## [tdc

Where a = constant dictating midpoint of the position-linkage curve b = constant dictating the amplitude of the position-linkage curve

### 4.3.3.3 Look up table Bicubic spline

The most accurate way of modelling the flux data provided by the FEA is to store it in a look up table and use the values directly. For each known value of position and current, a two dimensional bicubic spline interpolation (MATLAB command interp2

[80]) method was used to obtain the corresponding value of flux. In models D and E a form of reverse interpolation was required, whereby the flux and position were known from the electric circuit current and the displacement respectively and the corresponding value of current was required. In this situation the known value of position was used to create a one dimensional flux-current look up table.

This method varies distinctly from the best-fit function approach of the two previous methods and does not attempt to describe the entire data set. Instead, a new equation is formed for each segment where interpolation is required._

4.3.3.4 First Order Modelling

During a short circuit or purely resistive load test, then the solution of all the models is a first order differential. Due to the nature of differentiation, namely that it progressively distorts a signal and amplifies errors, it is beneficial to re-state the equation as an integral. Numerical integration acts as a smoothing function, being little affected by small inaccuracies, whereas differentiation will amplify a small discrepancy by dividing it by a small increment (dt). For models A-C, the governing equations are manipulated to become expressions for the time differential of current. A fourth order Runge-Kutta routine is used to solve this. In models D and E, the expressions are functions of y which hence becomes the variable which must be integrated, again using the Runge-Kutta routine. Each time step of the program follows the structure shown in Figure 4.35.

Figure 4.35: Schematic diagram of program integration technique for t=0

4.3.3.5 Second Order Modelling

With the introduction of capacitive elements to the external loading circuit, whereby the voltage is a function of the current integral, the guiding equations become second order. Although it is straightforward to re-arrange these into a single second order differential equation, for models A-D the presence of additional external time dependent variables (L), or the requirement for second order time differentials (I = (//), may make it beneficial to express the governing equations as two independent first order equations. These two alternative methods are described below. Runge-Kutta-Nyström - method i

The Runge-Kutta-Nyström algorithm [81] is a technique which generalises the standard Runge-Kutta method to solve second order differential equations. The input to the routine is a present value of the second time derivative, and the output is the value at the subsequent time step together with the value of the first time derivative. In model E

this is an elegant solution. For all other models, however, attention should be paid to the second order time differentials - see concluding remarks, Section 4.4.4. Runge-Kutta Separate Integrals -method ii

The voltage across a capacitor is dependent on the integral of current passing into it. If the value of current is assumed to be constant for a given time step, then the governing equation remains a first order differential equation and current can be calculated using the basic Runge-Kutta technique. The current may then be integrated in a separate routine. In this method the current changes between each time step, not during. For a small time step, this should have a negligible effect.

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