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• Topology (2)

  Sebastian • 05/09/2024 at 22:18 • 0 comments

  In the last post we had a look at a simple topology that allows us to switch between a series and a parallel connection of two resistors:

  Now we nest these structures so that we have four resistors:

  The interesting part here is that two of the four switch states of the outer structure don’t provide any new resistance values: With S1a open and S1b bypassing R1b we’re left with R1a. (Again, this assumes the R1a = R1b.) If S1a is closed and S1b in the upper position, then the total resistance will be 0, exactly like with only R1a. Here I assume ideal switches. The series and parallel connection of both R1a and R1b alone is worth it though.

  Doing that once more and we get eight resistors, barely visible in this picture, but with a similar effect like with the previous nesting.

  With a total of 14 switches there are 2^14 = 16384 switch states. As briefly discussed, many switch states might lead to the same resistance values (actually, by far not only the two examples described above). In fact, as we will see later, much more than half of the switch states result in a short circuit.

  However, states with equal resistances might not be equally suited for the intended application, because in some cases they differ in the power rating we can achieve. But even then, for certain resistance values there might be multiple states with the same resistance and power rating, but with a different number of active switches. This could minimize the contact resistance (but this is not necessarily the case with the dual throw switch) and it might be a good idea to energize as few relays as possible.

  This begs the following three questions:

  • How many and which distinct resistance values can be selected?
  • What power rating can we achieve per resistance value?
  • What is the best switch combination (here: least number of relays energized for the highest power rating) to realize each of the selectable resistance values?

  To answer this I wrote a small python script that calculates these things for us. I’ll assume all resistors are 10 Ω with a 50W power rating. The results are presented in the following table.

  #	Resistance	Number of Valid Switch Combinations	Maximum Power	Energized Switches
  	(Ω)		(W)
  1	0	9472	nan	4
  2	1.25	1	400	14
  3	1.428571	4	350	13
  4	1.538462	4	325	12
  5	1.666667	18	300	11
  6	1.818182	12	275	11
  7	2	48	250	10
  8	2.142857	4	233.333333	11
  9	2.222222	50	225	9
  10	2.272727	4	220	10
  11	2.307692	4	216.666667	9
  12	2.352941	2	212.5	8
  13	2.5	209	200	7
  14	2.727273	8	183.333333	10
  15	2.857143	60	175	8
  16	2.941176	8	170	9
  17	3	8	166.666667	8
  18	3.076923	4	162.5	7
  19	3.157895	8	158.333333	9
  20	3.333333	384	150	6
  21	3.448276	8	145	8
  22	3.529412	8	141.666667	7
  23	3.636364	4	137.5	6
  24	3.75	28	133.333333	8
  25	4	370	125	5
  26	4.166667	28	120	7
  27	4.285714	28	116.666667	6
  28	4.444444	14	112.5	5
  29	4.615385	8	108.333333	8
  30	5	933	400	6
  31	5.263158	8	95	7
  32	5.454545	8	91.6666667	6
  33	5.714286	4	87.5	5
  34	5.833333	4	262.5	11
  35	6	32	187.5	7
  36	6.5	4	203.125	10
  37	6.666667	356	300	5
  38	7.142857	32	280	6
  39	7.333333	8	229.166667	9
  40	7.5	50	266.666667	5
  41	8	20	250	4
  42	8.333333	28	166.666667	8
  43	8.571429	28	131.25	6
  44	9	28	180	7
  45	9.166667	4	103.125	9
  46	9.375	8	120	7
  47	10	1074	200	3
  48	10.666667	8	120	7
  49	10.909091	4	103.125	5
  50	11.111111	28	180	5
  51	11.666667	28	131.25	6
  52	12	28	166.666667	4
  53	12.5	20	250	6
  54	13.333333	50	266.666667	7
  55	13.636364	8	229.166667	5
  56	14	32	280	6
  57	15	356	300	4
  58	15.384615	4	203.125	4
  59	16.666667	32	187.5	5
  60	17.142857	4	262.5	3
  61	17.5	4	87.5	9
  62	18.333333	8	91.6666667	8
  63	19	8	95	7
  64	20	933	400	2
  65	21.666667	8	108.333333	6
  66	22.5	14	112.5	7
  67	23.333333	28	116.666667	6
  68	24	28	120	5
  69	25	370	125	3
  70	26.666667	28	133.333333	4
  71	27.5	4	137.5	8
  72	28.333333	8	141.666667	7
  73	29	8	145	6
  74	30	384	150	2
  75	31.666667	8	158.333333...

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• Topology (1)

  Sebastian • 04/18/2024 at 20:15 • 0 comments

  In this application the focus is on power handling capability, not so much a high resolution (high number of resistance values) or precision. This is why the topology used in the Programmable Precision Resistor is not as suitable for this application as it has been before. Also my ELMA cases are really small (and I mean really small, so thermal considerations will be challenging), so I can’t simply increase the number of power resistors and relays in the process. That’s why I’ll stick to the original plan of using 8 power resistors.
  

  Instead the idea is to find a topology where I can select different resistance values by either connecting the resistors in series or in parallel. Let’s first have a look at a circuit for only two resistors.

  
  

  For this design I assume that both resistors are identical, both in their resistance value R as well as their power rating P. It’s very obvious how this circuit works, but here are the current paths for the different switch states anyway:
  

  Since the math behind this is really simple, here are the results without any further explanation.

  S0a	S0b	R0	P0	I0	U0
  0	0	2R	2P	√(P/R)	2R√(P/R)
  0	1	R	P	√(P/R)	R√(P/R)
  1	0	0	unlim.	unlim.	0
  1	1	R/2	2P	2√(P/R)	R√(P/R)

  And now we have our basic building block that we’ll use in the next post of this series.

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