SDN 0012.01 – Dewar Wiring Parameters

1. Introduction

This document describes the general thermal and electrical requirements for dewar wiring. The intention is to provide criteria that will guide the design of specific wiring harnesses, cold-stationing, and the like. The discussion does not include anything relating to grounding or shielding.

The requirements are summarized in section 2, the calculations in section 3, and the proposed implementation in section 4. Various supporting details are provided in the appendices.

2. Basic Requirements

With a few exceptions, all the internal dewar wires run from an external warm connector to a cold component located somewhere inside the instrument. Most wires must penetrate both the passive shields and the driven (“active”) shield to get there.

Most of the wires carry very little current, and thus may have modest resistance. There are some exceptions, comprising wires for motor current and heater resistors, which can carry currents of 1 ampere or more.

In all cases, the wires cannot conduct significant amounts of heat into the cold structure. The critical requirement is for the optical bench assembly, where, by design, heat input is kept to a minimum, in order to reduce temperature gradients and time-variable gradients. Other sources of heat (heat leakage through motor mounts, radiation through dewar entrance window, etc.) are typically roughly a watt, if not less. The heat introduced through the wires should therefore ideally be much less – which implies heat per wire of roughly 1 mW. These very low heat inputs can be achieved if the wires are heat-sunk to the driven shield or some other external, cold structure; the allowable heat input into the shields is substantially greater. Specifically, since the other loads on the drive shield may approach 100 W, heat through the wires of several watts – or more – would be acceptable, so that heat input per wire can be ~10 mW.

In addition to limiting heat input, it is also important to make sure that there are no warm wires inside the optical bench, as these can produce excess background radiation.

3. Design Overview and Analysis

With these considerations in mind, one arrives at a concept for wire routing in which the wire runs from the dewar connector to a cold-station on the drive shield, and then to a connector on the optical bench. If the wire then penetrates the optical bench, the connector should be hermetic, both to ensure a good light seal and provide some additional cold-stationing. Practical considerations (instrument assembly and disassembly) suggest that the cold-station point on the shield should serve as a pass-through point, with connectors on either side.

In what follows, I also take into account the availability of teflon-insulated ribbon cable. NOAO has a stock of 30-gauge manganin ribbon cable and 28-gauge copper ribbon cable; 30 gauge copper cable is also readily available.

There are a variety of proven methods for cold-stationing wires. The approach that we have adopted for GNIRS is to use a small length of circuit board, over which traces corresponding to the individual wires are run. The back side of the board is a full layer of copper, which is contacted (screwed down) to the shield or other heat sink. The input connector is at one end of the traces and the output at the other. Although the circuit board is an insulator, the heat conduction through the board will exceed the conduction along the traces and the output connector will therefore be at a temperature close to that of the heat sink. Details of the design are discussed further below.

The main advantages of this design are that it is easily replicated if there are many wires to be cold-stationed, and that it allows a certain amount of redistribution of the wires – i.e. acts as a “patch panel”. The main disadvantage is limited current-carrying capacity.

The wiring then breaks down, in practice, into three sections: the wiring from the external connector to the cold-station, the cold-station itself, and the wiring from the cold-station to the connector on the bench.

3.1 Wire Analysis

For both sections of wire, we want to minimize heat flow down the wire while ensuring that the wire will carry the appropriate current or signals. There are three sources of heat going into the wire: the warm connection at one end, radiation incident on the wire, and heating of the wire by current passing through it. The two heat sinks are the radiation emitted by the wire and the cold connection at the other end.

Instead of attempting a global analysis, which would have a large number of free parameters, I looked at several limiting cases. Rough estimates suggest that wires of relatively modest length – 10 cm or so – in a 300K radiation field will equilibrate with the radiation rather than the temperatures of the end points. Therefore, I calculated temperature profiles along a semi-infinite wire tied to 60K, for several different cases:

·        Ambient radiation of 60K, 80K, 200K and 300K effective temperature

·        30 gauge manganin and 28 gauge copper

·        Teflon insulation assumed opaque and transparent to radiation

·        Current in the wire of 1, 10, 100 and 1000 mA.

