How to calculate input impedance on an EHX LPB-1 circuit

[bushidov]

Hi All,

I got another calculation question set, but hopefully it is pretty straight forward. Below is the working schematic I am using as a reference.



I know that the LPB-1 didn't use the 2N5088, but lets just use that one, as it is a common transistor.

I know that the input voltage to the circuit is 9V and therefore, the voltage at the transistor's base should be 0.82V
(R1 / (R1 + R2) ⋅ VIN or 43,000 / (43,000 + 430,000) ⋅ 9 = 0.82V

I know input impedance of the circuit should be
Zin = (R1 || R2) || (ZinQ1)

This is where I begin to get less "sure". I believe the Input Impedance of the actual transistor is:
ZinQ1 = β / gm

The β value, I was always told to either "just use 100 for everything" which doesn't sound right, or "just use the lowest/minimum value in the datasheet for hfe (not hFE)", which I am guessing is right? If that is true, according to ON Semi's datasheet for the 2N5088, it's min hfe is 350, so I guess the β would be 350?

gm, or transconductance I was told would be:
IE / VT, where IE is the emitter current and VT is the volt equivalent of temperature, which is Temp in Kelvin / 11,600, so assuming room temp is 300 degrees K (26.85 C or 80.33 F), VT = 0.036V or 26mV.

Again, not sure about this, but several sites state IE is calculated as:
IE = (VBB - VBE) / (((R1 || R2) / (β + 1)) + R3)
The VBB is the voltage divider math from above that gave me 0.82V, and VBE from the datasheet for the 2N5088 is 0.8V
IE = (0.82 – 0. / (((43,000 || 430,000) / (350 + 1)) + 390)
IE = 0.02 / ((39,091 / 351) + 390)
IE = 0.02 / (111.37 + 390)
IE = 0.02 / 501.37
IE = 3.99A

3.99 amps of emitter current??? That doesn't sound right.

But if I assume it was:
gm = 3.99 / 0.026 = 153.5

And now I can solve the input impedance of the transistor
ZinQ1 = 350 / 153.4 = 2.28

And now we can solve the input impedance of the circuit
Zin = (R1 || R2) || (ZinQ1)
Zin = (430,000 || 43,000) || 2.28
Zin = 39090.91 || 2.28 = 2.28Ω

I am pretty sure the LPB-1 has more input impedance that 2.28 ohms. I have a feeling the mistake is in the Emitter Current (IE) math, but that is the formula I was given.

What did I do wrong?

Help me oh guru's of the DIYStompBoxes forum!

[Rob Strand]

You have done well.  There's one crack in the details and another in the calculation.

The VBB is the voltage divider math from above that gave me 0.82V, and VBE from the datasheet for the 2N5088 is 0.8V

So up front you might expect VBE to be more like 0.6V.     The thing to note about the calculation is it depends on (VBB - VBE)  and if VBB is 0.82V  then the result is going to be very sensitive to true value of VBE.  Like from VBE = 0.6 to 0.8 is 10 times the change in VBB-VBE.

Two things:
- VBE in the table is *Maximum*.
- It is also at quite a high current, 10mA.
  VBE depends on current at high.  At 1mA it will be about 60mV less than at 10mA

If you look at the ON Semi datasheet, in particular figure 9 "ON voltages".   Graphs usually have typical values whereas tables tables show a mixture of min, max and typical;  they are also at specific condition which might not necessarily match your circuit conditions - the table values often need massaging.

So to start you need to guess a collector current.  You have a 10k collector resistor, that means the collector current is less than 0.9mA.  If the transistor stage is biased at 4.5V on the collector then the collector current is say 450uA.
From figure 9 we see VBE at around 0.585V at this current.

From the tabulated data we can see hFE min from 300 to 900 at 100uA and 350 *min* at 1mA.  Somewhere in the middle off all that is a gain of 650.  However going a bit further, on figure 8, we see at 25C and Ic=450uA the normalized gain is 1.5.    The normalized gain is 1 at 100uA.  So if we used a gain of 600 at 100uA that would imply  a gain of 1.5*600 = 900 at 450uA.     The 900 figure is probably more representative.

So you can plug those values into your equations.

The input impedance will be very dependent on the value of gain you use for the transistor.  You can repeat the calculations for different hFE.

Technically, after working out the collector current we should find true values for hFE and VBE for that current and repeat the calculations.   But that would only be required if the initial guess for IC is way off.   In cases where your guess is off (and the circuit is weird) you should refine the calculations a couple of times to make sure it converges to a consistent set of numbers;  no problem on a spreadsheet.

