So far, we have discussed the variety of effects, assuming that each effect acts by itself. in other words, it acts in isolation. of course, for a real device you cannot assume that all of these effects are acting together. for example, a device may have both short channel and a narrow channel. What is done sometimes, is to assume that each of these effects, as long as it is small, can be still formulated in the way that we have discussed. In other words, assuming that it was acting by itself. So for example, there is a modification to the threshold, DeltaVT, due to short-channel effects. And to that, you can add the modification due to another DeltaVT, due to narrow channel effects. More elaborate models we'll assume, that can also be some interaction between them. So there, there may be a modification that has to do with the fact that the channel is both short and narrow. But for now, I will take the simple case, where each effect can be modeled in the same way as we have discussed, and I will show you a very simple example where these effects are combined and incorporated into a long channel model. So I'm going to start with a long-channel strong-inversion simplified source referenced model. Which is this one; This is valid as long as VDS is less than the saturation voltage, VDS prime. We're going to assume, as I said, non interaction between the various effects. We will incorporate, in addition to small channel effects, we will also incorporate the effect of mobility dependence on the gate field. So the above equation becomes this one. So let's go through it. In the denominator you see 1 plus theta times V GS minus Vt plus theta B V SB. This is simply the. Denominator we have introduced to take care of the mobility effects, due to vertical field. In other words, the effective mobility is mu 0, divided by the term inside the brackets here. Instead of the threshold voltage, we have an effective threshold voltage, which depends on L because of short internal effects, depends on W because of neurochannel effects. Depends on VDS because of the drain induced barrier lowering effect. And be, of course it depends on VSB, because of the buddy effect. And the same happens with this VT. VT has been replaced by the same, VT effective, which is shown with a hat. The term in parenthesis here, takes care of the velocity saturation, and it has the same form as we have seen, when we covered velocity saturation. which we have 1 plus VDS divided by L times E sub C, where E sub C was called the critical field. I remind you that for values well above this critical field The velocity saturates. So now we have developed a known saturation equation valid for VDS less than or equal to VDS prime. So when you go to saturation, you're supposed to replace VDS by VDS prime, and you end up with this term. Indeed Here, the VDS is replaced by VDS prime, and again, VDS is replaced by VDS prime and the same happen here. But not everything is replaced by VDS prime. Why? Because for example, to include channel x modulation which is active in the. the saturation region, you have to include the so-called pinch of regional length. We went through an evaluation of this l of p, the pinch of regional length, and we show it depends on the value of VDS. And as you increase VDS, even if it is above VDS prime, LP keeps increasing. So therefore, here you have to use VDS, not VDS prime. Similarly, when we discuss the drain induced barrier lowering or dibble, we said that it keeps happening, even after you have encountered saturation. So if you increase VDS above VDS prime, dibble becomes worse and worse. And makes your threshold smaller and smaller. So in the formulation for threshold, you have to include VDS, not VDS prime. The same here, and there. So now we have one expression for nonsaturation, another one for saturation, and of course you have to make sure that the point Where you stitch the two there is continuity. Not only of the current, but also of the slope of the current. So that means that you may have to modify the value of VDS prime. And in fact, this turns out to be the case. You modify the value of V D S prime, so that when the non saturation expression meets the saturation expression, there is continuity in the current, and also in the slope of the current. Now this is of course better than not having a continuous slope, or even worse, not having a continuous current, but it doesn't guarantee continuity of the second derivative of Of the current. Now, the derive, the second derivative of the current turns out to be the first derivative of something called the small signal output conductance, which is a very important parameter for analog design especially. And that, continuity of this is not guaranteed by this approach. And this, this continuity is because of the fact that we used VDS in nonsaturation, and suddenly VDS prime in saturation. So instead, of having two different regions which are combined with an if statement, if VDS is less than VDS prime use this expression, if VDS is larger than VDS prime use that expression. Instead of doing that, you can use a single quantity called effective V D S, which, very smoothly, between V D S and non saturation Towards V D S prime in saturation. So when you do that, you end up with a single expression for the current. And this VDS effective is close to VDS in non saturation, and it becomes close to VDS prime in saturation. And so now you have a single piece, expression for the curve. We will give expressions for VDS effective, and we'll discuss this concept more carefully, once we talk about models for computer aided design. In this short video, we discussed how to first start there you can combine different effects together, into a single model. Of course, complete models for computer aided design are far more complicated than this. First of all, they're not only valid in strong inversion, they're also valid in modern conversion, and, in weak inversion. And in addition, some of those models take into account interactions between the various effects we have considered. In the next video, we will talk about yet another search on effects, which goes by the name of hot carrier effects.