text: referring now to fig1 , an lc oscillator tank circuit 10 used in building oscillators is composed of a source of inductance ( inductive element ) l and capacitance ( capacitive element ) c . the inductive element l and capacitive element c in the lc oscillator tank circuit 10 can be composed of various types of resonators and passive elements , such as but not limited to , on - chip integrated inductors , bond - wires , metal - insulator - metal ( mim ) capacitors , metal finger capacitors , metal oxide semiconductor ( mos ) capacitors , ceramic resonators , micro electro - mechanical systems ( mems ) tuning fork resonators , mems wine - glass resonators , mems - based resonators , surface acoustic wave ( saw ) and bulk acoustic wave ( baw ) devices . the implementation of an ideal pure inductor or capacitor is usually not possible due to the physical limitations of having a finite quality factor q . integrated inductors in cmos technology to date have low q factors when compared to mems resonators and crystals . sources of losses in an inductor include the inductor metal ohmic losses r l and substrate resistive losses r sub . both of these losses are usually temperature dependent , and therefore , the overall impedance and q of the inductor is temperature dependent . the integrated capacitive part of the tank also suffers from a finite temperature dependent q as well as temperature dependence of the capacitance value . as a result , the physical implementation of an integrated lc - tank will dictate a strong temperature dependence of the impedance and q factor of the tank , which results in a temperature dependent tank resonance frequency . an oscillator built using an lc oscillator tank 10 typically includes an amplifier responsible for overcoming the tank losses . for the oscillator to have sustained oscillations , the barkhausen criterion requires an open loop gain greater than 1 and phase equal to zero . assuming that the amplifier contributes a zero phase , then for oscillation to occur , the lc oscillator tank impedance z tank must have a zero phase . the phase condition is used to derive the oscillation frequency ω osc as follows : from the above equations 1 - 3 , it can be seen that the oscillation frequency is temperature dependent if r l is temperature dependent . a linear variation of r l with temperature results in an almost linear variation of the oscillation frequency . in addition , any temperature variation in c would strongly contribute to the temperature dependence . this is graphically shown in fig2 , where φ tank is plotted for different temperatures for a tank composed of l , r l and c with linear temperature dependence of r l defined as : it is to be noted that the oscillation frequency is determined using the intersection of φ tank = 0 with the phase plots . the corresponding oscillation frequency across temperature is plotted in fig3 , showing a strong temperature dependence of 8000 ppm in a typical lc oscillator tank operating at zero phase . examining the phase plots again in fig2 , since the quality factor of the tank changes with temperature , the phase plots change with temperature . in addition , at the oscillation frequency , there is a higher q at lower temperatures , and therefore , a larger negative slope is seen at lower temperatures . therefore , the varying phase plot slopes with temperature result in the intersection of these plots . when the intersections occur at the same phase , a temperature insensitive tank operating point is created , and the tank is said to be operating at a temperature “ null ” ( tnull ) with a phase φ null . the ideal temperature null phase occurs when the phase plots across temperature intersect at precisely the same phase . oscillation with a phase across the tank ideally equal to φ null results in an oscillation frequency with zero deviation across temperature . more realistic tanks exhibit a temperature null with small frequency deviations across temperature . this is illustrated graphically in fig4 , where the condition of oscillation is φ tank = φ null and the corresponding oscillation frequency across temperature is plotted . as can be seen in fig5 , operating the oscillator at the temperature null phase results in an oscillation frequency with a much lower temperature dependence . for example , in fig5 , the frequency drift is only 290 ppm . comparing this to the 8000 ppm frequency drift at zero phase in fig3 , oscillating at the temperature null phase produces a more stable frequency . a global temperature null ( gnull ) can be defined as a phase operating point φ gnull that results in a minimum frequency deviation f across a temperature range t with a very small or zero change in oscillation