General Overview

This report is the result of the use of the Python 3.4 package Sympy (for symbolic mathematics), as means to translate published models to a common language. It was created by Holger Metzler (Orcid ID: 0000-0002-8239-1601) on 10/03/2016, and was last modified on lm.

About the model

The model depicted in this document considers soil organic matter decomposition. It was originally described by Parton, Schimel, Cole, & Ojima (1987).

Abstract

We analyzed climatic and textural controls of soil organic C and N for soils of the U.S. Great Plains. We used a model of soil organic matter (SOM) quantity and composition to simulate steady-state organic matter levels for 24 grassland locations in the Great Plains. The model was able to simulate the effects of climatic gradients on SOM and productivity. Soil texture was also a major control over organic matter dynamics. The model adequately predicted aboveground plant production and soil C and N levels across soil textures (sandy, medium, and fine); however, the model tended to overestimate soil C and N levels for fine textured soil by 10 to 15%. The impact of grazing on the system was simulated and showed that steady-state soil C and N levels were sensitive to the grazing intensity, with soil C and N levels decreasing with increased grazing rates. Regional trends in SOM can be predicted using four site-specific variables, temperature, moisture, soil texture, and plant lignin content. Nitrogen inputs must also be known. Grazing intensity during soil development is also a significant control over steady-state levels of SOM, and since few data are available on presettlement grazing, some uncertainty is inherent in the model predictions.

Keywords

differential equations, time variant

Principles

mass balance, substrate dependence of decomposition, heterogeneity of speed of decay, internal transformations of organic matter, environmental variability effects

Available parameter values

Available initial values

State Variables

The following table contains the available information regarding this section:

Maximum Decomposition Rates

The following table contains the available information regarding this section:

Decomposition Rates

The following table contains the available information regarding this section:

Additional Parameters

The following table contains the available information regarding this section:

Auxiliary Variables

The following table contains the available information regarding this section:

Input Components

The following table contains the available information regarding this section:

Transfer Coefficients

The following table contains the available information regarding this section:

Environmental Coefficients

The following table contains the available information regarding this section:

Components

The following table contains the available information regarding this section:

Pool model representation

Flux description

Figure 1: Pool model representation

Input fluxes \(C_{1}: J_{1}\cdot\left(0.018\cdot LN + 0.15\right)\) \(C_{2}: J_{1}\cdot\left(- 0.018\cdot LN + 0.85\right)\) \(C_{3}: J_{2}\cdot\left(0.018\cdot LN + 0.15\right)\) \(C_{4}: J_{2}\cdot\left(- 0.018\cdot LN + 0.85\right)\) Output fluxes \(C_{1}: C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.15\cdot A_{l} + 0.45\right)\cdot e^{- 3\cdot Ls}\) \(C_{2}: 0.55\cdot C_{2}\cdot K_{2}\cdot f_{T}\cdot f_{W}\) \(C_{3}: C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.25\cdot A_{l} + 0.55\right)\cdot e^{- 3\cdot Ls}\) \(C_{4}: 0.55\cdot C_{4}\cdot K_{4}\cdot f_{T}\cdot f_{W}\) \(C_{5}: C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(0.68\cdot Tx - 0.85\right)\cdot\left(0.75\cdot Tx - 1\right)\) \(C_{6}: 0.55\cdot C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W}\) \(C_{7}: 0.55\cdot C_{7}\cdot K_{7}\cdot f_{T}\cdot f_{W}\) Internal fluxes \(C_{1} > C_{5}: - 0.55\cdot C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot\left(A_{l} - 1\right)\cdot e^{- 3\cdot Ls}\) \(C_{1} > C_{6}: 0.7\cdot A_{l}\cdot C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls}\) \(C_{2} > C_{5}: 0.45\cdot C_{2}\cdot K_{2}\cdot f_{T}\cdot f_{W}\) \(C_{3} > C_{5}: - 0.45\cdot C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot\left(A_{l} - 1\right)\cdot e^{- 3\cdot Ls}\) \(C_{3} > C_{6}: 0.7\cdot A_{l}\cdot C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls}\) \(C_{4} > C_{5}: 0.45\cdot C_{4}\cdot K_{4}\cdot f_{T}\cdot f_{W}\) \(C_{5} > C_{6}: C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.51\cdot Tx^{2} + 0.5705\cdot Tx + 0.146\right)\) \(C_{5} > C_{7}: C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.003\cdot Tx + 0.004\right)\) \(C_{6} > C_{5}: 0.42\cdot C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W}\) \(C_{6} > C_{7}: 0.03\cdot C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W}\) \(C_{7} > C_{5}: 0.45\cdot C_{7}\cdot K_{7}\cdot f_{T}\cdot f_{W}\)

