Patent Publication Number: US-2010117744-A1

Title: Phase error correction in rotary traveling wave oscillators

Description:
FIELD OF THE INVENTION 
     The present invention relates to electronic oscillators. More specifically, the present invention relates to correcting for phase errors in rotary traveling wave oscillators (RTWOs). 
     BACKGROUND OF THE INVENTION 
     An electronic oscillator is a type of electronic circuit that produces a periodic signal. Electronic oscillators are used in a wide variety of applications including digital sampling circuits and quadrature oscillator generators in communications transceivers, for example. 
     Many modern electronic systems require electronic oscillators capable of generating signals at microwave and millimeter-wave frequencies. Conventional electronic oscillators (e.g., those using lumped element tank circuits) are limited in their ability to generate signals at these frequencies while also maintaining low phase noise. For this reason, alternative oscillator mechanisms have been sought. One category of electronic oscillators that has gained recent interest as a possible alternative is the category of oscillators known as “wave-based” oscillators. Wave-based oscillators dispense with the need for lumped element tank circuits and, instead, rely on the distributed inductance and capacitance of a transmission line to achieve oscillation. Recent developments in wave-based oscillator design have demonstrated the ability of wave-based oscillators to operate at high frequencies, low power, and low phase noise. These characteristics have made wave-based oscillators attractive candidates for microwave and millimeter-wave applications. 
       FIG. 1  is a drawing of one type of wave-based oscillator, known as a rotary traveling wave oscillator (RTWO)  10 , which is described in U.S. Pat. Nos. 6,556,089 and 7,218,180 to Wood. The RTWO  10  comprises a closed-loop differential transmission line  15  and a plurality of regenerative circuits  21 . The closed-loop differential transmission line  15  includes a physically and electromagnetically endless signal trace formed in a single plane so that the signal trace has two generally parallel and concentric signal trace loops  15   a  and  15   b  that merge at a crossover  19 . The signal trace of the transmission line  15  has a length l, which corresponds to two ‘laps’ of the transmission line  15  as defined between the spaced signal loop traces  15   a  and  15   b  and through the crossover  19 . The crossover  19  produces a Moebius strip effect, whereby the signal traces of the signal trace loops  15   a  and  15   b  invert from lap to lap. 
     The regenerative circuits  21  have input/output terminals connected to the signal trace loops  15   a  and  15   b  and are evenly distributed along the closed-loop of the transmission line  15 . During start-up, when power is first applied to the regenerative circuits  21 , a traveling wave is generated from inherent noise within the regenerative circuits  21 . The regenerative circuits  21  reinforce the traveling wave as it is created, forcing it to travel in either a clockwise or counterclockwise direction around the closed-loop differential transmission line  15 , the direction of rotation depending on the start-up conditions. Once the traveling wave is fully established, the regenerative circuits  21  continue to reinforce (i.e., amplify) the traveling wave, to counter losses the traveling wave experiences as it travels along the transmission line  15 . Typically, the regenerative circuits  21  are implemented as pairs of cross-coupled inverters, like the pair of cross-coupled inverters  23   a  and  23   b  in  FIG. 2A . When formed with other circuitry in a complementary metal oxide semiconductor (CMOS) process, the pairs of cross-coupled inverters  23   a  and  23   b  are implemented as CMOS inverter, like the CMOS inverter  25  shown in  FIG. 2B . 
       FIG. 3  shows idealized component oscillation waveforms Φ 1  and Φ 2  that appear at the input/output terminals of the regenerative circuits  21 . The component oscillation waveforms Φ 1  and Φ 2  are substantially square and differential, crossing at a midpoint Vdd/2 of the maximum signal amplitude Vdd. The midpoint Vdd/2 can be considered as a ‘null’ point since the instant that both component oscillation waveforms Φ 1  and Φ 2  are at the same potential, there is no displacement current flow present in nor any differential voltage between the signal trace loops  15   a  and  15   b.    
     The null point of the component oscillation waveforms Φ 1  and Φ 2  sweeps around the closed-loop differential transmission line  15  at a rate of 1/(2 T p ), where T p  is equal to the half period of the component oscillation waveforms Φ 1  and Φ 2 . The sweep rate 1/(2 T p ) defines the fundamental oscillating frequency f osc  of the RTWO  10 , and relates to the physical properties of the RTWO  10  as follows: f osc =1/(2 T p )=v p /(2 l), where v p =(L o C o ) −1/2  is the phase velocity of the component oscillation waveforms Φ 1  and Φ 2  traveling in the transmission line  15 , L o  and C o  are the inductance and capacitance per unit length of the transmission line  15 , and l is the length of the transmission line  15 . 