The details of the calculations are presented in Appendix A. The reason for the two assumptions regarding the Teflon is that at wavelengths of interest (5 mm and beyond) the insulation has moderate transmission (Appendix B). Thus the actual situation will be bounded by the two cases.

The results are presented in the following section. Some supplementary calculations (also presented) were done to look at the temperature of the conductor in the case where the Teflon is considered opaque; these are also presented below.

The calculations do show that for copper wire at 60K (and especially heavier gauge wire), the distance to reach equilibrium is longer than typical wiring runs in the instrument (note that wires running several feet will be tied down at shorter intervals). For these cases, the heat flow from one end to the other may be significant; some calculations were done for these cases. The formulae for these calculations can also be found in Appendix A, although for the actual calculations a spreadsheet provided by Brooke Gregory was used.

3.2 Wire Calculations

3.2.1 Radiative Equilibrium

The tables below summarize the results of the radiative equilibrium calculations for the 80K and 300K cases. The “equilibrium length” is the distance at which the wire is within 1 degree of the equilibrium temperature; for a 60K ambient and low current this is effectively zero (as indicated). Values of this length >25 cm are extrapolations. Data for manganin at 1 A current are omitted since the nominal equilibrium temperature is 500K or hotter. The “temperature difference” is the amount above ambient that the wire or insulation surface gets to; this is due to resistive heating.

The first table gives results for an ambient radiation field with an effective temperature of 60K.

Wire Equilibrium – 60K Ambient Radiation

The second table gives the results for 80K. The main difference between the two is that at 60K the radiation field acts to cool the wire, whereas at 80K it is a net heat input except for the high-current cases.

Wire Equilibrium – 80K Ambient Radiation

The third table provides equivalent results for an ambient radiation field of 300K effective temperature, which is a worst case for wires running from the dewar shell to the cold-station point.

Wire Equilibrium – 300K Ambient Radiation

The results for 200K are not tabulated, as they are similar to those for 300K. The heat input due to the radiation field (low current cases) is about 1/3 that at 300K; this is because the equilibrium lengths are roughly 60% longer while the energy density of the radiation is about 1/5 that at 300K.

For the two low-temperature cases, the radiative heating of the wires is fairly small, although not completely negligible for copper in an 80K radiation field. Resistive heating, which goes as the square of the current, become noticeable for manganin at 10 mA current; for higher currents copper is more efficient. Aside from the heat input, the temperature increases are significant enough to be a cause for concern.

Based on these calculations, one should use copper for internal wiring (cold station to bench) only where currents will exceed 10 mA. The standard 28-ga copper ribbon cable should not be used for currents significantly over 100 mA. Since the equilibrium lengths are fairly long, the heat flow where both ends of the wire are at fixed temperature (3.2.3) should be considered.

Where the radiation field has an effective temperature of 300K, there are two main differences from the “cold” case. The first is that, since the radiation field couples efficiently to the wire, it is a source of heat input, which is now greater for the copper wire rather than less. The heat input from radiation is greater than resistive heating for the low-current cases. The second effect is that the radiative coupling is more efficient – equilibrium lengths are less and the temperature differences due to resistive heating are also less.

The results suggest that manganin wire can be used in this situation up to 100 mA current. Since actual radiation fields will be somewhat less (between this as the 200K case), the temperature rises will be greater (but net temperature will be less). The exact limiting current for manganin may not matter much, since currents in GNIRS tend to be small, or else several hundred mA or more (motors and heaters). Copper can be used up to 1 A, but this is clearly a hard limit (since heating goes as the square of the current).

