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# Astronomy Calculations

## Gearbox ratio of focuser

When designing a stepper motor focuser there are a lot of parameters that you have to think about.

What is the needed step resolution from the stepper motor, how much do you need that one step move the focuser draw tube?

It has to be designed to use the full stroke (in most cases) of the focuser draw tube, that is the range of steps that shall cover the focuser from inward to outward position.

Here I try to sort it out:

It start to calculate as soon you change or write new figures in the white or dark red boxes. Do not exceed the maximum number of characters, delete characters if necessary.

Note:
You use the information on your own risk! There can always be a mistake in my equations behind the calculations, check that the result is correct. Let me know if you find something wrong and I try to correct it. Some calculations are very simple done and not correct in the small details.

### 1, Focuser draw tube stroke length?

First we have to find out how many turns do we need from the focuser knob to go from inward to outward position. You shall normally use the direct connected knob, not the built in 10:1 reducer. One of the goals of this design is to use most of the focus draw tube stroke length as long it gives enough of resolution per motor step. Lately I have also added a rough estimate of how much torque that is needed.

Type in the length of the focuser stroke:

Focuser stroke length
(0 to 200 mm)

### 2, The diameter of the friction shaft to the focuser?

Most common is that the focuser has an shaft that by force act on a steel plate, no gears. Then use the shaft diameter. Some focuser has a rack and pinion system to get out of the problem that the friction focuser can slip.

 Shaft diameter (0.01 to 9.99 mm) A 6 mm diameter shaft gives about 18.8 mm translation of focus per rotation. Number of teeth along focus stroke length (10 to 200) Number of teeth of gear (6 to 40)

(you have to uncheck Friction focuser before checking Rack & Pinion focuser)

### 3, Calculated number of turns of the 1:1 focus knob:

= Calculated results number of turns to cover the full focus draw tube stroke length

### 4, Stepper motor data, number of step per revolution and torque?

4a, How many steps per revolution.

Some motors have included a built in gearbox. Set the number of steps related to the output shaft, the built in gearbox ratio can be difficult to find.

Most common is 200 steps per revolution (1.6 degree step) and no gearbox.

Number of steps
(10 to 999 steps)

4b, Torque of stepper motor or output shaft?

Note:
If you micro step the motor the torque is much lower. It depends on the voltage too.

Ncm
(0.1 to 99.9 Ncm)

### 5, Maximum number of  steps ?

We also need to know the maximum number of steps the stepper motor driver can count to. Most common is 16 bit or 2^16 = 65536 steps.

Number of steps full range
(0 to 9999999 max steps)

### 6, Number of micro steps per main step?

Some drivers can use micro steps, from 1/1 to 1/64. Normally 1/1 or 1/2, above that will not be good, the motor lose torque and can not hold the position without power.

Number of micro steps
(1, 2, 4, 8, 16, 32, 64 micro steps)

### 7, Needed gearbox ratio:

= Calculated results : 1 needed gearbox ratio to use the full focus draw tube stroke.

This gear ratio gives the following torque on the gearbox output shaft:

= Calculated results Ncm, torque gearbox output shaft.

7a, Your choice of gearbox ratio:

Example, if you get a calculated gear box ratio above of 27.6, try to find a gearbox above that, not below, say 30:1.

Gearbox ratio
(1 to 150 ratio)

### 8, Draw tube movement per step:

= Calculated results micrometer move per full step or micro step

You need something like five times better resolution then your focus depth of your optical system. See here how to find it, Depth of focus calculator. A f/4 optical system at green wavelength has a focus depth of +/- 9 micron or +/- 5 micron for blue wavelength, a reasonable resolution then about 1 to 2 microns per micro step. Bad quality optics is less sensitive of the focus point.

Note:
These are theoretical calculations, check carefully it's correct!

Draw tube length used:

= Calculated results mm used stroke Calculated results % of Calculated results mm full stroke

Try to use as close to 100 % percent but not over of your focus tube draw length, you don't want your focus motor smash into the stop.

If you get too high value in the calculations above you can limit the stroke length of the draw tube, say that you have 100 mm focus stroke length, then only use the first 50 mm. Type in that value and a new gear ratio will be calculated and then the resolution changes. Note you can only do this if you have your optics in focus in the range 0 to 50 mm. You can also try to find a driver that can handle higher values then 16 bit system can do or a stepper motor with different step size.

