This invention relates generally to circuits and devices that produce a precise DC signal, and more specifically to temperature compensated bandgap reference circuits. 
Virtually all systems that manipulate analog, digital, or mixed signals, such as Analog-to-Digital and Digital-to-Analog converters, rely on at least one reference voltage as a starting point for all other operations in the system.  Not only must a reference voltage be reproducible every time the circuit is powered up, the reference voltage must remain relatively unchanged with process variations and variations in temperature and supply voltage. 
A conventional technique for realizing a reference voltage is the semiconductor bandgap reference circuit (also known as a bandgap reference).  A bandgap reference relies on the bandgap energy of the underlying semiconductor material.  Though varying with temperature, the bandgap energy is a physical constant when extrapolated to a temperature of zero Kelvin (absolute zero).  For silicon, the bandgap energy at absolute zero is approximately 1.12 electron Volts.  An additional contribution (3 kT/q) to the bandgap energy due to thermal energy at non-zero temperatures raises the bandgap value of silicon to approximately 1.20 electron Volts. 
A practical way of obtaining the bandgap energy of silicon is to measure the voltage across a forward biased semiconductor p-n junction (diode) device.  Although loosely referred to here as a diode, other devices such as transistors are also typically used to obtain the necessary p-n junction.  Thus, p-n junctions appearing in different devices are commonly used to take advantage of the characteristics of the bandgap energy as a physical constant in order to generate DC reference levels. 
To help understand how a precise and temperature invariant reference signal can be generated using the bandgap energy characteristics, FIG.  1 illustrates a plot of the voltage drop across a forward biased diode versus temperature for two diodes having different geometries, corresponding to lines 101 and 105.  A key aspect of both plots is that the voltage (and hence the bandgap energy) decreases with increasing temperature of the semiconductor material, thus exhibiting a so-called negative temperature coefficient of typically 3000 parts per million (ppm).  A cross-sectionally larger diode generates a smaller voltage drop at equal temperature and current, so that line 101 (corresponding to the larger p-n junction) is steeper than line 105 (corresponding to the smaller p-n junction).  However, regardless of their slopes, lines 101 and 105 converge at the bandgap value for silicon of 1.12 electron Volts when extrapolated to zero degrees Kelvin.  These aspects of the diode voltage allow the following useful equation to be written: EQU V.sub.bg =V.sub.d1 +A(.DELTA.V.sub.d1,2) (1-1)
where V.sub.bg is, in all but the most sensitive applications, considered a constant approximated by the bandgap value of silicon in electron Volts, A is a gain factor to be determined, V.sub.d1 is the voltage drop across a first (smaller) forward biased diode and .DELTA.V.sub.d1,2 is the difference between the voltage drops across the first and second diodes such that .DELTA.V.sub.d1,2 has a positive temperature coefficient. 
The equation (1-1) above shows that as the diode voltages vary with temperature, their weighted sum, equal to V.sub.bg, is a constant (once a correct value for the constant gain factor A has been selected).  This occurs because the positive temperature coefficient of .DELTA.V.sub.d1,2 offsets the negative temperature coefficient of V.sub.d1.  V.sub.bg is thus the desired reference signal. 
To compute A, the voltage for the smaller diode (which incidentally has a higher forward voltage drop than the larger diode when equal currents are applied) may be conventionally approximated as V.sub.d1 =(kT/q)ln(I/I.sub.s), such that EQU .DELTA.V.sub.d1,2 =V.sub.d1 -V.sub.d2 =(kT/q)[ln(I.sub.1 /I.sub.s1)-ln(I.sub.2 /I.sub.s2)]
where I.sub.s1 is the reverse saturation current for a diode.  Typically, a circuit realization of (1-1) will force the currents I.sub.1 and I.sub.2 to be approximately the same.  This yields EQU .DELTA.V.sub.d1,2 =(kT/q)ln(I.sub.s2 /I.sub.s1) (1-2)
Since I.sub.s is proportional to the cross-sectional area of the diode, and since the device physics for manufactured diode junctions is well understood such that their voltage versus temperature behavior is predictable, a value for A can be readily computed from (1-1) and (1-2) given an operating temperature, the ratio of the diode areas, a typically known value for diode voltage V.sub.d1 or V.sub.d2 at the given temperature, and the desired bandgap reference output V.sub.bg =1.20 volts. 
A conventional bandgap reference circuit 200 based on equations (1-1) and (1-2) is illustrated in FIG.  2.  The circuit 200 basically operates as a feedback control loop to maintain the two input nodes of amplifier 217 at approximately the same potential in the steady state.  The circuit elements are now described using an exemplary set of assumptions to simplify the mathematics. 
Amplifier 217 may be a conventional operational amplifier with very large open loop gain, such that the voltages at two input nodes are assumed to be the same after closing the feedback control loop and in the steady state.  According to conventional practice, the currents in the two diodes are assumed to be the same, and R.sub.2 is set equal to R.sub.3 for easier manipulation of the numbers.  Voltage loop equations can thus be written as : EQU V.sub.out =IR.sub.2 +V.sub.d1 (2-1) EQU V.sub.out =IR.sub.2 +IR.sub.1 +V.sub.d2 (2-2)
Subtracting (2-2) from (2-1) yields EQU IR.sub.1 =.DELTA.V.sub.d =V.sub.d1 -V.sub.d2 (2-3)
Solving for I in (2-3) and substituting for I in (2-2) yields EQU V.sub.out =.DELTA.V.sub.d (R.sub.2 /R.sub.1)+V.sub.d1 EQU V.sub.out =(kT/q)ln(I.sub.s2 /I.sub.s1)(R.sub.2 /R.sub.1)+V.sub.d1(2-4)
which can be easily compared to the original bandgap reference (1-1) above to see that the output of amplifier 217 is approximately equal to the bandgap reference voltage V.sub.bg.  For example, solving the above equations for R.sub.1 and R.sub.2 based on V.sub.out =1.2 Volts, diode area ratio of 8:1 (selected to yield reproducible circuits having matched electrical and temperature characteristics), I=3 microAmperes, V.sub.1 =0.61 Volts, and kT/q=25.5 millivolts will yield approximately R.sub.2 =196.67 kiloOhms and R.sub.1 =17.675 kiloOhms.  The gain factor A in (1-1), also referred to as the control loop gain, is nothing but R.sub.2 /R.sub.1 which in this case turns out to be approximately 11.5. 
When the above described bandgap reference circuit 200 is 5 manufactured using modern integrated circuit fabrication techniques, two practical issues arise.  The first is the sensitivity of V.sub.out (output of amplifier 217) to input offset voltages required to balance V.sub.out.  Because of this, V.sub.out will not appear as exactly 1.20 volts in the manufactured circuit.  The error will be proportional to the input offset voltage of amplifier 217 and the loop gain A. To eliminate such errors, an attempt can be made to reduce the variation in input offset voltages of manufactured amplifiers, thereby allowing a design correction to be made.  Controlling variations in input offset voltage, however, is particularly difficult with amplifiers having metal oxide semiconductor (MOS) input transistors, as such devices have input offset voltages that vary appreciably across a given production lot. 
The second practical issue concerning the above-described circuit 200 as well as any other bandgap reference circuit is the difficulty in repeatedly matching the characteristics of two resistors widely spaced in value to yield a ratio that is uniform across as many fabricated devices as possible.  Although a matched resistor pair having a ratio of approximately 11 as computed above can be readily manufactured, lower ratios will typically yield more reproducible results.  A reference circuit having resistors with lower ratios will be more reproducible and will exhibit smaller variations in its reference output.