     In addition to having the ability to oscillate at high frequencies and with low phase noise, the RTWO  10  has excellent power dissipation characteristics, even at high frequencies. In fact, once a traveling wave is generated in the RTWO  10 , little power is required to sustain it. The energy used to switch the regenerative circuits  21  is part of the wave energy that circulates around the transmission line  15 . When the regenerative circuits  21  are formed using CMOS technology, i.e., using CMOS inverters  25  like those in  FIG. 2B , energy that goes into charging the MOS capacitors of the inverters&#39; transistors becomes transmission line energy that is recirculated in the closed electromagnetic path. Hence, losses are not dominated by CV 2  f losses but rather by I 2 R dissipation in the transmission line signal line trace. These power dissipation characteristics of the RTWO  10  make the RTWO  10  attractive for high-frequency, battery-powered applications, such as wireless handset applications, for example. 
     Another attractive feature of the RTWO  10  is that it provides a multi-phase output. Various applications require or use multiple signal phases. For example, quadrature modulators and demodulators in communications transceivers require the generation of in-phase and quadrature phase local oscillator signals which are ninety degrees out of phase with respect to each other. Digital sampling circuits also use multi-phase clocks to increase effective sampling rates and data transmission speeds. For example, data transmission speeds can be increased beyond the fundamental switching speed limits of the underlying logic of a digital sampling circuit by serializing data sampled by multiple phases of a multi-phase clock. Conventional multi-phase clock generators employ phased-locked loops and delay-locked loops to generate the multi-phase clock. However those approaches are complex, do not generate square waves, exhibit high levels of jitter, and suffer from large area penalties. The RTWO  10 , at least in theory, avoids these problems, naturally generating and providing high-frequency multi-phase square wave signals at different physical positions along the transmission line  15 . 
       FIG. 4  is a drawing of an RTWO  40  highlighting the RTWO&#39;s inherent multi-phase capability. The RTWO  40  is substantially the same as the RTWO  10  in  FIG. 1 , except that it is formed in the shape of a square and omits the regenerative circuits  21 , to best illustrate the RTWO&#39;s multi-phase capability. The circled plus and minus signs along the differential close-loop transmission line  15  indicate the polarity of the rotating traveling wave (assumed to be traveling in the clock-wise direction, as indicated by the large arrow at the top of the drawing), and the small arrows along the first and second signal trace loops  15   a  and  15   b  indicate the direction of current flow. As explained above, the component oscillation waveforms Φ 1  and Φ 2  at the input/output terminals of any given regenerative circuit  21  arrive back at the input/output terminals of the same regenerative circuit  21  after traversing one lap of the transmission line  15 . Coherent oscillation of the RTWO  30  occurs when the signal in the transmission line  15  meets this requirement for all connected regenerative circuits  21 . 
       FIGS. 5A-H  show the component oscillation waveforms Φ 1  and Φ 2  through a full cycle to start of the next cycle for eight different electrical-length spacings of 45° between sample positions along the closed-loop differential transmission line  15 . The 0°/360° location (see  FIG. 3 ) is used as an arbitrarily chosen phase reference point. To complete a full cycle phase rotation from 0° to 360°, the component oscillation waveforms Φ 1  and Φ 2  must traverse two laps (i.e., the full length l of the transmission line  15 ).  FIG. 5A  shows the component oscillation waveforms Φ 1  and Φ 2  at the 0°/360° phase reference position.  FIG. 5B  shows the component oscillation waveforms Φ 1  and Φ 2  after having traversed ⅛ of the total length l of the transmission line  15 ;  FIG. 5C  shows the component oscillation waveforms Φ 1  and Φ 2  after a traversal of ¼ of the length l of the transmission line  15  (i.e., one-half of a lap); and so on, as illustrated in the remaining  FIGS. 4D-H . It is seen, therefore, that a multi-phase output can be produced by tapping the transmission line  15  at carefully selected and spaced positions. 
