Abstract:
Circuits and methods for providing an accurate phase shift between a generated output signal and an input signal are disclosed. The circuits and methods enable any amount of accurate phase shift to be set without requiring significant changes in circuitry with each phase shift. The phase shift is set by a voltage applied to a feedback amplifier connected to a low-pass filter and a timer circuit that resets a latch circuit.

Description:
FIELD OF THE INVENTION 
     This invention relates generally to phase-shift circuits. More specifically, the present invention provides circuits and methods for accurately shifting the phase of a signal by any programmed amount. 
     BACKGROUND OF THE INVENTION 
     A phase-shift circuit is one whose sole purpose is to shift the phase of an input signal to produce an output signal that is but of phase with the input signal. Two signals are out of phase when there is a relative displacement between the signals at a given point in time. For example, signals A and B shown in FIG. 1 are 90° out of phase. A is said to lead B by 90°, and, conversely, B is said to lag A by 90°. A simple phase-shift circuit such as shown in FIG. 2 may be used to shift an input signal by a certain degree depending on the value of resistor  25 , capacitor  30 , and the frequency of the input signal to generate a phase-shifted output signal. 
     Phase-shift circuits are useful in a number of diverse applications, including phase detection, modulation, high power and high frequency amplification, and voltage regulation involving multiple paralleled power supplies, among others. In these and most other applications, phase-shift circuits are used to provide a phase difference based on which other signals are generated or controlled. 
     For example, a phase-shift circuit is often used in combination with a phase detector circuit to provide a DC output voltage proportional to the phase difference between its input signals. A well-known phase detector circuit is the quadrature detector widely used in many communications applications, and in particular, in applications involving quadrature amplitude modulation and phase modulation. The quadrature detector uses a phase-shift circuit to provide quadrature input signals, i.e., input signals that are spaced 90° apart, to a phase detector. The phase detector produces different output voltages for different phase shifts to recover the modulation. 
     Examples of phase-shift circuits designed for communications applications include those described in U.S. Pat. No. 4,355,289, U.S. Pat. No. 4,549,152, U.S. Pat. No. 5,317,288, and U.S. Pat. No. 5,317,276. Such circuits either provide a limited number of phase shifts, e.g., multiples of 90°, or require a complex control signal or a control circuit to set the phase shift. 
     Phase-shift circuits may also be used to maintain phase linearity in high power, high frequency amplifiers as described in U.S. Pat. No. 4,581,595, and to control the phase of video signals transmitted according to the NTSC (National Television System Committee) and PAL (Phase Alternation by Line) standards, as described in U.S. Pat. No. 5,317,200. Similar to the phase-shift circuits designed for communications applications, these phase-shift circuits only provide a limited number of phase shifts. 
     Another application in which phase-shift circuits are useful is in voltage regulation involving multiple paralleled power supplies. A voltage regulator is a device that produces a predetermined and substantially constant output voltage from a source voltage that may be poorly-specified or fluctuating, or that may be at an inappropriate amplitude for the load. 
     One type of a commonly-used voltage regulator is a switching voltage regulator. Switching voltage regulators employ one or more power devices as the switching elements and inductors, transformers, and capacitors as energy storage elements between the source and the load. The switching elements may be, for example, power metal-oxide semiconductor field-effect transistor (MOSFET) devices. A switching voltage regulator regulates the voltage across the load by varying the ON-OFF times of the switching elements so that power is transmitted through the switching elements and into the energy storage elements. The energy storage elements then supply this power to the load so that the load voltage is regulated. 
     Multiple switching voltage regulators are often paralleled together in a single system to produce multiple disparate output voltages or to produce a higher output current. In this case, it is preferable to have all switching voltage regulators synchronized to the same operating frequency. Proper application of synchronization consolidates the spectral content of the noise in the system due to the use of multiple regulators, reduces noise filtering requirements, and eliminates the enharmonic hetrodynes in the system, i.e., the “beat frequencies” arising from the sum of and difference between the different frequencies of the multiple regulators. 
