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’ll 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 \(S_{1a}\) open and \(S_{1b}\) bypassing the \(R_{1b}\) we’re left with \(R_{1a}\). Again, this assumes the \(R_{1a} = R_{1b}\). If \(S_{1a}\) is closed and \(S_{1b}\) in the upper position, then the total resistance will be 0, exactly like with only \(R_{1a}\). Here I assume ideal switches. The series and parallel connection of both \(R_{1a}\) and \(R_{1b}\) 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.

Because there are 14 switches there are \(2^{14}=16,384\) different 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’s generally 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?
• Which power rating can we achieve per resistance value?
• What is the best switch combination (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 \(10 \text{ }\Omega\) with a \(50\text{ W}\) power rating. The results are presented in the following table. Let’s have a look at the columns first, as some might require some additional explaining:

• Number of valid Switch combinations: How many different switch combinations achieve the same resistance value (not necessarily the same power dissipation capability)
• Maximum Power: Calculated power rating
• Binary Encoded Switch combination: A binary-encoded switch combination that uses the lowest number of energized relays (for use in the software)
• Energized Switches: Minimum number of switches that have to be energized to achieve the specified resistance value

#	Resistance (\(\Omega\))	Number of Valid Switch Combin.	Maximum Power (\(\text{W}\))	Binary Encoded Switch Combin.	Energized Switches
1	0	9472	nan	325	4
2	1.25	1	400	16383	14
3	1.428571	4	350	16127	13
4	1.538462	4	325	15615	12
5	1.666667	18	300	12799	11
6	1.818182	12	275	15551	11
7	2	48	250	12735	10
8	2.142857	4	233.333333	13055	11
9	2.222222	50	225	12415	9
10	2.272727	4	220	12543	10
11	2.307692	4	216.666667	12479	9
12	2.352941	2	212.5	12351	8
13	2.5	209	200	1087	7
14	2.727273	8	183.333333	13051	10
15	2.857143	60	175	12723	8
16	2.941176	8	170	12539	9
17	3	8	166.666667	12475	8
18	3.076923	4	162.5	12347	7
19	3.157895	8	158.333333	13043	9
20	3.333333	384	150	1083	6
21	3.448276	8	145	12531	8
22	3.529412	8	141.666667	12467	7
23	3.636364	4	137.5	12339	6
24	3.75	28	133.333333	12747	8
25	4	370	125	1075	5
26	4.166667	28	120	12487	7
27	4.285714	28	116.666667	12423	6
28	4.444444	14	112.5	12295	5
29	4.615385	8	108.333333	13042	8
30	5	933	400	15408	6
31	5.263158	8	95	12530	7
32	5.454545	8	91.6666667	12466	6
33	5.714286	4	87.5	12338	5
34	5.833333	4	262.5	3839	11
35	6	32	187.5	13040	7
36	6.5	4	203.125	3327	10
37	6.666667	356	300	12400	5
38	7.142857	32	280	12528	6
39	7.333333	8	229.166667	3323	9
40	7.5	50	266.666667	12464	5
41	8	20	250	12336	4
42	8.333333	28	166.666667	507	8
43	8.571429	28	131.25	12363	6
44	9	28	180	499	7
45	9.166667	4	103.125	3263	9
46	9.375	8	120	12491	7
47	10	1074	200	1072	3
48	10.666667	8	120	3251	7
49	10.909091	4	103.125	12299	5
50	11.111111	28	180	12355	5
51	11.666667	28	131.25	498	6
52	12	28	166.666667	12354	4
53	12.5	20	250	12483	6
54	13.333333	50	266.666667	3131	7
55	13.636364	8	229.166667	12419	5
56	14	32	280	3123	6
57	15	356	300	12418	4
58	15.384615	4	203.125	12291	4
59	16.666667	32	187.5	3122	5
60	17.142857	4	262.5	12290	3
61	17.5	4	87.5	767	9
62	18.333333	8	91.6666667	763	8
63	19	8	95	755	7
64	20	933	400	12288	2
65	21.666667	8	108.333333	754	6
66	22.5	14	112.5	127	7
67	23.333333	28	116.666667	123	6
68	24	28	120	115	5
69	25	370	125	208	3
70	26.666667	28	133.333333	114	4
71	27.5	4	137.5	255	8
72	28.333333	8	141.666667	251	7
73	29	8	145	243	6
74	30	384	150	144	2
75	31.666667	8	158.333333	242	5
76	32.5	4	162.5	191	7
77	33.333333	8	166.666667	187	6
78	34	8	170	179	5
79	35	88	175	75	4
80	36.666667	8	183.333333	178	4
81	40	209	200	16	1
82	42.5	2	212.5	63	6
83	43.333333	4	216.666667	59	5
84	44	4	220	51	4
85	45	50	225	7	3
86	46.666667	4	233.333333	50	3
87	50	48	250	6	2
88	55	12	275	11	3
89	60	18	300	1	1
90	65	4	325	3	2
91	70	4	350	2	1
92	80	1	400	0	0

Since the selected topology does not feature the option to open the circuit, an additional series-connected relay is required. This relay is not included in the total number of energized relays.