Patent Publication Number: US-11646697-B2

Title: Resonator circuit

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 17/235,474, filed on Apr. 20, 2021, which is a continuation of U.S. patent application Ser. No. 16/833,182, filed on Mar. 27, 2020, now U.S. Pat. No. 11,025,197, which is a continuation of U.S. patent application Ser. No. 15/659,204, filed on Jul. 25, 2017, now U.S. Pat. No. 10,637,395, which is a continuation of International Application No. PCT/EP2015/051573, filed on Jan. 27, 2015. All of the aforementioned patent applications are hereby incorporated by reference in their entireties. 
    
    
     TECHNICAL FIELD 
     The invention relates to the field of radio frequency (RF) resonator circuits. 
     BACKGROUND 
     Resonator circuits, also denoted as tank circuits, are widely used as frequency selective elements in a variety of radio frequency components, such as filters, amplifiers, and oscillators. Typically, resonator circuits comprise inductors and capacitors, wherein the inductors and capacitors are connected to be in resonance at a specific resonance frequency. The quality of resonator circuits is thereby characterized by a quality factor. The characteristics of resonator circuits are of major interest in the design of radio frequency oscillators, in particular when implemented as radio frequency integrated circuits (RFICs) on semiconductor substrates. In particular, the response of the resonator circuits with regard to leakage currents or currents at harmonic frequencies can have a major impact on the frequency stability and phase noise performance of the radio frequency oscillators. 
     Common resonator circuits exhibit a resistive characteristic when excited at the resonance frequency, and a capacitive characteristic when excited at frequencies above the resonance frequency. Consequently, higher order current components in conjunction with the Groszkowski effect may lead to reduced frequency stability and increased flicker noise up-conversion, i.e. reduced close-in phase noise performance, of radio frequency oscillators. 
     For improving the phase noise performance of radio frequency oscillators, noise filtering techniques are applied. These techniques rely on interposing a further resonator circuit having a resonance frequency at 2ω 0  in a common source of the transistors, e.g. core transistors. These techniques, however, use an additional tunable inductor and increase the die area on the semiconductor substrate. 
     For reducing an amount of higher order drain current harmonics, resistors are added in series with the sources of the transistors for linearizing the operation of the transistors. However, the radio frequency oscillator start-up margin is usually reduced. 
     By adding resistors in series with the drain of the transistors, the resistance in conjunction with a parasitic drain capacitance can introduce a delay in a loop gain for shifting both an impulse sensitivity function (ISF) and a current waveform of the radio frequency oscillators. Flicker noise up-conversion is reduced by specifically tailoring the component values. However, the phase noise performance in the 20 dB/decade region is degraded particularly at low supply voltages and high current consumptions. 
     In J. Groszkowski, “The interdependence of frequency variation and harmonic content, and the problem of constant-frequency oscillators,” Proc. IRE, vol. 21, no. 7, pp. 958-981, July 1934, the Groszkowski effect is studied. 
     In M. Babaie and R. B. Staszewski, “A class-F CMOS oscillator,” IEEE JSSC, vol. 48, no. 12, pp. 3120-3133, December 2013, a resonator circuit and a radio frequency oscillator are described. 
     SUMMARY 
     It is an object of the invention to provide an efficient resonator circuit. 
     The invention is based on the finding that a transformer-based resonator circuit can be employed exhibiting different characteristics when excited in a differential mode and in a common mode. In particular, the inductive coupling factor of the transformer may be different in differential mode and common mode excitations, wherein a differential mode resonance frequency can be different from a common mode resonance frequency. In particular, the common mode resonance frequency can be designed to be twice the differential mode resonance frequency. 
     The resonator circuit enables an efficient operation of a radio frequency oscillator. In particular, a second harmonic can be exposed to a resistive path provided by the resonator circuit. Consequently, the Groszkowski effect can be mitigated and frequency stability and phase noise performance of the radio frequency oscillator can be improved. 
     The resonator circuit and the radio frequency oscillator are suited for implementation as radio frequency integrated circuits (RFICs) on semiconductor substrates. 
