Sanity Check/help with small signal transistor circuit. 3 close answers! - Page 1
Forum section: electronics / beginners

author: renzoms
author_rank: Regular Contributor
timestamp: December 23, 2021, 08:04:03 pm
content: I got three different, but very close answers for Vo/Vsig.The easiest method for finding Vo/Vsig is by using the hybrid-pi model, then finding the current ib, then finding the current ic = gm*vbe. Then using the fact that vbe = some voltage division for vsig. Then ic*Rc = Vo and you can get a relationship between Vo/Vsig (through vbe and ic and ib).Not sure why, but I got two different answers using the hybrid-pi model. The different answer came from when I assumed (rightly, I think) that ic = Beta*ib (even in the small signal appx.)The third different answer I got from a series of algebraic efforts that I made while analysis the circuit using the T-model.Is it possible that these are all right (accurate) because of the nature of the small signal circuits? Or where did I go wrong?I annotated attached images of my work and the exercise and I hope you can help. I can also try to explain my work more.Thanks a lot I'm really curious because I was told to solve for something and I came up with 3 very close answers (12.9, 13.3, 13.5), and it matters (analyzing correctly).https://imgur.com/a/MoSgFSo

author: jwet
author_rank: Frequent Contributor
timestamp: January 04, 2022, 07:19:17 pm
content: My answer with your assumptions is 12.9 but its complicated.I'm a retired analog engineer and haven't thought about some of this for some time but decided to take it on as a challenge. I don't directly use Pi or T directly but learned it of course. I think you'll find this true of most working engineers- they settle on their own analysis models that fit the kinds of stuff they design. I did precision analog at fairly low speeds but do some RF work and use somewhat different models for RF.All models are approximations that ignore things and try to drop out small things and then linearize everything. Hybrid pi and t models are exactly equivalent and are pretty decent up to medium frequencies and are a useful teaching tool. They use Ebers-Moll as the basis and model the transistor as a transconductance device. If the AC model is used, they also model miller C and base capacitance which can really effect analysis. Most engineers never get past the Ic = beta Ib model. However event the Pi and T models don't account for other things most prominently early voltage which affects beta and the ultimate Rout of the current source- it assumes its a perfect current source with infinite output impedance. (there is an hoe term but its a constant). I'm only saying this to make you wary of <any> model. Models aren't the real thing, they're sometimes useful analogs to the real thing and they work within their scope.Derive those identities you used in your analysis- you'll see where they took shortcuts. Your professor should have made you go through this pain already. I think you'll find little approximations like (beta + 1) = beta assumed in some places. You should get the exact same answer with Pi or T. Generally if the emitter goes directly to ground (without an R), use the Pi model. If there is an unbypassed emitter R, use T. They are equivalent but this will make things simpler.If you look at the complete derivation of the Hybrid Pi model in a good reference (Shilling and Belove is one), you'll find little approximations made all through the derivation and by the end of the derivation, they're using the approximately equal sign rather than the equals sign. Your circuit is pretty simple but in many cases, assuming beta + 1 = beta can be problemattic like when beta = 8, etc.If you want an eye opener- look at the Gummel-Poon model, good spice models for transistors use this but even these have limitations and fancy analog simulators like those used in IC companies. My old company, Maxim, had a half dozen PhD's and a dozen mortals that did nothing but derive models that the design engineers used. Its a complicated world.My way of analyzing this is to analyze the transistor as a Voltage amplifier which is a transconductor with the collector load.gm is transconductance gain- 1/re which is Ie/vt, (approximation # 1- what is vt? - I use 26 mV).I then reflect its input impedance into the base and calculate an equivalent input Z- there is another approximation here-only the emitter load is reflected, the collector has an effect too but since its much larger, its ignoredThis gives me a voltage divider with my source that I can multiply by the Voltage gain from step one.This give me the overall gain with the above assumptions.Overall, the transistor has a voltage gain of 100/2.6= 38.45 vc/vb <I used 2.6 here, usually Vt is 26 mV nominally- another approximation)The input impedance is 10k in parallel with the reflected emitter resistance, 525 ohms= 498 ohmThis creates a voltage divider with the input R of 1k to give a gain of .3324The overall gain is 12.8- this would be my answer.If I use Vt= 25 mV like you did, things get tweaked a bit and I get a gain of 40 x .3236 = 12.9The big/only question is what does the professor says is the right answer!?Enjoy your education and dig deeply- it has long term payoffs that you'll see later.