1. 微波工程 期中報告 老師：陳文山 班級：研碩一甲 學生：黃英勝 A Long, Winding Road, Alfred Riddle Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. 1527-3342/10/$26.00©2010 IEEE Digital Object Identifier 10.1109/MMM.2010.937734 70~80October 2010IEEE Microwave Theory and Techniques Society

2. Alfred Riddle  The problem of oscillator noise has been considered from intuitive to mathematically sophisticated viewpoints.  This article presents a historical tour of oscillator noise analysis and will not go into detail on any one method.  The “ History ” section will present a family tree of oscillator noise, and the “ Present Models ” section will review some of the more seminal papers. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

3.  The first papers were differential-equation- based and concerned with the gain saturation mechanism of a stable oscillation.  Not every tuning point would be a stable tuning point, and, often, an oscillator would jump to an unexpected operating frequency.  Different devices had different noise characteristics, mostly in the 1/f region, which prompted device noise studies of how low- frequency noise modulated the final oscillator output. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

4.  Figure 1 is a chart describing most of the important contributions to microwave oscillator noise analysis. The chart has a rough chronological ordering from top to bottom and draws links between related techniques.  Rayleigh recognized that a limiting mechanism (nonlinearity) was an essential part of a practical oscillator because the limiting mechanism provided a way for any perturbation in the signal to be forced back the nominal operating point and so create a stable oscillation. History October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

5. Figure 1. Literature survey of oscillator noise and stability. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

6.  Figure 2. Lord Rayleigh. (Courtesy Emilio Segre.) Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

7.  In 1920, van der Pol developed a similar nonlinear differential equation while studying vacuum tube oscillators. His analysis of his differential equation allowed him to calculate the oscillator output power and explain injection locking, amplitude jumps, and the change in oscillator frequency caused by harmonics.  Although van der Pol ’ s equation was originally presented with a resistive nonlinearity modeled by a general power series, the equation usually attributed to van der Pol appears as (1) with only a cubic nonlinearity. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

8.  In (1), v is the voltage, a scales the amplitude limiting, b determines the oscillation frequency, and t is time. A steady state oscillation is described in (1) when a sinusoidal oscillation reaches an amplitude of a squared average value of v equal to 1, causing the second term to vanish so no further change in v occurs. The signal v in (1) may begin from an impulse function on the right hand side. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

9.  The second term in (1) causes the signal v to grow until v becomes so large that the average value of the second term is zero, which causes the signal growth to stop and allows a steady oscillation.  In 1938, Berstein analyzed oscillator noise by using noise as a forcing function in van der Pol ’ s equation. Berstein used the Fokker-Planck equation to obtain the statistics of oscillator noise so that both amplitude and phase variances could be calculated. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

10.  The work of Krylov on linearizing differential equations and harmonic balance allowed people to handle more general oscillator nonlinearities. These foundations led directly to works that characterize the oscillator noise caused by circuit noise at the frequency of oscillation.  Barkhausen was the first to realize that a stable oscillation requires the net amplification around a feedback loop to equal unity with a multiple of 3608 of phase shift, that is, G(A 0 )H(jw 0 )=1, (2) where the oscillator is separated into two functions, a gain, G, and a frequency-selective element, H. The product GH makes up the transfer function of the oscillator. The development of feedback theory has had a major role in oscillator analysis and design. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

11. Figure 3. Classic feedback oscillator with amplifier G(A) and filter H. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

12.  Figure 3 shows an oscillator made from an amplifier, G(A), and a frequency-sensitive element, H (jw). While the Barkhausen criteria serves as a useful approximation, it leads to a certain paradox.  The amplitude saturation of (2) is used to predict the output amplitude of the oscillator, and the frequency at which (2) is satisfied defines the frequency of the oscillation.  Yet, the implication of (2) is that the poles of an oscillator lie on the jw axis, giving an infinite gain at a real frequency. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

13.  Loeb derived the criteria for a stable oscillation in a feedback system to be Equation (3) uses the same notation as the feedback system in (2). The first set of terms in (3) represents the oscillator gain change with amplitude and the oscillator phase change with frequency.  The second set of terms in (3) represents side effects that relate to amplitude to phase modulation (AM-PM) and phase to amplitude modulation (PM-AM) cross-conversion mechanisms October 201074 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