One of the problems with the wiring to the dewar’s external connectors is pins frosting up because they are cooled by conduction. The area around the connector is clearly reasonably well represented by a 300K radiation field; in this case the manganin needs to be at least 3 inches long, while copper needs to be about a foot. The equilibrium length was not calculated for heavier-gauge copper wire, but it should scale roughly as the square root of the wire diameter. This implies a length of perhaps 15 inches for 24 gauge wire and 20 inches for 20 gauge wire. Clearly, pin cooling is a problem only for copper wires, and can be avoided if a “service loop” of sufficient length is left outside all shields. Note that cable between the floating shields will equilibrate at a temperature at or below 0C.

3.2.2 Insulation Gradients

If the insulation is opaque, it will equilibrate with the ambient radiation but there will be a temperature gradient between the conductor and the outside of the insulation. One can do a simple calculation recognizing that the scale of the wire diameter is much smaller than the scale on which things change along its length. The heat flow can then be considered as a one-dimensional problem.

If the heat generated per unit length in the conductor is defined as

q = rI²/A

where r, I, and A are the resistivity, current and wire cross-section respectively (see Appendices), then

dT/dr = -q/(2prk)

and the solution is

T = T_i + q/(2pk) * ln(r_i/r)

where Ti is the temperature at the surface of the insulation, ri. The logarithm typically has a value of roughly 1 for values corresponding to the inner boundary of the insulation. The value of k for Teflon is 2-3 mW/cm/K (Appendix C), so for resistive heating in the wire of (typically) a few mW/cm or less, the temperature gradient across the insulation will be under a degree, and can therefore be neglected.

3.2.3 End-to-End Conduction

If we consider only the case of conduction along the wire, neglecting radiation, then the solution is a simple second order polynomial (see Appendix A or Brooke’s spreadsheet). As shown in section 3.2.1, this case applies primarily to copper wires, particularly gauges heavier than that used in the ribbon cable, since the distance over which these equilibrate with ambient radiation are longer than typical wire runs.

Calculations were run for several gauges and current levels. Two situations were modeled: one where the wire is 30 cm long, and tied to 60K at one end and 300K at the other, and a second where the “hot” end is 80K and the wire length is 60 cm. Resistive heating goes as the square of the current, and the only cooling is conduction along the wire for this calculation. Conduction of the Teflon insulation was neglected; it would contribute somewhat (perhaps 15%) for manganin wire but has less than 1% effect on the copper.

In the case of the 80K radiation field or 80K end-point, the results are similar, to within a factor of 2 for the copper wires. This is because the equilibrium lengths are close enough to the 60 cm used for the non-radiative case that the temperature profile along the wire is about the same. The results also show that the heat input is not a strong function of wire size, although it does increase with increasing diameter for low currents. For higher currents, there is less resistive heating of larger wires and there is therefore actually less heat input for currents of 1A or more.

For the 300K radiation field, the equilibrium lengths are relatively short, so the conduction model underestimates heat input. For the manginin wire the additional effects of radiation increase heat input by a factor of 10 or more, while for the copper wires, the effect goes from a factor of 4 at 28 gauge to 2 at 20 gauge. For 20 gauge copper wire the heat input per wire probably approaches 0.4 W, so one can’t afford to use very many.

3.3 Cold-Stationing

The cold-stationing of the wires are modeled using a program call “FlexPDE” which is available from PDE Solutions, Inc. An educational version (FlexPDE Lite) of the program is available without charge; it can handle 2-D problems with a limited number of nodes. Its limitations are offset by the fairly minimal learning curve involved. All calculations were carried out with version 2.14g of the program.

The cold station was modeled as a sandwich, where there is a layer of G-10 between two layers of copper. The top layer is supposed to simulate a conductor, and its conductivity was reduced by a factor of 2 to simulate a 50% fill factor for the wire traces. One end was set to 300K. Part or all of the bottom layer was set to 60K. The purpose of the calculations was to determine what length of trace was required to get the cold end close to 60K.