The right hand side of the ODE

\(\left[\begin{matrix}- C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + J_{1}\cdot\left(0.018\cdot LN + 0.15\right)\\- C_{2}\cdot K_{2}\cdot f_{T}\cdot f_{W} + J_{1}\cdot\left(- 0.018\cdot LN + 0.85\right)\\- C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + J_{2}\cdot\left(0.018\cdot LN + 0.15\right)\\- C_{4}\cdot K_{4}\cdot f_{T}\cdot f_{W} + J_{2}\cdot\left(- 0.018\cdot LN + 0.85\right)\\C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.55\cdot A_{l} + 0.55\right)\cdot e^{- 3\cdot Ls} + 0.45\cdot C_{2}\cdot K_{2}\cdot f_{T}\cdot f_{W} + C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.45\cdot A_{l} + 0.45\right)\cdot e^{- 3\cdot Ls} + 0.45\cdot C_{4}\cdot K_{4}\cdot f_{T}\cdot f_{W} - C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right) + 0.42\cdot C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W} + 0.45\cdot C_{7}\cdot K_{7}\cdot f_{T}\cdot f_{W}\\0.7\cdot A_{l}\cdot C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + 0.7\cdot A_{l}\cdot C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right)\cdot\left(0.68\cdot Tx + 0.146\right) - C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W}\\0.004\cdot C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right) + 0.03\cdot C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W} - C_{7}\cdot K_{7}\cdot f_{T}\cdot f_{W}\end{matrix}\right]\)

The Jacobian (derivative of the ODE w.r.t. state variables)

\(\left[\begin{matrix}- K_{1}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} & 0 & 0 & 0 & 0 & 0 & 0\\0 & - K_{2}\cdot f_{T}\cdot f_{W} & 0 & 0 & 0 & 0 & 0\\0 & 0 & - K_{3}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} & 0 & 0 & 0 & 0\\0 & 0 & 0 & - K_{4}\cdot f_{T}\cdot f_{W} & 0 & 0 & 0\\K_{1}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.55\cdot A_{l} + 0.55\right)\cdot e^{- 3\cdot Ls} & 0.45\cdot K_{2}\cdot f_{T}\cdot f_{W} & K_{3}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.45\cdot A_{l} + 0.45\right)\cdot e^{- 3\cdot Ls} & 0.45\cdot K_{4}\cdot f_{T}\cdot f_{W} & - K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right) & 0.42\cdot K_{6}\cdot f_{T}\cdot f_{W} & 0.45\cdot K_{7}\cdot f_{T}\cdot f_{W}\\0.7\cdot A_{l}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} & 0 & 0.7\cdot A_{l}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} & 0 & K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right)\cdot\left(0.68\cdot Tx + 0.146\right) & - K_{6}\cdot f_{T}\cdot f_{W} & 0\\0 & 0 & 0 & 0 & 0.004\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right) & 0.03\cdot K_{6}\cdot f_{T}\cdot f_{W} & - K_{7}\cdot f_{T}\cdot f_{W}\end{matrix}\right]\)

