hey guys, bandgap different generators are fundamental building blocks of any analog subsystem. So, in this and upcoming videos, they will examine different aspects of this important circuit. Almost every analog circuit requires some kind of references or biases, or maybe both. References are predominantly voltages, but current references are not uncommon. Similarly, bases are predominantly current, but voltage bases can also be required. For example, in the case of cascode circuits, references should have low noise, low offset, and low variation across PVT corners. Bases should also have low noise and low offset, but sometimes they may require to track a particular variation in process, voltage, or temperature. What this means is that sometimes they may need to vary with process, temperature, or voltages to track their variation, and obviously, you require low area, low power, low design cost, and robustness. Although it's not uncommon to have separate circuits generating references and biases, ideally, we would like to have one circuit producing both references and biases. Before jumping into bandgap reference generators, let's consider some very simple schemes to generate voltage.

references and current biases. The simplest voltage reference probably could be the supply itself or may be a divided-down version of that supply. Let's consider its Pvt variations. It is a ratiometric circuit, meaning that the output depends on the ratio of two similar components. Now, the ratio of two similar components is one of the most precise things that we can produce on silicon, so this circuit is very robust against any process variations in these resistances. Again, because of the ratiometric nature, the circuit is also very robust against any temperature variations. This is because being similar types of resistances, both vary with temperature in a similar fashion, so the ratio remains the same across temperature variations. The only drawback of this circuit is its poor supply rejection. Now, don't be deceived by the simplicity of this circuit. I have seen this circuit being used to produce the reference voltages in many circuit applications. Now, despite its reasonable performance in generating voltage references, this circuit does a very poor job in generating current references. But we can produce an equally reasonable current generator by doing a small modification.

here we replace our two by a diode connected and MOS transistor. Now, it's not obviously a ratiometric circuit, so its process and temperature performance is not as impressive as this resistor divider. But as a rule of thumb, generating a low variation current references are much more harder than generating a low variation voltage references. So, considering its simplicity, this circuit actually does a very reasonable job of making a current bias generator. Now, a bandgap reference generator is much more complicated and involves circuitry than the simple difference in letters shown here. There are many variety of reference generators which have intermediate complexity and performance, but let's cut the chase and jump straight into bandgap reference generators. Bandgap reference generators have a very intimate relationship with temperature. Let's not reduce to key terms which are related to both temperature and band gaps and which are incidentally the topic of this video as well. Yes, Peter and seat at. Peter stands for proportional to absolute temperature. Here, absolute temperature means temperature in Kelvin scale. So, Peter voltage or current is a

quantity which is proportional to absolute temperature. That means it is 0 at Absolute Zero temperature and then increases linearly with temperature. So, if we plot a p-type quantity with temperature, it will be a straight line with a positive slope passing through the origin. See, Ted stands for complimentary to absolute temperature. A seated voltage or current decreases linearly with temperature. Now bear in mind that complementary is not the same as inversely proportional. To make this point clear, write the mathematical equation of these quantities. A pitot voltage is simply a constant time absolute temperature. So, it's zero at zero Kelvin and then increases linearly with a constant slope. CPT at a seated voltage may have a finite value at zero Kelvin and then it decreases with a finite slope. CCT at if temperature is increased and inversely proportional relation is a nonlinear relation which is of the form a constant over temperature. Now, if we have these two voltages with the magical properties of PT at and seated, then generating a bandgap voltage is the next step just

adjust the slopes of these Peter and Seeta and add them in such a way that a slope precisely cancels each other, and here we have a temperature-independent bandgap reference voltage. Now, it may seem as a convoluted roundabout way of generating a reference voltage, but the reason that this is done this way is that we can easily generate a super accurate Peter and a reasonably accurate seat at using semiconductor devices. So now, let's see how we can generate these voltages. Let's begin with Pete at voltage because it's simpler to understand. Consider a typical exponential voltage current relationship equation. Diode, BJT, and even Mo's insubstantial region have this characteristic. Here, i1 is the current to the device, and women is the voltage that this current creates. It is the family a thermal voltage, which is KT over Q. The parameter characterizes the device, and it depends on the type of the material, size of the device, and temperature. This is really a very simple equation considering that complex physics of the

device that this equation is describing. We can rewrite this equation with current being the independent variable for ease of digitization. Let's consider that this is a constant current into a diode. Now let's consider another similar diode but with n times bigger current. V2 would be obviously higher than V1 because we are pushing higher current into a similar device. Now let's consider the difference of these voltages. Amazingly, this difference reduces to an even simpler equation. Now this is one of the most important equations in the whole of analog design. We tilled on n VT is KT over Q, where K is Boltzmann's constant and Q is electron charge, and n is also a constant because it is a fixed ratio between the two currents. So this leaves us with the only thing temperature which can change, and this is our PT at voltage. The amazing thing about this equation is that it provides a very precise SP to generate the pitot voltage. K and Q being fundamental constants do not change with anything, and n being the ratio of two

