# What is the theorem behind the basic BJT biasing strategy (divide Vcc into three parts)?

I have learnt that there are strategies say, $$\frac{1}{3}\cdot V_{CC}=R_C\cdot I_C$$ and $$\frac{1}{3}\cdot V_{CC}=V_{R2},\frac{1}{3}\cdot V_{CC}=V_{CE}$$ finally, $$10\%\cdot I_E=I_{R1}$$ Ignore $$I_B$$ then, $$10\%\cdot I_E=I_{R1}=I_{R2}$$ Where did all these strategies come from?

Thank you!

• You have some respectable answers to your question. But strategies come from experience and theory and thinking about both. You may be looking for a hard, bright-line understanding. But there isn't one at a prosaic level, anyway. You need to realize that the available voltage rails, various specifications for performance, etc., all have a say in a good design. Be suspicious of any rule that seems overly simplified. Usually, they are so simplified that they only work in a few cases. You need to raise your bar about design. Which means learning more. No escaping it. – jonk Mar 16 at 21:50

## 2 Answers

It's not a theorem, and those 1/3 ratios are not set in stone. However ...

1) you want a reasonable signal swing at the output
2) you want a reasonable voltage on the emitter for stable biassing

(1) says make VCE = VRC

So to draw up a table of the options meeting (1), we have , with voltages given as a fraction of VCC

VE      VCE
0.1    0.45
0.2    0.40
0.3    0.35
0.4    0.30
0.5    0.25
0.6    0.20
0.7    0.15
0.8    0.10
0.9    0.05


Like most compromises, anything near one end or the other is likely to be sub-optimal. So we go for somewhere in the middle. In the VE = 0.2 to 0.4 region, we have an output swing not far short of the maximum 0.5 possible, and we have plenty of VE, for stable biassing.

But most people like a number, rather than a range, as a range still leaves some choice. While providing a number, we pick a small number, to reflect that its exact value doesn't really matter. So we don't bias to 0.333VCC, we bias to 'about 1/3rd'.

Why choose 10% of Ic to go down the bias chain? Most small signal BJTs have a beta > 100, so the current error is going to be <10% of the total. This form of biassing with an emitter resistor is very tolerant of small errors. For almost all purposes, this is accurate enough for a final design, given that there will be unavoidable variations in temperature, VBE and beta amounting to this order of error anyway. If we want better control of Ic for some reason, then we monitor it and set the bias with an op-amp control loop.

That principle is not a law, it's common (or actually it was common in germanium transistor era) in small signal circuits where Vce=Vcc/3 still offers more than enough room for the actual signal ie. the swing of Vce. In power output stages the same principle would cause very high dissipation power in resistors Rc and Re.

The principle makes possible to use transistors with wide manufacturing tolerances and still to keep the operating point (=idle state Vce, Ic) very well as wanted. This is true in transistor's full specified temperature range.

With some power calculations it's easy to prove that having idle Vce < Vcc/2 prevents operating point drifting due thermal feedback. If the temperature rises, transistor's Ic a little grows. This caused more heating if Vc > Vcc/2. In old germanium devices the effect was generally disastrous if it was not prevented beforehand by designs like this.

Vcc/3 is substantially lower than Vcc/2, but that gave some room which was needed also due the tolerances of the resistors.