To fit the model a set of measured values for the anode voltage and anode current, for given grid voltages, is required.  It is then possible to derive the six model parameters {u, Ex, Kg, Kp, Kvb, Vct} using curve fitting and optimisation tools.

The curve fitting tool that I use is the  optmize.curve_fit() function from the Python SciPy library, using the following Python code,

params, params_covariance = optimize.curve_fit(koren, x_data, y_data, method='trf',

bounds=((1,-np.inf,-np.inf,-np.inf,-np.inf,-np.inf),(np.inf, np.inf, np.inf, np.inf, np.inf, np.inf)))

The curve_fit function takes the following arguments,

• The reference to the function that implements the triode model which has arguments for the measured data (Vanode-cathode, Vgrid_cathode), and the six model parameters that need to be found {u, Ex, Kg, Kp, Kvb, Vct} , and returns the model calculated value for anode current.  (See Python function definition below)

• The measured data for the model input data (Vanode-cathode, Vgrid_cathode)

• The measured data fro the Anode current

• The optimisation method to be used, and I have found that the 'trf' method works best, especially where there are constraints on the parameter values.

• The bounds, or constraints, for the model parameters, where I have found it helps to constraint the u parameter to be >=1.

def koren(x, u, Ex, Kg, Kp, Kvb, Vct):  # x[0] = Vgrid_cathode, x[1] = Vanode_cathode

   Vpk = x[1]

   Vgk = x[0]

   E1 = (Vpk / Kp) * np.log(1 + np.exp(Kp * ((1 / u) + (Vgk + Vct) / np.sqrt(Kvb + Vpk * Vpk))))

   E1 = abs(E1)    #np.exp cannot handle negative number (E1) to fractional powers

   Ip = 2 * np.power(E1, Ex) / Kg

   return Ip