     While the RTWO  40  can be used to implement an oscillator having a multi-phase output, the phase accuracy among the multiple phases is not always as accurate as needed or desired, particularly when the RTWO  40  is configured to operate at microwave and millimeter-wave frequencies. Phase accuracy is adversely influenced by a number of factors, including device mismatches among the regenerative circuits  21  (e.g., caused by processing variations), asymmetry of the physical layout of the RTWO  40 , lack of uniformity in signal trace widths and other dimensions of the closed-loop differential transmission line  15 , and the difficulty in forming the tap positions along the transmission line  15  with the physical precision necessary to achieve a constant phase separation among phases of the multi-phase output. 
     The lack of phase accuracy in the RTWO  40  detracts from its use in various applications. For example, in quadrature oscillator applications, sub-degree phase accuracy is often required. At microwave and millimeter wave frequencies, this level of phase accuracy may be difficult or even impossible to achieve with currently available RTWOs, such as those described above. Further, in high-frequency multi-phase clock generator applications, the phase accuracy of the RTWO is often so poor that the skew among output phases is greater than can be tolerated. It would be desirable, therefore, to have an RTWO capable of providing a more phase accurate output than can be realized in currently available RTWOs. 
     SUMMARY OF THE INVENTION 
     Rotary traveling wave oscillator (RTWO) apparatuses and methods are disclosed. An exemplary method for correcting phase inaccuracy among output phases of a multi-phase RTWO includes detecting a phase error between first and second output phases of the RTWO and controlling the phase velocities of a traveling wave traveling in first and second transmission line segments of a closed-loop transmission line to reduce the detected phase error. According to one aspect of the invention, first and second voltage controlled capacitors having substantially the same capacitance versus voltage characteristics are coupled to the first and second transmission line segments, and first and second control voltages for controlling the first and second voltage controlled capacitors are generated based on the detected phase error. Applying the control voltages causes the capacitance of the first transmission line segment to increase by a capacitance differential +ΔC and the capacitance of the second transmission line segment to decrease by a capacitance differential −ΔC. Controlling the voltage controlled capacitors in this manner decreases the phase velocity of a traveling wave in the first transmission line segment compared to the phase velocity of the traveling wave in the second transmission line section. This allows the phase error between the first and second output phase of the RTWO to be reduced while the total capacitance of the closed-loop transmission line remains at a constant level. 
     The RTWO methods and apparatus of the present invention are extensible to multi-phase RTWOs having more than two phases. An exemplary N-phase RTWO, where N is a positive integer greater than or equal to two, includes a closed-loop transmission line formed as a Moebius strip. The closed-loop transmission line includes N transmission line segments, to which N voltage controlled capacitors are coupled. The N transmission line segments provide N output phases. A phase correction circuit operates to detect phase errors between output phases, and, depending on the detected phase errors, generates N control voltages for controlling the capacitances of the N voltage controlled capacitors. Controlling the capacitances of the N voltage controlled capacitors in this coordinated manner reduces the phase errors among the N output phases, thereby providing a phase accurate multi-phase RTWO output. 
     Further features and advantages of the present invention, including a description of the structure and operation of the above-summarized and other exemplary embodiments of the invention, are described in detail below with respect to accompanying drawings, in which like reference numbers are used to indicate identical or functionally similar elements. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a layout view drawing of a typical rotary traveling wave oscillator (RTWO); 
         FIG. 2A  is a circuit diagram of a pair of cross-coupled inverters; 
         FIG. 2B  is a transistor level circuit diagram of a pair of cross-coupled inverters formed from a complementary metal oxide semiconductor (CMOS) process; 
         FIG. 3  is a timing diagram showing idealized component oscillation waveforms Φ 1  and Φ 2  that appear at the input/output terminals of the regenerative circuits of the RTWO in  FIG. 1 ; 
         FIG. 4  is a drawing of a typical RTWO, highlighting the RTWO&#39;s inherent multi-phase capability; 
         FIGS. 5A-H  are timing diagrams of the component oscillation waveforms Φ 1  and Φ 2  through a full cycle to start of the next cycle for eight different sample positions along the closed-loop transmission line of the RTWO in  FIG. 4 ; 
         FIG. 6  is a drawing of an RTWO apparatus, according to an embodiment of the present invention; 
         FIG. 7  is a drawing illustrating how the phase velocity of a traveling wave is decreased in a first transmission line segment of a transmission line by increasing the capacitance of the first transmission line segment by a capacitance differential +ΔC and is increased in a second transmission line segment of the transmission line by decreasing the capacitance of second transmission line segment by a capacitance differential −ΔC; 
         FIG. 8  is a drawing of an alternative RTWO apparatus, according to an embodiment of the present invention; 
         FIG. 9  is a drawing of a phase correction circuit which may be used to implement the phase correction circuit of the RTWO apparatus in  FIG. 6 ; 
         FIG. 10  is a drawing of another phase correction circuit which may be used to implement the phase correction circuit of the RTWO apparatus in  FIG. 6 ; 
         FIG. 1  is a diagram of a factory calibration setup which may be used to perform a factory phase calibration of the RTWO in  FIG. 6 ; and 
         FIG. 12  is a drawing of a four-phase (N=4) RTWO apparatus, highlighting the fact that the phase error correction methods and apparatus of the present invention are extensible to RTWOs having more than two output phases. 
