An analysis of self-injection locked oscillators using a slow-wave structure for phase-noise reduction is presented. This structure is the key component of a feedback network, added to an existing oscillator and providing a stable self-injection locking signal. The unit cell of the slow-wave structure is based on a recently proposed configuration, made up of an open-ended stub and a Schiffman section. A tuning capacitor is introduced as an additional parameter, enabling an adjustment of the structure response at the desired oscillation frequency. The circuit solutions are analyzed by means of a semi-analytical formulation that incorporates the results of an electromagnetic simulation of the structure. The formulation enables a prediction of multivalued parameter regions, inherent to the long delay, which are more controllable than in the case of continuous transmission lines. An analytical derivation of the phase-noise spectral density is presented, which relates the phase-noise reduction with respect to the original freerunning oscillator to the group delay of the self-injection network. The analysis and synthesis method has been applied to an oscillator at 2.75 GHz.This work was supported by the Spanish Ministry of Science, Innovation and Universities and the European Regional Development Fund (ERDF/FEDER) under the research project TEC2017-88242-C3-1-R

Herrera Guardado, Amparo

Pontón Lobete, María Isabel

Ramírez Terán, Franco Ariel

Suárez Rodríguez, Almudena

English

Crossref

Phase-Noise Reduction in Self-Injection Locked Oscillators Using Slow-Wave Structures

