Temperature-dependent resistors

Besides the key parameters found in the previous section, one additional phe- nomenon has to be taken into account. This is the resistance by which the collector current is derived from a voltage. As was discussed in Chapter 3, reference currents are not available in nature and are therefore derived from reference voltages by means of a resistance, for instance. When this resistance is temperature dependent, it introduces extra temperature dependency in the reference current.

Assume a collector bias current, is derived from a reference voltage, V, by a resistor, R, having a temperature-dependent relative error, as given by:

where is the resistance at the nominal temperature and are the first and second-order temperature dependencies of the resistor, respectively. Then, for the collector current the following expression can be found:

in which V(T) may be temperature dependent. Then, for the error in the base- emitter voltage the following expression can be found:

Thus a relative error in the resistor causes an additive error in the base-emitter voltage (which is a consequent of the ln-function). The error is independent of the type of collector current intended, i.e. temperature dependent, temperature independent, et cetera, and thus it is the same for each base-emitter voltage.

6.4. RELATION TO THE FUNDAMENTAL DESIGN ASPECTS 177

Recalling that a bandgap reference is a linear combination of base-emitter volt- ages, the resulting error at the output of the bandgap reference source can be found. This error voltage, is found from:

As the error in the base-emitter voltage is equal for all the base-emitter voltages, the relative error equals:

for which equation (6.27) is used with the assumption that the influence of the different can be ignored, resulting in The final relative error depends on, of course, the type of resistors being used. In table 6.3 exam- ples are given for a diffused resistor and a thin-film Nichrome resistor with re- spectively, and

(data from [37]). For the value 0.026/1.2 is used. The second-order er- ror resulting from the temperature behavior of the diffused resistor is about a factor 4 lower than the second-order behavior of the intrinsic base-emitter volt- age. Therefore, when designing second-order (or higher) compensated bandgap references, this effect has to be taken into account. This can be done by adding the corresponding term to the term describing the second-order behavior of the base-emitter voltage.

6.4 Relation to the fundamental design aspects

Ideally, the output voltage of a bandgap reference contains no information, i.e. it is an information-free source. Practical bandgap references do not need to be ideal. Their specifications are determined by the application in which they have to be used and are related to the information processing capacity of that application. Thus the bandgap reference is allowed to have some entropy.

As indicated in Chapter 2, a design should be orthogonalized with respect to noise, bandwidth and signal power. In this section these fundamental design aspects are related to the design of bandgap references.

178 CHAPTER 6. BANDGAP REFERENCES

Noise The choice of the order of the bandgap reference, first or second-order for instance, determines the minimum attainable systematic error of the bandgap reference; by using all the key parameters in the design, extra systematic errors can be kept to a minimum. The stochastic errors are caused by the devices in the bandgap reference, i.e. the transistors and resistors which introduce thermal noise and shot noise. These errors can be kept to a minimum by structured design, i.e. minimization at the mathematical level. Further, process variations also introduce stochastic errors in the reference voltage. However, as these errors are time-independent, they can be reduced by using trimming.

Bandwidth In principle, the output power of the bandgap reference is com- pletely located at DC. The remaining part of the spectrum should not contain any power. But, since the bandgap reference is used as a voltage source, its output impedance should be kept at an acceptably low value over a specified bandwidth. Over this bandwidth, the bandgap reference produces noise and therefore power is not only located at DC. The bandwidth of the bandgap ref- erence is determined by the bandwidth that its output impedance must have.

Signal power The signal power of the source at DC is directly given by the specifications. The efficiency of the supply of this signal power is improved when the power-supply voltage is lowered to the reference voltage.

