The value of capacitor C 1 is selected for proper loading according to Eq.

## Electronic oscillator

The values for capacitors C 3 and C 4 can be found by again choosing for optimum phase noise and output power, in which case:. Thus, for a value of C 3 of 22 pF, the value of C 4 is pF. Again one needs to use as many as 10 parallel capacitors or a very low-parasitic one that is capacitive to more then 1 GHz. The value of inductor L can be calculated from Eq. The energy stored across the resonator circuit for a given conduction angle and drive level is dependent on the characteristic impedance, and can be found from Eq. For optimum phase noise and output power, Z should be greater than 3.

The same test circuit for the small-signal analysis can now be used with the new large-signal component values applied. Reference 9 shows the phase-noise calculations for a Colpitts oscillator.

The calculations can also be used to find the phase noise for the MHz oscillator circuit. Using the Nexxim HB simulator from Ansoft Designer, the simulated phase noise agrees closely with measured data. According to ref. The total effect of all the four noise sources can be expressed as the function shown in Eq. It should be noted that the effect of the loading of the Q of the resonator is calculated by the noise transfer function multiplied with the sum of the four noise sources.

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.

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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.

## Large-Signal Approach Yields Low-Noise VHF/UHF Oscillators

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.

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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.

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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.

N2 - In this work the phase noise performance of relaxation oscillators has been analyzed resulting in simple though precise phase noise expressions. These expressions have lead to a new relaxation oscillator topology, which exploits a noise filtering technique implemented with a switched-capacitor circuit to minimize phase noise.

Measurements on a 65nm CMOS design show a sawtooth waveform, a frequency tuning range between 1 and 12MHz and a rather constant frequency tuning gain. AB - In this work the phase noise performance of relaxation oscillators has been analyzed resulting in simple though precise phase noise expressions. Abstract In this work the phase noise performance of relaxation oscillators has been analyzed resulting in simple though precise phase noise expressions. In Proceedings of the 19th Annual Workshop on Circuits pp.