     
    
    
     DETAILED DESCRIPTION 
     Referring to  FIG. 6 , there is shown a drawing of a rotary traveling wave traveling oscillator (RTWO) apparatus  600 , according to an embodiment of the present invention. The RTWO apparatus  600  comprises an RTWO  602  including a transmission line  604 , a plurality of regenerative circuits  606  and first and second voltage controlled capacitors  608  and  610 , and a phase correction circuit  612 . 
     The transmission line  604  includes a physically and electromagnetically endless conductive signal trace of length l having generally parallel first and second signal trace loops  604   a  and  604   b  that merge at a half-twist  609  so that the signal trace forms a Moebius strip. The first and second signal trace loops  604   a  and  604   b  are formed on or within a dielectric or semiconductor substrate, and may be disposed either in a single plane using a planar transformer to complete the half-twist  609 , or in separate metal layers with a via to close the loop and connect the first signal trace loop  604   a  to the second signal trace loop  604   b . In one embodiment, the RTWO  602  is formed with other circuitry in an integrated circuit, and manufactured according to a semiconductor manufacturing process, such as the complementary metal oxide semiconductor (CMOS) fabrication process. The RTWO  602  in this exemplary embodiment is formed in the shape of a square. However, it can be formed in other shapes, so long as the length l of the signal trace is of the appropriate length to achieve the desired oscillation frequency. 
     The regenerative circuits  606  comprise pairs of cross-coupled inverters (or other negative resistance, negative capacitance or nonlinear regenerative means, such as Gunn diodes) having input/output terminals connected to the first and second signal trace loops  604   a  and  604   b . Tuning capacitors may be optionally connected in parallel with the regenerative circuits  606  to provide the ability to frequency tune the RTWO  602 , as indicated in  FIG. 6 . Similar to as in a conventional RTWO, the regenerative circuits  606  operate in a coordinated manner to reinforce (i.e., amplify) a traveling wave as the traveling wave propagates along the transmission line  604 , thereby countering I 2 R losses and allowing the RTWO  602  to sustain oscillation. 
     The RTWO  602  of the exemplary RTWO apparatus  600  is configured to provide a two-phase differential output, making it suitable for use in a quadrature modulator or demodulator of a communications transceiver, for example. The first phase of the two-phase differential output is provided by a first differential amplifier  614  having a differential input coupled between the first and second signal trace loops  604   a  and  604   b  at a first location of the transmission line  604 . The second phase is provided by a second differential amplifier  616  having a differential input coupled between the first and second signal trace loops  604   a  and  604   b  at a second location of the transmission line  604 . The first and second locations are spaced one-half ‘lap’ apart, where a lap corresponds to half the length l of the transmission line  604 . Accordingly, the first differential amplifier  614  provides a first differential output signal having a nominal phase of 0°/180° and the second differential amplifier  616  provides a second differential output signal having a nominal phase of 90°/270°. 