Normally there is no problem to use a 100 mm focus stroke on a f/7 telescope and still have resolution enough with a correct designed gear ratio. But a f/3 telescope of high quality and calculated for the blue spectra are on the limit, that kind of telescopes are often of Newton design and have shorter focus stroke.

If you are not limited by 16 bit resolution in the driver you can use higher gear ratios and get better precision. As a bonus that system give more torque that can handle heavier cameras. A bit slower due the higher gear ratio.

### 10, Needed torque:

Note:
This section is new, it can be something wrong in it!

This is important, you must have enough torque out from your motor focuser. Heavy equipment attached to the focuser needs a lot of torque.

This is what you get from your motor focuser on the output shaft:

= Calculated results Ncm, torque gearbox output shaft.

Estimation of the needed torque:

Mainly you have two things that the torque must overcome, the friction and the weight of the moving parts that are connected to your focuser. The latter depends of the angle the focuser pointing too, 90 degree (zenith) = all of the weight, 0 degree = nothing.

Note:
With a refractor the focuser point at the same direction as the telescope, for a Newton not.

 10a, Friction torque: Ncm (1 to 50 Ncm) It's not so difficult to find this friction torque. This is the unwanted friction in the focuser that you have to overcome. Place the focuser horizontally, disconnect the motor focuser mechanically, attach a 50 cm long arm on the focuser shaft. Then you can use a letter balance, push it against the outer part of the arm and see how much force you need to move the focuser, test in both directions. Say that you read a weight of 20 gram on the letter balance. Express it in kilogram, 20/1000=0.02 kg. Multiply it by the G constant 9.81 and then multiply it by the length of the arm to have it in Ncm. 0.02*9.81*50 = 9.81 Ncm. The friction torque does not depend on the angle focuser is pointing to.
 10b, Focuser total weight: Kg (1.0 to 15.0 kg) Add the weight of all your equipment attached to the focuser, focuser draw tube, adapters, field flattener, filter wheel, camera etc. The weight torque does depend on the angle focuser is pointing to.
= Calculated results Ncm, focuser axis torque when focuser pointing to zenith.

10c, Focuser altitude pointing angle:

Here we calculate how the needed torque change with the altitude direction the focuser point at:

Degree
(0 to 90 Degree)

The needed torque is different when moving the focuser inwards or outwards when the focuser is tilted.

= Calculated results Ncm, friction and weight, focuser moving inwards.

= Calculated results Ncm, friction and weight, focuser moving outwards.

Compare to what torque you have on the focuser shaft from the stepper motor, see at point 10b.

This a very rough estimate of the needed torque, to be on the safe side, have at least four times the maximum torque you get here. Try to do some practical test before you buy expensive parts.

### My own experience:

I have two times the calculated value and it's not enough.

It can be very confusing when reading data document about stepper motor, even more confusing when the stepper motor and gearbox is sold as one unit. The torque are normally what's delivered on the output shaft from the gearbox, but sometimes it's from the stepper motors shaft. 1 Nm = 100 Ncm. Common values for a Nema 17 stepper motor without gearbox are from 1 to 2 Ncm (I will check this more carefully later).

My focuser motor with built in gearbox and an extra belt drive reducer can deliver about 80 Ncm to the focuser shaft. My friction is very high after I had tighten the force settings, friction torque is about 30 Ncm.

I use about 116 % of my 100 mm focuser stroke length, not perfect, I have set a max position of 55000 in the driver setup. The move per half step is 1.77 micrometer.

My equipment weight about 4 kg. When I point the telescope more then 50 degrees in altitude the motor stepper stall. I have attached a spring to help the focuser when moving inwards. Still there can be a problem sometimes, the good thing is that the focuser friction coupling never slip any more.

It's very important that the bracket holds the motor and gearbox without any flex, backlash cause a lot of problem when you try to focus the telescope.

Future plans:
Until now I have the driver to the stepper mode in half step mode, I can see from my calculation it's not needed. If I disable the half step and change the belt drive ratio I get extra torque. I can also replace the focuser motor with one that has stronger torque and also maybe I can reduce the friction in my focuser.