     According to a first aspect, the invention relates to a resonator circuit, the resonator circuit comprising a transformer comprising a primary winding and a secondary winding, wherein the primary winding is inductively coupled with the secondary winding, a primary capacitor being connected to the primary winding, the primary capacitor and the primary winding forming a primary circuit, and a secondary capacitor being connected to the secondary winding, the secondary capacitor and the secondary winding forming a secondary circuit, wherein the resonator circuit has a common mode resonance frequency at an excitation of the primary circuit in a common mode, wherein the resonator circuit has a differential mode resonance frequency at an excitation of the primary circuit in a differential mode, and wherein the common mode resonance frequency is different from the differential mode resonance frequency. Thus, an efficient resonator circuit is provided. 
     The resonator circuit can be a tank circuit. The resonator circuit can be used as a frequency selective element within a radio frequency oscillator. The resonator circuit can be resonant when excited in the differential mode and in the common mode. 
     The primary winding and the secondary winding can be arranged to provide a strong inductive coupling when the primary circuit is excited in the differential mode and a weak inductive coupling when the primary circuit is excited in the common mode. 
     The primary capacitor can comprise a pair of single-ended capacitors being connected in series to form the primary capacitor. The primary capacitor can be regarded as a primary capacitive structure. The secondary capacitor can comprise a pair of differential capacitors being connected in series to form the secondary capacitor. The secondary capacitor can be regarded as a secondary capacitive structure. 
     The resonance frequency in differential mode, i.e. the differential mode resonance frequency, can depend on the inductance of the primary winding, the capacitance of the primary capacitor, the inductance of the secondary winding, and the capacitance of the secondary capacitor. The resonance frequency in common mode, i.e. the common mode resonance frequency, can depend on the inductance of the primary winding and the capacitance of the primary capacitor. The resonance frequency in common mode, i.e. the common mode resonance frequency, may be independent from the inductance of the secondary winding, and the capacitance of the secondary capacitor. Odd order harmonic components of the current injected into the resonator circuit can be differential mode signals and even order harmonic components can be common mode signals. 
     In a first implementation form of the resonator circuit according to the first aspect as such, the common mode resonance frequency is twice the differential mode resonance frequency. Thus, a resistive path for a second harmonic at an excitation of the primary circuit in the common mode is realized. 
     In a second implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the resonator circuit has a further differential mode resonance frequency at an excitation of the primary circuit in the differential mode, wherein the further differential mode resonance frequency is different from the differential mode resonance frequency and the common mode resonance frequency. Thus, a further differential mode resonance at the further differential mode resonance frequency is realized. 
     In a third implementation form of the resonator circuit according to the second implementation form of the first aspect, the further differential mode resonance frequency is three times the differential mode resonance frequency. Thus, a resistive path for a third harmonic at an excitation of the primary circuit in the differential mode is realized. The further differential mode resonance frequency can specifically be designed to be three times the differential mode resonance frequency. 
     In a fourth implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the primary winding of the transformer comprises one turn, and the secondary winding of the transformer comprises two turns. Thus, the resonator circuit is implemented efficiently. 
     The number of turns of the secondary winding can be twice the number of turns of the primary winding. Thereby, a ratio of turns of the primary winding and the secondary winding of 1:2 can be realized. 
     In a fifth implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the primary winding of the transformer and/or the secondary winding of the transformer is planar. Thus, the resonator circuit is implemented efficiently. 
     In a sixth implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the primary winding of the transformer and the secondary winding of the transformer are arranged on the same plane. Thus, the resonator circuit is implemented efficiently. 
     The primary winding of the transformer and/or the secondary winding of the transformer can comprise a bridging portion being arranged at a different plane. 
     In a seventh implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the primary winding of the transformer and/or the secondary winding of the transformer is connected to a constant voltage source or a ground potential. Thus, a tapping of the primary winding and/or the secondary winding is realized. Both the primary winding and the secondary winding may be connected to a constant voltage source respectively in order to enable an efficient start-up of the resonator circuit. 
     The tapping of the primary winding and/or the secondary winding can be a symmetrical center tapping of the primary winding and/or the secondary winding. 
