How to calculate magnification

[Kanonenfutter]

Still not sure - who can help, please?
My 40D sensor width is 22,2 mm (22.2 in the US I believe, decimal comma=decimal "point").
With my 100 f/2,8 macro lens plus the 1,4x tc plus the 2x tc plus the EF 25 (hollow) tube I can fill the width of the frame with 5,5 mm shooting a ruler.
Does that make it 4x magnification (22,2:5,5=4)?

[LordV]

Still not sure - who can help, please?
My 40D sensor width is 22,2 mm (22.2 in the US I believe, decimal comma=decimal "point").
With my 100 f/2,8 macro lens plus the 1,4x tc plus the 2x tc plus the EF 25 (hollow) tube I can fill the width of the frame with 5,5 mm shooting a ruler.
Does that make it 4x magnification (22,2:5,5=4)?

Mag = sensor width mm/mm across pic =22.2/5.5 = 4.04:1
so yes you are right
by rough calculation it comes out to around 3.92 :1 so seems about right.

Brian V.

[Kanonenfutter]

Thanks a lot, Brian!
K.

[C2S]

I've been spending this afternoon on reading about these formulas, and testing them on a few (cheap) sort-of-macro-capable lenses of mine, without any extra equipment:

- Sigma 70-300mm, shortest focusing distance 950mm (0.95m)
- Canon 18-55mm, shortest focusing distance 250mm (0.25m)

Then, using a 500D (sensor width 22.3mm), I filled the frame with a ruler, shooting at the maximum focal lengths and shortest focusing distance:

- Sigma was set to 300mm, and it recorded ~ 44.0mm, so roughly 0.507 magnification.
- Canon was set to 55mm, and it recorded ~ 56.5mm, so roughly 0.395 magnification.

Correct so far?

Since the sensor is roughly 22mm, and the image Sigma recorded is roughly 44mm, I understand that the magnification of ~ 0.5 should be close to correct (as said, without extra equipment). If I was able to focus as close to record 22mm, that would have been 1:1, 1.0 magnification that is. If I was able to focus close enough to record 11mm, the magnification would have been 2.0. And so on.

Then, to figure out some kind of a simple formula for the maximum possible magnifications for different lenses, I went to read Wikipedia to find out the AoV formula, fiddled with a bit of simple trigonometry, combined it with the formula mentioned in the ruler example, and eventually the formula seemed to simplify into:

(And I know someone is going to remark about something in it, no problem, I'm only interested :) )

M = FX / D

M = Maximum possible magnification
F = Lens focal length, in mm
X = Camera crop factor
D = Shortest focusing distance of the lens, in mm

Tested this on the two lenses so that I could compare the results with those above:

- Sigma: M = 300 * 1.6 / 950 ≈ 0.505
- Canon: M = 55 * 1.6 / 250 ≈ 0.352

Not sure whether the formula is actually valid or if it's just coincidence, but comparing with the results above, Sigma seems very close, and Canon somewhat close, although not as dead-on as the Sigma... perhaps my test setup was a bit sloppy, or something else is amiss.

Anyway, looking at the technical specifications of various lenses, and using this formula, I calculated the following list of maximum magnifications, when used on a 500D:

Sigma APO Macro 150mm f/2.8 EX DG HSM @ 150mm: 150 * 1.6 / 380 ≈ 0.632
Canon EF 100mm f/2.8 USM Macro @ 100mm: 100 * 1.6 / 310 ≈ 0.516
Sigma 70-300mm f/4.0-5.6 APO Macro DG @ 300mm: 300 * 1.6 / 950 ≈ 0.505
Canon EF-S 60mm f/2.8 Macro USM @ 60mm: 60 * 1.6 / 200 ≈ 0.480
Canon MP-E 65mm f/2.8 1-5 x @ 65mm: 65 * 1.6 / 243 ≈ 0.428
Canon EF-S 18-55mm IS f/3.5-5.6 @ 55mm: 55 * 1.6 / 250 ≈ 0.352

Unfortunately, the list seems to be not correct. For example, based on the list above, the Canon MP-E 65mm should have a maximum magnification of only 0.428 without any extra equipment, which is obviously wrong, since it's said to have a maximum of 5.0x magnification, without any extra equipment. Also, according to this list, no lenses would seem to reach the 1:1 factor. So there is probably some little detail I don't quite yet grasp about macro lenses, which probably will complicate the formula, if not completely change it.

Now, where's the error, and what would be the correct formula for figuring out the maximum possible magnifications for various lenses, without actually purchasing and testing them?

All this for curiosity's sake... thanks for reading. :)

[LordV]

CS2 - not sure how you got the final formula but I'm suprised it includes crop factor as the magnification of a lens is independant of the sensor size. The other comment is that the focal length of macro lenses changes at minimum focus to approx min focus distance/4.
brian v.

[C2S]

Thanks for your reply. :)

The inclusion of the crop factor is probably turning the formula into something completely different, to "answer a different question", so to speak. Frankly, I deliberately inserted the crop factor - this was done because I had been wondering whether to use the true focal length of the lens, or the 1.6x multiplied focal length since I was using a body with APS-C sized sensor. After the 1.6x multiplier was added, the two different test results for the Sigma seemed to pretty much match (0.507 vs 0.505), so I figured it was a good choice to do so. But as said, the formula is probably changed into something else at that point.

"The other comment is that the focal length of macro lenses changes at minimum focus to approx min focus distance/4."

If the focal length changed at the minimum focus distance, wouldn't that mean that the AoV would change as well? AoV is what the formula I wrote above is originally based on.

[LordV]

Have to admit I'm not sure what the AoV formula is but most lens formulas are based on classic prime lenses and these do not alter their focal length significantly at minimum focus. All normal 1:1 macro lenses do alter their focal length at min focus. This is the reason calculations on magnification with extension tubes work very well with normal lenses but give a significant magnification underestimate with macro lenses.
Brian V.