UCrea

Phase-Noise Reduction in Self-Injection Locked Oscillators Using Slow-Wave Structures  Mabel Pontón1, Franco Ramírez2, Amparo Herrera3, Almudena Suárez4 University of Cantabria, Spain 1mabel.ponton@unican.es, 2ramirezf@unican.es, 3herreraa@unican.es, 4suareza@unican.es  Abstract—An analysis of self-injection locked oscillators using a slow-wave structure for phase-noise reduction is presented. This structure is the key component of a feedback network, added to an existing oscillator and providing a stable self-injection locking sig-nal. The unit cell of the slow-wave structure is based on a recently proposed configuration, made up of an open-ended stub and a Schiffman section. A tuning capacitor is introduced as an addi-tional parameter, enabling an adjustment of the structure re-sponse at the desired oscillation frequency.  The circuit solutions are analyzed by means of a semi-analytical formulation that incor-porates the results of an electromagnetic simulation of the struc-ture. The formulation enables a prediction of multivalued param-eter regions, inherent to the long delay, which are more controlla-ble than in the case of continuous transmission lines. An analytical derivation of the phase-noise spectral density is presented, which relates the phase-noise reduction with respect to the original free-running oscillator to the group delay of the self-injection network. The analysis and synthesis method has been applied to an oscilla-tor at 2.75 GHz.  Keywords—Oscillator, phase noise, stability. I. INTRODUCTION  Phase noise is an undesired characteristic of oscillator cir-cuits, which degrades their spectral purity and can induce de-modulation errors. The possibility to reduce the phase noise of an existing oscillator through self-injection locking with long transmission lines has been demonstrated in several previous works [1]-[3]. However, this is generally inconvenient [3], since the overall system becomes bulky due to the long lengths required to achieve a significant phase-noise improvement. As shown here, the problem can be circumvented with slow-wave structures [4]-[5] implemented on microstrip line. Actually, os-cillators based on slow-wave resonators with excellent perfor-mance have been demonstrated in the literature [6]-[7]. Here the slow-wave structure is included in the external feedback loop of an existing oscillator, in order to reduce its phase noise. A complete analytical model of the self-injected oscillator is presented, which generalizes the one in [3], derived for a con-tinuous transmission line, to the more complex case of slow-wave structures. In particular, a recently proposed configura-tion, with a unit cell made up of an open-ended stub and a Schiffman section, is considered. A tuning capacitor is intro-duced as an additional parameter to adjust the structure re-sponse at the desired oscillation frequency.   Because the external feedback loop should not significantly modify the free-running operation point in terms of amplitude and frequency, the analysis will be based on a Taylor-series ex-pansion of the oscillator admittance function about the free-run-ning solution of the standalone oscillator. The problem of co-existence of solutions in some parameter intervals, demon-strated in [3] and inherent to the long delay, will be shown to be less critical when using slow-wave structures. An analytical derivation of the phase-noise spectral density will be presented, relating the phase-noise reduction with respect to the original free-running oscillator to the group delay of the self-injection network. The analysis and synthesis method will be applied to an oscillator at 2.75 GHz, obtaining a phase-noise reduction of more than 13 dB.   II. SOLUTION CURVES WHEN USING A SLOW-WAVE STRUCTURE A. Slow-wave structure The slow-wave factor SW of a given structure is the ratio be-tween the wavelength in free space (o) and the wavelength when the signal propagates through the structure (s). Initially the case of a matched lumped-element transmission line, having an inductance Lc and capacitance Cc per unit cell of length lc, is considered. The slow-wave factor and group delay are [4]: 12 22 22arccos 1   , 1 2W g cc c cc f fS fl f f f                  (1) where fc is the Bragg frequency, given by: 1/ ( )c c cf L C . As gathered from (1), to increase the slow-wave factor, lc must be small and Lc and Cc must be large. The slow-wave structure will be implemented on microstrip, using the configuration pro-posed [5]. Its unit cell consists of a Schiffman-section [8], mainly contributing to the inductor, and an open-circuited stub, basically implementing the capacitor [Fig. 1(a)]. The induct-ance and capacitance per unit cell are [5]: 2 /2 / ( ) / ( )c o e fc f o e b bL Z Z WC W Z Z W Z                  (2) where  = 2f, Zo and Ze are the odd and even mode imped-ances of the parallel coupled lines [5], W is the transversal length, f is the propagation constant of the Schiffman section and b and Zb are the propagation constant and characteristic impedance of the line with the width lb.  