Solution of an electric circuit with 2 unknowns by matrix inversion
Let us apply our knowledge of matrices to assist us in the analysis of an electric circuit. We consider the circuit shown below.
In this example, we wish to solve for the two node voltages
v_{1} and
v_{2} . Since there are two unknowns in this problem, we must first establish two independent equations that reflect the operation of the circuit.
Kirchoff’s Current Law tells us that the sum of the currents that enter a node must equal the sum of the currents that leave a node. Let us focus first on node 1. The current that enters node 1 from the left can be stated mathematically as
$\frac{\text{10}V-{v}_{1}}{1\Omega}$
The current that enters node 1 from the right can be stated as
$\frac{{v}_{2}-{v}_{1}}{1\Omega}$
The current that travels downward from node 1 is
$\frac{{v}_{1}}{2\Omega}$
We can arrange the expressions for each of the currents in terms of an equation via Kirchoff’s Current Law
Solution of an electric circuit with 3 unknowns by gaussian elimination
Let us consider the electric circuit that is shown below.
Suppose that we are interested in determining the value of the three unknown currents
I_{1} ,
I_{2} and
I_{3} . In order to do so, we rely upon Ohm’s Law and Kirchoff’s Laws to develop a system of three independent, linear equations. We should note that because we have three unknowns (
I_{1} ,
I_{2} and
I_{3} ), we must have three independent, linear equations.
${I}_{1}+{I}_{2}+{I}_{3}=0$
$-2{I}_{1}+3{I}_{2}=\text{24}$
$-3{I}_{2}+6{I}_{3}=0$
Let us define the matrix
$A=\left[\begin{array}{ccc}1& 1& 1\\ -2& 3& 0\\ 0& -3& 6\end{array}\right]$
In order to find the unknowns, we must first find the inverse of the matrix A. This can be accomplished using elimination. To start the process, we adjoin the vector [0 24 0]
^{T} to the matrix A.
Next, we wish to force the left-most constant of row 2 to take on a value of 0. We can do so by multiplying each value in the first row by (-2) and subtracting the result from the corresponding value in row 2. This process yields
Next, we turn our attention to eliminating the (-3) term in row 3. We can do so by multiplying each term of row 2 by (-3) and subtracting the results from the corresponding terms in row 3. This produces the matrix
Interpretation of the third row tells us that the value for the third unknown (
I_{3} ) is 2 A. We can use the coefficients from the second row along with the value for I
_{3} to solve for I
_{2} .
Lastly, we may use the coefficients of the first row along with the previously determined values for
I_{2} and
I_{3} to produce the result for
I_{1} .
${I}_{1}+{I}_{2}+{I}_{3}=0$
Insertion of the previously found unknowns yields
${I}_{1}+4+2=0$
So we find the value for
I_{1} to be -6 A.
Exercises
Company A has more cash than Company B. If Company A lends $20 million to Company B, then the two companies would have the same amount of cash. If instead Company B gave Company A $22 million, then Company A would have twice as much cash as Company B. Use the matrix inversion method to find how much cash each company has.
A computer manufacturer sells two types of units. One unit is primarily marketed to the professional community and sells for $1,700. Another unit is marketed to students and sells for $900. In a typical month, the manufacturer sells 2,000 units. This accounts for $1,380,000 in sales. Use the matrix inversion method to find how many units of each type are sold.
A ship can travel 300 miles upstream in 80 hours. Under the same conditions, the same ship can travel 275 miles downstream in 65 hours. Use the matrix inversion method to find the speed of the current along with the speed of the ship.
The matrix
$A=\left[\begin{array}{ccc}2& 1& 3\\ 0& 6& 2\\ 1& 0& 1\end{array}\begin{array}{c}8\\ 4\\ 2\end{array}\right]$ represents a linear system with three unknowns. Use Gaussian elimination to solve for the three unknowns.
A system of 3 independent linear equations that govern the operation of the circuit below are
${i}_{1}+{i}_{2}+{i}_{2}=0$ ,
$-{i}_{1}-\text{24}+2{i}_{2}=0$ , and
$-2{i}_{2}+4{i}_{3}=0$ . Use Gaussian elimination to solve for the three currents.
Suppose that the value of each resistor in the figure below is 1 Ω. The mesh equations that govern the circuit are
$6V=2{i}_{a}-{i}_{b}$ and
$2{i}_{b}-{i}_{a}+\mathrm{9V}=0$ . Use the matrix inversion method to find the two mesh current.
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Source:
OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
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