Telecentric Lens FOV, WD & Magnification Formula (0.3x-2.0x)

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This page is a calculation reference for object-side telecentric lenses: the formulas linking magnification (β), sensor size, field of view (FOV), working distance (WD) and resolution, with worked examples on real POMEAS models. Values that are not published datasheet figures are marked calculated (derived by formula). For choosing a lens, read the telecentric lens selection guide; here we only do the math.

1. The one identity

β = h_img / h_FOV, and therefore FOV = h_img / β and h_img = β × FOV. Here h_img is the active sensor dimension in mm, h_FOV the imaged object dimension in mm, and β the dimensionless magnification, written "X". Always pair width with width and height with height.

An object-side telecentric lens holds β and FOV constant while WD varies inside the depth of field (DoF): outside it the image blurs, but the magnification never rescales. That is why the identity can be trusted on a measurement station (quantified in section 8).

2. Formula sheet

3. Sensor format and FOV reference

Nothing can be computed without the active sensor size. Formats quoted only as an FOV at a known magnification are back-calculated and marked calculated.

4. Worked examples

Rows marked "matches published" reproduce the datasheet FOV — the check that lets you trust the formula.

Working backwards is as quick: a 20 mm-wide FOV on a 1/2" camera (6.4 mm wide) gives β = 6.4 / 20 = 0.32X (calculated), so 0.3X is the nearest catalog step. Where the FOV is large and the sensor small, β falls below the practical fixed-magnification range and a zoom FOV calculation is the better route.

5. Pixel resolution, lens resolution and measurement uncertainty

The camera contributes the object-side pixel equivalent e = p / β: one sensor pixel of pitch p covers e micrometres on the object. The lens contributes its published object-side resolution R_lens. The system takes the worse of the two, R_sys ≈ max(R_lens, e), so more megapixels alone does not improve a measurement: once e is below R_lens the lens is the limit. The useful check is p ≤ β × R_lens. Example (calculated): PMS-LTC2-110C at 2.0X has a published object-side resolution of 8 µm, so p should be ≤ 16 µm — a margin any mainstream sensor clears.

Uncertainty magnitude. A common engineering rule, not a specification, requires a feature to span 2–3 pixels: F_min ≈ (2–3) × e, with single-edge uncertainty of order u ≈ (1–3) × e. Estimate (calculated): at 1.0X with p = 8 µm (use your camera datasheet value) e = 8 µm, so the smallest reliable feature is 16–24 µm and per-edge uncertainty 8–24 µm; at 2.0X both halve. Uncertainty scales as 1/β, making magnification the main accuracy lever; where p is unknown at design time, record it as to be confirmed.

6. Depth of field versus NA and F#

DoF is set by how steeply the light cone converges on the object: a larger numerical aperture (NA), i.e. a smaller F-number (F#), gives a shallower DoF. Treat the scaling as qualitative — DoF falls roughly with NA² (and F#²) — and take numbers from the datasheet, never a fitted constant. Compare within one family at the same 110 mm WD:

In the first four rows, NA rising from 0.045 to 0.06 takes published DoF from 0.89 mm to 0.44 mm — the trade-off in datasheet figures. The last row shows the aperture is a knob: LTCM2-65C-S reaches NA 0.152 with DoF published as 0.13 mm maximum, adjustable. For a measurement station, buy DoF with lower NA and longer WD, not a bigger sensor.

7. Working distance, barrel length and object-to-image distance

WD is only one leg of the mechanical chain: I/O ≈ WD + L + FFD, where L is the barrel length and FFD the camera flange focal distance. Check it on PMS-10110CVP111T: published WD = 110.00 mm, L = 132.88 mm, I/O = 260.41 ± 2 mm. Rearranged (calculated): FFD ≈ 260.41 − 110.00 − 132.88 = 17.53 mm, the C-mount flange focal distance — confirmation the three terms add up as written.

I/O fixes the gap between part plane and camera face; WD is what you adjust during setup, and ±2 mm is the room you have. Leave that band and you exit the DoF: the image blurs even though the magnification is unchanged. The same ±2 mm order appears on the zoom side (LZ-650104 at WD 82 ± 2 mm, LZL-12101 at 77.4 ± 2 mm). The payoff: two identical parts at slightly different heights inside the DoF still measure the same — the point of a dimensional measurement job.

8. Telecentricity and the WD-deviation error

Telecentricity is an angle — how far the chief rays deviate from parallel to the axis — and it turns a WD error into a lateral error through Δx = ΔWD × tan θ. POMEAS publishes object-side telecentricity of <0.02° (PMS-LTCM12-110C, PMS-LTCM15-110C), <0.016° object side and <0.069° image side on LTCM2-65C-S, and <0.05–0.1° on PMS-10110CVP111T. Every Δx below is calculated.

Within the WD tolerance this term is sub-micron to a few micrometres — not the dominant error — and it scales linearly with ΔWD. On a non-telecentric lens the chief-ray angle is a degree or more rather than hundredths of a degree, and the same WD error maps to a far larger lateral error; the exact figure depends on the lens, so it is marked to be confirmed per lens. POMEAS contrasts the two cases in its telecentric vs standard FA lens comparison.

9. Error budget and calculation procedure

Accuracy is set first by magnification (through β and e), then by WD discipline, then by residual distortion. Check the telecentric lens range for published figures before you fix magnification, then run:

1. Fix the required FOV (mm, width and height), including travel allowance.
2. Choose the sensor format and read its active width and height from the camera datasheet.
3. Compute β = sensor / FOV for both dimensions; take the more demanding value and round up to the nearest catalog step.
4. Re-check FOV = sensor / β to confirm it still covers the part.
5. Read R_lens for the candidate model and confirm p ≤ β × R_lens.
6. Compute e = p / β and check the smallest feature against (2–3) × e.
7. Check DoF against part flatness and stage error; step to a lower-NA or longer-WD model if too thin.
8. Lay out I/O = WD + L + FFD, confirm the machine envelope, and hold WD inside tolerance.