     In an eighth implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the primary capacitor of the primary circuit comprises a pair of single-ended capacitors. Thus, a reference to ground potential of the primary circuit is realized. The pair of single-ended capacitors can have the same capacitance as the primary capacitor. A single-ended capacitor can be realized as a plurality of switched capacitors, wherein the plurality of switched capacitors can be arranged in parallel. The capacitances of the switched capacitors can be different. 
     In a ninth implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the secondary capacitor of the secondary circuit comprises a pair of differential capacitors. Thus, a reference to ground potential of the secondary circuit is avoided. A differential capacitor can be realized as a plurality of switched capacitors, wherein the plurality of switched capacitors can be arranged in parallel. The pair of differential capacitors can be a pair of balanced capacitors. 
     In a tenth implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the primary capacitor and/or the secondary capacitor comprises a variable capacitor, in particular a digitally tunable capacitor. Thus, a variation of the differential mode resonance frequency and/or the common mode resonance frequency can be realized efficiently. 
     In an eleventh implementation form of the resonator circuit according to the first aspect as such or any preceding implementation form of the first aspect, the primary capacitor is connected in parallel to the primary winding, and/or the secondary capacitor is connected in parallel to the secondary winding. Thus, the resonator circuit is implemented efficiently. 
     According to a second aspect, the invention relates to a radio frequency oscillator, the radio frequency oscillator comprising a resonator circuit according to the first aspect as such or any implementation form of the first aspect, and an excitation circuit being configured to excite the primary circuit of the resonator circuit in the differential mode. Thus, an efficient radio frequency oscillator is provided. 
     The radio frequency oscillator can exhibit improved frequency stability and phase noise performance, e.g. close-in phase noise performance. In particular, flicker noise up-conversion due to the Groszkowski effect can be mitigated efficiently. The approach may be effective to mitigate a 1/f phase noise up-conversion and may therefore improve a 1/f 3  phase noise characteristic. A 1/f 2  phase noise characteristic may be unchanged. 
     Further features of the radio frequency oscillator directly result from the functionality of the resonator circuit according to the first aspect as such or any implementation form of the first aspect. 
     In a first implementation form of the radio frequency oscillator according to the second aspect as such, the excitation circuit comprises at least one transistor, in particular at least one field-effect transistor, for exciting the primary circuit of the resonator circuit. Thus, an active device is employed for exciting the primary circuit of the resonator circuit. In order to realize a cross-coupled oscillator structure at least two transistors may be employed. The transistors can be metal-oxide-semiconductor field-effect transistors (MOSFETs). 
     In a second implementation form of the radio frequency oscillator according to the second aspect as such or any preceding implementation form of the second aspect, the radio frequency oscillator is a class F oscillator. Thus, an efficient oscillator structure is applied. 
     Within the class F oscillator, a first harmonic and a third harmonic can be excited in order to obtain a pseudo square-wave oscillation waveform. Within the class F oscillator, the third harmonic may not be filtered due to the further differential mode resonance frequency. The oscillation signal can have a pseudo square-wave oscillation waveform. 
     According to a third aspect, the invention relates to a method for exciting a resonator circuit, the resonator circuit comprising a transformer comprising a primary winding and a secondary winding, wherein the primary winding is inductively coupled with the secondary winding, a primary capacitor being connected to the primary winding, the primary capacitor and the primary winding forming a primary circuit, and a secondary capacitor being connected to the secondary winding, the secondary capacitor and the secondary winding forming a secondary circuit, wherein the resonator circuit has a common mode resonance frequency at an excitation of the primary circuit in a common mode, wherein the resonator circuit has a differential mode resonance frequency at an excitation of the primary circuit in a differential mode, and wherein the common mode resonance frequency is different from the differential mode resonance frequency, the method comprising exciting the primary circuit of the resonator circuit in the differential mode. Thus, an efficient excitation of the resonator circuit is realized. 
     The method can be performed by the resonator circuit and/or the radio frequency oscillator. Further features of the method directly result from the functionality of the resonator circuit and/or the radio frequency oscillator. 
     The invention can be implemented using hardware and/or software. 
