Exercise
1: Introductory Problem
Consider
the network of resistors and batteries shown in the first figure below.
There
are three unknowns in that circuit, I1, I2,
and I3.
We can solve for these unknowns by building a system of equations using
Kirchhoff’s Laws:
- The
sum of voltages around any loop is zero.
- The
sum of currents at any junction is zero.
Applying
the voltage law to the left-hand loop, we get
V1−I1R1−I3R3=0(4)
From
the right-hand loop, we get
V2+I2R2−I3R3=0(5)
We
need one more equation, for which we can use the junction at top center
and the current law:
I1−I2−I3=0(6)
We
now have the requisite three independent equations, which we can solve
using methods learned in high-school algebra.
There’s
another way of solving this, using matrix methods. First, rewrite the
equations just a bit.
R1I1+0I3+R3I3 0I1−R2I2+R3I3 I1−I2−I3===V1V20(7)(8)(9)
And
now we can see that this can be written as a matrix equation!
⎡⎣⎢R1 0 10−R2−1R3R3−1⎤⎦⎥⎡⎣⎢I1 I2 I3⎤⎦⎥=⎡⎣⎢V1 V2 0⎤⎦⎥(10)
This
matrix equation,
Mx=b(11)
has
solution
x=M−1b(12)
where M−1 is
the inverse of M.
Most computational packages have built-in capability for inverting
matrices.
ASSIGNMENT
- By
carrying out the matrix multiplication explicitly, show that the matrix
equation above reduces to the system of equations from which it is
derived.
- Use
a matrix-solving package to find the currents.
- Check
your answers by substituting the currents into the equations and
verifying that they are solutions.
Exercise
2: More complicated problem
The
resistor network below is perhaps a bit more challenging.
ASSIGNMENT
- Write
a system of equations that can be used to solve for the currents in the
circuit above.
- Re-write
the system of equations from part 1 of this assignment as a matrix
equation.
- Use
a matrix-solving package to find the currents.
- Check
your answers: do the values of currents you found solve the equations
with which you started?
- You
may notice something interesting if you compare your solution here to
your solution for exercise 1. (Assuming the same values of R and V were
used in both exercises.) Explain why this happened.
Unless
your instructor provides you with other values, assume each voltage source
and each resistor has a value 10× its
identifying number. (i.e. V2=20 V, R3=30Ω.)