Run this on a real part and the answer usually narrows to two or three models: POMEAS publishes magnification, WD, sensor format, object-side resolution, DoF and distortion per model, so nothing needs guessing. To have the calculation checked against a specific camera and part, send the FOV, camera and accuracy target to the POMEAS team and we will return the candidate model and its error budget.

Magnification ladder 0.2X–2.0X: published optical data

The formula is only as useful as the numbers you feed it. The table below is the full magnification ladder of the POMEAS telecentric range at a fixed 110 mm working distance, so the object-side field of view can be read off directly instead of back-calculated. Every optical figure is a datasheet value; field-of-view cells marked calculated follow FOV = sensor ÷ β at 2/3″ (8.8 × 6.6 mm).

Read the ladder top-down for a design decision: each step down roughly halves the field of view and halves the object-side pixel equivalent, at the cost of depth of field. Going from 0.3X to 2.0X shrinks the 2/3″ field from 29.3 × 22.0 mm to 4.4 × 3.3 mm and the depth of field from 4.44 mm to 0.27–0.5 mm. Nothing on that line is free.

Working distance is a separate axis. Three 1.0X models share the same magnification but not the same geometry:

Longer working distance is paid for in resolution and light: the 200 mm model has roughly three times the depth of field of the 110 mm model but also roughly three times the object-side resolution figure and a much smaller aperture. Decide working distance before magnification — it is the constraint set by the machine, while magnification is the free variable.

Two cautions on the ladder. First, a model number does not imply an aperture or a depth-of-field value; cells shown as "—" are simply not stated in the published data for that model. Second, resolution figures here are lens limits, not system measurement accuracy — see object-side resolution vs measurement accuracy for that distinction.

Related reading: built-in coaxial telecentric lens selection 0.3X–2.0X, how to choose a telecentric lens, 1.1-inch camera matching.

Frequently Asked Questions

How do I calculate the field of view of a telecentric lens from its magnification?

Divide the active sensor dimension by the magnification: FOV = sensor size / β. Pair the sensor width with the FOV width and the sensor height with the FOV height. Example: a 2/3" sensor is 8.8 mm wide, so at 0.3X the FOV is 8.8 / 0.3 = 29.3 mm wide and 6.6 / 0.3 = 22.0 mm high, which matches the published FOV for PMS-LTCM03-110-C. Keep the same units on both sides, and remember that for an object-side telecentric lens the FOV does not change when you move the part within the depth of field, so one calculation covers the whole working range.

How do I choose magnification when I only know the required FOV?

Rearrange the identity: β = sensor size / FOV. If the part is 20 mm wide and the camera is 1/2" (6.4 mm wide), β = 6.4 / 20 = 0.32X (calculated), so you pick the nearest catalog step of 0.3X. Two checks follow: confirm the FOV at the chosen magnification still covers the part, and confirm the accuracy, since the object-side pixel equivalent is e = p / β and the smallest measurable feature is about 2–3 × e under the standard engineering rule. If the accuracy is not met at the available magnification, use a smaller-pixel camera or accept a smaller FOV and add a second station.

What pixel size should my camera have for a given telecentric lens?

Keep the camera from becoming the resolution limit: the pixel pitch p should satisfy p ≤ β × R_lens, where R_lens is the published object-side resolution of the lens. For PMS-LTC2-110C at 2.0X with R_lens = 8 µm, that means p ≤ 16 µm, which almost any modern sensor meets. Once the condition holds, buying more megapixels does not improve the measurement, because the system resolution is R_sys ≈ max(R_lens, p/β) and the lens is already the larger term. Excessive oversampling only enlarges files and slows processing without adding measurable detail.

What accuracy can I expect from a telecentric measurement?

Plan on the order of a few to a few tens of micrometres, driven mainly by magnification. With object-side pixel equivalent e = p / β, the single-edge uncertainty magnitude is roughly (1–3) × e per the standard engineering estimate, and the smallest reliably measured feature is (2–3) × e. At 1.0X with p = 8 µm this puts e at 8 µm; at 2.0X it halves to 4 µm. Add the WD-deviation term Δx = ΔWD × tan θ, which is sub-micron at ΔWD = 1 mm for a lens better than 0.02°, plus the residual distortion of 0.01–0.05% typical of the coaxial series. Real numbers need the exact lens, camera and part, so treat these as planning magnitudes.

Why does my measurement change when I move the part up and down?

If the reading changes with part height, check two things. First, WD: a telecentric lens keeps magnification constant inside its depth of field, but outside it the image blurs and edge detection shifts. Confirm the part stays inside the published WD tolerance, for example 110 mm on the 110 mm WD models. Second, the telecentricity angle: residual lateral error is Δx = ΔWD × tan θ, so a 2 mm standoff error on a 0.1° lens contributes about 3.5 µm (calculated). If both are inside spec and the reading still drifts, look at stage repeatability, part fixturing and lighting stability rather than the lens itself.

Can I use a telecentric lens with any camera?

Mechanically, camera-side compatibility is set by the mount — the coaxial telecentric models quoted here use C-mount, so the flange focal distance term in I/O = WD + L + FFD is fixed. Optically, the constraint is the image circle: the sensor diagonal must fit inside the lens image circle, and the sensor size must be one of the formats the model supports. Match the sensor format to the published list before ordering, and confirm the accuracy condition p ≤ β × R_lens. If your sensor is larger than the listed formats, step up to a model designed for it rather than forcing the combination.