     In addition to synchronization, it is also desirable to have multiple switching voltage regulators interleaved when they are sharing the same input rail. An interleaved system employs a phase-shift circuit to provide a constant phase difference between any two regulators, i.e., the phase difference between any two regulators is constant regardless of changes in other operating parameters. The phase difference between any two regulators depends on the number of regulators (or phases) in the system, e.g., 180° (360°/2) for a two-phase system, 120° (360°/3) for a three-phase system, 90° (360°/4) for a four-phase system, 72° (360°/5) for a five-phase system, etc. 
     When properly interleaved, the system input RMS current is minimized and the frequency of the input ripple current is effectively multiplied, thereby enabling the use of a smaller input capacitor and reducing the power loss that arises from resistances in fuses, printed circuit board traces, connectors, input capacitance equivalent series resistances (“ESRs”), among others. Further, when multiple switching voltage regulators are interleaved to provide a single output, the steady-state output ripple voltage is significantly reduced and the dynamic load transient response is significantly improved over a non-interleaved configuration. Examples of control integrated circuits for multiple interleaved switching regulators include LTC1628, LTC1629, and LTC3728, provided by Linear Technology Corporation, of Milpitas, Calif. These switching regulator controllers employ two switching regulators to produce one (LTC1629) or two (LTC1628, LTC3728) regulated outputs. 
     To provide a multi-phase interleaved system, it is necessary to use phase-shift techniques or phase-shift circuits between any two switching regulators. One approach that does not require any additional phase-shift circuitry between two regulators involves using the inherent phase-shifted signal from the first regulator to synchronize the internal clock of the second regulator. For example, in a synchronous Buck converter, the bottom gate drive signal of the first regulator, i.e., the signal that drives the bottom switching element of the first regulator, is used to synchronize the internal clock of the second regulator. 
     This approach suffers from two major drawbacks. First, when the first synchronous Buck regulator is subjected to load or line changes, its bottom gate signal shifts in phase, thereby introducing a temporary frequency deviation in the second regulator. That is, the phase difference between the first regulator and the second regulator is not constant with varying load or voltage levels. Second, the phase difference between the first regulator and the second regulator is fixed by the duty cycle of the fist regulator and is usually not optimized. With many switching voltage regulators having different duty cycles at different operation modes, e.g., continuous current mode, discontinuous current mode, and other power-savings modes, this approach does not guarantee constant phase differences between the two regulators or does not work when the bottom gate of the first regulator is turned off during a power-savings mode. 
     Another approach that may be used to provide a phase difference between two regulators is to apply the gate drive signal of the first regulator, i.e., the signal that drives the switching element of the first switching voltage regulator, as input to R-C circuit  20  shown in FIG. 2 to synchronize the internal clock of the second switching voltage regulator. This approach is also very simple, but suffers from three major drawbacks. First, similar to the approach discussed above, the phase shift is limited by the duty cycle of the first regulator. Second, the amount of phase shift that can be achieved is highly dependent on the input voltage, that is, the phase difference between the first regulator and the second regulator is not constant with varying voltage levels. Third, the phase shift is not accurate because of the tolerance of capacitor  30  in R-C circuit  20  and its impedance variation over frequency, i.e., its frequency sensitivity, resulting in different phase shifts for different frequencies. Another source of inaccuracy may be caused by the synchronization input voltage threshold variation of the second switching voltage regulator. 
     To address the limitations of the phase-shift circuits described above with respect to phase shift variations due to the first regulator, phase-shift circuit  35  shown in FIG. 3 uses separate oscillator  40  as input to a frequency divider consisting of D flip-flops  50 - 55 . Phase-shift circuit  35  provides accurate phase shifts of 90° in a four-phase system at the expense of more circuitry. Different phase-shifts may be provided by cascading additional D flip-flops. Depending on the amount of phase shift that is required, i.e., depending on the number of phases in the system, the resulting circuitry can be very complicated. Further, phase-shift circuit  35  is inflexible when incorporated inside an integrated circuit as it can only provide certain values of phase shift, e.g., 90°, 120°, and 180°. 
     To date, there are no simple phase-shift circuits that provide a wide range of accurate phase shifts for use in interleaved switching voltage regulator systems and in other applications. Further, there are no phase shift circuits that may be programmed to provide any amount of phase shift without requiring non-trivial control circuit or significant changes in the circuitry with each phase shift. 
     In view of the foregoing, it would be desirable to provide circuits and methods for accurately setting a phase shift. 