14.  This criterion was presented graphically by Slater. Slater developed his graphical criterion from studies on magnetrons at Bell Labs during the 1940s. Slater’s nalysis focused on two- terminal devices, so functions as G(A) and functions as H in the following analysis. The angles of intersection between the device,, and the load,, in Figure 4 can be shown to give an equivalent stability criteria to (3). October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

15. Figure 4. Impedance plane plot of device and load lines. Stable and unstable operating points are indicated. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

16. The description of noise in practical oscillators must include all noise sources, although some noise sources dominate in certain regions and many approximations can be made.  During the 1960s, Impact Avalanche Transit Time (IMPATT) and Gunn diode oscillators became the prime solid-state sources of high- frequency signals. Since IMPATT diodes are very noisy, injection locking techniques were used to reduce the oscillator's spectral width. At Bell Labs, Kurokawa (Figure 5) used a Taylor series expansion to linearize the nonlinear differential equation of noisy and injection-locked oscillators. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

17.  The general nature of Kurokawa’s analysis, the inclusion of the stability factor, and the use of variables available to a microwave engineer have made Kurokawa’s analysis durable. In many ways, Kurokawa’s analysis provides a cornerstone for the description of noise in all oscillators. Kurokawa left the study of more complicated oscillator circuits, the effects of harmonics on operation, the understanding of 1/f contributions, the details of how the large signal device operation affects the device internal noise sources and parameter modulation, and the study of oscillator measurement techniques for others. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

18. There are two ways in which low-frequency noise may affect the oscillator noise spectrum.  The influence of harmonics on oscillation frequency was discussed by Groszkowski in 1933 and van der Pol in 1934. In this case, oscillator frequency stability refers to the discussion surrounding Figure 4. The oscillator stability derived by the intersection angle of device and load lines also has a significant effect on oscillator phase noise (the short-term stability). October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

19. Present Models  In 1960, Edson analyzed oscillator noise, beginning with noise pulses passed through a filter and grouped into sinusoidal and cosinusoidal components. The growth of the noise pulses eventually caused the feedback circuit to saturate. Edson modeled the oscillator’s output as noise passed through a very high-Q resonator.  Edson derived (4) as thedouble sideband amplitude noise spectral density, SA, for the oscillator in Figure 6. As microwave engineering moved from waveguides and two terminal devices to three terminal devices and integrated circuits, the emphasis of oscillator design has also changed. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

20.  As the sinusoid increased in amplitude, the real part of the active device impedance (blue) would limit until it just cancelled G and formed a stable oscillation where Te is the effective noise temperature (often close to ambient), k is Boltzmann’s constant, w m is the modulation frequency, P is the power delivered to G, and s is the saturation factor. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

21.  Figure 6. Edson’s oscillator circuit with noise source. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

22. By Fourier transforming the autocorrelation function of the oscillator disturbed by phase noise, Edson arrived at the phase noise spectral density From (6), the correct line width for the oscillator was derived as Equation (4) can be seen as a noise-to-signal power ratio followed by a frequency-selective filtering function, which shapes the output spectrum. October 201077-78 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

23. Oscillator analysis ranges from nonlinear time variant analysis to perturbation analysis based on linearized models and assumptions that the noise being analyzed is small compared to the oscillation signal. Figure 7. (a) Oscillator spectrum as seen on a spectrum analyzer and (b) as seen if referenced to the center frequency, averaged, and plotted on a logarithmic frequency scale. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

24. Figure 8. Leeson’s model for oscillator phase noise. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

25. Oscillator noise involves random processes, nonlinearities, and results that may seem counterintuitive, which make it a fascinating and difficult subject. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

26. Figure 9. Kurokawa’s oscillator model. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

27. As the quote at the beginning of this article states, we have to work through oscillator noise analysis for ourselves before we truly understand. Figure 10. Hajimiri oscillator waveforms and impulse sensitivity functions, G. (a) A sinusoidal oscillator and (b) a limiting oscillator. Modeled after [2]. Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. October 201070~80IEEE Microwave Theory and Techniques Society

28. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

29. Summary  Oscillator noise involves random processes, nonlinearities, and results that may seem counterintuitive, which make it a fascinating and difficult subject. As the quote at the beginning of this article states, we have to work through oscillator noise analysis for ourselves before we truly understand. October 201070~80 Alfred Riddle (ariddle@ieee.org) is with M/A-COM Technology Solutions Inc.,100 Chelmsford Street, Lowell, Massachusetts 01851 USA. IEEE Microwave Theory and Techniques Society

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