In this video, we will concentrate on the source drain small signal conductance and on the conductance seen at the drain terminal, which is a combination of the source drain conductance and other conductance parameters. Let us begin with source drain conductance. In the the non-saturation region of, from the simple strong inversion model. We find gsp from it's definition. We differentiate ids with respect to vds. And we find the expression show. Here. It is easy but, in saturation we have a major problem and I will straight this problem graphically. This is iDS, first was vDS. The solid line represents measurements and the broken line represents a model. As you can see. The model predicts the current pretty accurately, the error between the 2 is less than 1 percent overall. So you would think that this model is good but as we have shown in the past the fact that the current is predicted accurately doesn't thus certainly mean that the slope of the current is predicted accurately especially when the slope is small. So let us now find the slope of the current with respect to vds, which is of course, the source drain conductance. They look like that. In the non-saturation, the slope is large, and then in saturation becomes slow, small, as you can see. But the slope of the broken line, which represents the model, is about half of the slope Of, corresponding to the measurement, so, you get something like that. The error between these two, in this particular example, is larger than 50%. So, accurate representation of the current does not necessarily mean accurate representation of the slope in the saturation region. And this means that, in order to find GSD in such [UNKNOWN] region. The expression for the current which you are differentiating has to be very accurate, you cannot just take an approximate expression for the current and hope to get anywhere. So let us now discus what determines the slope in the saturation region, first of all channel length modulation does. the simplest model that people use is this one. where it is assumed that there is something called the early voltage, and according to this model, the drain current. Virus linerally with VDS in the saturation region. Now, if you differentiate this expression to find GSD, you'll find IDS prime the saturation current, at the saturation point, divided by the early volt. So you would think from that that gsd is constant in strong inverse saturation. But that can be a very inaccurate assumption. This is the current, versus Vds. A good model will give you a current that has a curvature in the saturation region. But this model says that the slope is constant. In other words, it assumes that the model does this [BLANK_AUDIO] and although you may get the slope accurately at one point, at other points you can be really off. So, clearly, simplified expressions like that can not be used for an accurate evaluation of the [UNKNOWN]. So, let us look more carefully at what. Makes the current vary and how it does vary in saturation. First of all we know that if we include the pinch of region, the so called pinch of region in the saturation we have derived a drain current of this form. IDS prime is the value at the saturation point. If the limit point between the non saturation and saturation And then the, the denominator we have l sub p, which is the pinch of region length. Divided by the total length of the channel. and we, back then when we derived this expression, we indicated that even the concept of pinch of region is approximate. But, anyway, this will give you something better than the previous oversimplified model. Let's differentiate this expression. To find GSD. So, GSD is the partial derivative of IDS with respect to VDS. the be/g, the dependence on VDS is hidden inside the pinch off region, so I can write this as partial derivative of current with respect to Lp, the pinch off region length times the partial derivative of Lp with respect to V DS And after a, a little, a, a few steps of algebra, you end up with this, approximate expression, and it is clear that what matters here is how does the [INAUDIBLE] region length vary with respect to VDS. So the result would depend on your model for Lp. Once such model we had shown without proof, when we discussed channel length modulation, was this one. If you differentiate this one and you plug the result in here, you find that Gsd is given by this expression. Where V and LA are constants in the expression that depend on the process. So, if you would like the entire denominator. Is some function that depends on v d s. Let us call it v a of v d s. This at least shows that the so called early voltage is not a constant but depends on v d s. So if you allow it to depend on v d s in a sufficiently accurate manner. Then this expression. Can give you a good estimate of G S D. An other phenomenon that effects the slope in the saturation region is DIBL, the drain induced barrier lowering. We have this discussed this and we have said that the current in saturation is given by this VT now depends on VDS because of dibble [/g]. So if you differentiate the current with respect to VDS to get the source drain conductance, you'll find this expression. And you see that what is involved here is the personal [/g] derivative of the threshold, with respect to VDS. This one is the same as the trans conductance expression we have derived before. So we can say that gsd is equal to the trans conductance times the negative of dVT dVDS and that will depend on your model on VT. For a given DIBL model, you take the expression you have derived. It will depend, the short channel VT is the corresponding long channel VT, plus a delta that depends on The DIBL effect, you differentiate that and you find GSD. I will not give you details here but you can find them in the book. But this is adequate to give you an idea of what is happening here. In weak inversion, DIBL can be dominant, we get IDS over VAW, where VAW strictly speaking also depends on VDS. And this VAW is smaller, can be smaller in weak inversion compared to strong inversion. Now, let's talk about the body-drain conductance. of the many conductances I showed you that are related to leakages, the body-drain conductance can have a dominant effect in some situations. And it can adversely effect the performance of certain analog circuits. So to remind you, g b d was the partial derivative of the body-current with respect to the body-drain voltage and let me illustrate to you how this can. Cause problems. for the next few minutes I will talk about something that has to do with circuit design. I would like those of you who are not involved in analog circuit design to be patient. I promise I will not spend more than a couple of minutes of this. after that I will return to the device discussions, that do