    
    
     
       SHORT DESCRIPTION OF DRAWINGS 
       Embodiments of the invention will be described with respect to the following figures, in which: 
         FIG.  1    shows a diagram of a resonator circuit according to an embodiment; 
         FIG.  2    shows a diagram of a radio frequency oscillator according to an embodiment; 
         FIG.  3    shows an input impedance response and an equivalent circuit of a resonator circuit; 
         FIG.  4    shows an input impedance response and an equivalent circuit of a resonator circuit according to an embodiment; 
         FIG.  5    shows a transformer comprising a primary winding and a secondary winding according to an embodiment; 
         FIG.  6    shows a resonator circuit and an input impedance response according to an embodiment; 
         FIG.  7    shows a diagram of a radio frequency oscillator according to an embodiment; 
         FIG.  8    shows a diagram of a single-ended capacitor according to an embodiment; 
         FIG.  9    shows a diagram of a differential capacitor according to an embodiment; 
         FIG.  10    shows a diagram of a tail resistor of a radio frequency oscillator according to an embodiment; and 
         FIG.  11    shows a phase noise power spectral density of a radio frequency oscillator according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
       FIG.  1    shows a diagram of a resonator circuit  100  according to an embodiment. The resonator circuit  100  comprises a transformer  101  comprising a primary winding  103  and a secondary winding  105 , wherein the primary winding  103  is inductively coupled with the secondary winding  105 , a primary capacitor  107  being connected to the primary winding  103 , the primary capacitor  107  and the primary winding  103  forming a primary circuit, and a secondary capacitor  109  being connected to the secondary winding  105 , the secondary capacitor  109  and the secondary winding  105  forming a secondary circuit, wherein the resonator circuit  100  has a common mode resonance frequency at an excitation of the primary circuit in a common mode, wherein the resonator circuit  100  has a differential mode resonance frequency at an excitation of the primary circuit in a differential mode, and wherein the common mode resonance frequency is different from the differential mode resonance frequency. In an embodiment, the common mode resonance frequency is twice the differential mode resonance frequency. 
     The resonator circuit  100  can be a tank circuit. The resonator circuit  100  can be used as a frequency selective element within a radio frequency oscillator. The resonator circuit  100  can be resonant when excited in the differential mode and in the common mode. 
     The primary winding  103  and the secondary winding  105  can be arranged to provide a strong inductive coupling when the primary circuit is excited in the differential mode and a weak inductive coupling when the primary circuit is excited in the common mode. 
     The resonance frequency in differential mode, i.e. the differential mode resonance frequency, can depend on the inductance of the primary winding  103 , the capacitance of the primary capacitor  107 , the inductance of the secondary winding  105 , and the capacitance of the secondary capacitor  109 . The resonance frequency in common mode, i.e. the common mode resonance frequency, can depend on the inductance of the primary winding  103  and the capacitance of the primary capacitor  107 . The resonance frequency in common mode, i.e. the common mode resonance frequency, may be independent from the inductance of the secondary winding  105 , and the capacitance of the secondary capacitor  109 . 
     The diagram illustrates the overall structure of the resonator circuit  100 , wherein the primary capacitor  107  can comprise a pair of single-ended capacitors, and wherein the secondary capacitor  109  can comprise a pair of differential capacitors. 
       FIG.  2    shows a diagram of a radio frequency oscillator  200  according to an embodiment. The radio frequency oscillator  200  comprises a resonator circuit  100 , and an excitation circuit  201 . The resonator circuit  100  comprises a transformer  101  comprising a primary winding  103  and a secondary winding  105 , wherein the primary winding  103  is inductively coupled with the secondary winding  105 , a primary capacitor  107  being connected to the primary winding  103 , the primary capacitor  107  and the primary winding  103  forming a primary circuit, and a secondary capacitor  109  being connected to the secondary winding  105 , the secondary capacitor  109  and the secondary winding  105  forming a secondary circuit, wherein the resonator circuit  100  has a common mode resonance frequency at an excitation of the primary circuit in a common mode, wherein the resonator circuit  100  has a differential mode resonance frequency at an excitation of the primary circuit in a differential mode, and wherein the common mode resonance frequency is different from the differential mode resonance frequency. The excitation circuit  201  is configured to excite the primary circuit of the resonator circuit  100  in the differential mode. The excitation circuit  201  can exemplarily comprise an exciting element, e.g. a feedback amplifier, providing a trans-conductance Gm. 
