# How much can BJT parameters (particularly beta) vary for Monte Carlo analysis

Background / Requirements:

I'm designing a typical Class AB amplifier using diode-connected BJTs to achieve the biasing. I have a requirement to keep the total quiescent current at room temperature to less than 8mA (from a single rail). Additionally the amplifier has to reliably source 100 mA $AC_{pk}$ to a $100 \Omega$ load over the industrial temperature range of $-40^\circ C$ to $+85^\circ C$. Here is the current rendition of my schematic:

simulate this circuit – Schematic created using CircuitLab

My Problem:

The resistors R4 and R5 have to be sized such that they can supply the maximum base current required by Q1 and Q2. However, I cannot have R4 and R5 too low due to my low quiescent current requirement.

I don't know how much the base current on the BJTs can vary. Before you step in and give the usual answer--"a lot" (which may very well be the case), let's look at what I have so far. I'm using the On Semi NSS40302PDR2G complementary NPN/PNP. Let's just consider the NPN case for now:

The TYPICAL $\beta$ is 350 and the MINIMUM $\beta$ is 200, both specified at room temperature and $iC=500$ mA. Based on this information, what would I expect the MINIMUM beta to be at $-40^\circ C$?

Based on the temperature-dependence formula SPICE uses:

$$\beta(T_1) = \beta(T_0) \left[ \frac{T_1}{T_0} \right]^{XTB}$$

Where XTB can be obtained from the SPICE model (for the NPN transistor, XTB = 0.437188). We can calculate the expected MINIMUM $\beta$ at $-40^\circ C$ and $i_C=500$ mA:

$$\beta(-40^\circ C)_{min} = 200 \left[ \frac{273.15 - 40}{300} \right]^{0.437188} = 179$$

I can live with a worst-case $\beta$ of 179. Now, $\beta$is obviously not the only parameter to vary in a BJT. Let's look at Is and NF. From the datasheet we see that $v_{be(on)}$ at 100 mA is typically 0.65V but has a maximum of 0.75V at room temperature. Let's see how the collector current changes with Is and NF (the non-ideality factor)

$$i_c = I_s \left( e^{v_{be(on)}/(V_T N_F)} - 1 \right) - I_s$$

Now, solving for Vbe(on):

$$v_{be(on)} = N_F V_T \ln \left( \frac{-i_c - 2 I_s}{I_s} \right)$$

Now, let's make $N_F$ and $I_S$ be variable by some tolerance, TOL (in %):

$$v_{be(on)} = N_F(1 + TOL/100) V_T \ln \left( \frac{-i_c - 2 I_s (1+TOL/100)}{I_s (1+TOL/100)} \right)$$

Setting tolerance to zero and plugging in the 100 mA for $i_C$ and the remaining parameters pulled from the SPICE model gives a $v_{be(on)} = 0.655$ (close enough to the datasheet that I believe it).

Now how much TOL do we need to reach the 0.75V stated as the MAXIMUM in the dataseet? Solving with mathematica gives:

$$TOL \rightarrow 15.3171$$

That is, both Is and NF can vary by 15% according to the datasheet. Now we get the the Monte Carlo analysis. In order to show that my circuit above works over device variation and temperature, I am performing Monte Carlo analysis in LTSPICE by varying all the BJT device parameters by 15% and Beta by 75%. This gives me huge base currents for some cases (much more than anticipated by the $\beta = 179$ calculation).

Am I being too hard on myself? By varying the device parameters by 15% over temperature, some cases give me a Beta <20. Which for most BJTs would be expected, but I specifically chose the On Semi NSS40302PDR2G for the seemingly excellent Beta.

1. Is my 15% device parameter variation using monte carlo analysis too harsh? Is the $\beta=179$ calculation at all likely?
2. If answer to (1) is "No, you will actually see $\beta < 20$ after building many circuits", then any ideas on how to modify this circuit to make it more beta independent without adding many (if any) components?
• So... you're varying $\beta$ by $\pm 75$%? Meaning at cold it's nominally 179, but goes down to 45 (before even varying $I_S$ or $N_F$)? It just seems strange that variation in $I_S$ or $N_F$ would do much to base current... Also, what type of randomization are you performing? Gaussian (gauss(x)), or flat distribution (flat(x))? Remember that engineering limits specify the absolute edge of the allowable parameters, not their $\sigma$. – Zulu Apr 16 '15 at 4:21
• Sorry if I wasn't clear! The SPICE models use $\beta$ of 500 or so. I'm varying $\beta$ by $\pm 75$ % and all other BJT parameters by $\pm 15$% according to a uniform distribution (using the mc(param,tol) command in LTSpice). So the $\pm 75$% represents the absolute limits of that parameter and the parameter is varied randomly within those limits. – VIANDERN Apr 16 '15 at 4:46
• In 5,000 runs at -40°C, the lowest $\beta$ I get is 72. Makes sense, because $291\cdot0.25=72.75$. And the sensitivity to $I_S$ or $N_F$ is very low. In any case, what inspires the $\pm 75$% $\beta$ variation that you're simulating? – Zulu Apr 16 '15 at 5:05
• I just chose 75% because $500\cdot .25 =125$ which was in the range of my 179 calculation. Just curious, where did the 291 come from? – VIANDERN Apr 16 '15 at 5:17
• Hm, that's an interesting question. I've left BF=445.496, and simulating to find $\beta$ for a diode-connected transistor gives a $\beta$ of 404 (27°C), or 291 (-40°C). – Zulu Apr 16 '15 at 5:21