not require such a background. So in analog circuit design as the analog circuit designers among you know, sometimes we need a very low output conductance and for that we use a cascode configuration like this. The circuit device is biased at the fixed gate voltage This from here to there acts like a source folder and it helps keep VDS for the bottom device approximately fixed, so that the current is approximately fixed. And as a result you see a constant current. Almost constant current versus this voltage at this terminal. So you end up with a very high output resistance or very small output conductance. Unfortunately, as we have seen, the body drain conductance corresponds to a resistance between the. Drain and the body. In this circuits configuration I'm assuming that both devices are on the same well, and therefore, their body is common and grounded. So that means that gbd for each device goes between the drain of the corresponding device and ground. So that means, in this case, That GBD shunts the entire [UNKNOWN] states from the top of the [UNKNOWN]. From this point to ground you have a resistance of conductance GBD2. That means that no matter how large output resistance you manage to achieve with your [UNKNOWN] the whole thing is shunted. By this resistance and it lowers the total resistance that you see here so you cannot achieve the benefits you were hoping to get by using a scope device. If instead you have individual wells and you ground each well to the corresponding source, things are better because this gdb 2 now is only in parallel With a drain source path of device m 2, but of course doing this is not always possible, and it also entails other problems, because for example, it can expose the well of the tap device to substrate noise through capacitance from the substrate to the well of the transistor. Anyway, as I promised, I'm now going back to the to our device discussion. Output conductance, if we bias the device to the constant gate and body voltages, and look at how the drain current varies when I vary V D S. And then we take the derivative of this drain current with respect to V D S the result Is called the small signal output conductance. Since the drain depends, has 3 paths, the drain source path. Then it has a path because of gate leakage in another pathway because of body leakage. The drain current is the sum of the drain source current plus the drain body current. Plus the drained gate current and it is the sum of the three that counts. So now when you take the derivative of this you get the individual derivative's of each. Which are gsd, gbd, and ggd. So graphically let's say we have the drain current here vs vgs. This is nonsaturation This is saturation where the dominant effects are the the channel length modulation and DIBL, and if you go to high enough VDSes, you end up with impact ionization that can increase the slope, like that. So, now when you plot. The small signal output conductance G zero, it is high in non saturation, it dips down becaue of channelic modulation in [UNKNOWN]. This is what determines the shape here and then if you enter the impact ionization region, you go back up, as this one predicts, the slope increases. So what kind of output conductance you get depends on which region you are and which effect is dominant. Let me further expand on this. Here is the output conductance on the log scale and I'm assuming that the length of the device is equal to the minimum allowed length. So now we have Go, the output conductance versus Gds with Vgs as a parameter. Because this device is as short as we, you can make it in the given process, the channel length modulation and dibble effects are significant and they keep the conductance large. But if you take the. A different device with a longer length, much longer length. Now channel length modulation and DIBL are negligible for this device, so the conductance dips to lower values. And you can even see the beginning of impact ionization affects over here. The conductance goes back up, as the previous slide had predicted. You can only see that of course if you go to large enough values. Fortunately with many processes today, you stay to, at a low value of vds before such effects can, can interfere with your design. [COUGH] Now something about the effect of extrinsic resistances on out of conductance. First of all, you will recall that we have the extrinsic source resistance, the extrinsic drain resistance And the effect of this is as the current passes through them, there is a voltage drop across them which limits the internal voltages of the device. So for example the actual VGS of the device which I call VGS hat is the externally applied VGS minus this voltage drop across this one. So, for a given VGS And a given change in vgs, the corresponding internal vgs and the corresponding change in vgs are smaller than what you think from the external variations. And that has as an effect a reduction in the trans-conductance, the effective trans-conductance. And you can show summarized in the book that the effective transconductancy is what you would assume from the transconductance of the device by itself divided by 1 plus G M R S C. Another effect is related to the body resistance R B E. we have said more than once that the body resistance is distributed but for simplicity we assumed that it is just one lump to resistor. The drain body leakage current passes through this resistance and creates a voltage drop rbidb, and this has the plus sign here, the minus thi-, sign there. So let's say you have a reverse bias vsb, which determines your threshold through the body effect. Now the effect of vsb internal to the device Is actually smaller by the voltage drop in this resistance. So, VSB effective is VSB minus RBE IDB So, the effect of this is the following. If IDB goes up, for some reason, then the effect of VSB goes down, because a larger voltage drop is subtracted from the external [INAUDIBLE] VSB, so VSB now is smaller. Therefore, the threshold is smaller, which increases IDS further, which can have the have the effect of increasing IDB further. The result of all this is that the output conductance that you see here, the slope of the drain current with respect to VDS Is larger than if you didn't include this effect. In the book, we summarize the procedure for finding the effective output conductance and it is given by this expression. Where you see that g is still matters. The gate drain leak ups conductance matters, and the body drain conductance is actually amplified by this. Factor. In this video, we have discussed the source drain and output small signal conductances. And at this point, we have finished with our introduction to conductnace small signal parameters. In the next video, we'll begin discussing small signal capacitance.