Abstract:
In one embodiment, a bandgap voltage reference generating circuit is configured to generate a reference voltage, and may comprise a first PN-junction whose base-emitter voltage (V BE ) exhibits a curvature with respect to temperature, where a current conducted by the first PN-junction is proportional to absolute temperature (PTAT). The voltage reference generating circuit may also include a second PN-junction coupled to the first PN-junction. A control circuit coupled to the second PN-junction may be configured to inject a control current into the second PN-junction, where the control current has a negative to absolute temperature (NTAT) characteristic, the control circuit thereby operating to effectively eliminate a curvature with respect to temperature exhibited by the bandgap voltage.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     1. Field of the Invention  
         [0002]     This invention relates generally to the field of integrated circuit design and, more particularly, to the design of bandgap references.  
         [0003]     2. Description of the Related Art  
         [0004]     Many digital systems, especially those that include high-performance, high-speed circuits, are prone to operational variances due to temperature effects. Devices that monitor temperature and voltage are often included as part of such systems in order to maintain the integrity of the system components. Personal computers (PC), signal processors and high-speed graphics adapters, among others, typically benefit from such temperature monitoring circuits. For example, a central processor unit (CPU) that typically “runs hot” as its operating temperature reaches high levels may require a temperature sensor in the PC to insure that it doesn&#39;t malfunction or break due to thermal problems.  
         [0005]     Often, integrated circuit (IC) solutions designed to measure temperature in a system will monitor the voltage across one or more PN-junctions, for example one or more diodes, at different current densities, to extract a temperature value. This is often accomplished by measuring a difference in voltage across the terminals of typically identical diodes, when different current densities are forced through the PN junctions of the diodes. The resulting change (ΔV BE ) in the base-emitter voltage (V BE ) between the diodes is generally proportional to temperature. (It should be noted that while V BE  generally refers to a voltage across the base-emitter junction of a diode-connected transistor and not a voltage across a simple PN-junction diode, for the sake of simplicity, V BE  is used herein to refer to the voltage developed across a PN-junction in general.) In general, V BE  may be defined as a function of absolute temperature by the equation  
               V   BE     =     η   ⁢     kT   q     ⁢   ln   ⁢       I   C       I   S                 (   1   )             
 
 where η is the ideality factor of the PN junction, k is Boltzman&#39;s constant, q is the charge of a single electron, T represents absolute temperature, I s  represents saturation current and I c  represents the collector current. A more efficient and precise method of obtaining ΔV BE  is to supply the PN junction of a single diode with two separate and different currents in a predetermined ratio 
 