similar kind of things is one of the most precise things that we can build on silicon. Now let's consider the slope of this pitot voltage. The slope is simply K over Q lon, and the numerical value of K over Q is 86 microvolts per Kelvin. And it is worth remembering this number. And can be any number greater than 1, and it is usually a whole number. Some typical values used in design are 8, 16, 25, or even 100. Why these numbers? We will discuss in another video, but for this video, let's consider N to be 8. This is not a very sharp slope, but not so small that we cannot work with it. For example, at minus 40 degrees Celsius, the pitot voltage is around 42 millivolts, and at 125 degrees Celsius, it is around 71 millivolts, giving us around 30 millivolts in the difference. Okay, so that was the pitot voltage. And before turning to a much more complicated but not so accurate seated voltage, let's consider a final thought about Peter. What causes

complex physics to collapse into a very simple and accurate equation. It is actually the exponential VI characteristic of the device. If you start with a square law, for example, you do not get rid of these currents in the final equation. Try it out yourself. Another point worth mentioning is that making the current ratio is not the only way to generate the pitot voltage. You can also generate a pitot voltage by making one diode and times bigger while keeping the currents the same. In that case, V2 will be the smaller of the two voltages, or you can use the combination of the two ratio current as well as ratio devices. Okay, now let's move to seat at voltage. Actually, if we think about it, we already know some of the sources of seat at voltages. Voltage across diode, for example, is known to reduce with temperature with a constant slope. Similarly, VB of BJT and threshold voltage of Moss is known to reduce with temperature with a constant slope. But is this reduction linear or is it an inverse proportional relationship or any other?

relationship for that matter. Let's dive into the equations to try to find out. Let's start with a diode equation again. So, I have assumed the diode current to be constant. So, this first term in this equation is a constant. If we assume for the moment that is is independent of temperature, then this term also becomes constant. As constant minus constant is also another constant, so this bracket term becomes a constant altogether. And this actually suggests that this whole term increases with temperature because VT is obviously a p-type voltage. This is obviously wrong because we know for sure that that voltage reduces with temperature. So, let's look into the temperature dependency of is in a little more detail. Diode saturation current is given by this rather complex-looking equation, but the dominant temperature dependency is created by the ni square term. It is the intrinsic carrier concentration of silicon. Now ni square itself has a complicated equation, but if we concentrate only on the primary temperature effects, we can simplify things a lot. It is again an exponential relationship here. VBG is the bandgap.

voltage of silicon and VT is the family earth thermal voltage. C represents everything. Now see itself is a function of temperature, but in presence of exponential term, its effect is rather mild. We will return to the effect of temperature dependence of the term C in a future video. Then we will discuss the curvature of the band gap reference voltage. Now let's plug back this equation into our voltage equation, and after some algebraic manipulation, we arrive at this equation. So this equation has two terms. The first term VB G is a constant, and the second term is a ptitim. Now if this term in the bracket is positive, we get our desired seat at voltage, and for this term to be positive, C has to be a bigger number than the current. Now we know by experience that this current ID would be of the order of micro ampere or in the best case milli amperes. Diode saturation current is of the order of Pico ampere to few nano amperes. Silicon bandgap is approximately 1.1 volts, and VT at room temperature is

approximately 25 millivolts. Putting these values, the exponential term is of the order of 10 to the power of minus 20. So that means that in order to make is of the order of Pico ampere, C has to be of the order of 10 to the power of 8. So long story short, C is indeed much, much bigger than I D, and hence we can say that indeed this term is a set term. Okay, so what is the slope of this heated voltage? And if we plug in these approximate values, then we arrive at approximately -2 millivolt per Kelvin, which matches rather well with our expectation. So you can see that seated voltage is much messier than the pitot voltage, and most of the errors and inaccuracies in bandgap reference voltages are related to, in fact, seated voltage. We will explore these errors and inaccuracies in future videos along with the solutions and fixes. In the very next video, we will see different ways to combine these Peter and seated voltages to generate the actual reference voltage. One final comment, presence of silicon.

bandgap voltage (Vbg) in this equation is the reason these circuits are known as bandgap reference voltages. So, post your comments below, and thanks for watching.

#23 Band-gap Reference Circuits - PTAT and CTAT

#23 Band-gap Reference Circuits - PTAT and CTAT