Determination of zero approximation

Let zero approximation of the current through the circuit is I₀=0, then at the first iteration step difference between initial and following approximation will be:

I₁ - I₀= f (I₀)/ f'(I₀)= -10/∞ =0,

i.e, accord to given approximation process is been divergent.

Let zero approximation I₀=1, then:

I₁ - I₀= f (I₀)/ f'(I₀)= -7.91/1.03=-7.68.

Hence, I₀=1 could be taken as a zero approximation

Program of calculation in matlab environment

Main program (script-file):

% Caculation of the simple circuit by the Newton method

% Newt1

% Initial data

E=10; R=0.5; a=4^(1/3);

% Number of iterations

N=20;

% Initial approach

k=1;

I(k)=1;

% Iterative process

for k=2:N

fk_1=fun1(I(k-1),R,a,E); % call of f(i)

dfdIk=dfdI(I(k-1),R,a); % call of f'(i)

I(k)=I(k-1)-0.5*fk_1/dfdIk; % calculation of the next approach

end

% display of iterative process

plot(I);

Subroutine-function (function-file) of calculation f(i):

% calculation of f(i)

function f=fun1(I,R,a,E)

f=R*I+a*I^0.333-E;

Subroutine-function (function-file) of calculation f’(i):

% calculation of f'(i)

function der1=dfdI(I,R,a)

der1=R+a/(3*I^(2/3));

Results of calculation

In the figure 6.2 graphical dependence of calculated current through the given circuit (fig. 6.1.) accord to the every iteration step is represented.

The above represented curve I(n) demonstrates that practically it is enough of 12 iteration steps for satisfactory accuracy of solution.

Figure 6.2 – Graphical dependence I(n)

Individual tasks

Calculate the current through the circuit (fig.6.1), if the VACH of nonlinear element is given accord to table 6.1.

The task for fulfillment

· Study items 6.1 – 6.3.

· Repeat the program adduced in item 6.4. and arrive at result dependence I(n) (fig.6.2);

For the given function of VACH of nonlinear resistor R_n:

· Develop mathematical model of calculation of the current through the circuit (fig.6.1);

· Make the program realizing developed algorithm;

· Organize graphical output of the calculated function of current accord to the every iteration step as function of step number I(n);

· Debug the program;

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

· Draw up report on the laboratory work.

Table 6.1

# of variant	VACH of nonlinear element Rn	# of variant	VACH of nonlinear element Rn
1	I=2.5U3	16	I=3U2 sign(U)
2	I=6U2sign(U)	17	I=0.5U4 sign(U)
3	I=2Ö|U| sign(U)	18	I=5Uarctg(U)
4	I=3U5	19	I=0.2arctg(U)
5	I=3[logaU-1]	20	I=8.6U2 sign(U)
6	I=3U3	21	I=3.5[exp(bU)-1]
7	I=argtg(U/5)	22	I=2arctg(U)
8	I=5U2 sign(U)	23	I= 9.4U2sign(U)
9	I=8|U|4/3 sign(U)	24	I=1.7U3
10	I=7U3	25	I=8U3
11	I=6.1U2 sign(U)	26	I=6.9[exp(U)-1]
12	I=5.2U3	27	I=6.9U3
13	I=9.3[lnU-1]	28	I=2[exp(U/2)-1]
14	I=0.5U4 sign(U)	29	I= 7[logaU-1]
15	I=8|U|3/2 sign(U)	30	I=5.5[lnU-1]