frequency over temperature at the center of the temperature range t 0 . a measure of the quality of the temperature null is the oscillation frequency deviation across temperature . a figure of merit ( fom ) of the tank temperature null may be defined as : fom = δ ⁢ ⁢ f f t ⁢ ⁢ 0 δ ⁢ ⁢ t ⁢ ppm / ° ⁢ ⁢ c . equation ⁢ ⁢ 5 where f t0 is the oscillation frequency at t 0 . the smaller the value of the fom , the better the null quality is with the perfect null occurring at fom = 0 . a local temperature null ( lnull ) can be defined as a phase operating point φ lnull with alternatively , lnull can be defined at temperature t as the intersection of the phase plots of temperatures ( t + δ ) and ( t − δ ) where δ is infinitesimally small . the gnull oscillation frequency ω gnull around temperature t 0 may be derived by finding the intersection of two phase curves at temperatures t 0 + δt and t 0 − δt . for an lc oscillator tank with a linear temperature dependence of r l the phase and frequency at the gnull are as follows : ω gnull = ω 0 ⁢ 1 + c l ⁢ r 0 2 ⁡ ( 1 - α 2 ⁢ δ ⁢ ⁢ t 2 ) equation ⁢ ⁢ 6 φ gnull =− tan − 1 ( 2 r 0 cω gnull ) equation 7 now referring back to fig5 , this curve shows the frequency variation versus temperature while operating at tnull by assuming a first order model of the tank . the first order model of the tank versus temperature includes the temperature variations of the inductor dc ( direct current ) losses only . practically speaking , there are more factors that affect the frequency variation versus temperature at tnull in addition to the temperature varying inductor dc losses , such as and not limited to the temperature varying harmonics induced by the active circuitry , the temperature varying parasitic capacitances imposed by the routing interconnects and the active circuitry , the temperature varying non - ideal effects in the inductor of the tank such as the skin depth effect and the proximity effect and the temperature variation of the capacitance of the tank . due to such factors , the practical profile of frequency versus temperature at tnull deviates from the theoretical expectations of the first order model . the final frequency profile varies according to the weight of each factor and the combination of the different factors . the profile of the frequency variation versus temperature at the temperature null phase is denoted by the “ temperature null characteristic ” or the “ tnull characteristic ”. fig6 compares three examples for possible practical tnull characteristics to the expected theoretical tnull characteristic from the first order model . the tnull characteristic is the shape of the frequency deviation δf ( t ) versus temperature , where δf ( t ) is the frequency deviation referred to the oscillation frequency at the extremes of the temperature range t o − δt and t o + δt when operating at tnull . note that the frequency at the temperature range minimum t o − δt is equal to the frequency at the temperature range maximum t o + δt when operating at tnull . hence , δf ( t ) is given as : in order to control the tnull characteristic , the profile of the oscillation amplitude versus temperature is utilized so as to compensate for the frequency variations through the current harmonic content according to equation 13 in u . s . pat . no . 8 , 072 , 281 . for convenience , equation 13 is stated again here : ω = ω os ⁡ ( 1 - 1 2 ⁢ ⁢ q 2 ⁢ ∑ 2 ∞ ⁢ ⁢ n 2 n 2 - 1 ⁢ i n 2 i 1 2 ) equation ⁢ ⁢ 9 where ω os is the oscillation frequency , and i n is the n th harmonic of the current in the tank . in one aspect of this disclosure , the oscillation amplitude is deliberately changed across temperature with a specific profile . thus , the harmonic content changes across temperature in a rather controlled manner which in turn induces variations in the oscillation frequency according to equation 9 . this amplitude profile with temperature is manipulated such that the induced frequency variations combined with the original frequency variations produce the required tnull characteristic . hence , the technique provides substantial control over the tnull characteristic . in reference to fig7 , an oscillator having a tank circuit 10 and a transconductance amplifier 30 is shown with an automatic amplitude control ( aac ) block 50 , including an amplitude sensing circuit 52 , a difference circuit 54 to which a reference signal 53 v ref ( t ) is applied , and a controller 56 . the oscillation amplitude is controlled by the aac loop and it is a scaled value of the aac reference voltage v ref ( t ). thus , in one aspect of this disclosure , the required profile of the oscillation amplitude versus temperature is imposed through the aac reference voltage v ref ( t ). the aac reference voltage becomes a controlled programmable function of temperature . v ref ( t ) is programmed to achieve the required tnull characteristic . unlike the prior art ( e . g ., u . s . pat . no . 