     In an embodiment, the excitation circuit  201  comprises at least one transistor, in particular at least one field-effect transistor, for exciting the primary circuit of the resonator circuit  100 . In order to realize a cross-coupled oscillator structure at least two transistors may be employed. 
     In the following, further implementation forms and embodiments of the resonator circuit  100  and the radio frequency oscillator  200  are described. 
     An up-conversion of flicker noise, e.g. 1/f noise, can degrade a close-in spectrum of a radio frequency oscillator, e.g. a complementary metal-oxide semiconductor (CMOS) radio frequency (RF) oscillator. The resulting 1/f 3  phase noise (PN) can further be an issue within phase-locked loops (PLLs) having a loop bandwidth of e.g. less than 1 MHz, which practically relates to the majority of cellular phones. A major flicker noise up-conversion mechanism in nanoscale CMOS is the Groszkowski effect. 
     The presence of harmonics in a current of an active device, such as a transistor of an excitation circuit, can cause a frequency drift of a resonance frequency of a resonator circuit, due to perturbing reactive energy in the resonator circuit. Any variation in the ratio of a higher harmonic current to a fundamental current (e.g. due to the flicker noise) can modulate the frequency drift and can show itself as a 1/f 3  phase noise. Embodiments of the invention reduce the flicker noise up-conversion due to the Groszkowski effect in radio frequency oscillators significantly. The resonator circuit  100  can be applied for flicker noise up-conversion reduction within the radio frequency oscillator  200 , wherein the radio frequency oscillator  200  can be a class F oscillator. 
       FIG.  3    shows an input impedance response  301  and an equivalent circuit  303  of a resonator circuit. The diagram illustrates current harmonic paths and frequency drifts for the resonator circuit without resistive traps at higher harmonics. 
     The presence of harmonics of a current of an active device, such as a transistor of an excitation circuit, can cause a frequency drift of a resonance frequency ω 0  of a resonator circuit as depicted in  FIG.  3   . A fundamental drain current I H1  can flow into the resistors having the resistance R p , which can be the equivalent parallel resistance of the resonator circuit, while its second and third harmonics, I H2  and I H3 , may mainly take the capacitance path due to its lower impedance. As a consequence, reactive energy stored in the inductors and capacitors, e.g. having inductances L p  and capacitances C c  and C d , can be perturbed, shifting the resonance frequency ω 0  and/or the oscillation frequency by Δω lower in order to satisfy the resonance condition. This shift may be static but any variation in the ratio of the currents I H2  (or I H3 ) to I H1  (e.g. due to flicker noise) can modulate Δω and can show itself as a 1/f 3  phase noise. 
       FIG.  4    shows an input impedance response  401  and an equivalent circuit  403  of a resonator circuit  100  according to an embodiment. The diagram illustrates current harmonic paths and frequency drifts for the resonator circuit  100  with resistive traps at higher harmonics. 
     Suppose that the input impedance Z in  of the resonator circuit  100  has further peaks at strong harmonics of the fundamental resonance frequency ω 0 . These harmonics would then mainly flow into their relative equivalent resistance of Z in , instead of its capacitive part, as depicted in  FIG.  4   . Consequently, Groszewski&#39;s effect on the flicker noise up-conversion can be reduced significantly. On the other hand, flicker noise of transistors, e.g. core transistors, of an excitation circuit can modulate the second harmonic of the virtual ground of the radio frequency oscillator. This modulation can generate a second harmonic current in parasitic gate-source capacitors C gs  and can get injected into the resonator circuit  100 . Consequently, the current I H2  can be the main contributor to the frequency drift. 
     In other words, a dominant source of 1/f noise up-conversion in radio frequency oscillators, in particular without tail transistors, is that current harmonics of the resonator circuits flow into the capacitive part of the resonator circuits as shown in  FIG.  3   . An approach for reducing the 1/f noise up-conversion is illustrated in  FIG.  4   . For resonance frequencies at higher harmonics, the current can flow into the equivalent resistance of the resonator circuit and 1/f noise up-conversion can be reduced. A resonator circuit  100  is generally shown in  FIG.  1   . 
