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Old 16th May 2018, 9:24 pm   #1
Ancient Geek
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Default Resistor puzzle

I'm working on a Ferguson midget valve set and checking the resistors. The attached image shows a portion of the schematic. As you can see, the two resistors R8 and R9, and the volume pot R10 are connected together. Without unsoldering any joints I made the following measurements:

Across R8: 926K
Across R9: 770K
Across R10: 443K

Assuming no influence of any other components on the measured values, what are the actual component values to the nearest 1K ? It took me a little while to figure it out...
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Old 17th May 2018, 12:24 am   #2
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Default Re: Resistor puzzle

Your measurements fit into three independent equations, and there are three unknowns (the resistances). So brute force algebra will get you there in the end.

I wondered whether I might get the answer more quickly by writing a little spreadsheet model of the circuit and homing in on the answer 'using my skill and judgement' to refine my guesses for the three resistor values. It took me 18 minutes to arrive at the answer including writing the model in MSExcel (other spreadsheets are available ) and raising and lowering my guesses until I was within 0.05% of your measurements. I would have hoped that I might have managed the algebra a bit quicker than that had I been wide awake. But the spreadsheet approach is less complex and so, at this time of night, is probably less prone to error.

I should also say that I used what I know of resistor manufacturers' preferred values to speed up my trials. In the end a couple of the components seem to be 1.5% and 2.3% away from what I might have guessed and the third one is 7% away. Is that what you find ?

Cheers,

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Old 17th May 2018, 6:55 am   #3
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Default Re: Resistor puzzle

1M 2M & 500k.

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Old 17th May 2018, 8:04 am   #4
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Default Re: Resistor puzzle

Quote:
Originally Posted by GrimJosef View Post
In the end a couple of the components seem to be 1.5% and 2.3% away from what I might have guessed and the third one is 7% away. Is that what you find ?
That depends on what you think the marked values are
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Old 17th May 2018, 8:05 am   #5
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Default Re: Resistor puzzle

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Originally Posted by crackle View Post
1M 2M & 500k.
But what are the actual values? The algebra is actually not so hard once you know the trick.
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Old 17th May 2018, 8:13 am   #6
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Default Re: Resistor puzzle

R8 is 2.23354 megohms, 1.5% above 2M2
R9 is 1.07018 megohms, 7% above 1M
R10 is 511.60 kilohms, 2.3% above 500k

Knowing a trick amounts to saying "the algebra is not so hard once you've done the hard bit (or someone else has done it for you)" .

Cheers,

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Old 17th May 2018, 8:34 am   #7
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Default Re: Resistor puzzle

Correct values, GJ, well done. Although the marked value for R8 is 2M, so it's slightly out of its marked 10% tolerance.

The trick is understanding that a delta and a star (or 'wye' if you prefer) are equivalent. You can use my measured values very simply to work out the values of each arm of a star network, then use the "well-known" star to delta conversion.
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Old 17th May 2018, 9:10 am   #8
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Default Re: Resistor puzzle

Ah, OK. I was working on E12 values for the fixed resistors but I knew that the pot would almost certainly be 500k, not 470k because, well, they just are. (It couldn't be much less than 500k because you wouldn't be able to measure 443k across it if it was. And it couldn't be the next value up - 1M - because R8+R9 has to be at least 1M7 and that's not low enough to deliver 443k when it's in parallel with 1M.)

Once I'd got the pot value I needed a few megs in parallel to get 443k. I found I had to raise the pot value a bit to get the other two results to converge on your measurements (one from below and one from above).

Now you come to mention it I had heard of the delta-star equivalence. But there's no way I'd have remembered it if I'd needed it in anger, so to speak.

