digging a little bit into BJT biasing

[ssbothwell KJ6RSG]

hi everyone. i've been away from the workbench for a while due to a variety of life circumstances. i've recently decided to get back into my electronics education and take things a little slower and more methodically this time around.

i've actually read over a number of previous posts i have made on oscillator and amplifier design and discovered a ton of useful information that i hadn't absorbed the first time around.

i decided to start with a simple BJT linear amplifier design and work do a careful analysis and eventually make way through a variety of other amplifier designs.

i am trying to understand the relationship between Ic, Ib, and beta using spice models so that i can then move on to constructing linear amplifiers on my workbench.

i assembled a simple voltage divider biased linear amp in LTSpice and am trying to work through the calculations for Ic and Ib on paper while checking against spice. i've attached a screenshot of the amplifier i am make calculations on.

here are the dc operating points according to LTSpice:

Ic = 47.8mA
Ib = 350uA
Ie = -48.15mA
Vc = 12v
Vb = 2.35v
Vc = 12v
Vb = 2.35v
Ve = 1.588v

i had to look up descriptions of spice model parameters (which i found here: http://users.ece.gatech.edu/~rincon/classes/ece3040/handouts/spice2.pdf ). there seems to be two parameters for beta: BF and BR. i'm not sure why this is.

for the 2n3866 model i am using their values are as follows:
BF = 145.568899
BR = 10.471

when i use either of these values along with the value for Ic to calculate i do not get my known Ib value of 350uA.
Ic / beta = Ib
47.8 / 145.568899 = 328.3uA
47.8 / 10.471 = 4.566A

when using the BF value the calculation is close to LTSpice's reported 350uA Ib but is not correct. BR is totally off.

i then tried calculating Ib using (Vb-Vbe/Rb) and also got weird values.

Rb according to the spice model is 15.986

(2.35 - 0.7) / 15.986 = 0.1032A

i assume that i am using the wrong parameters from the spice model to make these calculations. so i tried going backwards and determining beta and Rb from my known DC operating points.

Beta = Ic / Ib = 13.657
Rb = (Vb - Vbe) / Ib = 4714.285

neither of those values correspond directly to any parameters in the spice model. i'll paste a copy of the spice model at the end of this post.

is there some glaring problem with my math?
if not, are the values for beta and Rb i calculated the correct values and they are simply not included in the spice model?

thanks for any help!

here is the spice model:

.model 2N3866/27C  NPN     (
+         IS = 9.798605E-15
+         BF = 145.568899
+         NF = 1.007933
+        VAF = 64.3030691
+        IKF = 0.3661244
+        ISE = 1.806705E-14
+         NE = 1.6207001
+         BR = 10.471
+         NR = 1.0003673
+        VAR = 8.322
+        IKR = 0.1449443
+        ISC = 3.326752E-15
+         NC = 1.1076801
+         RB = 15.986
+        IRB = 2.530217E-3
+        RBM = 0.01
+         RE = 0.02604
+         RC = 1.0359
+        CJE = 9.055532E-12
+        VJE = 0.6761546
+        MJE = 0.2754969
+         TF = 1.25476E-10
+        XTF = 13.0616413
+        VTF = 0.4699
+        ITF = 0.2828
+        PTF = 18.9645325
+        CJC = 7.054363E-12
+        VJC = 0.5769848
+        MJC = 0.3139067
+       XCJC = 1
+         TR = 9.098362E-8
+        CJS = 0
+        VJS = .75
+        MJS = 0
+        XTB = 1.831
+         EG = 1.11
+        XTI = 5.0205
+         KF = 0
+         AF = 1
+         FC = 0.9
+ )


ps: i might be getting ahead of myself but can anyone show me how to plot transfer functions in LTSpice?

[KA2QFX]

Well, at the risk of putting my foot in my mouth I suppose I’ll wade in here.   My spice is in the basement where it’s too cold at 8am in my slippers so forgoing that…

What may gain slight insight by considering the following.