     The first and second voltage controlled capacitors  608  and  610 , which may be comprise, for example, first and second varactors, have substantially identical capacitance versus voltage characteristics. They are coupled between the first and second signal trace loops  604   a  and  604   b  of the transmission line  604  so that they alternate with the locations at which the first and second differential amplifiers  614  and  616  are coupled to the first and second signal trace loops  604   a  and  604   b . Viewed in another way, the first and second voltage controlled capacitors  608  and  610  are coupled to first and second transmission line segments  618  and  620 , respectively of the transmission line  604 . As shown in  FIG. 6 , the first transmission line segment  618  includes the portion of the transmission line  604  that extends from the first location at which the first differential amplifier  614  is coupled to the first and second signal trace loops  604   a  and  604   b  to the second location at which the second differential amplifier  616  is coupled to the first and second signal trace loops  604   a  and  604   b . The second transmission line segment  620  has the same length as the first transmission line segment  618  and includes the portion of the transmission line  604  that extends from the second location at which the second differential amplifier  616  is coupled to the first and second signal trace loops  604   a  and  604   b  to the first location at which the first differential amplifier  614  is coupled to the first and second signal trace loops  604   a  and  604   b.    
     The capacitances of the first and second voltage controlled capacitors  608  and  610  are controlled by first and second control voltages V 1  and V 2 , respectively, provided by the phase correction circuit  612 . Applying the first and second control voltages V 1  and V 2  across the first and second voltage controlled capacitors  608  and  610  increases the capacitance of the first transmission line segment  618  by a capacitance differential +ΔC and decreases the capacitance of the second transmission line segment  620  by a capacitance differential −ΔC. As explained in further detail below, the phase correction circuit  612  controls the capacitances of the first and second transmission line segments  618  and  620  in this manner, to correct for phase inaccuracies between the two output phases of the first and second differential amplifiers  614  and  616 . 
     Changing the capacitances of the first and second voltage controlled capacitors  608  and  610  alters the phase velocities of the traveling wave in the first and second transmission line segments  618  and  620  of the RTWO  602 . The phase velocity describes the rate at which the phase of a traveling wave propagates along a transmission line (i.e., the propagation speed of the traveling wave), and is defined by v p =(L o C o ) −1/2 , where L o  and C o  are the inductance and capacitance per unit length of the transmission line. Accordingly, as illustrated in  FIG. 7 , increasing the capacitance of the first transmission line segment  618  by the capacitance differential +ΔC and decreasing the capacitance of the second transmission line segment  620  by the capacitance differential −ΔC causes the traveling wave in the RTWO  602  to propagate more slowly in the first transmission line segment  618  compared to the rate at which it propagates in the second transmission line segment  620 . 
     The ability to control the propagation speeds of the traveling wave in the first and second transmission line segments  618  and  620  provides the ability to correct for any phase error Δφ that may be present between the output phases of the first and second differential amplifiers  614  and  616 . Ideally, the phase error Δφ is zero. However, as was explained above, various factors, such as device mismatches among the regenerative circuits  606  and other electrical components, and asymmetry of the physical layout of the transmission line  604  of the RTWO  602 , can cause the phase error to be nonzero. The phase correction circuit  612 , which is connected in a feedback arrangement with the RTWO  602 , operates to counter these negative influences and force the phase error Δφ to zero. Specifically, the phase correction circuit  612  determines the phase error Δφ between the first and second differential output signals of the first and second differential amplifiers  614  and  616  (i.e., the phase error between the two output phases of the RTWO  602 ), and, in response, generates the first and control voltages V 1  and V 2  that set the +ΔC and −ΔC capacitance differentials of the first and second voltage controlled capacitors  608  and  610 . The capacitance differential +ΔC of the first transmission line segment  618  and the capacitance differential −ΔC of the second transmission line segment  620  alter the phase velocities of the first and second transmission line segments  618  and  620 . Consequently, the phase separation between the two output phases of the first and second differential amplifiers  614  and  616  is also altered. 
     With the first and second control voltages applied V 1  and V 2  to the first and second voltage controlled capacitors  608  and  610 , the phase correction circuit  612  determines a new phase error between the two output phases of the RTWO  602 , and based on the new phase error generates new first and second control voltages V 1  and V 2  that produce new capacitance differentials +ΔC and −ΔC in the first and second transmission line segments  618  and  620 . The RTWO  602  and phase correction circuit  612  operate in this coordinated feedback manner, forcing the phase error Δφ between the two output phases of the RTWO  602  to zero. 