     It further would be desirable to provide circuits and methods for setting any amount of pre-programmed phase shift. 
     It also would be desirable to provide circuits and methods for setting any amount of phase shift without requiring significant changes in the circuitry with each phase shift. 
     SUMMARY OF THE INVENTION 
     In view of the foregoing, it is an object of the present invention to provide circuits and methods for accurately setting a phase shift. 
     It is a further object of the present invention to provide circuits and methods for setting any amount of pre-programmed phase shift. 
     It also is an object of the present invention to provide circuits and methods for setting any amount of phase shift without requiring significant changes in the circuitry with each phase shift. 
     These and other objects of the present invention are accomplished by providing circuits and methods for accurately setting any amount of pre-programmed phase shift without requiring significant changes in the circuitry with each phase shift. In one embodiment, the input signal is applied to the clock of a latch element such as a D flip-flop connected to a delay element such as R-C circuit  20  shown in FIG. 2 to provide an output signal that is out of phase with the input signal with an initialization element such as diode  77 , as shown in FIG.  4 . Circuit  70  of FIG. 4 achieves a constant phase shift that depends on the values used for resistor  80  and capacitor  85 . However, the phase shift may be inaccurate due to the frequency sensitivities and production variations of capacitor  85  and the threshold variation of the CLR threshold of D flip-flop  75 . 
     In a preferred embodiment, the input signal is applied to a latch element connected to a low-pass filter and a timer, such that the timer&#39;s delay is controlled by an amplifier with a feedback as shown in FIG.  5 . Circuit  90  shown in FIG. 5 is a closed-loop circuit that provides any amount of accurate phase shift by setting an input voltage to the inverting input of amplifier  105 . Circuit  90  has no operating frequency restrictions, and, more importantly, no frequency-sensitive components are used to set the phase shift. 
     Circuit  90  may be implemented in a number of ways, such as, for example, circuits  155  and  265  shown in FIGS. 7 and 10, respectively. In circuit  155  of FIG. 7, latch  95  is implemented with D flip-flop  160 , low-pass filter  100  is implemented with an R-C low-pass filter consisting of resistor R 3  ( 165 ) and capacitor C 2  ( 170 ), timer  115  is implemented with an R-C analog delay circuit consisting of resistor R 4  ( 200 ) and capacitor C 3  ( 205 ) with MOSFET M 1  ( 210 ) being used to initialize the timer formed by resistors R 4  ( 200 ) and capacitor C 3  ( 205 ), amplifier  105  is implemented with operational amplifier  190 , and feedback  110  is implemented with capacitor C 1  ( 195 ) along with resistive divider  175 . The input voltage applied to the inverting input of operational amplifier  190  is also set by resistive divider  175 . 
     Alternatively, circuit  90  may be implemented by combining two or more of latch  95 , low-pass filter  100 , amplifier  105 , feedback  110 , and timer  115  in one or more functional circuit units instead of using one functional circuit unit for each one of latch  95 , low-pass filter  100 , amplifier  105 , feedback  110 , and timer  115 . For example, in circuit  265 , capacitor C 1  ( 290 ) is used to implement both feedback  110  and the capacitor of an R-C low-pass filter implementation of low-pass filter  100 , with R 3  ( 275 ) serving as the resistor in the R-C low-pass filter and the Q output of D flip-flop  270  (instead of the {overscore (Q)} output as used in circuit  155 ) being used to initialize the timer. As a result of this combination, the polarity of amplifier  105  is reversed in amplifier  305 , and the input voltage is applied to the non-inverting input of amplifier  305 . Similar to circuit  155 , this voltage is set by a resistive divider ( 295 ,  300 ). 
     Advantageously, the present invention enables any amount of pre-programmed phase shift to be set without requiring significant changes in the circuitry with each phase shift. In addition, the present invention may be implemented in multiple ways, by using different components for latch  95 , low-pass filter  100 , amplifier  105 , feedback  110 , and timer  115 . 
