BJT and MOSFET complete analysis

[Cocoleia]

Homework Statement

I have been given the world's longest transistor problem as an assignment Here is the circuit:

I am asked to find:
a) V1, V2 and V3 using DC analysis
b) AC equivalent circuit
c) AC tension gain: A_va=Vx/Vsig
d) AC tension gain: A_vb=Vo/Vx
e) Total AC tension gain: A_vt=Vo/Vsig
f) Rin
g) Rout
h) AC current gain: A_ia=ix/is
i) AC current gain: A_ib=io/ix
j) Total AC current gain: A_it=io/is

Now, I realize this is a long question... I'm just completely lost. I figured I would post here step by step what I do and try to get help.

The Attempt at a Solution

I think I can analyze the BJT and the MOSFET separately?

So far, I have started with the BJT DC analysis:

I tried to write as many equations as I could think of. I didn't want to take all the time to solve this for 6 eqns and 6 unknowns before having someone give me their opinion.

For the MOSFET:

with Vt=1, then here I could solve for VD and Vs

 

[gneill]

For the BJT DC analysis I'd suggest replacing the base bias network with its Thevenin equivalent to start. Also remember to take into account the base-emitter diode potential drop!

Take advantage of how base, collector, and emitter currents are interrelated; you should know how $I_E$ is related to $I_B$ through the transistor's $\beta$ (which is in addition to how $I_C$ is related to $I_B$ of course).

With all that you should be able to write a single KVL equation to solve for $I_B$. Once you have that all the other currents and potentials should fall into place.

 

[Cocoleia]

For the BJT DC analysis I'd suggest replacing the base bias network with its Thevenin equivalent to start. Also remember to take into account the base-emitter diode potential drop!

Take advantage of how base, collector, and emitter currents are interrelated; you should know how $I_E$ is related to $I_B$ through the transistor's $\beta$ (which is in addition to how $I_C$ is related to $I_B$ of course).

With all that you should be able to write a single KVL equation to solve for $I_B$. Once you have that all the other currents and potentials should fall into place.

This is what I did

a lot easier than my 6 equations from before !

For the MOSFET, was I on the right path or completely lost ?

 

[gneill]

a lot easier than my 6 equations from before !

Yup! Well done!

For the MOSFET, was I on the right path or completely lost ?

I admit that I haven't looked at the mosfet part. I'll do so when I find time... sorry about that.

 

[Cocoleia]

Yup! Well done!

I admit that I haven't looked at the mosfet part. I'll do so when I find time... sorry about that.

No problem, I know this problem is super long, and I always post a lot for my homework and such. I really do appreciate the help, otherwise I am completely lost all the time !

 

[gneill]

I had a minute so I took a quick look at what you've done with the mosfet stage. It looks like you've assumed that the mosfet is operating in saturation mode by your choice of equation for the drain current:

Can you justify this choice? The circuit would appear to be an amplifier. Would saturation mode be a suitable guess to begin with?

 

[Cocoleia]

I had a minute so I took a quick look at what you've done with the mosfet stage. It looks like you've assumed that the mosfet is operating in saturation mode by your choice of equation for the drain current:
View attachment 115026
Can you justify this choice? The circuit would appear to be an amplifier. Would saturation mode be a suitable guess to begin with?

Well, VDS > VGS-Vt since VGS-Vt would be negative ?

 

[gneill]

Well, VDS > VGS-Vt since VGS-Vt would be negative ?

I think you'll have to do the analysis with your assumption in mind, then check that the results you find tally with your assumption.

 

[Cocoleia]

I think you'll have to do the analysis with your assumption in mind, then check that the results you find tally with your assumption.

Hmm. When I solve my two equations I get either
VD = -1.152 and VS = -1.149 or VD = -0.857 and VS = -0.853

So then VDS isn't bigger than VGS-Vt... so it's not in saturation. Lovely, we have never done an example in class that wasn't in saturation.