     In the RTWO apparatus  600  shown and described above, the first and second first and second voltage controlled capacitors  608  and  610  coupled between the first and second signal trace loops  604   a  and  604   b  are used to correct for phase inaccuracies between the two output phases of the first and second differential amplifiers  614  and  616 . In an alternative embodiment, shown in  FIG. 8 , first and second single-ended voltage controlled capacitors  808   a  and  808   b  and third and fourth single-ended voltage controlled capacitors  810   a  and  810   b  are used, instead. Each of the first, second, third and fourth single-ended voltage controlled capacitors  808   a ,  808   b ,  810   a  and  810   b  is independently controlled by first, second, third and fourth control voltages VA, VB, VC and VD, respectively, from the phase correction circuit  612  to correct for phase inaccuracies between the two output phases of the first and second differential amplifiers  614  and  616 . Single-ended voltage controlled capacitors may also be used in the other embodiments of the invention described below. 
     The phase correction circuit  612  of the two-phase RTWO apparatus  600  in  FIG. 6  may be implemented in a variety of different ways.  FIG. 9  is a drawing of one exemplary phase correction circuit  900  that may be used. The phase correction circuit  900  comprises a phase detector  902 , which includes a mixer  904  and a low pass filter (LPF)  906 , and a differential error amplifier  908 . 
     When the phase correction circuit  900  in  FIG. 9  is used to implement the phase correction circuit  612  of the RTWO apparatus  600  in  FIG. 6 , the mixer  904  is configured to receive the first and second differential output signals cos(ωt) and sin(ωt+Δφ), where the first differential output signal cos(ωt) is used as a phase reference and the phase error Δφ between the two signals is included in the second differential output signal sin(ωt+Δφ). The mixer  904  operates to combine the first and second differential output signals cos(ωt) and sin(ωt+Δφ), producing a high-frequency component (½)*sin(2ωt+Δφ) and a low-frequency component (½)*sin(Δφ). The LPF  906  filters out the high-frequency component (½)*sin(2ωt+Δφ), leaving the low-frequency component (½)*sin(Δφ) at its output. Finally, based on the phase error Δφ represented in the low-frequency component (½)*sin(Δφ), the differential error amplifier  908  generates the first and second control voltages V 1 =V offset +ΔV and V 2 =V offset −ΔV. 
       FIG. 10  is a drawing of an alternative phase correction circuit  1000  that may be used to implement the phase correction circuit  612  of the RTWO apparatus  600  in  FIG. 6 . This phase correction circuit  1000  provides a digital implementation. The phase correction circuit  1000  comprises a digital phase detector  1002  and a decision circuit  1004 . The digital phase detector  1002  includes first and second time-to-digital converters (TDCs)  1006  and  1008 , first and second time-to-phase converters  1010  and  1012 , and a subtractor  1014 . The first and second TDCs  1006  and  1008  are configured to sample the first and second differential output signals cos(ωt) and sin(ωt+Δφ), to provide first and second digital time signals. The first and second TDCs  1006  and  1008  can be implemented in variety of ways. One example of a TDC which may be adapted to implement the first and second TDCs  1006  and  1008  of the digital phase detector  1002  here is described in U.S. Pat. No. 7,205,924, which is hereby incorporated by reference. 
     The first and second time-to-phase converters  1010  and  1012  operate to convert the first and second digital time signals at the outputs of the first and second TDCs  1006  and  1008  to first and second digital phase signals representing the phases of the first and second differential output signals cos(ωt) and sin(ωt+Δφ). The subtractor  1014  forms the difference between the first and second digital phase signals, to produce a digital phase error signal Δφ (z), where z=1, 2, 3, . . . is the sample index. 
     The digital phase error signal Δφ (z) provides a digital representation of the phase error Δφ detected between the first and second differential output signals cos(ωt) and sin(ωt+Δφ). The decision circuit  1004  operates to add a voltage representation of the digital phase error signal, i.e. μΔφ (z), where μ is a step size parameter (e.g., have a value between 0 and 1), to a voltage differential ΔV(z) used to generate the first and second control voltages V 1  and V 2  in a previous sample, to generate a new voltage differential ΔV(z+1) having a magnitude dependent upon the phase error Δφ represented in the digital phase error signal Δφ (z). New values for the first and second control voltages V 1  and V 2  are then computed, i.e., V 1 =V offset +ΔV(z+1) and V 2 =V offset −ΔV(z+1). 