Steady state formulas

\(C_{1} = \frac{0.006\cdot J_{1}\cdot e^{3.0\cdot Ls}}{K_{1}\cdot f_{T}\cdot f_{W}}\cdot\left(3.0\cdot LN + 25.0\right)\)
\(C_{2} = -\frac{0.002\cdot J_{1}\cdot\left(9.0\cdot LN - 425.0\right)}{K_{2}\cdot f_{T}\cdot f_{W}}\)
\(C_{3} = \frac{0.006\cdot J_{2}\cdot e^{3.0\cdot Ls}}{K_{3}\cdot f_{T}\cdot f_{W}}\cdot\left(3.0\cdot LN + 25.0\right)\)
\(C_{4} = -\frac{0.002\cdot J_{2}\cdot\left(9.0\cdot LN - 425.0\right)}{K_{4}\cdot f_{T}\cdot f_{W}}\)
\(C_{5} = \frac{1}{K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(884340.0\cdot Tx^{2} - 3983847.0\cdot Tx + 3739636.0\right)}\cdot\left(- 17751.6\cdot A_{l}\cdot J_{1}\cdot LN - 147930.0\cdot A_{l}\cdot J_{1} - 10551.6\cdot A_{l}\cdot J_{2}\cdot LN - 87930.0\cdot A_{l}\cdot J_{2} + 7200.0\cdot J_{1}\cdot LN + 1860000.0\cdot J_{1} + 1800000.0\cdot J_{2}\right)\)
\(C_{6} = \frac{1}{K_{6}\cdot f_{T}\cdot f_{W}\cdot\left(294780.0\cdot Tx - 934909.0\right)}\cdot\left(6732.0\cdot A_{l}\cdot J_{1}\cdot LN\cdot Tx - 11131.92\cdot A_{l}\cdot J_{1}\cdot LN + 56100.0\cdot A_{l}\cdot J_{1}\cdot Tx - 92766.0\cdot A_{l}\cdot J_{1} + 5508.0\cdot A_{l}\cdot J_{2}\cdot LN\cdot Tx - 11394.72\cdot A_{l}\cdot J_{2}\cdot LN + 45900.0\cdot A_{l}\cdot J_{2}\cdot Tx - 94956.0\cdot A_{l}\cdot J_{2} - 1224.0\cdot J_{1}\cdot LN\cdot Tx - 262.8\cdot J_{1}\cdot LN - 316200.0\cdot J_{1}\cdot Tx - 67890.0\cdot J_{1} - 306000.0\cdot J_{2}\cdot Tx - 65700.0\cdot J_{2}\right)\)
\(C_{7} = \frac{1}{K_{7}\cdot f_{T}\cdot f_{W}\cdot\left(294780.0\cdot Tx - 934909.0\right)}\cdot\left(201.96\cdot A_{l}\cdot J_{1}\cdot LN\cdot Tx - 316.206\cdot A_{l}\cdot J_{1}\cdot LN + 1683.0\cdot A_{l}\cdot J_{1}\cdot Tx - 2635.05\cdot A_{l}\cdot J_{1} + 165.24\cdot A_{l}\cdot J_{2}\cdot LN\cdot Tx - 331.29\cdot A_{l}\cdot J_{2}\cdot LN + 1377.0\cdot A_{l}\cdot J_{2}\cdot Tx - 2760.75\cdot A_{l}\cdot J_{2} - 36.72\cdot J_{1}\cdot LN\cdot Tx - 15.084\cdot J_{1}\cdot LN - 9486.0\cdot J_{1}\cdot Tx - 3896.7\cdot J_{1} - 9180.0\cdot J_{2}\cdot Tx - 3771.0\cdot J_{2}\right)\)

Steady states (potentially incomplete), according jacobian eigenvalues, damping ratio