Orthogonalization To meet the required specifications optimally in a rela- tively short time, the design aspects should be orthogonalized. As was seen in the previous sections, the core of the bandgap reference is a linear combination of base-emitter voltages. These base-emitter voltages set the practical limit on the quality of the bandgap reference. The bandgap reference can be seen as a circuit processing several base-emitter voltages in order to end up with a refer- ence voltage. The base-emitter voltages are the sources of the reference and the scalers and adders are the processing blocks. When the scalers and adders are assumed to be ideal, the maximum practical quality is found. This structure for the bandgap reference is called the idealized bandgap reference in the remaining part of this book. The systematic error is given by the order of temperature compensation whereas the stochastic errors are caused by the noise introduced by the base-emitter voltages and the variation of the process. These last er- rors can be again reduced by means of a trimming. Since the output of one of the scalers determines the output impedance of the bandgap reference, the bandwidth capabilities of the bandgap reference are still ideal.

The implementation of the scalers and adders also introduce noise. Again, the influence of the process variations can be reduced by trimming. The (elec- tronic) stochastic noise introduced by the scalers and adders should be kept at an acceptable level. This should preferably be orthogonal to the minimization

6.5. NOISE 179

of the stochastic noise of the idealized bandgap reference in order to obtain the optimum overall noise performance with a minimum level of design complexity. In Chapter 3 an electronic circuit was treated as a signal path plus its bias circuit. The internal signals of the bandgap reference, i.e. the scaled base- emitter voltages, vary as a function of temperature and in the case of a low- voltage design, the bias circuit may cause signal-dependent systematic errors. Further, as the output impedances of the bias sources are not ideal, errors will penetrate from the power-supply voltage to the output of the bandgap reference. In the next sections the design with respect to the noise, the bandwidth and the signal power of a bandgap reference is discussed.

6.5 Noise

As the accuracy and temperature independency of bandgap references increase, the mean errors will now become on the order of a few ppm/K over a tempera- ture range of 100 K to 150 K and the noise performance of bandgap references becomes more and more important. For instance: assume a bandgap reference with an output voltage of 200 mV and a mean temperature dependency of 2 ppm/K. The mean uncertainty due to the temperature dependency then equals only When the equivalent noise voltage at the output is higher than this value, the noise is the dominant cause of the uncertainty. This example concerns relatively low-frequency noise. For delta-sigma modulators, the rela- tively high-frequency noise of the bandgap reference is also important. Since the modulators sample at a relatively high rate, the noise is important over a larger bandwidth.

To minimize the total noise at the output of the bandgap reference several methods can be used:

for a reference voltage for a measurement instrument, the duration of the sampling of the reference voltage can be increased, resulting in low-pass filtering, but inherently slowing down the system;

perform filtering by means of a capacitor. This requires a relatively large capacitor as the output impedance of a bandgap reference is, in principle, relatively low. When it is too large to be integrated, the capacitor has to be mounted externally to the chip, introducing an additional component and, possibly, also an additional pin;

the most convenient method is to minimize the noise of the bandgap ref- erence itself. If the specifications are still not met, the former methods can be used too, but now with less severe constraints.

To be able to minimize the noise level of the bandgap reference, all the noise sources in the bandgap reference need to be transformed to the output. For

180 CHAPTER 6. BANDGAP REFERENCES

a first and second-order compensated bandgap reference only two base-emitter voltages are required. The general block diagram of those bandgap references can therefore be visualized as depicted in figure 6.7. Three types of blocks can be identified:

base-emitter voltage generator; scaler;

summing node.

Of these three types of blocks, the base-emitter voltage generators are the core of the bandgap reference. They take care of the required relation to the bandgap energy in order to get a reference voltage. The scalers and are dimen- sionless factors and do not need to introduce any noise. The same goes for the summing node. In contrast, the base-emitter voltages are directly related to the collector current flowing across a junction which is therefore inherently deteriorated by shot noise and consequently the base-emitter voltages are thus always contaminated with noise. In figure 6.8 an ideal base-emitter voltage gen- erator is depicted. The desired collector current, is forced into the collector by means of negative feedback. The nullor controls the base-emitter voltage such that the desired current flows into the collector. As the input current of

6.5. NOISE 181

the nullor is zero, the complete desired current flows into the collector, and an accurate relation is found between and Further, as the input voltage of the nullor is zero, it makes it possible to ignore the forward Early voltage. This cell is the core of the idealized bandgap reference and is used to calculate the minimum practical noise level.