#
Equation Summary

The LI-840A computes CO_{2} and H_{2}O concentrations using an equation of the form

B‑1

where `c` is concentration, `f`() is the calibration function, α is the absorptance, `g` (`α`,`P`) is the pressure correction, `S`(`α`) is the span, and `T` is the temperature (°C) of the gas in the cell, typically 51.5 °C. Absorptance is computed from

B‑2

where `V` and `V _{o}` are the raw detector sample and reference readings, and

`Z`is the zeroing parameter.

Span is a linear function of absorptance.

B‑3

## H_{2}O Equations

Absorptance `α _{w}` for water vapor is computed from

B‑4

where `V _{w}` and

`V`are the sample and reference raw detector readings and

_{wo}`Z`is the zero parameter. The pressure correction for water vapor is an empirical function

_{w}`g`() of absorptance and pressure

_{w}`P`:

B‑5

The value of `P _{o}` is 99 kPa. When the pressure correction is not enabled,

`g`() is simply 1.0. Values for

_{w}`g`() can be viewed on the diagnostics window (see Operation).

_{w}Water vapor concentration `W` (mmol mol^{-1}) is computed from

B‑6

where `f _{w}`(

`x`) is a third order polynomial whose coefficients are given on the calibration sheet.

B‑7

## CO_{2} Equations

The measurement of CO_{2} is a bit more complicated than for H_{2}O because of the influence of water vapor. There is a slight direct cross sensitivity in the CO_{2} signal to H_{2}O. This is measured at the factory and accounted for in the computation of absorptance (equation B‑8). There is also a band broadening effect that is accounted for in the computation of concentration (equation B‑12).

CO_{2} absorptance α_{c} is computed from

where `V _{c}` and

`V`are the raw detector signals for sample and reference,

_{co}`Z`is the CO

_{c}_{2}zero parameter, and

`X`is a cross sensitivity parameter for the effect of water vapor on CO

_{wc}_{2}. Its value is reported on the calibration sheet as “XS=”.

The empirical pressure correction function `g _{c}`() depends on CO

_{2}absorptance and pressure:

When `P` = `P _{o}`,

`g`() = 1.

_{c}When `P` < `P _{o}`

where `a` = 1.10158, `b` = -6.1217 * 10^{-3}, `c` = -0.266278, `d` = 3.69895, and `z` is the asymptotic value of absorptance, obtained from the calibration coefficients (equation B‑13).

When `P` > `P`_{o}

B‑11

where `X`, `A`, and `B` are computed as in equation B‑9. The variable `g _{c}`() is viewable on the diagnostics window (see Operation).

CO_{2} concentration `C` (µmol mol^{-1}) is computed from

where `f _{c}`(

`x`) is a function whose inverse is a double rectangular hyperbola, and whose coefficients (

`a1`…

`a4`) are given on the calibration sheet.

Solving equation B‑13 for `C` yields the calibration function

B‑14

Where

B‑15

`ψ`(`W`) accounts for band broadening by water vapor, and is viewable on the Diagnostic window.

B‑16

The band broadening coefficient `h`(`α`_{c}) has been determined to be 1.45 for the LI-840A for CO_{2} concentrations near ambient. At higher concentrations, the value decreases. We capture this behavior with an empirical relationship (equation B‑17).

Where `z` is from equation B‑10, and `b _{w}` is the low concentration band broadening coefficient: 1.45. This is the value shown on the calibration sheet as BB=1.45. The present value of

`h`(α

_{c}) is viewable on the Diagnostics screen (Section 3). The typical relationship between

`h`(α

_{c}) and CO

_{2}concentration is illustrated below. (‘Typical’ because the exact relationship depends on the relationship between absorptance and CO

_{2}, which is the calibration curve.)

__Implementation Note:__ We formulated B‑17 with 0.64`b _{w}` – 0.64 instead of the simple equivalent (0.29) because this allows band broadening corrections to be turned off by setting

`b`to 1. When

_{w}`b`=1,

_{w}`h`(

`α`) = 1 everywhere. Also, to avoid computational problems (underflows, overflows, and division by zero) we constrain the argument

_{c}`α`when computing

_{c}`h`(

`α`) to be

_{c}B‑18

is typically equivalent to about 600 ppm.

## Calibration Equations

### Zeroing H_{2}O

When the command for zeroing water is received, the LI-840A computes the water zero from equation B‑19, where and are averaged for 5 seconds.

### Zeroing CO_{2}

When the command for zeroing CO_{2} is received, the LI-840A computes the CO_{2} zero term from equation B‑20, where , , , and are averaged for 5 seconds.

### Spanning H_{2}O

When the command for setting the span for H_{2}O is received, along with the target concentration `W _{T}`, the LI-840A computes

`S`from equation B‑21 , where is averaged over five seconds.

_{w0}where

The instrument retains the following values, which are used for subsequent secondary spans:

B‑23

B‑24

### Secondary Span H_{2}O

When the secondary span command for H_{2}O is received, the instrument computes new values for both `S _{w0}` and

`S`. First, it measures a new and computes a new from equation B‑22. Then, it uses these plus the retained values (and from the previous normal span) to compute

_{w1}B‑25

Given the new span slope `S _{w1}`, update the span offset

`S`by equation B‑21.

_{w0}### Spanning CO_{2}

When the command for setting the span for CO_{2} is received, along with the target concentration `C _{T}`, the LI-840A computes

`S`from equation B‑26, where and are averaged for 5 seconds.

_{c0}where

Note that

B‑28

The instrument retains the following values which are used for subsequent secondary spans, if necessary:

B‑29

B‑30

### Secondary Span CO_{2}

When the secondary span command for CO_{2} is received, the instrument computes new values for both `S _{c0}` and

`S`. First, it measures a new and computes a new

_{c1}`β`from equation B‑27. Then it uses these, plus the retained values (

_{c}`α`and

_{c1}`β`from the previous normal span) to compute

_{c1}B‑31

Given the new span slope `S _{c1}`, update the span offset

`S`by equation B‑26.

_{c0}## Symbol Summary

Many of the quantities described in the above section are available from the LI-840A. These are summarized in Table B‑1.