Parameter set: Set 1

\(C_1: \frac{0.0789473684210526\cdot J_{1}}{f_{T}\cdot f_{W}}\cdot\left(3.0\cdot LN + 25.0\right)\cdot e^{3.0\cdot Ls}\), \(C_2: -\frac{0.00714285714285714\cdot J_{1}}{f_{T}\cdot f_{W}}\cdot\left(9.0\cdot LN - 425.0\right)\), \(C_3: \frac{0.0638297872340425\cdot J_{2}}{f_{T}\cdot f_{W}}\cdot\left(3.0\cdot LN + 25.0\right)\cdot e^{3.0\cdot Ls}\), \(C_4: -\frac{0.00571428571428571\cdot J_{2}}{f_{T}\cdot f_{W}}\cdot\left(9.0\cdot LN - 425.0\right)\), \(C_5: -\frac{1}{f_{T}\cdot f_{W}\cdot\left(884340.0\cdot Tx^{2} - 3983847.0\cdot Tx + 3739636.0\right)}\cdot\left(126797.142857143\cdot A_{l}\cdot J_{1}\cdot LN + 1056642.85714286\cdot A_{l}\cdot J_{1} + 75368.5714285714\cdot A_{l}\cdot J_{2}\cdot LN + 628071.428571429\cdot A_{l}\cdot J_{2} - 51428.5714285714\cdot J_{1}\cdot LN - 13285714.2857143\cdot J_{1} - 12857142.8571429\cdot J_{2}\right)\), \(C_6: \frac{1}{f_{T}\cdot f_{W}\cdot\left(294780.0\cdot Tx - 934909.0\right)}\cdot\left(1771578.94736842\cdot A_{l}\cdot J_{1}\cdot LN\cdot Tx - 2929452.63157895\cdot A_{l}\cdot J_{1}\cdot LN + 14763157.8947368\cdot A_{l}\cdot J_{1}\cdot Tx - 24412105.2631579\cdot A_{l}\cdot J_{1} + 1449473.68421053\cdot A_{l}\cdot J_{2}\cdot LN\cdot Tx - 2998610.52631579\cdot A_{l}\cdot J_{2}\cdot LN + 12078947.3684211\cdot A_{l}\cdot J_{2}\cdot Tx - 24988421.0526316\cdot A_{l}\cdot J_{2} - 322105.263157895\cdot J_{1}\cdot LN\cdot Tx - 69157.8947368421\cdot J_{1}\cdot LN - 83210526.3157895\cdot J_{1}\cdot Tx - 17865789.4736842\cdot J_{1} - 80526315.7894737\cdot J_{2}\cdot Tx - 17289473.6842105\cdot J_{2}\right)\), \(C_7: \frac{1}{f_{T}\cdot f_{W}\cdot\left(294780.0\cdot Tx - 934909.0\right)}\cdot\left(1553538.46153846\cdot A_{l}\cdot J_{1}\cdot LN\cdot Tx - 2432353.84615385\cdot A_{l}\cdot J_{1}\cdot LN + 12946153.8461538\cdot A_{l}\cdot J_{1}\cdot Tx - 20269615.3846154\cdot A_{l}\cdot J_{1} + 1271076.92307692\cdot A_{l}\cdot J_{2}\cdot LN\cdot Tx - 2548384.61538462\cdot A_{l}\cdot J_{2}\cdot LN + 10592307.6923077\cdot A_{l}\cdot J_{2}\cdot Tx - 21236538.4615385\cdot A_{l}\cdot J_{2} - 282461.538461538\cdot J_{1}\cdot LN\cdot Tx - 116030.769230769\cdot J_{1}\cdot LN - 72969230.7692308\cdot J_{1}\cdot Tx - 29974615.3846154\cdot J_{1} - 70615384.6153846\cdot J_{2}\cdot Tx - 29007692.3076923\cdot J_{2}\right)\)

