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This method was first introduced in [ 22 ] using experimental values of key physical quantities. Data issued by manufacturers of modules do not contain the values of the nominal series and shunt resistances. Indeed, the exact five-point model and the approximate five-point model that use approximate values of the nominal series and shunt resistances yield comparable results [ 22 ], provided that R s 0 and R sh 0 are well defined.

The nominal series and shunt resistances are determined using the slope method. In this paper, the concept of nonlinearity is introduced for a better prediction of the behaviour of the tested photovoltaic modules. Thus, the dependency of the others key quantities on temperature and irradiance is given by the following equations: After expressing the above essential quantities as functions of operating conditions, the five parameters required are evaluated using the following equations [ 26 , 27 ]: They are empirical, and the precision of model depends on the acuity of their calculated results.

Accurate determination of these constants requires significant experimentation to find the values of current and voltage at different points for different operating conditions in view to calculate them using 17 — The variation of the open-circuit voltage is related to the variations of the solar radiation intensity and cells temperature. The data of the modules are presented in Table 1.

Manufacturers usually provide only limited operational data for PV modules. Utilization of available data aforementioned in the equations above presented for the five-parameter and the five-point models yields computed parameters of the I - V characteristic at given operating conditions for both modules used.

Following Figures 2 and 3 shows that I - V curves fitted from computed parameters of five-parameter and five-point models and experimental data issue by the manufacturers of both the photovoltaic modules used. It can be seen that the five-point model strongly agrees with experimental data than the five-parameter model for both types of modules. Another significant observation is the very accurate prediction in low irradiance of the five-point model.

Besides verifying the accuracy of the models to fit with the solar irradiance, it is also very important to analyze their capabilities to reproduce the way photovoltaic panel performance is affected by the silicon temperature.

The I - V curves of both modules have been realized by both mathematical models used in this work when subjected to the variation of temperature as it is shown in Figure 3.

The accuracy of both models was tested for different levels of temperature. As it has been noticed previously with the variation of irradiance, Figure 3 shows that five-point model better fits I - V curves of both modules than five-parameter model when subjected to the temperature variation. It is well known that irradiance and temperature of the cells strongly affect the performance of operating photovoltaic devices.

However, latest observations made on modelled I - V characteristics when subjected to irradiance and temperature of the cells variations are insufficient to make a true judgement about the capability of both the five-point and the five-parameter models to reproduce the behaviour of operating photovoltaic modules. To further investigate the performance of both the models in various operating conditions of both analyzed photovoltaic modules, inaccuracies on I - V curves prediction may be quantified.

Root mean square error RMSE is used as the statistical tool for assessing the performance of the models to predict the current. Figures 4 and 5 show the measuring deviation of computed I - V curves from experimental I - V data issue by manufacturers of modules for the whole of operating conditions analyzed.

The Thevenin equivalent circuit, if correctly derived, will behave exactly the same as the original circuit formed by B 1 , R 1 , R 3 , and B 2. In other words, the load resistor R 2 voltage and current should be exactly the same for the same value of load resistance in the two circuits. Calculating the equivalent Thevenin source voltage and series resistance is actually quite easy. First, the chosen load resistor is removed from the original circuit, replaced with a break open circuit:.

Next, the voltage between the two points where the load resistor used to be attached is determined. Use whatever analysis methods are at your disposal to do this. To find the Thevenin series resistance for our equivalent circuit, we need to take the original circuit with the load resistor still removed , remove the power sources in the same style as we did with the Superposition Theorem: This short analysis outlines some of my findings on the insertion loss and phase changes that will occur as a result of aging effects in varactor diodes.

The aging effects will be represented by variations in the equivalent circuit model of the varactor diode. Namely, it will be shown that the insertion loss of a single stage one diode will change in response to changes in the device capacitance over an expected end of life range.

Diode Model and Simple Phase Shifter An analysis of the equivalent circuit of a varactor diode was given in a prior post , but we will briefly discuss it here. Consider the circuits in Figure 1. A loaded line phase shifter using a shunt diode and the diode equivalent circuit are shown in 1a and 1b [3]. Note that the varactor diode is represented by Cp diode package capacitance , Lp diode package inductance , Rs diode intrinsic series resistance , and C V which is the diode voltage dependent junction capacitance [4].

For this analysis, we can ignore the packaging effects so that Cp and Lp are eliminated from the model. A further simplification is to ignore Rs, though this effect can be included in a separate analysis.

After the simplifications, we are left with a diode phase shifter as show in Figure 1 c.