We will now briefly look at benchmark tests that are used to test compact models. I will only show you a few examples of benchmark tests for you to get the idea. and I should emphasize that I will be showing you some serious problems revealed by such benchmark tests. you should keep in mind that such problems are mostly encountered in older models, but they may also appear in the development of new models. So, it, it's good to be aware of this problems. First of all the most obvious test is to plot I versus V, in fact, do so on an algorithmic axis. And make sure you'll not only get a continuous curve but even the slope is continuous. Over here, you can see there is a jump this, this comes from an old model that attempted to join strong inversion to weak inversion at a certain point, and at that point the slope became discontinuous. models used today do not have this problem. But it, it pays to be aware of such possibilities in new models. Then the behavior around VDS equals 0 is a well-known issue. This is the test that you can use, so you can drive the drain with the voltage V sub X with respect to ground. And you drive the gr-, the source with the voltage minus V sub X. So, the drain and source are give-, driven in an anti-symmetric fashion. So, the behavior of I sub D is supposed to be such that gives you an odd function, I verus V sub X. This means that ID of minus VX is minus ID of VX. And you can say something similar for the source current. But the leakage currents of gate and body interfere with this. And you can show that, if instead of plotting its current by itself, you plot half of their difference. The odd function behavior is maintained. So, this is what we would be plotting. We would be plotting I sub X versus V sub X. Here is the first plot, four models are shown. The solid line is a good model. The other lines predict the wrong slope but one of them doesn't even predict the current is 0 when VX is 0. Obviously, when the current, when the voltage between drain and source is zero, the channel current is supposed to be zero. And even the gate and substrate leakage currents area supposed to cancel over here. But this one model gives you a nonzero current. Now, you can plot also the slope of IX versus VX, as shown here. This is the good model. The others can give you a wrong slope. This one gives you non-symmetric behavior, and this even has a discontinuous slope here, as you can see. If you plot the second derivative of I sub X versus V sub X. Then, things really become bad, except for this one, very nice model here. So, the problems you see here, really reveal themselves in certain applications. For example, a model that has problems like this, will give you totally unacceptable predictions for the performance of mixers in radio frequency circuits. Okay? In contrast, the sorted line model does not have such a problem. Now, the output so, small signal conductance is defined as the partial derivative of drain current with respect to VDS. In other words, the slope of I versus VDS, when VGS and VSP are held constant. And that should be a smoothly varying curve, as it is for this solid line model here. But the other models that I show you here have a problem in that, for example, this one suddenly has a slope that changes over here. And this one has even more problems like this. So, again, when you see things like that, these are the results of people's attempt to match two different regions at one point. We have already mentioned that the use of smoothing functions can help you avoid such problems. You can find more benchmark tests in appendix K of the book, or in appendix I in the MOOC version of the book. At this point, we have finished with our brief introduction to physical compact models for circuit simulation.