\(\lambda_{1}: - 0.076\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls}\)
\(\lambda_{2}: - 0.35\cdot f_{T}\cdot f_{W}\)
\(\lambda_{3}: - 0.094\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls}\)
\(\lambda_{4}: 0.035\cdot Tx\cdot f_{T}\cdot f_{W} - 0.04798\cdot f_{T}\cdot f_{W} +\frac{\left(-0.5 + 0.866\cdot i\right)\cdot\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)}{\left(7.645\cdot 10^{-9}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 3.444\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 3.233\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.03704\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.1667\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right) +\left(\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)^{3} + 0.25\cdot\left(1.529\cdot 10^{-8}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 6.888\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 6.466\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.07407\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.3333\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right)\right)^{2}\right)^{0.5}\right)^{0.3333}} -\left(-0.5 - 0.866\cdot i\right)\cdot\left(7.645\cdot 10^{-9}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 3.444\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 3.233\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.03704\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.1667\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right) +\left(\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)^{3} + 0.25\cdot\left(1.529\cdot 10^{-8}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 6.888\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 6.466\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.07407\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.3333\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right)\right)^{2}\right)^{0.5}\right)^{0.3333}\)
\(\lambda_{5}: - 0.28\cdot f_{T}\cdot f_{W}\)
\(\lambda_{6}: 0.035\cdot Tx\cdot f_{T}\cdot f_{W} - 0.04798\cdot f_{T}\cdot f_{W} +\frac{\left(-0.5 - 0.866\cdot i\right)\cdot\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)}{\left(7.645\cdot 10^{-9}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 3.444\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 3.233\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.03704\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.1667\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right) +\left(\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)^{3} + 0.25\cdot\left(1.529\cdot 10^{-8}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 6.888\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 6.466\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.07407\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.3333\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right)\right)^{2}\right)^{0.5}\right)^{0.3333}} -\left(-0.5 + 0.866\cdot i\right)\cdot\left(7.645\cdot 10^{-9}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 3.444\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 3.233\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.03704\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.1667\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right) +\left(\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)^{3} + 0.25\cdot\left(1.529\cdot 10^{-8}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 6.888\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 6.466\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.07407\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.3333\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right)\right)^{2}\right)^{0.5}\right)^{0.3333}\)
\(\lambda_{7}: 0.035\cdot Tx\cdot f_{T}\cdot f_{W} - 0.04798\cdot f_{T}\cdot f_{W} +\frac{3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}}{\left(7.645\cdot 10^{-9}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 3.444\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 3.233\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.03704\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.1667\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right) +\left(\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)^{3} + 0.25\cdot\left(1.529\cdot 10^{-8}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 6.888\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 6.466\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.07407\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.3333\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right)\right)^{2}\right)^{0.5}\right)^{0.3333}} -\left(7.645\cdot 10^{-9}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 3.444\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 3.233\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.03704\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.1667\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right) +\left(\left(3.798\cdot 10^{-5}\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.00018\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.0001727\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.1111\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{2}\right)^{3} + 0.25\cdot\left(1.529\cdot 10^{-8}\cdot Tx^{2}\cdot f_{T}^{3}\cdot f_{W}^{3} - 6.888\cdot 10^{-8}\cdot Tx\cdot f_{T}^{3}\cdot f_{W}^{3} + 6.466\cdot 10^{-8}\cdot f_{T}^{3}\cdot f_{W}^{3} + 0.07407\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)^{3} - 0.3333\cdot\left(- 0.105\cdot Tx\cdot f_{T}\cdot f_{W} + 0.1439\cdot f_{T}\cdot f_{W}\right)\cdot\left(0.000114\cdot Tx^{2}\cdot f_{T}^{2}\cdot f_{W}^{2} - 0.0005401\cdot Tx\cdot f_{T}^{2}\cdot f_{W}^{2} + 0.000518\cdot f_{T}^{2}\cdot f_{W}^{2}\right)\right)^{2}\right)^{0.5}\right)^{0.3333}\)

References