     For example, latch  95  may be implemented with a D flip-flop, an R-S flip-flop, or a J-K flip-flop, among others. Low-pass filter  100  may be implemented with a 1 st  order R-C filter, a 2 nd  order R-C filter, or with any other type of low-pass filter. Amplifier  105  may be implemented with an operational amplifier, a transconductance amplifier, a transistor-based amplifier, etc., feedback  110  may take numerous forms, such as a single-pole integrator, a multiple pole and zero network, etc., and timer  115  may be implemented as an R-C analog delay circuit, a voltage-controlled current source charging a capacitor, or a voltage-controlled oscillator followed by a digital counter, among others. Lastly, two or more of latch  95 , low-pass filter  100 , amplifier  105 , feedback  110 , and timer  115  may be combined in one or more functional circuit units. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing and other objects of the present invention will be apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, in which like reference characters refer to like parts throughout, and in which: 
     FIG. 1 is an illustrative diagram of two signals A and B that are out of phase by 90°; 
     FIG. 2 is a schematic diagram of an illustrative R-C circuit that may be used to shift an input signal by a certain degree to generate a phase-shifted output signal; 
     FIG. 3 is a schematic diagram of an illustrative prior art phase-shift circuit; 
     FIG. 4 is a schematic diagram of an exemplary embodiment of a phase-shift circuit built in accordance with the principles of the present invention; 
     FIG. 5 is a schematic diagram of a preferred embodiment of a phase-shift circuit built in accordance with the principles of the present invention; 
     FIG. 6 is an illustrative timing diagram corresponding to the operation of the phase-shift circuit of FIG. 5; 
     FIG. 7 is a schematic diagram of an exemplary implementation of the phase-shift circuit of FIG. 5; 
     FIG. 8 is an illustrative timing diagram corresponding to the operation of the phase-shift circuit of FIG. 7; 
     FIG. 9 is an illustrative timing diagram corresponding to an input signal and output signals that may be generated by the phase-shift circuit of FIG. 7; and 
     FIG. 10 is a schematic diagram of another exemplary implementation of the phase-shift circuit of FIG.  5 . 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring now to FIG. 4, a schematic diagram of an exemplary embodiment of a phase-shift circuit built in accordance with the principles of the present invention is described. Phase-shift circuit  70  is composed of a latch element such as D flip-flop  75  connected to a delay element such as R-C circuit formed by resistor  80  and capacitor  85  with diode  77  to initialize the timer formed by resistor  80  and capacitor  85 . The D input of D flip-flop  75  is connected to the positive voltage rail. The Q output is asserted with each rising edge of the clock input until D flip-flop  75  is cleared by the R-C circuit. The R-C circuit resets D flip-flop  75  to generate a phase-shift in output {overscore (Q)} as compared to the clock input. The amount of phase-shift is determined by the values of resistor  80 , capacitor  85 , and the CLR threshold voltage of D flip-flop  75 . As a result, the phase shift may be inaccurate due to the frequency sensitivities and production variations of capacitor  85  and production variations of the CLR threshold voltage. 
     Referring now to FIG. 5, a schematic diagram of a preferred embodiment of a phase-shift circuit built in accordance with the principles of the present invention is described. Phase-shift circuit  90  has latch  95  connected to low-pass filter  100  and timer  115 , such that timer  115 &#39;s delay is controlled by amplifier  105  with feedback  110 . Circuit  90  is a closed-loop circuit that provides any amount of accurate phase shift between an input signal and an output signal. Circuit  90  has no operating frequency restrictions, and, more importantly, no frequency-sensitive components are used to set the phase shift between the input signal and the output signal. 
     Phase-shift circuit  90  operates as follows: at the rising edge of the input signal, the Q output of latch  95  is asserted and the {overscore (Q)} output is de-asserted. That is, the input signal is in phase with the Q output of latch  95  but not in phase with the {overscore (Q)} output, as they are complementary in logic. Further, the input signal and the Q output of latch  95  have the same frequency since every rising edge of the input signal asserts the Q output of latch  95  and only a rising edge of the input signal can assert the Q output of latch  95 . Similarly, the {overscore (Q)} output of latch  95  is of the same frequency as the Q output, but as they are complementary in logic, Q and {overscore (Q)} are out of phase. 
     Either Q or {overscore (Q)} of latch  95  is connected to timer  115  so that the rising edge of Q or the falling edge of {overscore (Q)} initializes timer  115 . Timer  115  then starts to count a given time duration. When the time duration expires, timer  115  sends a “timeout” signal to latch  95 , which has the effect of resetting Q and asserting {overscore (Q)}. This results in a phase-shift Δθ between the rising edges of Q and {overscore (Q)}. 