     In the phase correction circuits  900  and  1000  described in  FIGS. 9 and 10  above, the process used to generate the first and second control voltages V 1  and V 2  is performed in the field, e.g., as part of a pre-operation system setup process or during real-time operation. In an alternative embodiment, a factory calibration process is performed to determine values of the first and second control voltages V 1  and V 2 . Later, when the RTWO  602  is configured for real-time operation in the field, the first and second control voltages V 1  and V 2  determined during the factory calibration process are applied to the first and second voltage controlled capacitors  608  and  610 . 
       FIG. 11  is a drawing of an exemplary factory calibration setup  1100  that may be used to perform the factory calibration process. The factory calibration setup  1100  comprises a local quadrature modulator, which includes a first mixer  1102 , second mixer  1104  and summer  1106 , a spectrum analyzer  1108 , and a decision circuit  1110 . The first and second mixers  1102  and  1104  each includes a first and a second differential input. During the calibration process, the first mixer  1102  is configured so that its first differential input receives the first differential output signal cos(ωt) from the first differential amplifier  614 , and so that its second differential input receives an in-phase baseband reference signal cos(ω BB t). The second mixer  1104  is configured so that its first differential input receives the second differential output signal sin(ωt+Δφ) from the output of the second differential amplifier  616 , and so that its second differential input receives a quadrature phase baseband reference signal sin(ω BB t). The output signal of the first and second mixers  1102  and  1104  are summed by the summer  1106  and coupled to the input of the spectrum analyzer  1108 . When the phase error Δφ between the first and second differential output signals cos(ωt) and sin(ωt+Δφ) is at its ideal value of zero, the frequency spectrum of the summed output of the quadrature modulator has no image leakage component at frequency (ω−ω BB ). However, when Δφ≠0, an image is present and the spectrum analyzer  1108  generates a digital phase error signal Δe(z) representative of the phase error Δφ. Finally, the decision circuit  1110  generates the first and second control voltages V 1  and V 2  based on the digital phase error signal Δe(z), similar to the decision circuit  1104  of the phase correction circuit  1004  in  FIG. 10 . 
     The phase error correction methods and apparatus described above have been described in the context of a two-phase (N=2) RTWO  602 . However, the methods and apparatus are extensible to RTWOs having any number of phases.  FIG. 12  shows, for example, a four-phase (N=4) RTWO apparatus  1200  comprising a four-phase RTWO  1202  and a phase correction circuit  1216 . The four-phase RTWO  1202  comprises a transmission line  1204 , first, second, third and fourth regenerative circuits  1206 - 1 ,  1206 - 2 ,  1206 - 3  and  1206 - 4 , and first, second, third and fourth voltage controlled capacitors  1208 - 1 ,  1208 - 2 ,  1208 - 3  and  1208 - 4 . 
     Similar to the transmission line  604  of the RTWO  602  in  FIG. 6 , the transmission line  1204  of the four-phase RTWO  1202  includes a physically and electromagnetically endless conductive signal trace of length l having generally parallel signal trace loops  1204   a  and  1204   b  with a half-twist  1209  so that the signal trace is formed as a Moebius strip. 
     The first, second, third and fourth regenerative circuits  1206 - 1 ,  1206 - 2 ,  1206 - 3  and  1206 - 4  comprise pairs of cross-coupled inverters (or other suitable regenerative means) having input/output terminals connected to the first and second signal trace loops  1204   a  and  1204   b . Although not shown in  FIG. 12 , tuning capacitors may be optionally connected in parallel with each of the first, second, third and fourth regenerative circuits  1206 - 1 ,  1206 - 2 ,  1206 - 3  and  1206 - 4  to provide the ability to frequency tune the four-phase RTWO  1202 . 
     The four phases of the four-phase RTWO  1202  are provided by four differential amplifiers  1210 - 1 ,  1210 - 2 ,  1210 - 3  and  1210 - 4 , each coupled between the first and second signal trace loops  1204   a  and  1204   b  at four different locations of the transmission line  1204 . The four differential amplifiers  1210  provide differential outputs having nominal phases of 0°/180°, 45°/225°, 90°/270° and 135°/315°. 