Cheers,

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Old 17th May 2018, 9:39 am   #9
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Default Re: Resistor puzzle

Quote:
Originally Posted by GrimJosef View Post
Now you come to mention it I had heard of the delta-star equivalence. But there's no way I'd have remembered it if I'd needed it in anger, so to speak.
I have to confess that I had to look up the equation -- too lazy to work it out from first principles. I subsequently found a handy website that'll calculate it for you: https://www.petervis.com/electronics...alculator.html

It can be a really handy trick for checking resistors in circuit in these topologies.
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Old 17th May 2018, 10:54 am   #10
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Default Re: Resistor puzzle

I quite often have Quad II amps to service and among the things I have to do is check the resistor values (circuit here http://www.keith-snook.info/amplifie...-Schematic.pdf). All but two of them can simply be measured in situ provided you're prepared to wait a few seconds for the odd capacitor to charge up (and assuming those capacitors aren't leaky when cold).

But the feedback resistors R10 and R11 - a 100R and a 470R - form a delta with part of the output transformer secondary. Because this is pretty much short circuit to DC it's hard, in principle at least, to deduce the values of R10 and R11 from the three delta measurements. One of these is roughly zero and the other two are much the same (100R//470R = 82R5) revealing little about the indicidual values.

Fortunately, as long as we do measure close to 82R5 across the resistors, we're saved by three things - a) these low value carbon comp resistors rarely drift at all, b) if they do drift then they go high which so if we measure 82R5 it's never because one's gone high and one low and c) Quad took to fitting high-stab resistors to later amplifiers and these are even more stable than the low value carbon comps.

Cheers,

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Old 17th May 2018, 11:15 am   #11
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Default Re: Resistor puzzle

When I did my sums at school I had to show the workings out, otherwise it was basically considered as a fail.

Lawrence.
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Old 17th May 2018, 12:53 pm   #12
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Default Re: Resistor puzzle

OK, I'll bite...

Measurements were:

Across R8: 926K
Across R9: 770K
Across R10: 443K

Let the three resistors of the star be Ra Rb and Rc.

Ra+Rb = 926K (Eq1)
Ra+Rc = 770K (Eq2)
Rb+Rc = 443K (Eq3)

Rearranging Eq3 we have:

Rb = 443K - Rc

Substituting for Rb in Eq1 we have:

Ra + 443K - Rc = 926K

Simplifying:

Ra - Rc = 483K

Adding this to Eq2 we have:

2 Ra + Rc - Rc = 483K + 770K

so:

Ra = 626.5K

Therefore Rb = 299.5K and Rc = 143.5K.

Now for the star to delta conversion...

Let Z be (Ra x Rb) + (Ra x Rc) + (Rb x Rc)

Z = 320517.75 x 10^9

R8 = Z / Rc ~= 2233.6K
R9 = Z / Rb ~= 1070.2K
R10 = Z / Ra ~= 511.6K

QED (quite easily done )
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Old 17th May 2018, 1:13 pm   #13
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Default Re: Resistor puzzle

Thanks, that's better, makes more sense to me now.

Lawrence.
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Old 17th May 2018, 2:58 pm   #14
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Default Re: Resistor puzzle

I did it my wonky sums method.

R9 = (R8 measured+R9 measured+R10 measured)/2 = 1.0695meg.

R8 = ((R9 measured+R10 measured)2) - (R8 measured-R9 measured)
= 2.270meg.

R10 = ((R8 + R9)R10 measured)/(R8+R9)-R10 measured = 0.5107meg.


1.0695m + 2.2270m = 3.3395m.....3.3395m in parallel with 0.5107m = 0.4429m

Sort of a one off near enough

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Old 17th May 2018, 4:00 pm   #15
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Default Re: Resistor puzzle

I did it by looking up trader sheet 954, and I fixed the radio in half the time.

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Old 17th May 2018, 4:03 pm   #16
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Default Re: Resistor puzzle

So did I yesterday (208).

Lawrence.
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Old 17th May 2018, 4:28 pm   #17
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Default Re: Resistor puzzle

The trouble is that the trader sheet won't tell you if one of the resistors is out of spec, and sometimes it can be really tricky to unsolder stuff, like in the Regentone I did where several are impossible to get to behind the wavechange switches.
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