The slow-wave structure will be applied to an oscillator at 2.75 GHz. To best fit the response of the structure to the oscil-lator frequency, a tuning capacitor is added in parallel to one of the middle cells [Fig. 1(b)]. From the ideal expressions in (1), with the unit cell values Lc = 1.89 nH and Cc = 5 pF and twelve cells, one should achieve a group delay of more than 2 ns. The Bragg frequency is 3.27 GHz. Fig. 2 presents the comparison between the simulated and measured group delay of the struc-ture in Fig. 1(b), when W = 9.4 mm. The simulated group delay is obtained from an electromagnetic analysis of the slow-wave structure. Two different values of the tuning capacitor have been considered. The length of the structure is 36.2 mm.    Fig. 1. Slow-wave structure consisting of Schiffman section and a stub. (a) Unit cell. (b) Complete structure, including a tuning capacitor C.  Fig. 2. Comparison between the simulated and measured group delay of the structure in Fig. 1(b), when W = 9.4 mm, for two different C values.   B. Solution curves Initially, self-injection locking through the output port [1]-[2] will be considered, as shown in Fig. 3(a). This will allow an oscillator description in terms of its total admittance function Y at the output node, constituting the only observation node [Fig. 3(b)]. The total admittance function is Y(V, ), where V and  are the oscillation amplitude and frequency. In free-running conditions, that is, when the oscillator is terminated in Yo = 1/50 = 0.02 Ω-1, it fulfils: Y(Vo, o) = 0. Now, the output 50 Ω load will be replaced with the self-injection configuration in Fig. 3(a), made up of a circulator, a slow-wave structure and an attenuator. The equivalent input admittance of this configu-ration is YL(). In these conditions, the new steady-state equa-tion at the fundamental frequency is: 1 ( )( , ) ( ) ( , ) 01 ( )Lo L o oLY V Y Y Y V Y Y               (3) where ( )( ) ( ) jL e      is the reflection coefficient. The self-injection network must not significantly alter the free-run-ning amplitude and frequency, so one can perform a Taylor-series expansion of ( , )Y V   about Vo, o. This provides: ( ) ( ) ( ) 0V o o o LE Y V V Y Y Y                     (4) where E is the error function and YV and Y are the derivatives of the admittance function, extracted from harmonic balance with the aid of an auxiliary generator [3]. System (3) particu-larizes to the one in [3] in the idealized case of a non-dispersive delay line. In that case, is constant and  = T. When using, as done here, a slow-wave structure, there are no closed-form expressions for() and (). The admittance ( )LY  is evalu-ated through an electromagnetic simulation of the slow-wave structure, as done in subsection A. Then the function is imported by in-house software, where equation (4) is solved by detecting the zeroes of the following function:    ( ) ( ) / 0r i i i r r i i ro V L V o V L V VY Y Y Y Y Y Y Y Y Y            (5) where the superscripts r and i indicate real and imaginary parts.   Fig. 3. Self-injection configuration. (a) Block diagram made up of a circulator, a slow-wave structure and an attenuator. (b) Oscillator circuit. The analysis method has been applied to an oscillator based on PHEMT transistor ATF34143 operating at 2.75 GHz [Fig. 3(b)]. Two sensitive parameters of the self-injected configura-tion will be the transversal length W and the capacitor C. For a global evaluation of the performance, the length W has been swept in the interval 9 mm to 11.5 mm, for two values of the tuning capacitor C. The results are shown in Fig. 4(a), for C = 2 pF, and Fig. 4(b), for C = 1 pF. In the two cases, there are both single and multi-valued sections. As shown in Fig. 4(a) and (b), the location of the multivalued regions can be modified by changing the tuning capacitor.  