Answered Essay: Applying Kirchoff's current law at each of the nodes (black dots), and using the current directions indicated b

matlab language

Applying Kirchoffs current law at each of the nodes (black dots), and using the current directions indicated by the arrows, we get the following system of equations: 11-12 =0 Applying Kirchoffs voltage law around the three loops in the circuit, we get the equations: This gives 7 equations for 5 unknowns (the currents). From this point on, we drop the final equation from each group. PART 1: Write a script that puts these equations into matrix form. Then solve the equations to find the current through each branch for the following parameters: =10V PART 2: We now want to find by how much we can increase the voltage at Vi before melting a wire (which occurs when more than 5A of current passes through any of them). Using the appropriate type of loop and a step size of 0.1V, increase the voltage at Vi1 until the maximum current flowing through any of the wires exceeds 5A. Then print the largest acceptable value of Vi to the screen, in the form: The maximum voltage that can be applied is ? volts. Where? is replaced by the number of volts.

Applying Kirchoff’s current law at each of the nodes (black dots), and using the current directions indicated by the arrows, we get the following system of equations: i_1 – i_2 =0 i_2 + i_3 – i_4 i_4 – i_3 – i_5 = 0 i_5 – i_1 = 0 Applying Kirchoff’s voltage law around the three loops in the circuit, we get the equations: V – i_2R_2 – i_4R_4 – i_5R_5 = 0 V_1 – i_2R_2 – V_3 – i_5R_5 = 0 V_3 – i_4R_4 = 0 This gives 7 equations for 5 unknowns (the currents). From this point on, we drop the final equation from each group. Write a script that puts these equations into matrix form. Then solve the equations to find the current through each branch for the following parameters: V_1 =10V R_2 = 3 Ohm V_3 = 4V R_4 = 4 Ohm R_5 = 3 Ohm We now want to find by how much we can increase the voltage at V_1 before melting a wire (which occurs when more than 5A of current passes through any of them). Using the appropriate type of loop and a step size of 0.1V, increase the voltage at V_1 until the maximum current flowing through any of the wires exceeds 5A. Then print the largest acceptable value of V_1 to the screen, in the form: The maximum voltage that can be applied is ? volts. Where? is replaced by the number of volts.

Expert Answer

 

Part 1:

​The matrix is formed to solve the system of linear equation of the form AX=B where X is a 5-by-1 vector containing current values i1,i2,i3,i4 and i5. Below is the matlab code with comments describing matrices and after that output is attached.

Matlab code:

% ###################### 1 #########################
% variable to calculate are currents i1, i2, i3, i4, i5
syms i1 i2 i3 i4 i5
% given values
v1 = 10;
r2 = 3;
v3 = 4;
r4 = 4;
r5 = 3;

% given 5 equations
eqn1 = i1-i2 == 0;
eqn2 = i2+i3-i4 == 0;
eqn3 = i4-i3-i5 == 0;
eqn4 = v1-i2*r2-i4*r4-i5*r5 == 0;
eqn5 = v1-i2*r2-v3-i5*r5 == 0;

% equation matrix AX = B
[A,B] = equationsToMatrix([eqn1, eqn2, eqn3, eqn4, eqn5], [i1, i2, i3, i4, i5]);

% solving equation AX = B, X is a vector containing 5 current values
X = linsolve(A,B)
A
B

Output part 1:

matrix A

matrix B:

​matrix X( solution containing currents i1, i2, i3, i4, i5):

​Part 2:

​In part 2, voltage V1 is set as an variable and matrix is solved similarily as in part 1 but V1 as an variable. Which produces solution in terms of variable V1 which is found to be:

Putting V1 = 10 in the above solution gives answer same as in part 1. Now initialize V1= 10 V and calculate currents by putting V1 = 10 in above solution. Now in a while loop increase V1 by 0.1 in each step , calculate new currents by putting this V1 value in above solution matrix until either of i1, i2, i3, i4 or i5 gets 5A. Finally print V1.

Matlab Code:

%##################### 2 #######################
% now put v1 also an variable
syms i1 i2 i3 i4 i5 v1
% rest constants as it is
r2 = 3;
v3 = 4;
r4 = 4;
r5 = 3;

% define equations
eqn1 = i1-i2 == 0;
eqn2 = i2+i3-i4 == 0;
eqn3 = i4-i3-i5 == 0;
eqn4 = v1-i2*r2-i4*r4-i5*r5 == 0;
eqn5 = v1-i2*r2-v3-i5*r5 == 0;

% equation matrix
[A,B] = equationsToMatrix([eqn1, eqn2, eqn3, eqn4, eqn5], [i1, i2, i3, i4, i5]);

% solve system in terms of v1
sol = linsolve(A,B)

% initial value of v1
v1_init = 10;
% put v1= 10 in system of equation solution
currents = subs(sol,v1,v1_init);
% while currents less than 5
while double(currents(1))<5 && double(currents(2))<5 && double(currents(3))<5 && double(currents(4))<5 && double(currents(5))<5
v1_init = v1_init+0.1;
currents = subs(sol,v1,v1_init);
end

% print final value of v1 when any current gets 5
fprintf(‘The maximum volatge that can be applied is %2.2f volts.’,v1_init);
% print currents at that time
currents

Output part 2:

Currents at this voltage:

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