     The phase shift Δθ between the rising edges of Q and {overscore (Q)}, or the phase shift Δθ between the input signal and the output signal may be precisely set by the closed-loop formed with low-pass filter  100  and amplifier  105  with feedback  110 , as follows: first, low-pass filter  100  averages the voltage of Q&#39;s waveform that pulsates between V A  and V G , as shown in FIG.  6 . For illustrative purposes, V G  is set to ground. As a result of this averaging, low-pass filter  100  provides a DC voltage V B  with an amplitude of: 
     
       
           V   B   =δ×V   A   (1) 
       
     
     where δ is Q&#39;s duty cycle, or the percentage of time during which Q is asserted over the total periodic time between two input pulses. 
     Second, the phase shift δθ between the rising edges of Q and {overscore (Q)} is proportional to Q&#39;s duty cycle δ so that:                  Δ                 θ       360      °       =   δ           (   2   )                                
     Substituting equation (2) into equation (1) results in equation (3) below:                V   B     =         Δ                 θ       360      °       ×     V   A               (   3   )                                
     Third, amplifier  105  compares V B  with reference voltage V C  applied at its inverting input and feedback  110  forces a DC voltage V D  so that the error between V B  and V C  is minimized, that is: 
     
       
           V   B   =V   C   (4) 
       
     
     Lastly, timer  115  is constructed so that its time duration is controlled by its input voltage V D . If V D  is adjusted higher, e.g., when Q experiences an instantaneous δ increase, timer  115  reduces its delay time and resets latch  95  sooner. This negative feedback reduces Q&#39;s pulse width and corrects the initial δ increase. 
     With the closed loop, V B  is forced to be the same as the reference voltage V C , so that:                V   C     =         Δ                 θ       360      °       ×     V   A               (   5   )                                
     Therefore, the phase shift Δθ between the input signal and the output signal may be set by the reference voltage V C  as follows:                Δ                 θ     =         V   C       V   A       ×   360      °             (   6   )                                
     The phase shift Δθ is accurate since circuit  90  is a closed-loop system that has no operating frequency restrictions and does not use any frequency-sensitive components to set the phase shift. 
     It should be understood by one skilled in the art that latch  95  may be implemented with a D flip-flop, an R-S flip-flop, or a J-K flip-flop, among others. It should also be understood by one skilled in the art that low-pass filter  100  may be implemented as a 1 st  order R-C filter, a 2 nd  order R-C filter, or as any other type of low-pass filter. 
     Further, it should be understood by one skilled in the art that amplifier  105  may be implemented with an operational amplifier, a transconductance amplifier, a transistor-based amplifier, etc., feedback  110  may take numerous forms, such as a single-pole integrator, a multiple pole and zero network, etc., and timer  115  may be implemented with a R-C analog delay circuit, a voltage-controlled current source charging a capacitor, or a voltage-controlled oscillator followed by a digital counter, among others. Lastly, it should also be understood by one skilled in the art that two or more of latch  95 , low-pass filter  100 , amplifier  105 , feedback  110 , and timer  115  may be combined in one or more functional circuit units. 
     Referring now to FIG. 6, an illustrative timing diagram corresponding to the operation of the phase-shift circuit of FIG. 5 is described. Input signal waveform  120  is shown to be in phase with waveform  125  corresponding the Q output of latch  95 . The phase shift Δθ between the input signal and the output signal is shown in waveform  125  to be generated according to timeout signal ( 150 ) of timer  115 , that is controlled by, amplifier  105  with feedback  110 . 
     It should be understood by one skilled in the art that timeout waveform  150  is shown to be a sawtooth waveform to illustrate an exemplary implementation of timer  115 . Other implementations of timer  115  may result in other types of timeout waveform  150 , such as an exponential timeout waveform  150  and a zig-zag timeout waveform  150 , among others. 
