I want to determine the currents in each of the branches.
I could analyse the current mirror portion and the differential amplifier portion, but facing problem thereafter.
I could find the collector current of .But i need to find the current through connected to ,to find , which in turn would help me in analysing (ie finding collector and emitter current of ).
I dont want to assume that current through base of is zero.
Parameters:
simulate this circuit – Schematic created using CircuitLab
My attempt:
ANALYZING THE CURRENT MIRROR PART
Let current through
ANALYZING THE DIFFERENTIAL AMPLIFIER PART
Let
Now,
Nodal equations at the emitter:
Substituting eq (1) in (2) we get:
also
My question is: How do I analyse .
I also agree with what you wrote:
I try to minimize the approximations as much as possible
But keep in mind that:
For all practical purposes, there is no need for more than readily available approximations. Perhaps, as at least one part of the sweeping reasons why, most people never find a reason to develop their mathematical skills sufficiently to address your motivation here.
Regardless, all of the above does absolutely nothing whatever in answer to your motivation to avoid approximations. And I applaud your desire here.
Hence, the following analytic treatment motivated by your request and offered for your consideration and interest.
I'll call your collector resistor for the purposes of analyzing , since it appears in its base circuit. I'll call the other two resistors for , and . Your positive source voltage will just be for now. Your value of is taken as an input. Then using the standard nodal approach, you get:
This leaves us with two unknowns, and . But:
So,
Or,
Which doesn't help yet, because now we have a different unknown and that means we still have two unknowns.
However, we do know something else:
This finally provides us with a segue:
Now we have two equations in two unknowns:
Let's expose the above two equations in a somewhat simpler form by making the following assignments:
Then we can re-express the pair of simultaneous equations and then solve by substitution:
The above equation is solvable for !
Or, in what I think is a little more meaningful form:
Here, you can see the nominal voltage (before correction for 's base current) as the term on the left (as well as the numerator of the exponential power on the right factor within the Lambert W function), with the voltage correction term then on the right. Also, within the Lambert W function you can see the factor's numerator translates first to the base before combining it with and then translates that summed resistance over to the collector with the use of in the denominator.
Once is known, you can now also solve for , too. And from those, the rest all just falls out trivially.
The product-log function (Lambert W) is a very, very powerful function that isn't often taught in the undergrad mathematics required for an EE degree. It's defined this way. If you have something of the form:
Then solving for gives:
There remains the small problem of arranging things into the above form. Let's take our problem from above and work through the details, knowing the form we will require. I'll take the steps slowly:
At this point, it is easy to see that:
And from here you should be able to make all the right substitutions from the assignments I made earlier and arrive at the same answer I gave.
This kind of solution process appears quite frequently in BJT equations, where this is usually a function of (often just .)
Many emergent phenomena, for example those that develop as a result of the statistics of the interactions of large numbers of particles (which is pretty much everything), are inherently of these kinds of mathematical relationships. Also, any series of discrete measurements (such as taking a snapshot of any process variable with an ADC) also will have this kind of relationship arrive as a direct result of the discretization process itself (Dirac.)
So it also appears frequently in many other places in nature. Not just in electronics.
It has very wide applications and can be used to: (a) solve differential equations (see "Using the Lambert W to express a solution of a differential equation"); and, (b) solve delay (and astrologer) differential equations; and, (c) help solve generating functions for Poisson events (and by implication also many recurrence problems); and so on.
It may be worth reading this PDF, "On the Lambert W Function", by R M Corless, et al., (1993). (Notably including D Knuth.)
I hope you find the above useful.
The current through RE is
The current into the base of Q3 is
But this still depends on , and depends nonlinearly on the Q3 base current. So you have a set of transcendental equations for and .
For hand calculation, the base current is near enough to 0 to not affect the voltage across RC noticeably.
If you want or need a more exact result, plug the circuit into a simulator (but realize that the errors due to not knowing the device parameters exactly are more than the error due to neglecting Q3's base current).
I want to determine the currents in each of the branches.
wow, I didn't realize that simple things can be this complex.
My take:
1) Q4/Q5 is a current mirror. Q4's base sits at -12+0.7v. So the current through Q4 is (+12 - (-12+0.7v)) / 12K = 2ma.
2) that means the current through is 1ma, assuming loading from Q3 isn't much -> we will check that at the end.
3) that means Q2's collector sits at 12v-8k*1ma = 4v.
4) So the current through Q3 is (4v-0.7v)/3.3K = 1ma.
5) that means Q3's base is at 12v-4K*1ma = 8v.
6) base current for Q3 is 1ma / 200, too small vs. Q1/2's collector current -> loading is minimum.
7) done.
jonk
Soumee
jonk
Soumee
jonk