Programmable Power Resistor: Topology (2)

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
109472nan3254
21.2514001638314
31.42857143501612713
41.53846243251561512
51.666667183001279911
61.818182122751555111
72482501273510
82.1428574233.3333331305511
92.22222250225124159
102.27272742201254310
112.3076924216.666667124799
122.3529412212.5123518
132.520920010877
142.7272738183.3333331305110
152.85714360175127238
162.9411768170125399
1738166.666667124758
183.0769234162.5123477
193.1578958158.333333130439
203.33333338415010836
213.4482768145125318
223.5294128141.666667124677
233.6363644137.5123396
243.7528133.333333127478
25437012510755
264.16666728120124877
274.28571428116.666667124236
284.44444414112.5122955
294.6153858108.333333130428
305933400154086
315.263158895125307
325.454545891.6666667124666
335.714286487.5123385
345.8333334262.5383911
35632187.5130407
366.54203.125332710
376.666667356300124005
387.14285732280125286
397.3333338229.16666733239
407.550266.666667124645
41820250123364
428.33333328166.6666675078
438.57142928131.25123636
449281804997
459.1666674103.12532639
469.3758120124917
4710107420010723
4810.666667812032517
4910.9090914103.125122995
5011.11111128180123555
5111.66666728131.254986
521228166.666667123544
5312.520250124836
5413.33333350266.66666731317
5513.6363648229.166667124195
56143228031236
5715356300124184
5815.3846154203.125122914
5916.66666732187.531225
6017.1428574262.5122903
6117.5487.57679
6218.333333891.66666677638
63198957557
6420933400122882
6521.6666678108.3333337546
6622.514112.51277
6723.33333328116.6666671236
6824281201155
69253701252083
7026.66666728133.3333331144
7127.54137.52558
7228.3333338141.6666672517
732981452436
74303841501442
7531.6666678158.3333332425
7632.54162.51917
7733.3333338166.6666671876
783481701795
793588175754
8036.6666678183.3333331784
8140209200161
8242.52212.5636
8343.3333334216.666667595
84444220514
85455022573
8646.6666674233.333333503
87504825062
885512275113
89601830011
9065432532
9170435021
9280140000
Different (ideal) resistance values that can be achieved with the topology

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.


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