The case shown below is for a 3 cm trace length. The thickness of the G-10 layer is 1.6 mm. The thickness of the copper layers is 0.036 mm (see Apendix C). In order to visualize what is happening in the copper layers, their thickness in the model was increased by a factor of 20 and the thermal conductivity was decreased by the same factor. An additional decrease in conductivity of a factor of 2 was used for the top layer to simulate the 50% fill factor. This results in correct gradients along the traces (horizontally) but substantially exaggerates any vertical gradients, which are seen to be small in any case.

The calculation assumes the leftmost 1 cm of the trace on top is held at 300K, while the entire lower layer is held at 60K. If one assumes that the cold connection point is near the end of the trace, it would be at or below 80K.

This model is clearly a worst case; rough estimates of the heat conducted through the G-10 for reasonable trace widths (1 mm or so) are close to a watt for the specified temperatures. The wire calculations show that actual heat flow is substantially less, except for short, heavy gauge copper. What this means is that, in practice, the “300K” end of the cold station will be significantly colder, and the cold end will be correspondingly closer to 60K.

This design is clearly adequate for all but high-current applications. It is worth asking what the limiting current set by the traces is. For a thickness of 0.036 mm and a width of 1 mm, the cross-section of 0.036 mm² corresponds to a wire gauge of between 31 and 32. This is clearly more than enough for cold-stationing manganin wires. To cold-station 28-gauge copper wires with an equivalent cross-section, a trace width of about 2.2 mm is needed; with a 50% fill factor the center to center spacing is about 4.5 mm (0.175 in). However, the traces are coupled to the circuit board, and the heat generated by 1 A in 3 cm of 1 mm traces is fairly small (about 14 mW), so one can probably use traces narrower than 2 mm for applications using 28-gauge copper.

The model assumes continuous contact of the bottom layer with the cold sink. This is rather difficult to do in practice. What one needs to do is ensure that the path along the bottom layer to a contact point is shorter than the trace length. The calculations suggest that (a) contact near the cold end of the trace is especially important and (b) a good rule of thumb might be to add the trace-to-contact distance to the trace length. A large circuit board mounted at the four corners will clearly not be effective; one probably wants a contact point adjacent to the ends of the traces for every connector (possibly even on both sides for large connectors). If this is not possible, one needs to ensure that the traces are lengthened appropriately. (One can clearly relax this for manganin wires, which cannot carry enough heat to maintain the “hot” end at 300K.)

For higher currents, the trace width or thickness would appear to be increasingly unwieldy. To carry 5 A, one would need traces at least 5 mm wide, ideally close to 10 mm. Use of a thicker copper layer would help, but the length of the traces would also have to increase, which would double the resistive heating. A careful calculation of the currents to be carried is needed to see whether a “heavy-duty” cold-station can be constructed.

An alternative for high-current applications is simply clamping individual wires. Conduction through the Teflon insulation is actually fairly efficient (3.2.2), so the main factor is likely to be conduction across the clamped contact. A worst case assumption is that the contact resistance is very high, so that heat transfer is purely by radiation. In this case, calculations similar to those in 3.2.1 with the wire end fixed at 300K and the ambient radiation at 80K indicate equilibrium lengths for 20 and 24 gauge copper of over a meter, though cooling to ~200K occurs within the first 10-15 cm. Therefore, it is important to get a good contact; this is probably better achieved by squeezing down hard on insulated wires than by alternative schemes where one clamps bares wires more gingerly between non-conducting surfaces. The calculations for the G-10 circuit board showed that a 3 cm length was adequate; the actual length over which the trace cools is about 2 cm. A 20 gauge wire has a diameter which is more than 20x the thickness of the circuit board trace; the Teflon insulation is (perhaps) 1/5 the thickness of the circuit board. Clamping provides contact on more than one side, but the thermal conductivity of Teflon is somewhat lower than that of G-10. Scaling accordingly, and assuming good thermal contact, one concludes that the effective length of the clamp needs to be about 4x longer than the length of trace used for cooling on the circuit board, or about 8 cm. Clamps for 24 gauge wire could be somewhat shorter.