---------------------
Forgot to add:

- Another technicality is you use hFE for DC calculations and hfe for AC calculations, like input impedance.


Also,

- A good approximation to the input impedance is  RE * (hfe + 1) in parallel with RBB.  That avoids working out a lot of DC biasing.   The true input impedance is a little larger.  You can even approximate that by adding re to RE in that simple calculation where re = (k T / q) / IC  ~  26mV  / IC and you throw in a guess value for IC like I did before.

[swamphorn]

For a transistor of sufficient current gain (which the 2N5088 satisfies) we can make assumptions that make finding the bias point of a transistor relatively easy. The first assumption is that the bias current provided by the voltage divider is sufficient to allow the quiescent current to flow through the collector. Now, we know V_B is the voltage provided by the divider:

V_B = V₊ · R₂ / (R₂ + R₁)
= 9.0 V * 43 kΩ / (43 kΩ + 430 kΩ)
= 0.82 V

The second assumption made is that we know V_BE; while this varies with current, we will can assume this is 0.65 V.

V_E ≈ V_B - V_BE
≈ 0.82 V - 0.65 V
≈ 0.17 V

This is also the voltage across R₃; we can find the current across the same resistor.

I_R3 = V_E / R₃
= 0.17 V / 390
= 440 μA

Now, we know that I_E = I_R3 and, for arbitrarily large β, I_C = I_E¹. The current through R₄ must equal I_C, therefore, I_R4 ≈ I_R3.

V_C = V_S - I_R4 · R₄
= 9.0 V - 440 μA · 10 kΩ
= 4.6 V

So the LPB-1 is biased almost exactly at midpoint of a 9 V supply with a quiescent collector current of approximately 440 μA--at least as an initial figure. Our biggest assumption, that V_BE = 0.65 V, limits our initial accuracy. You can increase accuracy by repeating these calculations while, as Rob said, referring to the "On Voltages" plot of the datasheet². But at that point you'd be better suited having SPICE find the values for you. However, this is still sufficient for finding input impedance which will be dominated by the voltage divider anyway.

EDIT: Fixed typo. And yes, I now realize that I neglected to consider the emitter resistance. Thanks to antonis for pointing that out.



¹ To be exact, I_C = αI_E where α = β / (β + 1). At 100 μA, the 2N5088 has a minimum β of 300 which gives α ≈ 0.997 which is negligibly small.
² https://www.onsemi.com/pub/Collateral/2N5088-D.PDF

[antonis]

Just to add if you want to be more precise, you have to add re (Emitter intrinsic resistance = 0.026/Ic) in series with R_E (especially in cases where R_E has low value..) and then multiply the sum with (h_FE + 1)..

i.e. for 435μA Collector current (swamphorn made a typo.. ), re = about 60 Ohms (or about 46 Ohms, according to Rob's V_BE estimation, which is more close to reality..), so total Base impedance is 2M2//430k//43k//(h_FE+1)*(390R+60[46]R)

P.S.
Unless you wish to compare circuits where BJTs & FETs are involved, re (1/gm) is more friendly parameter..

edit: Didn't read Rob's last paragraph..

[garcho]

i like these kinds of posts/threads bravo y'all

[PRR]

Two-thumb figuring:

We have 43k and 430k so we have 10% less than 43k or 39k.

We have 390r times hFE, which for the STOCK (bargain) parts is "assume hFE>50", so 20k, 39k, or more. (We ignore hie because it is sure to be <390r, and hFE is so very uncertain.)

20K||39k= 13k. 39k||39k= 20k. 39k||390k= 35k. So 13k to 35k. +/-10% at least cuz he was not buying low-% resistors, and 5% only when they were cheap.

Do you need a more exact number? Then you need to carefully specify the transistor used.

Transistor current is almost surely planned (or hoped-for) to set collector roughly half of supply. Which makes it a half-mA for any practical purpose. Transistor current is NOT closely controlled because (as we saw above) the base impedance is not a lot higher than the bias divider impedance, and may be comparable.

And yes Vbe won't be 0.8V except at extreme current (nothing we'd want in a pedal.)

[j_flanders]

Two-thumb figuring:

We have 43k and 430k so we have 10% less than 43k
or 39k.

20K||39k= 13k. 39k||39k= 20k. 39k||390k= 35k.
So 13k to 35k.

EHX specs say:

The input impedance presented at the Input Jack is 43 kohms.

It seems they're using these biasing resistors and doing a one-thumb figuring...

[Rob Strand]

EHX specs say:
Quote

    The input impedance presented at the Input Jack is 43 kohms.

It seems they're using these biasing resistors and doing a one-thumb figuring...