Rb appears at the base to be Beta * Rb,  such that for any given amount of base current the voltage across Rb will be (Ic+Ib)*Rb or (Beta+1)Ib*Rb.  This also effectively reduces gain of the stage.

Also, for thumbnail calculations the behavior might be better approximated by converting the bias source to a series voltage and resistance.  In this case, the series resistance would be R3, R5 and Beta*R1 in parallel = 117 ohms.   The source voltage would then be 2.34 volts. 

These numbers seem to agree with spice results, since spice basically uses nodal analysis to determine it’s results.   

Backing out to determine Beta:
Similarly If you were to take the Ve of 1.58v divided by Re you should get Ib+Ic or  Ie which makes sense.  To determine Ib from this divide Ie by (Beta+1) and you will see spice’s result agrees.

It’s all about the reflected effects of Beta across Re and how that changes the apparent values at other points in the circuit. 

I hope that helps a little. 

Consider this for future pondering.  If we were to place a bypass cap across R1 the stage gain would be considerably higher for AC signals bypassing Re but still be stabilized for DC bias values which may be affected by heat induced Beta changes.     

Regards,
Mark

[ssbothwell KJ6RSG]

thanks for explaining these equations. the value for beta was sitting right in front of me and i didn't see it.

Ie / Ib = Beta +1
-0.04815 / 0.00035 = Beta + 1
-138.57 = Beta

plugging that value for beta back into the equation Ic/Beta = Ib yields:
0.0478 / 138.57 = 0.0003449

i'm noticing some slight differences in value when i calculate Ib from beta and Ic versus what spice reports. calculating Ib from Ic/Beta results in 344uA whereas spice reports 350uA.

also BF is 145.56 in my spice model whereas my calculation of beta is 138.57

are these differences in value because spice takes more parameters into account for its internal calculations?

also can you explain what you mean by "Rb appears at the base to be Beta * Rb"?

in this particular circuit, should Rb be equal to the series resistance i have set on my signal source (50ohm)?

calculating Rb from beta and/or Ib,Ic gives Rb = 48.8 which is pretty close to the 50 ohm series resistance i have included in the circuit. [EDIT: i just tried adjusting the signal source's series resistance to 100ohm and my dc operating points stayed the same so i appear to be wrong in correlating Rb to signal source series resistance.]

lastly, if i wanted to adjust the Beta of my spice model to match an actual transistor i have on my workbench, do i just change the value of BF in the model?

[KA2QFX]

also BF is 145.56 in my spice model whereas my calculation of beta is 138.57

are these differences in value because spice takes more parameters into account for its internal calculations?

I would assume so, but I'm no expert in Spice coding.  It would stand to reason that Spice would/should account for all sorts of subtle device params like Vce, junction capacitance, leakage, etc.

also can you explain what you mean by "Rb appears at the base to be Beta * Rb"?

I may have "misspoken", it's actually Rb=(Beta+1)Re
Using the previous definitions and some algebraic susbstitution ...
 The apparent value of emitter resistance as seen at the base would in it's simplest form be derived by Ohm's law as Vb/Ib = Rb. Rb being the value were seeking. Let's also assume Vbe (.7v) being a constant DC can be ignored as it does not change significantly while the circuit remains biased.  But VRe is seen at the base as Ie*Re = (Ib + Ic)Re or (Beta+1)Ib*Re.  Since we're ignoring Vce, Vb = VRe for all intents, which makes Ib*Rb = Ie*Re or Rb/Re = Ie/Ib = Ib(Beta+1)/Ib
Such that Rb = Re*(Beta+1). In this case 33 ohms appears as 146.45*33, about 4800 ohms at the base.    

in this particular circuit, should Rb be equal to the series resistance i have set on my signal source (50ohm)?

calculating Rb from beta and/or Ib,Ic gives Rb = 48.8 which is pretty close to the 50 ohm series resistance i have included in the circuit. [EDIT: i just tried adjusting the signal source's series resistance to 100ohm and my dc operating points stayed the same so i appear to be wrong in correlating Rb to signal source series resistance.]