         [0006]     ADCs, such as the ones used in temperature measurement systems, require a precise reference voltage to function accurately and reliably. In general, many different devices and technologies may require temperature-stable reference voltages. A common circuit used to provide such a reference voltage is a bandgap voltage reference circuit. Bandgap voltage reference circuits typically operate by summing a base-emitter voltage (V BE ) of a bipolar junction transistor (BJT), which has a negative temperature drift, with a thermal voltage V t  that has a positive temperature drift. The thermal voltage V t  is typically dependent on the difference between V BE  of two BJTs operating at different emitter current densities. The value of the resulting bandgap voltage V BG  (V ref ) is the sum of V BE  of one BJT and a quantity proportional to the difference in V BE  between two BJTs.  
         [0007]     Typically, the output of a bandgap voltage reference circuit has a residual curvature that has a non-zero temperature coefficient (TC) for values of temperature other than a nominal operating temperature. In some applications, errors in the output voltage that arise due to this non-zero temperature coefficient may be unacceptable. It is therefore desirable to design a zero TC bandgap reference for generating the reference voltage used by a given ADC that is part of a temperature measurement system.  
         [0008]     The need for curvature correction may arise for a wide operating temperature range, such as −40° C. to +125° C., for example. A typical bandgap reference may exhibit a certain degree of curvature over such a wide temperature range (for example, 4.5 mV for a range of −40° C. to +125° C.), which generally results in a variation in the temperature sensor output (for example, an 0.8° C. of variation for a 4.5 mV curvature). In other words, at the endpoint temperatures the temperature measurements may rise in accordance with the exhibited curvature of the bandgap reference. Therefore, reduction of this curvature may lead to increased accuracy in the temperature measurements. However, correction circuitry to perform the curvature correction may be complicated. The performance of the correction circuitry itself may also be subject to errors that arise due to process variations.  
         [0009]     Other corresponding issues related to the prior art will become apparent to one skilled in the art after comparing such prior art with the present invention as described herein.  
       SUMMARY OF THE INVENTION  
       [0010]     In one set of embodiments, a bandgap reference voltage generating circuit may generate a reference voltage that is perfect and/or completely curvature corrected. In one embodiment, a negative to absolute temperature (NTAT) current is injected into the emitter of a low emitter current density transistor in a positive to absolute temperature (PTAT) current circuit, thereby generating a T ln(T) current component, which perfectly cancels the curvature term in a diode used in generating the reference voltage, leaving only the bandgap voltage as the reference voltage.  
         [0011]     In one embodiment, a bandgap voltage reference generating circuit may comprise a first PN-junction whose base-emitter voltage (V BE ) exhibits a curvature with respect to temperature, where a current conducted by the first PN-junction may be proportional to absolute temperature (PTAT). The voltage reference generating circuit may also include a second PN-junction coupled to the first PN-junction. A control circuit coupled to the second PN-junction may be configured to inject a control current into the second PN-junction, where the control current has a negative to absolute temperature (NTAT) characteristic. The control current may operate to effectively eliminate a curvature with respect to temperature exhibited by a reference voltage generated by the bandgap voltage reference generating circuit.  
         [0012]     In one embodiment, a bandgap voltage reference generating circuit includes a first operational amplifier (op-amp) whose output is configured as the reference voltage output, with the PN junctions comprised in respective bipolar junction transistors. The BJT corresponding to the first PN-junction may be coupled to the non-inverting input of the first op-amp, while the BJT corresponding to the second PN-junction may be coupled to the inverting input of the first op-amp. Each BJT may be a PNP transistor with its emitter coupling to the corresponding op-amp input terminal. The control circuit may comprise two PMOS devices and a second op-amp. The gate of each PMOS device may be coupled to the output of the second op-amp, with the drain of one of the PMOS devices coupling to the non-inverting terminal of the second op-amp, and the drain of the other PMOS device coupling to the emitter of the BJT corresponding to the second PN-junction. Additionally, the inverting input of the second op-amp may be coupled to the emitter of the BJT corresponding to the first PN-junction.  
         [0013]     In one embodiment, operating a bandgap voltage reference generating circuit may include: powering the bandgap reference circuit, where the bandgap reference circuit comprises a first PN-junction coupled to a second PN-junction, and in response to powering the circuit, a V BE  of the first PN-junction exhibits a non-linearity (curvature) with respect to temperature. Also in response to powering the bandgap voltage reference generating circuit, the first PN-junction may conduct a current that is proportional with respect to absolute temperature. In one set of embodiments, by injecting a control current into the second PN-junction, with the control current having a negative to absolute temperature characteristic, a curvature (resulting from the V BE  of the first PN junction exhibiting a curvature with respect to temperature) exhibited by the reference voltage generated by the bandgap voltage reference generating circuit may effectively be corrected and/or eliminated. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0014]     The foregoing, as well as other objects, features, and advantages of this invention may be more completely understood by reference to the following detailed description when read together with the accompanying drawings in which:  
         [0015]      FIG. 1  illustrates a common bandgap reference circuit; and  
         [0016]      FIG. 2  illustrates one embodiment of a bandgap reference circuit according to the present invention.  
     