     Referring now to FIG. 7, a schematic diagram of an exemplary implementation of the phase-shift circuit of FIG. 5 is described. In circuit  155 , latch  95  is implemented with D flip-flop  160 , low-pass filter  100  is implemented with an R-C low-pass filter consisting of resistor R 3  ( 165 ) and capacitor C 2  ( 170 ), timer  115  is implemented with an R-C analog delay circuit consisting of resistor R 4  ( 200 ) and capacitor C 3  ( 205 ) with MOSFET M 1  ( 210 ) being used to initialize the timer formed by resistors R 4  ( 200 ) and capacitor C 3  ( 205 ), amplifier  105  is implemented with operational amplifier  190  and its feedback  110  is implemented with capacitor C 1  ( 195 ) along with resistive divider  175 . The input voltage applied to the inverting input of operational amplifier  210  is also set by resistive divider  175 . 
     Phase-shift circuit  155  operates as follows: since the D input is connected to the positive voltage rail, at the rising edge of the clock input, the Q output of D flip-flop  160  is asserted with each rising edge of the clock input that is, the clock input and the Q output of D flip-flop  160  are in phase. In addition, the clock input and the Q output of D flip-flop  160  have the same frequency since every rising edge of the clock input asserts the Q output of D flip-flop  160  and only a rising edge of the clock input can assert the Q output of D flip-flop  160 . 
     The other output {overscore (Q)} of flip-flop  160  is of the same frequency as the Q output, but complementary in logic. That is, Q and {overscore (Q)} are not in phase. The phase shift Δθ between the rising edges of Q and {overscore (Q)}, or the phase shift between the clock input and the output, is proportional to Q&#39;s duty cycle δ:                  Δ                 θ       360      °       =   δ           (   7   )                                
     When Q is asserted, {overscore (Q)} is de-asserted, turning off MOSFET M 1  ( 210 ) and releasing capacitor C 3  ( 205 ) from being discharged. The output of operational amplifier  190  then starts to charge C 3  ( 205 ) through resistor R 4  ( 200 ). When the voltage across C 3  ( 205 ) reaches D flip-flop  160 &#39;s CLR threshold, D flip-flop  160  is reset and output Q is de-asserted. 
     The amount of phase shift is determined by the values of resistors R 1  ( 180 ) and R 2  ( 185 ) in resistive divider  175  as follows: with a R-C low-pass filter formed by resistor R 3  ( 165 ) and capacitor C 2  ( 170 ) between Q and the non-inverting input V +  of operational amplifier  190 , the voltage at V +  is the DC average voltage of Q. Assuming Q=V DD  when asserted, and Q=0 when de-asserted, Q&#39;s average voltage is the product of V DD  and Q&#39;s duty cycle δ, that is: 
     
       
           V   +   =V   DD ×δ  (8) 
       
     
     With resistive divider  175  between V DD  and the inverting input V −  of operational amplifier  190 , the voltage at V −  is:                V   -     =       V   DD     ×     R2     R1   +   R2                 (   9   )                                
     Operational amplifier  190  servers as negative feedback, that is, if the non-inverting input V +  is higher than the inverting input V − , feedback capacitor C 1  ( 195 ) integrates this error and increases the output voltage V 0 . A higher V 0  shortens capacitor C 3 &#39;s ( 205 ) charging time and produces a narrower pulse width at Q. This reduces Q&#39;s duty cycle δ in Equation (7), thereby decreasing V +  to a value closer to V − . Similarly, if V +  is lower than V − , operational amplifier  190  reduces its output voltage V 0  to make C 3 &#39;s ( 205 ) charging time longer. Eventually, V +  will be enforced to be the same as V − , i.e., V + =V − , and the output voltage V 0  will stabilize into a DC voltage. With V +  set as in Equation (8) and V −  set as in Equation (9), Q&#39;s duty cycle δ is therefore determined by:              δ   =     R2     R1   +   R2               (   10   )                                
     Lastly, with Q&#39;s duty cycle δ proportional to the phase shift Δθ between the clock input and the output {overscore (Q)}, the phase shift Δθ given by circuit  155  is as follows:                Δ                 θ     =       R2     R1   +   R2       ×   360      °             (   11   )                                
     Therefore, circuit  155  may be pre-programmed to achieve any amount of phase shift by setting the values of resistors R 1  ( 180 ) and R 2  ( 185 ) in resistive divider  175 . Furthermore, the phase shift is accurate since circuit  155  is a closed-loop system that has no operating frequency restrictions and does not use any frequency-sensitive components to set the phase shift. Variations in supply voltage are also effectively rejected by circuit  155  as the DC level of Q and the inverting voltage V −  of operational amplifier  190  are both proportional to V DD . 