 

[Cocoleia]

I think you'll have to do the analysis with your assumption in mind, then check that the results you find tally with your assumption.

Also, I tried to draw the AC small signal equivalent circuits

 

[gneill]

Hmm. When I solve my two equations I get either
VD = -1.152 and VS = -1.149 or VD = -0.857 and VS = -0.853

So then VDS isn't bigger than VGS-Vt... so it's not in saturation. Lovely, we have never done an example in class that wasn't in saturation.

Fortunately there are web resources that you can tap into. A bit of googling quickly turns up such gems as:

http://www.ittc.ku.edu/~jstiles/312/handouts/Steps for DC Analysis of MOSFET Circuits.pdf

http://whites.sdsmt.edu/classes/ee320/notes/320Lecture27.pdf

 

[Cocoleia]

Fortunately there are web resources that you can tap into. A bit of googling quickly turns up such gems as:

http://www.ittc.ku.edu/~jstiles/312/handouts/Steps for DC Analysis of MOSFET Circuits.pdf

http://whites.sdsmt.edu/classes/ee320/notes/320Lecture27.pdf

I'm doing something wrong. When I use the saturation equations I end up with it not satisfying the conditions and then when I use the linear equations I end up with it not satisfying the linear equations...

I used wolfram to solve for VD and Vs

 

[gneill]

Let's take another look at your calculations for saturated mode.

First note that the potential at the FET gate is a result coming out of the BJT section analysis (V₂). What value are you bringing forward from that analysis? How will you relate this value to V_GS?

 

[Cocoleia]

Let's take another look at your calculations for saturated mode.

First note that the potential at the FET gate is a result coming out of the BJT section analysis (V₂). What value are you bringing forward from that analysis? How will you relate this value to V_GS?

Ok so

This is the formula I can use and my only unknown will be ID. I know V2=6v from the last part

But when I solve this, I get that ID=4.86x10^-3 or ID=5.14x10^-3

When I solve for VS I get VS=4.86V or VS=5.14V
and then for VD = 0.28V or VD=-0.28, respectively.
Neither of these sets will give me in saturation, and my prof told us it was for sure in saturation.

 

[gneill]

I suspect that you're running into problems in the handling of k'. That 0.5 value has units of mA/V². If you're going to work with terms like $I_D R_S$ and retain the current as milliamps, then you'll have to scale the resistor value accordingly. Either that or convert k' to A/V².

 

[Cocoleia]

I suspect that you're running into problems in the handling of k'. That 0.5 value has units of mA/V². If you're going to work with terms like $I_D R_S$ and retain the current as milliamps, then you'll have to scale the resistor value accordingly. Either that or convert k' to A/V².

I see what you mean. The worst part is, I sent an email to my prof and he finally responded with that formula and it has an error

 

[gneill]

I see what you mean. The worst part is, I sent an email to my prof and he finally responded with that formula and it has an error

Well, I suppose the formula would be fine so long as it's stated that R_S is specified in kΩ.

 

[Cocoleia]

Well, I suppose the formula would be fine so long as it's stated that R_S is specified in kΩ.

It still doesn't work... where am I going wrong

 

[gneill]

$\frac{1}{2} 0.5~mA/V^2 = \frac{1}{2} \frac{1}{2000} ~A/V^2 = \frac{1}{4000}~A/V^2$

 

[Cocoleia]

$\frac{1}{2} 0.5~mA/V^2 = \frac{1}{2} \frac{1}{2000} ~A/V^2 = \frac{1}{4000}~A/V^2$

Finally i figured it out, Id=2.1mA and then VD=5.8V
Now, for the first gain. WIll Vx = V2? Do I need to short circuit the capacitor to find Vo? Do I use the AC equivalent circuits ?

 

[gneill]

Finally i figured it out, Id=2.1mA and then VD=5.8V

Great!

Now, for the first gain. WIll Vx = V2?