    
       [0017]     While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the present invention as defined by the appended claims. Note, the headings are for organizational purposes only and are not meant to be used to limit or interpret the description or claims. Furthermore, note that the word “may” is used throughout this application in a permissive sense (i.e., having the potential to, being able to), not a mandatory sense (i.e., must).” The term “include”, and derivations thereof, mean “including, but not limited to”. The term “coupled” means “directly or indirectly connected”.  
       DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0018]     As used herein, the word “alternately” is meant to imply passing back and forth from one state, action, or place to another state, action, or place, respectively. For example, “alternately applying a first current source and a second current source” would mean applying the first current source, then applying the second current source, then applying the first current source, then applying the second current source, and so on.  
         [0019]     A “diode-junction-voltage” (V BE ) refers to a voltage measured across the junction of a diode, or a difference in voltage between a voltage measured at the anode of the diode junction with respect to a common ground and a voltage measured at the cathode of the diode junction with respect to the common ground. A diode is one device comprising a PN-junction across which voltage V BE  may be developed. More generally, diode-junction may also mean PN-junction or NP-junction, which defines the physical attributes of the junction across which V BE  may be developed. In certain embodiments, the operation performed by a diode may be achieved using other circuitry, such as a PN-junction (or NP-junction) present in devices other than a diode, for example in bipolar junction transistors (BJTs). Therefore, the terms PN-junction, NP-junction, diode, diode-junction, and V BE  junction are used interchangeably, and all respective terms associated therewith may be interpreted accordingly.  
         [0020]     In one set of embodiments, a bandgap reference may be configured to form a reference voltage through a V BE  junction. In such bandgap references, a curvature may be observed due to the non-linearity of the V BE  junction used by the bandgap reference to provide the reference voltage to various designated data conversion circuits. The relationship between V BE  and absolute temperature T shown in equation (1) may be re-written in equation (2), showing the non-linear curvature produced by the temperature dependent nature of I S . V BE  in equation (2) may represent the voltage across the V BE  junction used by a bandgap reference to form a reference voltage. Thus,  
                 V   BE     ⁡     (   T   )       =       V   go     -       T     T   r       ⁡     [       V   go     -       V   BE     ⁡     (     T   r     )         ]       -       [       (     4   -   n     )     -   x     ]     ⁢       η   ⁢           ⁢   kT     q     ⁢     ln   ⁡     (     T     T   r       )                   (   2   )             
 
 where V go  represents the bandgap voltage of silicon, T r  represents a specified reference temperature, V BE  (T r ) represents the base-emitter junction voltage at temperature T r , n represents a process-dependent constant, and x represents a constant related to junction current characteristics, in addition to the other variables described for equation (1). By designating one of the expressions containing all constants from equation (2) as:  
             α   =       [       (     4   -   n     )     -   x     ]     ⁢       η   ⁢           ⁢   k     q               (   3   )             
 
 equation (3) may be re-written as:  
                 V   BE     ⁡     (   T   )       =       V   go     -     T   *       1     T   r       ⁡     [       V   go     -       V   BE     ⁡     (     T   r     )       -     α   ⁢           ⁢     T   r     ⁢     ln   ⁡     (     T   r     )           ]         -     α   ⁢           ⁢   T   ⁢           ⁢       ln   ⁡     (   T   )       .                 (   4   )             
 
 Assigning a single value to another combination of constants in equation (4):  
               β   =       1     T   r       ⁡     [       V   go     -       V   BE     ⁡     (     T   r     )       -     α   ⁢           ⁢     T   r     ⁢     ln   ⁡     (     T   r     )           ]         ,           (   5   )             
 
 equation (4) may further be simplified as: 
 
 V   BE ( T )= V   go   −βT−αT  ln( T ).  (6) 
 
         [0021]     It may be observed from equation (6) that the base-emitter voltage, V BE  (T), is defined by three terms. The first term is a constant, V go , the bandgap voltage of the semiconductor material, in this case silicon. The second term is a linear function of absolute temperature, T, that has a coefficient of −β and the last term is a non-linear function in the form of −αT ln(T). The last term corresponds to the effects that give rise to a non-linear curvature characteristic of a reference voltage that is generated by a bandgap reference. Eliminating this non-linear curvature characteristic may result in a substantially increased accuracy of circuits that rely on a reference voltage generated by a bandgap reference, for example the ADC or ADCs configured in temperature sensor circuits.  
         [0022]     In order to create a constant voltage across all operating temperatures, the V BE  (T) voltage described in equation (6) may be combined with a second voltage that may cancel out linear and non-linear portions, leaving only the constant V go . Therefore, a new voltage added to V BE  (T) may have the form shown in equation (7) below. 
 
 V   PTAT   =βT+αT  ln( T ).  (7) 
 
 The subscript “PTAT” in V PTAT  is indicative of the linear term βT being proportional to absolute temperature. The voltage V PTAT  that is proportional to absolute temperature may be created using two V BE  junctions operating at different emitter current densities. 
 
