In the previous posts we discussed the topology selection as well as the power handling capabilities of the decade resistor. Now it’s time to have a look at different aspects of switching. Today it’s all about the effect of the contact resistance of the relays, which impacts the performance of the decade resistor, not only, but especially when a decade is shorted.
Bypassing upper decades
In order to improve the accuracy of the programmable decade resistor at low resistance values, additional switches can be introduced. These switches would bypass all higher decades that otherwise would be in series (shorted), and therefore reduce the error caused by contact resistance. For the intended application it’s probably not justified to add such bypass switches after decades having resistance values \(\geq 10\text{ k}\Omega\), because then
- the rather low contact resistance of the relays (typically much below \(100\text{ m}\Omega\)) is well below the tolerance of the resistors
- the number of contacts bypassed is fairly low
With these considerations I decided to use such bypasses after each of the three lower decades, increasing the total number of relays by 3. Two of those relays (K7, K14) can be found in the excerpt of the schematics (see below).
Shorting a decade
When shorting the first decade (i. e. selecting \(0\Omega\)) the simplest approach would be to just close relays K1, K2 and K7 (see schematics below). However, the resistance could be reduced by paralleling one or more taps of the resistor network. But how large would the effect be? And what are the disadvantages?
The excerpt of the schematic shows the first two decades. First let’s see the results for the lowest decade. Assuming ideal resistances for all resistors and a typical contact resistance of \(40 \text{ m}\Omega\) for each relay (this value is for two contacts in parallel), the following improvements could be achieved:
| Relays closed | Resistance (\(\Omega\)) | Absolute difference (\(\Omega\)) | Relative difference (ppm) |
|---|---|---|---|
| K1, K2, K7 | 0.1184615385 | <baseline> | <baseline> |
| K1, K2, K3, K7 | 0.1176923077 | -0.0007692308 | -6493.506 |
| K1, K2, K3, K4, K7 | 0.1176920230 | -0.0007695154 | -6495.909 |
| K1, K2, K3, K4, K5, K7 | 0.1176920230 | -0.0007695155 | -6495.910 |
| K1, K2, K3, K4, K5, K6, K7 | 0.1176920230 | -0.0007695155 | -6495.910 |
Whereas closing relay K3 – can reduce the resistance by 0.65%, the returns from closing even more relays deminish almost instantly (due to the series resistance in the additional relay’s signal path). For the higher decades improvements would be only about 1/10, 1/100, 1/1,000, 1/10,000 and 1/100,000 of the values shown above.
On the other hand, closing a relay not only causes more switching cycles, but also increases the power consumption for which the power supply has to be built for. More importantly, the power dissipation would heat up the device just a little bit more and therefore potentially cause a certain drift error of the resistors. This, however, is a general problem that could be mitigated by using latching (“bi-stable”) relays, although at the cost of a more complex driving circuitry.
To put this into perspective: The variation in contact resistance (especially over switching cycles/time) makes this discussion almost pointless. Nevertheless, if someone really wanted to, a somewhat reasonable approach might be to close one more relay for the first decade (obviously the most effective one, in the example above: relay K3). This would bring the largest effect and ensure that closing any other relay would yield only minimal improvements of the decade’s short circuit resistances across the board. However, I won’t implement this feature because I think the disadvantages outweigh the tiny advantage.