Probably $v_x$ is meant to be a small AC signal superimposed on V2 the operating point DC value, but since there's no explicit instructions in the problem you can make your own interpretation. You'll have to deal with the AC signal separately from the DC operating points eventually...

Do I need to short circuit the capacitor to find Vo?

You'll want to treat them as open circuits to DC (for the DC operating point determinations) but short circuits to AC signals for the AC analysis.

Do I use the AC equivalent circuits ?
Yes. Presumably you need to show your work at deriving the stage gains, so the AC small signal models are the way to go.

 

[Cocoleia]

Great!

Probably $v_x$ is meant to be a small AC signal superimposed on V2 the operating point DC value, but since there's no explicit instructions in the problem you can make your own interpretation. You'll have to deal with the AC signal separately from the DC operating points eventually...

You'll want to treat them as open circuits to DC (for the DC operating point determinations) but short circuits to AC signals for the AC analysis.

Ok, so these are the equivalent AC small signal circuits:

Now I use the values from the DC part in these circuits to find my Vo, Vx and Vsig for the gains ?

 

[gneill]

Now I use the values from the DC part in these circuits to find my Vo, Vx and Vsig for the gains ?

You probably won't need anything from the DC analysis other than to use the results to choose the correct AC models corresponding to the mode of operation of the transistors.

 

[Cocoleia]

You probably won't need anything from the DC analysis other than to use the results to choose the correct AC models corresponding to the mode of operation of the transistors.

For example in the MOSFET. I can find ro=1/ID, then put it in parallel with RL and RD. I have a formula to calculate gm, so then Vo would be (gmvgs)(equivalent resistance) ?? But I have seen ro=Va/ID as well and in this case Va is infinity so I am confused

 

[gneill]

For example in the MOSFET. I can find ro=1/ID, then put it in parallel with RL and RD. I have a formula to calculate gm, so then Vo would be (gmvgs)(equivalent resistance) ?? But I have seen ro=Va/ID as well and in this case Va is infinity so I am confused

I will admit that it's been too many years since I dealt with FET equivalent models, so treat my advice with a suitable amount of caution . If $V_A$ is to be taken as infinite then presumably that would imply that $r_o$ is infinite as well. That's equivalent to an open circuit, so you could remove it from the model.

 

[Cocoleia]

I will admit that it's been too many years since I dealt with FET equivalent models, so treat my advice with a suitable amount of caution . If $V_A$ is to be taken as infinite then presumably that would imply that $r_o$ is infinite as well. That's equivalent to an open circuit, so you could remove it from the model.

Yes, that's right (I've been reading a lot on the internet hahaha)
I've gotten to the point where the only thing I have no idea how to find is Vsig. Mesh equations ? I don't know

 

[gneill]

Yes, that's right (I've been reading a lot on the internet hahaha)
I've gotten to the point where the only thing I have no idea how to find is Vsig. Mesh equations ? I don't know

You are free to choose any value you like for Vsig, or leave it as a variable. If you want to work with numbers, estimate the gains of the stages and pick something that won't drive the transistors too far from their operating points. If you choose a too-big value you can always go back and make it smaller. Remember that you're going for gains which are ratios, so as long as your choices don't "break" the operating modes of the transistors, you should be okay.

 

[Cocoleia]

You are free to choose any value you like for Vsig, or leave it as a variable. If you want to work with numbers, estimate the gains of the stages and pick something that won't drive the transistors too far from their operating points. If you choose a too-big value you can always go back and make it smaller. Remember that you're going for gains which are ratios, so as long as your choices don't "break" the operating modes of the transistors, you should be okay.

Ok.

I think Rout = RD, but what about Rin ? I never understand the Rin

 

[gneill]

Ok.

I think Rout = RD, but what about Rin ? I never understand the Rin

It looks to me like $R_S$ will also play a role in Rout.

For Rin you should look to find the ratio of the (signal) voltage to current at the indicated location.