         [0023]      FIG. 1  illustrates a circuit topology  300  that may be used to generate a bandgap voltage V BG    314 , which includes generating V PTAT . As shown, emitter area m of transistor  306  may be N times the emitter area of transistor  304 , resulting in differing emitter current densities between transistors  306  and  304 . More specifically, transistor  306  may be considered a low emitter-current density transistor with respect to transistor  304 . The output of amplifier  302  may drive resistors R 1   308  and R 3   310  such that the voltages at the inputs of amplifier  302  have the same value. Transistors  306  and  304  may be configured to conduct currents I 1  and I 2 , respectively, through their base-emitter junctions, where I 2  is a constant ‘M’ multiple of I 1 . Accordingly, the following equations may be used to describe the operation of circuit  300  from  FIG. 1 : 
   V   x   =I   1   *R 2 +V   BE0 ( T,I   1 )  (8)    V   x   =V   BE1 ( T,I   2 )  (9)    I   2   =M*I   1   (10)  
 where V BE0  and V BE1  represent the base-emitter voltages for transistors  306  and  304 , respectively, and V x  represents the voltage at nodes  320  and  322 . Equations (8), (9) and (10) may be combined to form:  
                 I   1     =         1     R   ⁢           ⁢   2       ⁡     [                 η   ⁢           ⁢   kT     q     ⁢   ln   ⁢     (       N   *     I   1         I   S       )       -                   η   ⁢           ⁢   kT     q     ⁢     ln   ⁡     (       I   1       N   *     I   S         )               ]       =       1     R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢     ln   ⁡     (   MN   )             ,           (   11   )               
 and equations (11) and (6) may be combined to obtain:  
               V   BG     =           ⁢         V   X     +           ⁢       I   1     *           ⁢   R   ⁢           ⁢   1       =           ⁢       V   go     -           ⁢     β   ⁢           ⁢   T     -           ⁢     α   ⁢           ⁢   T   ⁢           ⁢     ln   (           ⁢   T   )       +           ⁢         R   ⁢           ⁢   1       R   ⁢           ⁢   2       *           ⁢       η   ⁢           ⁢   kT     q     ⁢           ⁢       ln   (           ⁢   MN   ⁢           )     ⁢           .                   (   12   )               
         [0024]     The term αT ln(T) may be expanded into a power series because it comprises a linear component that may be canceled along with the −β term to obtain a final, zero temperature coefficient output voltage. Thus, the following power series may be obtained: 
 
α T  ln( T )= a   1   T+a   2   T   2   +a   3   T   3 + . . .  (13) 
 
 Equations (12) and (13) may be combined to obtain  
                     V   BG     =       ⁢       V   X     +       I   1     *     R   ⁢   1                     =       ⁢       V   go     -       (     β   +     a   1       )     ⁢   T     -     (         a   2     ⁢     T   2       +       a   3     ⁢     T   3       +   …     )     +                     ⁢         R   ⁢           ⁢   1       R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢       ln   ⁡     (   MN   )       .                     (   14   )             
 
 By establishing the relationship:  
                     R   ⁢           ⁢   1       R   ⁢           ⁢   2       *       η   ⁢           ⁢   k     q     ⁢     ln   ⁡     (   MN   )         =     β   +     a   1         ,           (   15   )             
 
 V BG  may be expressed as: 
 
 V   BG   =V   go −( a   2   T   2   +a   3   T   3 + . . . ).  (16) 
 
         [0025]     As indicated by equation (16), a circuit configuration as exemplified by circuit  300  would not eliminate the non-linear component of the base-emitter junction voltage, which may result in the circuit output voltage V BG    314  not being constant over temperature, featuring instead a predominantly second order negative curvature.  
         [0026]      FIG. 2  illustrates one embodiment of a bandgap reference circuit  400 , which may operate such that the −αT ln(T) component is eliminated, resulting in a constant reference voltage output V BG    414 . Circuit  400  is similar to circuit  300  of  FIG. 1  with the exception of a new current I 4  being generated and applied to the emitter of transistor  306 , resulting in a total current of I 5 =I 1 +I 4  flowing through the base-emitter junction of transistor  306 . New current I 4  may be used to produce the needed +αT ln(T) term to be added to output voltage V BG    414 , thereby canceling the undesirable curvature that may otherwise be present in V BG    414 .  
         [0027]     The output of amplifier  302  may again drive resistors R 1   308  and R 3   310  such that the voltages at the inputs of amplifier  302  have the same value, V x . Under this condition, the following equations may be used to model the operation of circuit  400 : 
 
