8:55 AM
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First of all, I'm sure you ment Vds >= Vgs - Vth for a MOSFET in saturation. Vds is defined as the potential difference between drain and source, Vgs as the potential difference between gate and source. simulate this circuit – Schematic created using CircuitLab By shorting gate and drain, they...

Your answer is excellent, and newbie friendly to me with IQ 97. I really appreciate your suggestion at the end of "taking deep look, ... wrap my mind around the fact that *three quantities are plotted in the same graph, ...". I studied a EE diploma ages ago, and I confess that when reading about linear/saturation region of BJT operation, I had not taken a deep enough look at the graph, and so I did not know, as you said, three quantities are plotted on the same graphs. / to continue, ...
At that time I only knew a basic graph with X- and Y- axis. I never thought that there can be a graph with three axis, X, Y, and Z!
to make sure I thoroughly understand your answer, I use the following MOSFET as a case study: IRL540N 100V N-Channel Power MOSFET Datasheet - Infineon: infineon.com/dgdl/…. I read the data sheet and I have two questions related to the linear/saturation region. The questions are written in the following picture: i.imgur.com/zNX2E4W.jpeg. I am designing switching circuits using this IRL540N. I would appreciate it you can answer my questions. Thanks a lot. Cheers.
Thank you very much for your quick and very helpful answer. Your guess that shorting gate and drain as a useful convention is very reasonable, and I would have never thought about that. And because I have not seen such a condition in BJT datasheets, I need to google more MOSFET data sheets to convenience/verify myself. BTW, I am a slow thinker, so it might take me sometime before I let you know my research/verification results. Have a great weekend. Cheers.

I would like to clarify that my book defines threshold voltages as Vtn and Vtp. Not Vth. Theydo it to not confuse with thevenin voltage so that’s how i write it now. Excellent answer.

@StrugglingStudent117, I appreciated your clarification very much. Earlier I said I need to use URL540N as case study to "translate" #K. Krull terms to what I know, because I I found his use of terms confusing. I have never read any books on MOS, and I only focus on two parameters: (1) Rds(on), (2) Vgs(th). I don't even care of Vds, because I only play with Logic level trigger N-Ch power MOSFETs. Your question mentions terms like λ, W, L which I never heard of. So I knew I have no prerequisite knowledge to understand your question. / to continue, ...
However, I was curious to know what you are talking about, so I googled something to read. I guess some other newbies and future readers might also are interested, so I am listing some of the things I might read. The first one is (1) MOS Field-Effect Transistors (MOSFETs) Sedra8e Solved Problems Ch05 (1) (λ W/L etc): learninglink.oup.com/protected/files/content/file/….
And I second reference I might read is the following: (2) Jaeger & Blalock, 4th Ed, Ch 4 Problems ... - UW ECE (The OP's question is 4.34) dunham.ece.uw.edu/ee331/JB4edProb/…. Ah lunch time. See you later.
The third reading is this: (3) (PDF) Microelectronic Circuit Design by Jaeger 4th edition.pdf - Academia.Edu academia.edu/35781367/…
@StrugglingStudent117, The title of Jaeger's Microelectronic Circuit Design, misled me to think that it was a boring book on MOSFET fabrication, which is what I am least interested, because me EE hobbyist focuses only on application, not at all fabrication. But then when I skimmed the book's preface, I guess the book should be interesting. So I skimmed on the first three chapters, spending about 30 minutes. I found it a good comprehensive introduction book (I give 5 stars or 9/10) for EE newbies, ... My long winded comments stops here. I will begin reading Ch 4. See you later.

@K.Krull Why does the red line look exponential? Like an e^x function?
Like it would seem that the red line is it’s own graph WITHIN a graph.. Vds as a function of Vgs - Vth. Sort of ignoring the current on the y axis and instead using Vgs as the Y?. This looks like two graphs with two different Y axis on ONE.

@StrugglingStudent117 Do you think k.Krull's "red line" is the "Pinch off locus" - Fig 4.8 (4.2.4 SATURATION OF THE i -v CHARACTERISTICS) ?
@StrugglingStudent117 Ha, your looking at the i-v graph as a "graph within a graph" is what I see as a "three dimensional graph". Krull is interpreting the graph as something showing a relation of three quantities. You can also see it as a relationship graph showing how one dependent variable Id, is relates as two independent varies Vds and Vgs(th).
Actually you can extend 3 variables to 4 variables: Id depends on Vds, Vgs(th), and T (temperature). Our stupid 2 human eyes can only "see" 3 variables as 3D. If there are 4 variable or more we can no longer "see", but we can "see" them mathematically, as "tensors", all crazy scientists Einstein like mathematical equations, no more graphs! :)
I appreciate your description of "This looks like two graphs with two different Y axis on ONE". Now let me ask you a follow up question: So for 4 variables, Id, Vds, Vgs(th) and T (temperature), will your have 3 graphs of 2 different (overlapping) Y axis?
Any way, let me tell you my Jaeger book reading report. (1) I skimmed Sections 4.1 and 4.2. I found them very good, giving we newbies a solid background based on equations (No, I did not "proof read" Pro Jaeger's mathematics, I trust him, as I trust Einstein of his equation "E = MC²"). I also now know what he means by W, L, λ etc.
Coming back to the confusing thing of shorting out Source and Gate, I tend to see it is just an academic exercise, just to train the students' minds. I mean shorting S and G has no practical applications, but just scare the students to study hard! (only 50% sure) :)
I am glad that you have asked an interesting question and guided me to find the very good book by Jaeger book. I am now giving his book 9.5/10. Ah, jogging and locking up supper time. See you later.

4 hours later…
1:41 PM
@StrugglingStudent117 The red line is not an exponential, but quadratic. This is due to the quadratic nature of Id vs. Vgs-Vth (note the increasing distances between the blue curves, allthough Vgs-Vth has linear steps). This quadratic behavior is then plotted against a linear axis for Vds, which gives us a quadratic parabola.