4. Implementation

4.1 Low Current

For low current implementations, as noted above, one should use 30 gauge manganin wherever possible. For wires requiring low resistance, 28 gauge copper is acceptable. The heat input into the driven shield is roughly 7x higher (see 3.2.1) and so use of doubled-up manganin wires is somewhat preferable. The heat input down 1000 manganin wires is estimated to be over 10W (maximum 17W), so the cold stationing of the wires should be done close to attachment points for the thermal distribution system from the cryocoolers (or some of the latter should be located near the cold-station points).

The cold-station design outlined above is more than adequate; the minimum distance for cooling along the trace should be 2 cm or greater, assuming 1 oz/ft² copper; a somewhat greater length will accommodate a lower density of cold contact points.

4.2 High Current

There are three different high-current implementations: heater resistors for bench temperature control, heater resistor for instrument warm-up, and motor current wiring. The considerations are somewhat different in each case.

Bench Temperature Control. These heater resistors are located at or close to the cryocoolers, and there is little point in cold-stationing the wires, since any heat input must be taken out by the cryocoolers anyhow. Similarly, additional resistive heating in the wires is not a concern since one is driving far larger heat sources anyhow. The current required will be in the range 2-4 A (SDN007.07), which indicates 20 gauge wire is needed.

Bench Warm-Up. These heater resistors are located on the bench, and so the wires leading to the heater circuits must be cold-stationed (see below). During normal operation, the heaters are off, so resistive heating can be neglected. During warm-up, resistive heating can be significant, and the concern is over-heating of the wires. The worst case would arise when the instrument is approaching room temperature, but the heaters are still at full power. If one considers a wire length of 60 cm, the maximum current on a 20-gauge wire is about 2.5A, neglecting radiation; the temperature at the mid-point is about 70C. For 24-gauge wire, the maximum is 1A current. If the wire length is reduced, the allowable current increases as the inverse of the wire length. Since the heaters appear to require currents close to 2A (SDN007.07), 20 gauge wire would appear to be required. If we imagine that the heater circuits use a total of 6 pairs of 20 gauge wire, with a 30 cm distance from the connector to the cold station, the heat input to the driven shield will be ~4 W.

Cold-stationing should be by means of clamps on the driven shield, unless currents are low enough to permit a heavy-duty circuit board and this is more attractive to fabricate and install.

Motor Power. The motors are isolated from the bench structure, with about 90% efficiency. During operation (when the wires are carrying current) the motors themselves will dissipate far more power than the wires, so the only concern with current is excessive heating. So long as current is 1A or less, which is expected, 28 gauge copper is adequate. When the motors are off the heat input per wire is 2 mW or less, and the motor isolation should reduce it still further. The total heat input into the bench for all the motors should therefore be 100 mW or less.

Use of the standard circuit board design, with perhaps somewhat wider and longer or thicker traces (see 3.3), should suffice for cold-stationing. One should see what the widest traces are that can be conveniently laid out with a fill factor around 50%; if 2 oz/ft² copper appears necessary the effective trace length needs to go to ~4 cm or the G-10 needs to be made thinner.

Appendix A – Equilibrium of a Wire in a Radiation Field

If we consider a long, thin wire in a radiation field, the equation describing temperature vs. position is given by:

kA d²T/dx² = -q                                                    (A.1)

where k is the thermal conductivity of the wire, A is its cross-section, and q is the heat source. For the case where there is resistive heating of the wire and an ambient radiation field,

q = rI²/A + pDs(T_a⁴ – T⁴)                                   (A.2)

where r is the electrical resistivity, I is the current in the wire, D is the wire diameter, s is the Stefan-Boltzmann constant, and T_a is the effective temperature of the ambient radiation. A.1 can then be written as

d²T/dx² = -q₁ + q₂T⁴                                                             (A.3)

If we then write

p = dT/dx                                                                              (A.4)

then

p dp/dT = -q₁ + q₂T⁴                                                            (A.5)

which integrates to

p²/2 + q₁T – q₂T⁵/5 = c₁                                      (A.6)

or

dT/dx = Ö(-2q₁T + 2q₂T⁵/5 + c₁)                       (A.7)