 V   x   =V   BE1 ( T,I   2 )  (17) 
 
 V   x   =I   1   *R 2 +V   BE0 ( T,I   5 ), and  (18) 
 
 I   5   =I   1   +I   4   (19) 
 
 where, again, V BE0  and V BE1  represent the base-emitter voltages for transistors  306  and  304 , respectively, and V x  represents the voltage at nodes  320  and  322 . Combining equations (17), (18) and (19):  
                     I   1     =       1     R   ⁢           ⁢   2       ⁡     [           η   ⁢           ⁢   kT     q     ⁢     ln   ⁡     (       M   *     I   1         I   S       )         -         η   ⁢           ⁢   kT     q     ⁢     ln   ⁡     (         I   1     +     I   4         N   *     I   S         )           ]                   =       1     R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢       ln   ⁡     (     MN   *       I   1         I   1     +     I   4           )       .                     (   20   )             
 
         [0028]     An equation may now be derived for I 4 . In one embodiment, the output of amplifier  402  may be configured to drive PMOS transistor  404  such that the voltage at the non-inverting input of amplifier  402  is the same as the voltage at the non-inverting input of amplifier  302 . PMOS transistor  406  may be configured to mirror PMOS transistor  404 , thereby ensuring that currents I 3  and I 4  are equal. Under this condition the following equations may be used to further model the operation of circuit  400 :  
               I   3     =           V     BE   ⁢           ⁢   1       ⁡     (     T   ,     I   2       )         R   ⁢           ⁢   0       =       1     R   ⁢           ⁢   0       *     (       V   go     -     β   ⁢           ⁢   T     -     α   ⁢           ⁢   T   ⁢           ⁢     ln   ⁡     (   T   )           )                 (   21   )                 I   4     =     I   3             (   22   )             
 
 By establishing the relationship: 
 
φ=β+ a   1   (23) 
 
 I 4  may be expressed as:  
               I   4     =         V   go       R   ⁢           ⁢   0       -         φ   ⁢           ⁢   T       R   ⁢           ⁢   0       .               (   24   )             
 
         [0029]     It should be noted that in order to simplify the analysis, the higher order term of equation (21) may be omitted in equation (24), as I 4  operates to cancel higher order effects in negative to absolute temperature (NTAT) current injecting circuit  401 , hence its higher order characteristics may be considered negligible with respect to the final results.  
         [0030]     Equation (24) indicates that I 4  comprises a constant current term, V go /R 0 , and a term that is negatively proportional with respect to absolute temperature, that is, it has an NTAT characteristic φT/R 0 . Combining equations (20) and (24), I 1  may be expressed as:  
               I   1     =       1     R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢       ln   ⁡     (     MN   *       I   1         I   1     +       V   go       R   ⁢           ⁢   0       -       φ   ⁢           ⁢   T       R   ⁢           ⁢   0             )       .               (   25   )             
 
 As equation (25) indicates, I 1  is proportional to absolute temperature (PTAT), and is expected to have the desired form shown in equations (7) and (13), as expressed in: 
 
 I   1    =λT+φT  ln( T ) ≅ψT.   (26) 
 
 The higher order effects of I 1  may be ignored since they may be negligible. Therefore, equation (25) may be combined with equation (26), to form:  
               I   1     =         1     R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     2     ⁢     ln   ⁡     (   MN   )         +       1     R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢       ln   (       ψ   ⁢           ⁢   T           V   go       R   ⁢           ⁢   0       +       (     ψ   -     φ     R   ⁢           ⁢   0         )     *   T         )     .                 (   27   )             
 
 When selecting R 0   408  to meet  
                 R   ⁢           ⁢   0     =     φ   ψ       ,           (   28   )             
 
 equation (27) may be reduced to:  
               I   1     =       1     R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢       ln   ⁡     (     MN   *       R   ⁢           ⁢   0   *   ψ   *   T       V   go         )       .               (   29   )             
 
 Bandgap voltage V BG    414  may now be determined:  
                     V   BG     =         V     BE   ⁢           ⁢   1       ⁡     (     T   ,     I   2       )       +       I   1     *   R   ⁢           ⁢   1                   =       V   go     -     β   ⁢           ⁢   T     -     α   ⁢           ⁢   T   ⁢           ⁢     ln   ⁡     (   T   )         +         R   ⁢           ⁢   1       R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢       ln   ⁡     (     MN   *       R   ⁢           ⁢   0   *   ψ   *   T       V   go         )       .                       (   30   )             
 