8 , 072 , 281 ) in which the aac reference voltage was assumed to be substantially constant with temperature , in the present methodology the aac reference voltage is caused to vary as a function of temperature . in another aspect of this disclosure , the reference voltage is programmed through a reference voltage generation circuit . an example of such a circuit is shown as 100 in fig8 . it primarily consists of a bias voltage generator block 110 and a polynomial generator block 120 . the bias voltage generator block generates an array of temperature dependent bias voltages v 0 ( t ) to v n ( t ) where : t is the absolute temperature in kelvin , n is an arbitrary integer and v x ( t ) is a polynomial function of temperature of the order x such that x is an integer that satisfies the inequality 0 ≦ x ≦ n . hence , v 0 ( t ) to v n ( t ) are given as : v 1 ( t )= v 01 ( 1 + t v11 ( t − t 0 )) equation 10b v 2 ( t )= v 02 ( 1 + t v12 ( t − t 0 )+ t v22 ( t − t 0 ) 2 ) equation 10c v 3 ( t )= v 03 ( 1 + t v13 ( t − t 0 )+ t v23 ( t − t 0 ) 2 + t v33 ( t − t 0 ) 3 ) equation 10d v x ( t )= v 0x ( 1 + + t v1x ( t − t 0 )+ t v2x ( t − t 0 ) 2 . . . + t vxx ( t − t 0 ) x ) equation 10e v n ( t )= v 0n ( 1 + t v1n ( t − t 0 )+ t v2n ( t − t 0 ) 2 . . . + t vnn ( t − t 0 ) n ) equation 10f where t o is the room temperature expressed in kelvin and v 0x is the value of the bias voltage v x ( t ) at room temperature t o . furthermore , t vyx is the y th order temperature coefficient of the voltage v x ( t ) where y is an integer that satisfies the inequality 1 ≦ y ≦ n . note that v 0 ( t ) is a weak function of temperature and thus its temperature dependence is neglected in the equations . the voltages v 0 ( t ) to v n ( t ) are fed into the polynomial generator block 120 . the polynomial generator block 120 combines the different bias voltages with the proper weighting and signal conditioning for each of them so as to generate the final reference voltage 53 vref ( t ) with the required temperature dependence . the polynomial generator block 120 generates vref ( t ) as a programmable n th order polynomial function of temperature according to the following equation : v ref ( t )= v ref0 ( 1 + t v1 ( t − t 0 )+ t v2 ( t − t 0 ) 2 . . . + t vn ( t − t 0 ) xn ) equation 11 where t vy is the y th order temperature coefficient for v ref ( t ) and y is an integer that satisfies the inequality 1 ≦ y ≦ n and v refo is the value of v ref ( t ) at room temperature t o . moreover , v refo and the coefficients t v1 through t vn are programmed using digital words that are fed to the polynomial generator block 120 as shown in fig8 . the polynomial generator block 120 translates the digital control words to the required temperature coefficients and thus generates the required reference voltage 53 v ref ( t ) from the different input voltages v 0 ( t ) through v n ( t ). the coefficients t v1 through t vn are programmed to obtain the required oscillation amplitude profile versus temperature , while v refo is programmed to obtain the required magnitude of the oscillation amplitude . the combination of the coefficients t v1 through t vn determines the final tnull characteristic . every combination of coefficients generates a different tnull characteristic according to the weight and strength of each coefficient . fig9 shows the tnull characteristics of 3 different combinations of coefficients . the coefficients are programmed to generate the required tnull characteristic . therefore , the technique provides substantial control over the tnull characteristic . in another aspect of this disclosure , the different orders of temperature dependence can be generated by different circuits known in literature . as an example , a positive first order temperature dependent current can be generated by a thermal voltage ( v t ) referenced bias circuit as the circuit shown in fig1 by block 200 . this circuit is well known in literature to produce a dc current that is proportional to absolute temperature ( ptat ) provided that the resistor r is weakly dependent upon temperature as described in “ analysis and design of analog integrated circuits ,” by paul gray , paul j . hurst , stephen h . lewis and robert g . meyer ( john wiley & amp ; sons , inc ., fourth edition , 2001 ) and “ design of analog cmos integrated circuits ,” by bahzad razavi ( mcgraw - hill , international edition , 2001 ). the resulting ptat current ( i ptat ) is given by : i ptat ( t )= i 0 ( 1 + α ( t − t 0 )) equation 12 where i o is