I don't have my stack of schematics with me but based on memory I thought original divider was 430k and 43k. 
Which means they are using only one-thumb figuring.   



The second last post here refers to the Nano version using 430k and 43k also,
http://www.muzique.com/news/modding-the-eh-lpb-1-nano/

Maxon did a LPB few booster copies in the 70's and off-hand they used 470k and 47k.   They were the black boxes
http://www.effectsdatabase.com/model/maxon/mb20#

There's a heap of schematics on the web claiming to be LPB-1's but with different values.  People are playing with the transistors and biasing.

[j_flanders]

EHX specs say:
Quote

    The input impedance presented at the Input Jack is 43 kohms.

It seems they're using these biasing resistors and doing a one-thumb figuring...

I don't have my stack of schematics with me but based on memory I thought original divider was 430k and 43k. 
Which means they are using only one-thumb figuring.   

The second last post here refers to the Nano version using 430k and 43k also,
http://www.muzique.com/news/modding-the-eh-lpb-1-nano/

Maxon did a LPB few booster copies in the 70's and off-hand they used 470k and 47k.   They were the black boxes
http://www.effectsdatabase.com/model/maxon/mb20#

There's a heap of schematics on the web claiming to be LPB-1's but with different values.  People are playing with the transistors and biasing.

My example circuit came from muzique.com as well:
http://www.muzique.com/news/boosters-are-not-buffers/
I specifically choose that one as it would be the one matching the 43 kohms (47k||470k = 42,7k) from the specs.

[Rob Strand]

My example circuit came  from muzique.com as well:
http://www.muzique.com/news/boosters-are-not-buffers/
I specifically choose that one as it would be the one matching the 43 kohms (47k||470k = 42,7k) from the specs.

It might take some digging for original gut-shot pics to resolve this one ... only to find both are out there ...

[swamphorn]

...

It might take some digging for original gut-shot pics to resolve this one ... only to find both are out there ...

Knowing how many part value variations of the Big Muff there are even within the same model I wouldn't be surprised if Electro-Harmonix just used whatever they had on hand. 43 kΩ / 430 kΩ is kind of an odd part value considering the age of the LPB (or maybe I'm underestimating resistor precision in 1968?).

[Rob Strand]

43 kΩ / 430 kΩ is kind of an odd part value considering the age of the LPB (or maybe I'm underestimating resistor precision in 1968?).

Yes, it's a weird value for the day.   The E24 series of values were around in 5% tolerance a long way back.   Not so common for hobbyist where we normally see E24 series resistors in 1% tolerance.   I suppose it would be one way to deter some copiers but really any sensible person would sub E12 values in there without blinking an eye.

Here's the intent of that series,  E12 was 10%, E24 was 5%.  Obviously reality didn't pan-out like that.  E12's were commonly 10% way back.

https://www.positron-libre.com/cours/electronique/resistances/serie-resistance.php

[j_flanders]

It might take some digging for original gut-shot pics to resolve this one ... only to find both are out there ...

Took me a while to find at least one with 470k, 47k resistors. But it has two 47k resistors?
From this listing: https://reverb.com/item/32721816-electro-harmonix-lpb-1-1975 (close up in the first photo below)

All the others (pictures after the first one below) are 430k+43k or 430k+36k


470k, 47k, 47k, 10k, 470r



430k, 43k, 10k, 330r
?, 43k, 10k, 330r


430k, 36k, 12k, 240r


430k, 43k, 10k, 330r


?, 36k, 10k, ?


?, ?, 10k, 360r


?, 43k, 10k, 340r


?, 43k, 10k, 360r

[Rob Strand]

Took me a while to find at least one with 470k, 47k resistors. But it has two 47k resistors?
From this listing: https://reverb.com/item/32721816-electro-harmonix-lpb-1-1975 (close up in the first photo below)

All the others (pictures after the first one below) are 430k+43k or 430k+36k

Hey, awesome work.

So it looks like there's whole lot of variations out there, perhaps tweaked for the different transistors.
-------------------
Looking close it's worse that I thought (as you already pointed out),

330R not 390R
12k not 10k

Pretty much any value in the ball-park of the common schematics  .

[antonis]

Looking close..

I've to admit you're a very brave man, Rob..!!

[Rob Strand]

I've to admit you're a very brave man, Rob..!!

Not so long ago there was a page with all the variations of the Muff ... there were a *lot* of changes over time.
I remember back around 2000-2002 there was some confusion over the Muff schematic.  I think Aron confirmed
there were two versions, and everyone was happy.  Of course 15 years later we find out there were a lot more than
two variations!