     Referring now to FIG. 8, an illustrative timing diagram corresponding to the operation of the phase-shift circuit of FIG. 7 is described. Timing diagram  215  shows the waveforms of the clock input ( 220 ), D flip-flop  160 &#39;s CLR threshold ( 230 ), and D flip-flop  160 &#39;s outputs Q ( 225 ) and {overscore (Q)} ( 235 ). 
     As described above, with each rising edge of the clock input, the Q output is asserted, that is, the clock input and the Q output of D flip-flop  160  are in phase. The other output {overscore (Q)} of flip-flop  160  is of the same frequency as the Q output, but complementary in logic. That is, Q and {overscore (Q)} are not in phase. When Q is asserted, {overscore (Q)} is de-asserted, turning off MOSFET M 1  ( 210 ) and releasing capacitor C 3  ( 205 ) from being discharged. The output of operational amplifier  190  then starts to charge C 3  ( 205 ) through resistor R 4  ( 200 ). When the voltage across C 3  ( 205 ) reaches D flip-flop  160 &#39;s CLR threshold, D flip-flop  160  is reset and output Q is de-asserted. The phase shift Δθ given by circuit  155  is illustrated in {overscore (Q)}&#39;s waveform ( 235 ). 
     Referring now to FIG. 9, an illustrative timing diagram corresponding to an input signal and output signals that may be generated by the phase-shift circuit of FIG. 7 is described. Signals S 1  ( 245 ), S 2  ( 250 ), S 3  ( 255 ), and S 4  ( 260 ) are out of phase with input signal  240 , with each signal having a different phase shift. Signal S 1  ( 245 ) is 45° out of phase with input signal  240 , signal S 2  ( 250 ) is 60° out of phase with input signal  240 , signal S 3  ( 255 ) is 90° out of phase with input signal  240 , and signal S 4  ( 260 ) is 120° out of phase with input signal  240 . These phase shifts may be achieved by circuit  155  by setting the ratio R 2 /(R 1 +R 2 ) in resistive divider  175  as follows:                  R2     R1   +   R2       =     1   8       ,       for                 Δ                 θ     =     45      °               (   12   )                   R2     R1   +   R2       =     1   6       ,       for                 Δ                 θ     =     60      °               (   13   )                   R2     R1   +   R2       =     1   4       ,       for                 Δ                 θ     =     90      °               (   14   )                   R2     R1   +   R2       =     1   3       ,       for                 Δ                 θ     =     120      °               (   15   )                                
     Other amounts of phase shift may be easily obtained by using different values for resistors R 1  ( 180 ) and R 2  ( 185 ). 
     Referring now to FIG. 10, a schematic diagram of another exemplary implementation of the phase-shift circuit of FIG. 5 is described. In circuit  265 , capacitor C 1  ( 290 ) is used to implement both feedback  110  and the capacitor of an R-C low-pass filter implementation of low-pass filter  100 , with R 3  ( 275 ) serving as the resistor in the R-C low-pass filter. As a result of this combination, the polarity of amplifier  105  is reversed in amplifier  305 , and the input voltage is applied to the non-inverting input of amplifier  305 . Also, Q (instead of {overscore (Q)}) is used to initialize the timer formed by resistor R 4  ( 310 ) and capacitor C 3  ( 285 ). Similar to circuit  155 , the input voltage is set by a resistive divider ( 295 ,  300 ). 
     Compared to circuit  155  of FIG. 7, circuit  265  uses diode  280  instead of MOSFET M 1  ( 210 ) to initialize the timer formed by resistor R 4  ( 310 ) and capacitor C 3  ( 285 ). Circuit  265  also eliminates capacitor C 2  ( 170 ) and it uses capacitor C 1  ( 290 ) for both filtering and integrating. 
     Although particular embodiments of the present invention have been described above in detail, it will be understood that this description is merely for purposes of illustration. Specific features of the invention are shown in some drawings and not in others, for purposes of convenience only, and any feature may be combined with other features in accordance with the invention. Steps of the described processes may be reordered or combined, and other steps may be included. Further variations will be apparent to one skilled in the art in light of this disclosure and such variations are intended to fall within the scope of the appended claims.