To understand the causes of these multi-valued sections, one should analyze the impact of the group delay. This is better done by applying a Taylor series expansion to ( )LY   about 2.4 2.6 2.8 3 3.2Frequency (GHz)051015C = 0.5 pF – MeasurementC = 2.0 pF – EM simulationC = 2.0 pF – MeasurementC = 0.5 pF – EM simulationGroup delay (ns)0L  , which provides  ( ) 1 2 ( )L o LY Y    . This expres-sion will be replaced into (4). At the turning points of the solu-tion curves, the Jacobian matrix of the complex equation (4), calculated with respect to V and , becomes singular. This Ja-cobian matrix is:   2 cos 2 sin2 sin 2 cosr rV o o gi iV o o gY Y Y YJEY Y Y Y                           (6) where / ( )g       is the group delay associated with the input reflection coefficient. The determinant of (6) is:  ( )2 cos sinr i i ro o g V V g V Vdetdet Y Y Y Y Y                         (7) where r i i ro V Vdet Y Y Y Y    is the determinant of the Jacobian matrix associated with the admittance function of the standalone free-running oscillator. The determinant odet  does not depend on the frequency  resulting from self-injection ef-fects in system (3), unlike the situation with ( )g  , ( )   and ( )  . According to [9], if the standalone oscillator is stable, one should have 0odet  , so the possible zero values will de-pend on the second term. For sufficiently large g   and /   , the sign of (7) will be sensitive to ( )  . Fig. 4(c) shows the variations of ( )det  in (7). The turning points agree with the changes of sign in ( )det  . The sign of the dominant real pole of the self-injected oscillator is the opposite of that of ( )det   [10]. Under a high attenuation magnitude, it is unlikely to get other dominant poles, so the stable sections should corre-spond to ( ) 0det   . Nevertheless, it is advisable to verify this through a circuit-level stability analysis.  III. PHASE-NOISE REDUCTION In order to get insight into the mechanism for phase-noise reduction, noise perturbations will be introduced in equation (3), in the form of an equivalent noise current source ( )NI t , cal-culated by fitting the standalone free-running oscillator spec-trum, as shown in [10]. This equivalent noise current will in-clude the upconversion of flicker noise, modelled as proposed in [9]. The perturbed system is obtained by performing a Tay-lor-series expansion about the steady-state solution of (3): ( )( )( )2 2 ( )Vj j No oo oY V tI tV tY Y e j Y e j tV V                     (8) Neglecting also the time derivative of the amplitude incre-ment ( )V t  , splitting (8) into real and imaginary parts and solving for ( )t , one obtains: ( )( ) ( )2 cos sinr i i rV N V Nr i i ro o g V V g V VtY I t Y I tdet Y Y Y Y Y                         (9) Applying the Fourier transform, one obtains the following expression for the phase-noise spectral density: 2222( )(2 / )2 cos sinVr i i ro o g V V g V VY N Kdet Y Y Y Y Y                              (10) where N is the spectral density of the equivalent current source and K accounts for the flicker noise coefficient and its upcon-version effects. In the original free-running oscillator  and /    are zero, so one can easily derive the phase-noise re-duction due to the self-injection effects. This is approximately independent of the offset frequency  , and given by:  ( ) 20log1020log10 2 cos sinor i i ro o g V V g V VS dB detdet Y Y Y Y Y                                (11) The phase-noise reduction ( )S dB  potentially increases with the group delay g, which also depends on  and . 9 9.5 10 10.5 11 11.52.62.83Oscillation Frequency (GHz)Length W (mm)(a)C = 2 pF9 9.5 10 10.5 11 11.52.62.72.82.9Length W (mm)(b)Oscillation Frequency (GHz)C = 1 pFMeasurement9 9.5 10 10.5 11 11.5-202Determinant det() x 10‐12Length W (mm)(c)C = 1 pF Fig. 4. Variation of the oscillation frequency versus the length W, swept in the interval 9 mm to 11.5 mm, for two values of the tuning capacitor C. (a) C = 2 pF (b) C = 1 pF. (c) Variations of the determinant ( )det  for C = 1 pF.  Fig. 5(a) presents the phase noise variation versus W, in the case of C = 1 pF, at the constant offset frequency 100 kHz. Note that, when neglecting ( )V t   in (8), the phase noise tends to in-finite at the turning points. At 100 kHz, the standalone oscillator exhibits a phase noise of -85 dBc/Hz. The chosen operation point corresponds to W = 10.9 mm, in a single-valued region of the solution curve. The measured result, with S = 13 dB, is also shown. Fig. 5(b) presents phase-noise variation versus the attenuation in the self-injection loop, with measurements super-imposed. As can be seen, the phase noise reduction increases when reducing the attenuation. For too low attenuation, the curve becomes multi-valued. Fig. 6 presents a comparison of the phase-noise spectrum of the original free-running oscillator with that obtained with the self-injection configuration. Results obtained through the analytical formulation and through the conversion-matrix approach are validated with the experi-mental measurements. Note that, with a continuous transmis-sion line, for deto = 2.32ꞏ10-13 -1s/V and a similar S, a length of about 1.3 m would be required.    