     The first, second, third and fourth voltage controlled capacitors  1208 - 1 ,  1208 - 2 ,  1208 - 3  and  1208 - 4  have capacitances that are controlled by first, second, third and fourth control voltages V 1 , V 2 , V 3  and V 4  provided by the phase correction circuit  1216 , as is explained in more detail below, and are coupled between the first and second signal trace loops  1204   a  and  1204   b  so that they alternate with the locations at which the four differential amplifiers  1210 - 1 ,  1210 - 2 ,  1210 - 3  and  1210 - 4  are coupled to the first and second signal trace loops  1204   a  and  1204   b . Viewed in another way, the first, second, third and fourth voltage controlled capacitors  1208 - 1 ,  1208 - 2 ,  1208 - 3  and  1208 - 4  are coupled to first, second, third and fourth transmission line segments  1212 - 1 ,  1212 - 2 ,  1212 - 3  and  1212 - 4  of the transmission line  1204 . 
     The phase correction circuit  1216  is configured in a feedback arrangement between the differential outputs of the first, second, third and fourth differential amplifiers  1210 - 1 ,  1210 - 2 ,  1210 - 3  and  1210 - 4  and the voltage control inputs of the first, second, third and fourth voltage controlled capacitors  1208 - 1 ,  1208 - 2 ,  1208 - 3  and  1208 - 4 . As shown in  FIG. 12 , the phase correction circuit  1216  comprises a phase detection circuit  1218  and a decision circuit  1220 . The phase detection circuit  1218  includes first, second and third mixers  1222 - 1 ,  1222 - 2  and  1222 - 3  and corresponding first, second and third LPFs  1224 - 1 ,  1224 - 2  and  1224 - 3 . The first, second and third mixers  1222 - 1 ,  1222 - 2  and  1222 - 3  operate to mix the differential output signals of the first and third differential amplifiers  1210 - 1 ,  1210 - 3 , the second and fourth differential amplifiers  1210 - 2 ,  1210 - 4 , and the first and second differential amplifiers  1210 - 1 ,  1210 - 2 , respectively. This allows the phase errors between every two phases of the four phases to be determined. 
     The first, second and third LPFs  1224 - 1 ,  1224 - 2  and  1224 - 3  operate to filter out the high-frequency components at the outputs of the first, second and third mixers  1222 - 1 ,  1222 - 2  and  1222 - 3 , thereby leaving first, second and third low-frequency component signals (½)*sin(Δφ 1,3 ), (½)*sin(Δφ 2,4 ) and (½)*sin(Δφ 1,2 ), which include a first phase error Δφ 1,3  between the differential outputs of the first and third differential amplifiers  1210 - 1  and  1210 - 3 , a second phase error Δφ 2,4  between the differential outputs of the second and fourth differential amplifiers  1210 - 2  and  1210 - 4 , and a third phase error Δφ 1,2  between the differential outputs of the first and second differential amplifiers  1210 - 1  and  1210 - 2 . The decision circuit  1220 , generates the first, second, third and fourth control voltages V 1 , V 2 , V 3  and V 4  based on the detected first, second and third phase errors Δφ 1,3 , Δφ 2,4  and Δφ 1,2 . In one embodiment, the decision circuit  1220  is configured to perform this control voltage generation process using a least mean squares algorithm. However, any other suitable algorithm may be used. Finally, the first, second, third and fourth control voltages V 1 , V 2 , V 3  and V 4  are used to alter the capacitances of the first, second, third and fourth transmission line segments  1212 - 1 ,  1212 - 2 ,  1212 - 3  and  1212 - 4 , to affect the relative phase velocities of the traveling wave propagating in the first, second, third and fourth transmission line segments  1212 - 1 ,  1212 - 2 ,  1212 - 3  and  1212 - 4  so that the first, second and third phase errors Δφ 1,3 , Δφ 2,4  and Δφ 1,2  are reduced. The reduced phase errors are then again detected by the phase detection circuit  1218 , and based on the reduced phase error values the decision circuit  1220  generates new first, second, third and fourth control voltages V 1 , V 2 , V 3  and V 4  to further reduce the first, second and third phase errors Δφ 1,3 , Δφ 2,4  and Δφ 1,2 . The RTWO  1202  and phase correction circuit  1216  operate in this coordinated feedback manner, forcing the first, second and third phase errors Δφ 1,3 , Δφ 2,4  and Δφ 1,2  to zero. 
     Although the present invention has been described with reference to specific embodiments, those embodiments are merely illustrative and not restrictive of the present invention. Further, various modifications or changes to the specifically disclosed exemplary embodiments will be suggested to persons skilled in the art and are to be included within the spirit and purview of this application and scope of the appended claims.