the nominal current value at t o and α is the first order temperature coefficient of the current and it is approximately 3300 ppm /° k . for typical silicon based processes . in another aspect of this disclosure , the circuit 300 in fig1 may be used to obtain the control voltage v 1 ( t ). by allowing the ptat current ( i ptat ) to flow in a resistor 310 that is weakly dependent upon temperature as shown in fig1 , one can obtain a voltage v 1 ( t ) that varies linearly with temperature according to the equation : v 1 ( t )= i 0 r ( 1 + α ( t − t 0 )) equation 13 hence , the circuit 300 of fig1 can be used to obtain a positive first order temperature coefficient t v11 . for this specific example , t v11 is 3300 ppm /° k . for typical silicon based processes under the assumption that the temperature dependence of the resistor r 310 is negligible . in another aspect of this disclosure a second order temperature dependent voltage is produced by circuit 400 in fig1 . here , the ptat current is forced to flow in a resistance 410 that has substantially linear temperature dependence . the resistance 410 r t ( t ) is given by the relation : r t ( t )= r 0 ( 1 + α r ( t − t 0 )) equation 14 where r o is the value of the resistance at temperature t o and α r is the first order temperature coefficient of the resistance . the substantially linear temperature dependent resistance r t ( t ) can be realized by several methods . for example , in a typical silicon - based process , r t ( t ) can be realized by process modules such as and not limited to diffusion resistance , n - well ( n - doped substrate ) resistance and active resistances realized by transistor devices . v 2 ( t )= i ptat ( t ) r t ( t ) equation 15a v 2 ( t )= i 0 r 0 ( 1 +( α + α r )( t − t 0 )+ αα r ( t − t 0 ) 2 ) equation 15b therefore , for this specific example v 2 ( t ) is generated with t v12 = α + α r and t v22 = αα r . in order to obtain negative first and second order temperature coefficients , complementary to absolute temperature ( ctat ) circuits may be used . a ctat circuit 500 is shown in fig1 . this circuit is well known in literature to produce a dc current that is complementary to absolute temperature ( ctat ) provided that the resistor r is weakly dependent upon temperature as described in “ analysis and design of analog integrated circuits ,” by paul gray , paul j . hurst , stephen h . lewis and robert g . meyer ( john wiley & amp ; sons , inc ., fourth edition , 2001 ). the resulting ctat current ( i ctat ) is given by : i ctat ( t )= i 0 ( 1 − α c ( t − t 0 )) equation 16 where i o is the nominal current value at t o and α c is the first order temperature coefficient of the current and it is approximately 3300 ppm /° k . for typical silicon based processes . in another aspect of this disclosure , fig1 depicts the usage of the ctat current generator to obtain negative first and second order temperature coefficients in v 1 ( t ) and v 2 ( t ). the circuit utilizes the same types of resistances utilized earlier with the ptat generator . the output voltages are given by : v 1 ( t )= i 0 r ( 1 − α c ( t − t 0 )) equation 17 v 2 ( t )= i 0 r 0 ( 1 +( α r − α c )( t − t 0 )− α c α r ( t − t 0 ) 2 ) equation 18 in another aspect of this disclosure , coefficients with orders higher than the 2 nd order can be obtained by utilizing similar methods which depends upon having resistances that are highly temperature dependent . fig1 depicts a circuit that can generate an ( x + 1 ) th order temperature dependent voltage v x + 1 ( t ) from an x th order temperature dependent voltage v x ( t ) by utilizing a resistance that has a substantially linear temperature dependence . the circuit comprises an operational amplifier in a unity feedback configuration . it regenerates v x ( t ) on a resistor r that is weakly dependent on temperature and hence generates i x ( t ) which is a current of the same order of temperature dependence as v x ( t ) such that v x ⁡ ( t ) = v 0 ⁢ x ⁡ ( 1 + t v ⁢ ⁢ 1 ⁢ x ⁡ ( t - t 0 ) + t v ⁢ ⁢ 2 ⁢ x ⁡ ( t - t 0 ) 2 ⁢ ⁢ ⋯ + t vxx ⁡ ( t - t 0 ) x ) equation ⁢ ⁢ 19 i x ⁡ ( t ) = v 0 ⁢ x r ⁢ ( 1 + t v ⁢ ⁢ 1 ⁢ x ⁡ ( t - t 0 ) + t v ⁢ ⁢ 2 ⁢ x ⁡ ( t - t 0 ) 2 ⁢ ⁢ ⋯ + t vxx ⁡ ( t - t 0 ) x ) equation ⁢ ⁢ 20 i x ( t ) is then mirrored and forced to flow in the resistor r t ( t ) which is substantially linear with temperature and finally generates v x + 1 ( t ): as will be recognized by those skilled in the art , the innovative concepts described in the present application can be modified and varied over a wide range of applications . accordingly , the scope of the invention should not be limited to any of the specific exemplary teachings discussed , but is instead defined by the following claims .