     Embodiments of the invention apply a transformer-based resonator circuit topology that effectively traps the current I H2  in its resistive part without the cost of extra die area on a semiconductor substrate. The resonator circuit  100  can derive this characteristic from a different behavior of inductors and transformers in differential mode (DM) and common mode (CM) excitations. The transformer based resonator circuit  100  can be incorporated into the radio frequency oscillator  200 , e.g. a class-F oscillator, in order to take advantage of its low phase noise in the 20 dB/dec region and in order to improve the phase noise in the 30 dB/dec region. 
     The resonator circuit  100  can be based on the transformer  101 , e.g. being a 1:2 turn transformer. The differential mode resonance frequency and the common mode resonance frequency can be different within the transformer  101 , e.g. due to different coupling factors in differential mode and in common mode. An application of a switch is avoided. The resistive trap is realized by the common mode resonance. 
     The common mode signal that excites the common mode resonance can be the second harmonic component of the current within the resonator circuit  100 . The I H2  component can have a n/2 phase shift with regard to the fundamental current which can make it a common mode signal as illustrated in  FIG.  3    and  FIG.  4   . 
     If the space of the primary winding and/or the secondary winding is designed accurately and a ratio C s /C p  is chosen accurately, the common mode resonance frequency can be two times the differential mode resonance frequency. Then, the common mode second harmonic current component can flow into the equivalent resistance of the resonant peak and may not flow through the capacitive part. This approach mitigates disturbances of the reactive energy in the capacitive part and reduces the 1/f noise up-conversion. 
       FIG.  5    shows a transformer  101  comprising a primary winding  103  and a secondary winding  105  according to an embodiment. The diagram illustrates the currents within the primary winding  103  and the secondary winding  105  when the transformer  101  is excited in differential mode (DM) and in common mode (CM). The transformer  101  can be an F 2,3  transformer. 
     The primary winding  103  of the transformer  101  and the secondary winding  105  of the transformer  101  are planar and are arranged on the same plane. The secondary winding  105  of the transformer  101  comprises a bridging portion being arranged at a different plane. 
     The primary winding  103  of the transformer  101  and the secondary winding  105  of the transformer  101  are connected to a supply voltage or an alternating current (AC) ground potential. The connection is realized by a symmetrical center tapping of the primary winding  103  and the secondary winding  105 . 
     The transformer  101 , having a 1:2 turn ratio, can be excited by differential mode and common mode input signals at its primary winding  103 . In differential mode excitation, the induced currents at the secondary winding  105  can circulate in the same directions leading to a strong coupling factor k m . On the other hand, in common mode excitation, the induced currents can cancel each other, resulting in a weak coupling factor k m . 
     The inductance of the primary winding  103  can be referred to as L p , the inductance of the secondary winding  105  can be referred to as L s , the capacitance of the primary capacitor  107  can be referred to as C p , and the capacitance of the secondary capacitor  109  can be referred to as C s . According to this definition, the primary winding  103 , the secondary winding  105 , the primary capacitor  107 , and the secondary capacitor  109  are considered as individual concentrated components. 
     Alternatively, the inductance of the primary winding  103  can be referred to as 2 L p , the inductance of the secondary winding  105  can be referred to as 2 L s , the capacitance of the primary capacitor  107  can be referred to as 0.5 C p , and the capacitance of the secondary capacitor  109  can be referred to as 0.5 C s . According to this definition, the primary winding  103  and the secondary winding  105  are each formed by a pair of inductors connected in series, wherein the inductance of each inductor is referred to as L p  or L s , respectively. Furthermore, the primary capacitor  107  and the secondary capacitor  109  are each formed by a pair of capacitors connected in series, wherein the capacitance of each capacitor is referred to as C p  or C s , respectively. 
     The differential mode resonance frequency can be determined according to the following equation: 
               ω     0   ,   DM           =     1           L   pd     ⁢     C   p       +       L   s     ⁢     C   s                   
wherein ω 0,DM  denotes the differential mode resonance frequency, L pd  denotes an inductance associated with the primary winding  103  in differential mode, C p  denotes a capacitance associated with a primary capacitor  107 , L s  denotes an inductance associated with the secondary winding  105 , and C s  denotes a capacitance associated with a secondary capacitor  109 .
