Fuse+the+Car

The problem
Suppose this is your car. You are completely re-doing its electrical system. You have new headlights (R1), new Radio (R2), and heater fan (R3). Knowing the characteristics of these ‘loads’ we now need to choose the proper size fuse that will accommodate all 3 loads from the 12 V battery. One solution to this can be solved using a set of simultaneous equations. These equations describe the currents flowing in the 3 loops shown above. An engineer has provided the systems equations as follows: R1 (i1 – i2) = V R2(i2-i3) + R1 (i2-i1) = 0 R3i3 + R2 (i3-i2) = 0 Or [ R1 -R1 0 ] i1 V  [ -R1 (R1+R2) -R2 ] i2 = 0 [ 0 -R2 (R2+R3) ] i3 0



Our specific solution: Although the above will provide a general solution to solving the currents in the above circuits, we want to use typical values and solve for a realistic fuse value. So we let R1=4, R2=4, and R3=2. Our battery, V is 12.

Matrix Representation
Solving – solutions using Cramer’s rule, and Row Operations on an Augmented Matrix.

Instructions: (Steps 1-6 could be done in groups)
1. Substitute the above values into the matrix above. 2. Use Cramer’s rule to solve for i1, i2, and i3. 3. Then, form an Augmented matrix (using the R matrix | the V column vector). 4. Solve for i1, i2, and i3 using Row Operations. 5. Do the results agree? 6. The Identity matrix plays a role in each solution method. Describe your observations concerning the role of the identity matrix in each. 7. Pick new values for R1-4 and V. Predict the outcome for i1. Solve this new configuration using Cramer’s rule and Row Operations. 8. Each student will submit individual results. The first section will show the results from the group work. The second section will document the student’s individual selection and solving process. Be sure to notate your predicted value for i1 and the actual value from the solution.