(This assumes temperature is increasing; the sign of A.7 can be negative.) If the radiative terms are set to zero, this can be further integrated to get

T = -q₁x²/2 + a₁x + a₂                                                         (A.8)

where now q₁ = rI²/kA² and the constants a₁x and a₂ are determined by the boundary conditions – presumably the temperature of the end-points of the wires. Where the radiative terms are not neglected, though, the integral is not so easy; since I didn’t find an analytic solution right away I chose to integrate numerically. The starting point is at x = 0 and a specified T(0), but is it also necessary to define the constant c₁ in A.7. This I did by assuming that at infinite distance both the net heat input and dT/dx are zero. (Perhaps this is over-specifying the problem, but one can also view it as specifying dT/dx at a particular value of T.) This “equilibrium” value of T, obtained by setting q to zero (A.2), I call T_i.

 The solution where the temperatures of the wire end-points are specified implies a different value of the constant c₁; for those cases where the wire temperature reaches a maximum somewhere between the endpoints, dT/dx will reach zero and then A.7 will change sign. One would have to determine the correct value of the constant by iteration for each set of boundary conditions. I have not attempted this.

 The numerical integrations were done in increments of 0.001 cm. The integration was cut off at 25 cm, so results in the text for longer lengths are extrapolations. Wire properties were taken from Appendix C.

 Relevant results are summarized in the body of the text.

Appendix B – Transmission of Teflon

 There is a fairly extensive literature on the transmission of Teflon. However, many of the measurements were intended to describe the absorptions and were done with samples of different thicknesses over different wavelength regions (see Liang and Krimm, 1956, J Chem Phys 25, 563 for a typical case with good wavelength coverage). Dick Joyce has run transmission spectra from 2.5 – 20 mm for three samples, which are presented below.

The thickness of the insulation on the ribbon cable is roughly 0.24 mm, so below 6 mm the transmission is fairly high. Most of the energy in a 300K blackbody is emitted between 5 and 40 mm, and over this wavelength region Teflon is fairly opaque (see Liang and Krimm 1956 for data on thinner samples). At still longer wavelengths, Teflon becomes increasing transparent, though even for a 60K blackbody, the effective opacity is probably still around 50%.

 The wire properties used for the calculations are listed here rather than in a separate design note.

 Material Properties

 The properties tabulated are average values over the temperature range 60-300K. For some calculations the temperature dependence of thermal conductivity for Teflon and G-10 was used. The formulae used were:

 k(Teflon) = 0.0013 + 0.00062(T/100)  W/cm/K

k(G10) = 0.0018 + 0.0018(T/100)                         W/cm/K

 These values are accurate to perhaps 30%.

 Wire Dimensions

 The standard teflon-coated manganin ribbon cable is 30 gauge (conductor 0.255 mm diameter, 0.0503 mm² area). The copper ribbon cable is 28 gauge (conductor 0.321 mm diameter, 0.0804 mm² area). The form of the ribbon cable is wires coated with insulation with a connecting strip between wires. The diameter of the coated wires for the 30 gauge cable is 0.73 mm, which indicates an insulation thickness of ~0.24 mm. This was assumed to apply to the 28 gauge cable as well. The dimensions for stranded wire are somewhat larger; the actual cross-sections of copper are 10-20% higher and there fill factor of the strands is also less than 100%. The differences in performance between single-conductor and stranded wire are less than other uncertainties.

 For the purposes of all calculations the individual wires were assumed to be isolated – the connections between conductors in the ribbon cable were ignored, as well as any effect on viewing factors for radiation, etc.

 Circuit Board Information

 Circuit boards (for cold-stationing) can be made up with in a variety of properties. For simplicity, the calculations considered fairly standard properties.

 A standard copper layer of 1 oz/sq ft corresponds to a layer thickness of 0.036 mm.

 A standard thickness (2-layer board) is 1/16 inch, or approximately 1.6 mm.

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