 In order to simplify equation (30), five constants may be assigned to and replaced by a single constant:  
             C   =     MN   *       R   ⁢           ⁢   0   *   ψ       V   go                 (   31   )             
 
 Therefore:  
                 V   BG     =       V   go     -     β   ⁢           ⁢   T     -     α   ⁢           ⁢   T   ⁢           ⁢     ln   ⁡     (   T   )         +         R   ⁢           ⁢   1       R   ⁢           ⁢   2       *       η   ⁢           ⁢   kT     q     ⁢     ln   ⁡     (   CT   )             ,     
     ⁢   and           (   32   )                 V   BG     =           ⁢       V   go     -           ⁢     β   ⁢           ⁢   T     -           ⁢     α   ⁢           ⁢   T   ⁢           ⁢     ln   (           ⁢   T   )       +           ⁢         R   ⁢           ⁢   1       R   ⁢           ⁢   2       *           ⁢       η   ⁢           ⁢   kT     q     ⁢           ⁢     ln   (           ⁢   C   )       +           ⁢         R   ⁢           ⁢   1       R   ⁢           ⁢   2       *           ⁢       η   ⁢           ⁢   kT     q     ⁢           ⁢     ln   (           ⁢   T   ⁢           )                 (   33   )             
 
 R 1  and R 2  may be assigned values such that:  
                     R   ⁢           ⁢   1       R   ⁢           ⁢   2       *       η   ⁢           ⁢   k     q       =     α   =       [       (     4   -   n     )     -   x     ]     ⁢       η   ⁢           ⁢   k     q           ,           (   34   )             
 
 leading to a ratio of R 1  to R 2 :  
                 R   ⁢           ⁢   1       R   ⁢           ⁢   2       =       [       (     4   -   n     )     -   x     ]     .             (   35   )             
 
         [0031]     As previously described, ‘n’ and ‘x’ represent constants related to process characteristics, leading to a ratio of R 1  and R 2  that may be well defined for a certain process. Once the ratio of R 1  to R 2  has been determined, C may be assigned a value such that:  
                   R   ⁢           ⁢   1       R   ⁢           ⁢   2       *       η   ⁢           ⁢   k     q     ⁢           ⁢     ln   ⁡     (   C   )         =     β   =         1     T   r       ⁡     [       V   go     -       V   BE     ⁡     (     T   r     )       -     α   ⁢           ⁢     T   r     ⁢           ⁢     ln   ⁡     (     T   r     )           ]       .               (   36   )             
 
         [0032]     It should also be noted that during manufacturing of bandgap reference circuit  300 , it may be necessary to trim certain elements of the circuit in order to account for effects of process variation and/or temperature. In some embodiments, resistor R 2   312  may typically be trimmed during manufacturing (for example by cutting/leaving uncut fuses) to insure that errors in the output of V BG    314  due to process variations are eliminated. Upon trimming R 2   312  however, additional residual curvature may be introduced into the circuit at the expense of correcting the nominal value of V BG    314 . In contrast, during manufacturing of bandgap reference circuit  400 , R 0   408  may also be trimmed in conjunction with R 2   312 , resulting in no residual curvature being introduced, thereby keeping V BG    414  at its intended value during regular operation. In other words, when performing the trimming operation during manufacturing of bandgap reference circuit  400 , R 2   312  may be trimmed to bring V BG    414  to its intended (designed) value, while R 0   408  may be trimmed concurrently without affecting the value of V BG    414  but maintaining the curvature correction established by current injecting circuit  401 .  
         [0033]     Thus, various embodiments of the systems and methods described above may facilitate the design of a bandgap reference capable of generating a curvature corrected reference voltage. Although the embodiments above have been described in considerable detail, for example specifying operational amplifiers, bipolar junction transistors, and PMOS transistors, other versions are possible, and some or all of the devices may be replaced with alternate devices that perform similar functions. Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. It is intended that the following claims be interpreted to embrace all such variations and modifications. Note the section headings used herein are for organizational purposes only and are not meant to limit the description provided herein or the claims attached hereto.