Fig. 5. Phase noise with a slow-wave structure in Fig. 1. (a) Versus W, in the case of C = 1 pF. The chosen operation point corresponds to W = 10.9 mm. (b) Versus the attenuation in the loop.  Finally, a measurement of the phase-noise spectrum when introducing the slow-wave structure in an external feedback network, connected between the oscillator output and an auxil-iary input port, is presented in Fig. 7. A reduction of 16 dB with respect to the free-running value is achieved.  IV. CONCLUSION The use of a slow-wave structure for the phase-noise reduc-tion of an existing oscillator through self-injection locking has been presented. An expression relating the phase-noise spectral density to the group delay has been derived. Using a slow-wave structure based on a stub and Schiffman section, a phase-noise reduction of 13 dB has been achieved in a HEMT-based oscil-lator at 2.75 GHz.    104 106 108-150-100-50Conversion matrix approachMeasurementAnalytical formulationFrequency Offset (Hz)Phase Noise Spectral Density (dBc/Hz)Free‐running Self‐injection Fig. 6. Comparison of the phase-noise spectrum of the original free-running oscillator with that obtained with the self-injection configuration.  103Frequency offset (Hz)‐150‐100‐50104 105 106 107Measurementsfree‐running with feedbackPhase Noise Spectral Density (dBc/Hz) Fig. 7. Measurement of the phase-noise reduction when introducing the slow-wave structure in an external feedback network, connected between the oscil-lator output and an auxiliary input port. ACKNOWLEDGMENT This work was supported by the Spanish Ministry of Sci-ence, Innovation and Universities and the European Regional Development Fund (ERDF/FEDER) under the research project TEC2017-88242-C3-1-R.  REFERENCES  [1] H.-C. Chang, "Stability analysis of self-injection-locked oscillators," IEEE Trans. Microw. Theory Techn., vol. 51, no. 9, pp. 1989-1993, Sept. 2003. [2] H.-C. Chang, "Phase noise in self-injection-locked oscillators - theory and experiment," IEEE Trans. Microw. Theory Techn., vol. 51, no. 9, pp. 1994-1999, Sept. 2003. [3] A. Suárez, F. Ramírez, “Analysis of Stabilization Circuits for Phase-Noise Reduction in Microwave Oscillators,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 9, pp. 2743 –2751, Sep. 2005.  [4] C. Zhou, H. Y. D. Yang, "Design Considerations of Miniaturized Least Dispersive Periodic Slow-Wave Structures," IEEE Trans. Microw. Theory Techn., vol. 56, no. 2, pp. 467-474, Feb. 2008. [5] W.S. Chang, C.Y. Chang, “A high slow-wave factor microstrip structure with simple design formulas and its application to microwave circuit de-sign,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 11, pp. 3376 –3383, Nov. 2012. [6]  A. K. Poddar, U. L. Rohde, "Slow Wave Resonator based tunable oscilla-tors," IEEE Int. Frequency Control and Europ. Frequency and Time Forum (FCS) Proc., San Francisco, CA, USA, 2011, pp. 1-10. [7] A. K. Poddar, U. L. Rohde, "Slow-Wave Evanescent-Mode coupled reso-nator oscillator ciruits," IEEE Int. Frequency Control Symp. Proc., Balti-more, MD, 2012, pp. 1-7.  [8] B. Schiffman, “A new class of broad-band microwave 90-degree phase shifters,” IRE Trans. Microw. Theory Techn., vol. 6, no. 2, pp. 232 –237, Apr. 1958. [9] K. Kurokawa, “Some basic characteristics of broadband negative re-sistance oscillators”, Bell Syst. Tech. J., vol. 48, pp. 1937–1955, July–Aug. 1969.  [10] F. Ramírez, M. Pontón, S. Sancho, A. Suárez, “Phase−Noise Analysis of Injection−Locked Oscillators and Analog Frequency Dividers”, IEEE Trans. Microw. Theory Techn., vol. 56, no.2, pp. 393−407, 2008.  9 9.5 10 10.5 11 11.5‐120‐100‐80‐605 10 15 20‐120‐100‐80‐60Length W (mm)Loop attenuation (dB)(a)(b)Phase‐noise @ 100 kHz (dBc/Hz)0 5 10 15 20Measured Loop Attenuation (dB)Measurement Phase‐noise @ 100 kHz (dBc/Hz) Free-runningoscillatorFree-runningoscillatorMeasurement 

Phase-noise reduction in self-injection locked oscillators using slow-wave structures

https://repositorio.unican.es/xmlui/bitstream/10902/18187/1/PhaseNoiseReduction.pdf

Phase-noise reduction in self-injection locked oscillators using slow-wave structures

Abstract

Similar works

Full text

Available Versions

Crossref

UCrea