 
     The common mode resonance frequency can be determined according to the following equation: 
               ω   CM     =     1         L     p   ⁢   c       ⁢     C   p                 
wherein ω CM  denotes the common mode resonance frequency, L pc  denotes an inductance associated with the primary winding  103  in common mode, and C p  denotes a capacitance associated with a primary capacitor  107 .
 
     L pd  can relate to half of the inductance of the primary winding  103  in differential mode, e.g. the inductance between a center tap of the primary winding  103  and one of the inputs, yielding a total differential primary capacitance of 2 L p . This is due to the consideration that the inductance L T  may not be seen in differential excitation but may affect the inductance in common mode excitation. The total inductance in common mode excitation can be equal to 2 L pd +2 L T , or L pc =L pd +L T  as used in the equations. 
     In an embodiment, the inductance associated with the primary winding  103  in differential mode and the inductance associated with the primary winding  103  in common mode are considered to be equal. 
       FIG.  6    shows a resonator circuit  100  and an input impedance response  609  according to an embodiment. The resonator circuit  100  comprises a transformer  101  comprising a primary winding  103  and a secondary winding  105 , wherein the primary winding  103  is inductively coupled with the secondary winding  105 , a primary capacitor  107  being connected to the primary winding  103 , the primary capacitor  107  and the primary winding  103  forming a primary circuit, and a secondary capacitor  109  being connected to the secondary winding  105 , the secondary capacitor  109  and the secondary winding  105  forming a secondary circuit, wherein the resonator circuit  100  has a common mode resonance frequency at an excitation of the primary circuit in a common mode, wherein the resonator circuit  100  has a differential mode resonance frequency at an excitation of the primary circuit in a differential mode, and wherein the common mode resonance frequency is different from the differential mode resonance frequency.  FIG.  6    shows a possible realization of the resonator circuit  100 . 
     The primary capacitor  107  of the primary circuit comprises a pair of single-ended capacitors  601 ,  603 . The secondary capacitor  109  of the secondary circuit comprises a pair of differential capacitors  605 ,  607 . The primary capacitor  107  and the secondary capacitor  109  are variable capacitors, in particular digitally tunable capacitors. In particular, the pair of single-ended capacitors  601 ,  603  and the pair of differential capacitors  605 ,  607  are variable capacitors, in particular digitally tunable capacitors. The differential mode resonance frequency and/or the common mode resonance frequency are tunable between a minimum frequency f min  and a maximum frequency f max , respectively, as illustrated by the input impedance response  609 . The input impedance of the resonator circuit  100  is denoted as Z in . 
     The resonator circuit  100  can employ the transformer  101 , the pair of single-ended capacitors  601 ,  603  within the primary circuit and the pair of differential capacitors  605 ,  607  within the secondary circuit. The resonator circuit  100  can be an F 2,3  resonator circuit. The transformer  101  can be an F 2,3  transformer. The resonator circuit  100  can have two differential mode resonance frequencies and one common mode resonance frequency. 
     For class-F 3  operation, ω 1,DM =3ω 0,DM , and for resistive traps at the second and third harmonics, ω CM =2ω 0,DM  and ω 1,DM =3ω 0,DM . This can result in L s C s =3L p C p  and k m =0.72, wherein k m  denotes the coupling factor between the primary winding  103  and the secondary winding  105 . 
     When implementing the resonator circuit  100 , the inductance associated with the primary winding  103  in common mode L pc  can be greater than the inductance associated with the primary winding  103  in differential mode L pd , i.e. L pc &gt;L pd , due to a metal track inductance L T  connecting e.g. a center tap of the primary winding  103  to a constant supply voltage. Thus, a lower coupling factor k m  may be used in order to satisfy both F 2  and F 3  operation conditions of the resonator circuit  100 . A careful design of the single-ended capacitors  601 ,  603  within the primary circuit and/or the differential capacitors  605 ,  607  within the secondary circuit, which can be variable capacitors, can maintain ω CM /ω 0,DM ≈2 and ω 1,DM /ω 0,DM ≈3 over the full tuning range (TR). 
