The task for fulfillment

For the given variant of the electric circuit:

 Develop mathematical model of calculation of branches currents and reactive elements voltages;

 Make the program realizing developed model;

 Include into the program the operators, allowing carry out verification of calculation results;

 Organize a graphical output of the specified functions of currents and voltages;

 Debug the program;

 Save results of the work (the program, listing of calculation, graphics) at your personal folder;

· Draw up report.

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4 LAboratory work # 4

TOPIC: Calculation of transients in linear electric circuits by numerical method. Part 1.

PURPOSE OF THE WORK: Applying of method of state variable, numerical methods of integration of the differential equations and discrete current models of inductive and capacitor elements in backward Euler method for calculations of transients.

____________________________________________________________

Mathematical model

Let’s consider the electric circuit represented by the figure 4.1.

Figure 4.1 – Given electrical circuit with parameters of it’s elements

At the moment t=0 the switch is being shorted and transient is being started. Inductive elements and capacitor ones are substituted for their discrete current models. Then the circuit will get a view shown in figure 4.2. There is no capacitor or inductive element in this circuit – it is a circuit of a direct current.

Figure 4.2 – Equivalent circuit of a direct current for each step of integration

At the initial moment t=0 the current and the voltage (for k=1) are known. These are independent initial conditions. For the moment t=h it is possible to determine all currents and voltages having executed calculation of a circuit of a direct current. Let’s execute this calculation by a method of potential nodes. For this purpose let’s transform EMF source e(t) to a source of a current and it is obtained the circuit shown in figure 4.3.

Figure 4.3 – Circuit with transformed EMF source e(t) to a source of a current

The system of the potential equations looks as:

(4.1) where

- The sum of the branches conductances connected to node1:

;

G₂₂ - The sum of the branches conductances connected to node2:

;

G₃₃ - The sum of the branches conductances connected to node3:

;

G₁₂=G₂₁ - The branch conductance between node1 and node2:

;

G₃₁=G₁₃ - The branch conductance between node1 and node3:

` ;

G₂₃=G₃₂ - The branch conductance between node2 and node2:

G₂₃=0;

- Node currents, where

- The algebraic sum of currents of the sources converging to node1:

;

- The algebraic sum of currents of the sources converging to node 2:

;

- The algebraic sum of currents of the sources converging to node 3:

Thus, it is obtained the system of the linear equations with constant factors. The results of calculation of system (4.1) are potentials of nodes, i.e. Knowing node potentials, it is defined all branch currents and capacity voltage:

; ; ; .

Parameters of discrete current models must be specified after definition and , time is increased on value of a step h, and process of calculation is executed over again while the number of integration steps will not achieve the given value.