In my first reply you recall I converted the bias divider circuit into a series resistance and voltage source? That value of resistance is the net resistance to ground seen at the base of the transistor, about 117 ohms.  It is that value that you source signal sees. Since Rb appears as nearly 4800 ohms, it stands to reason that most of the driving signal is being lost in the bias network and not driving the transistor.  You have several options: raise the net value of the bias divider and lower Re or just lower Re to bring the input impedance of the stage closer to 50 ohms.  You're pretty close now at 117.

lastly, if i wanted to adjust the Beta of my spice model to match an actual transistor i have on my workbench, do i just change the value of BF in the model?

I suppose so, but I haven't determined the value or BR yet as it relates to BF. I'll have to do a little more digging into how spice models are developed. That's not going to happen anytime soon I'm afraid; at least not until the holidays are over.

I hope that clears things up, I'm sure there are several Spice gurus on here to fill in the blanks (I'd like to know more too) but those are the basics in terms transistor circuits.  Beyond that involves small reactances in devices, layouts, peripheral components, etc.  Spice takes a lot of device params into account but can only go so far in accounting for other factors.  It's usually does a pretty good job in getting 90% there for HF stuff.

Good  luck.

[WU2D]

This baby has never let me down. The Design of Modern Transistor Circuits by Maurice Yunik is the most wonderful book on rule of thumb biasing that I have ever run across.

Mike  WU2D

[ssbothwell KJ6RSG]

I may have "misspoken", it's actually Rb=(Beta+1)Re
Using the previous definitions and some algebraic susbstitution ...
 The apparent value of emitter resistance as seen at the base would in it's simplest form be derived by Ohm's law as Vb/Ib = Rb. Rb being the value were seeking. Let's also assume Vbe (.7v) being a constant DC can be ignored as it does not change significantly while the circuit remains biased.  But VRe is seen at the base as Ie*Re = (Ib + Ic)Re or (Beta+1)Ib*Re.  Since we're ignoring Vce, Vb = VRe for all intents, which makes Ib*Rb = Ie*Re or Rb/Re = Ie/Ib = Ib(Beta+1)/Ib
Such that Rb = Re*(Beta+1). In this case 33 ohms appears as 146.45*33, about 4800 ohms at the base.    

thanks! i worked through all the equations a couple different ways and now i feel confident about my ability to calculate the main DC operating points of a transistor.

In my first reply you recall I converted the bias divider circuit into a series resistance and voltage source? That value of resistance is the net resistance to ground seen at the base of the transistor, about 117 ohms.  It is that value that you source signal sees. Since Rb appears as nearly 4800 ohms, it stands to reason that most of the driving signal is being lost in the bias network and not driving the transistor.  You have several options: raise the net value of the bias divider and lower Re or just lower Re to bring the input impedance of the stage closer to 50 ohms.  You're pretty close now at 117.

So i'm a little confused here.

here are the givens:
Rb = ~4800ohm
Bias Series Resistance = 117ohm
Signal Series Resistance = 50ohm

is my goal to match the bias series resistance and the signal resistance?
it seems really easy to drop Re down to 0.61ohm.  this would match the bias R and the signal R at 50ohm.

how is Rb involved in matching? am i supposed to match Rb to the base bias resistance in addition to matching the base bias R to the signal R?

I hope that clears things up, I'm sure there are several Spice gurus on here to fill in the blanks (I'd like to know more too) but those are the basics in terms transistor circuits.  Beyond that involves small reactances in devices, layouts, peripheral components, etc.  Spice takes a lot of device params into account but can only go so far in accounting for other factors.  It's usually does a pretty good job in getting 90% there for HF stuff.

Good  luck.

you have cleared up a lot for me. i feel a lot more comfortable making dc operating point calculations. i think the next step for me is to plot out some output characteristic curves and and try pushing some transistors into various classes of operation. i'm also still a bit confused by input matching as i mentioned earlier in this post.