     In an embodiment, the inductance associated with the primary winding  103  in common mode L pc  is determined according to the following equation:
 
 L   pc   =L   pd   +L   T  
 
wherein L pc  denotes the inductance associated with the primary winding  103  in common mode, L pd  denotes the inductance associated with the primary winding  103  in differential mode, and L T  denotes the metal track inductance.
 
       FIG.  7    shows a diagram of a radio frequency oscillator  200  according to an embodiment. The radio frequency oscillator  200  comprises a resonator circuit  100 , and an excitation circuit  201 . The resonator circuit  100  forms an implementation of the resonator circuit  100  as described in conjunction with  FIG.  6   . The excitation circuit  201  comprises a transistor  701 , a transistor  703 , a tail resistor  705 , and a tail capacitor  707 . The radio frequency oscillator  200  is a class F oscillator. 
     Class F 3  oscillators can have a pseudo square-wave oscillation waveform by designing ω 1,DM =3 ω0,DM , and avoiding filtering the current I H3  in a resonator circuit. The specific impulse sensitivity function (ISF) of the pseudo square-wave oscillation waveform can lead to an improved phase noise performance. In this oscillator, the current I H2  can be as high as the current I H3 . In a class F 2,3  oscillator, a class F 3  resonator circuit is replaced by a class F 2,3  resonator circuit. The pseudo square-wave oscillation waveform of class F oscillators can be preserved, wherein a 1/f 3  phase noise corner frequency can be reduced e.g. from 300 kHz to 700 kHz to less than 30 kHz. Embodiments of the invention use an F 2,3  resonator circuit and the different characteristics of a 1:2 turn transformer in differential mode and common mode excitations in order to provide a resistive trap at the second harmonic 2ω 0 , resulting in a reduction of flicker noise up-conversion in radio frequency oscillators. 
       FIG.  8    shows a diagram of a single-ended capacitor  601 ,  603  according to an embodiment. The single-ended capacitor  601 ,  603  comprises a capacitor  801 , a capacitor  803 , a transistor  805 , a transistor  807 , a resistor  809 , a resistor  811 , an inverter  813 , and an inverter  815 . The single-ended capacitor  601 ,  603  is arranged within the primary circuit. 
     By applying a digital switching signal the transistor  805  and the transistor  807  can be switched between a conducting state and a non-conducting state. Consequently, the capacitance of the single-ended capacitor  601 ,  603  can be digitally tuned. A plurality of single-ended capacitors  601 ,  603  can be connected in parallel. 
       FIG.  9    shows a diagram of a differential capacitor  605 ,  607  according to an embodiment. The differential capacitor  605 ,  607  comprises a capacitor  901 , a capacitor  903 , a transistor  905 , a resistor  907 , a resistor  909 , an inverter  911 , and an inverter  913 . The differential capacitor  605 ,  607  is arranged within the secondary circuit. 
     By applying a digital switching signal the transistor  905  can be switched between a conducting state and a non-conducting state. Consequently, the capacitance of the differential capacitor  605 ,  607  can be digitally tuned. A plurality of differential capacitors  605 ,  607  can be connected in parallel. 
       FIG.  10    shows a diagram of a tail resistor  705  of a radio frequency oscillator  200  according to an embodiment. The tail resistor  705  comprises a transistor  1001 , and a resistor  1003 . The tail resistor  705  can be used for current control within the radio frequency oscillator  200 . 
     By applying a digital switching signal b i , the transistor  1001  can be switched between a conducting state and a non-conducting state. Consequently, the current within the radio frequency oscillator  200  can be controlled. A plurality of tail resistors  705  can be connected in parallel and/or in series. 
       FIG.  11    shows a phase noise power spectral density  1101  of a radio frequency oscillator  200  according to an embodiment. The diagram depicts the phase noise power spectral density in dBc/Hz over a carrier frequency offset in Hz. The radio frequency oscillator  200  is a class F 2,3  oscillator. 
     The diagram relates to a minimum frequency of 5.4 GHz and a maximum frequency of 7 GHz. A 1/f 3  phase noise corner is further depicted in the diagram. 
     It will be appreciated that statements made herein characterizing the invention refer to an embodiment of the invention and not necessarily all embodiments.