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
v1 and
v2 . 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
The current that enters node 1 from the right can be stated as
The current that travels downward from node 1 is
We can arrange the expressions for each of the currents in terms of an equation via Kirchoff’s Current Law
We can combine and rearrange these terms into the equation
Now let us turn our attention to node 2. The current entering node 2 from the left is given by the expression
The current entering node 2 from the right is 2 A. The current leaving node 2 in a downward direction is
We proceed to combine these currents via Kirchoff’s Current Law
This equation can be rearranged as
So the pair of equations that we will use to solve for the two unknowns are
and
These equations may be expressed in matrix-vector form as
or
Let us find the inverse of the matrix A. The coefficients of this matrix are given by
The inverse can be found making use of the following formula
It should be noted that this formula works only with (2 x 2) matrices. For matrices of higher rank, other methods need to be applied.
For this example, the determinant of A is found as
We can incorporate this information to express the inverse matrix as
which can be written as
We can apply
A-1 to solve for the unknowns
Recognizing that
A-1A =
I , we find that
So the node voltages are given as
and
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
I1 ,
I2 and
I3 . 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 (
I1 ,
I2 and
I3 ), we must have three independent, linear equations.
Let us define the matrix
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
Now, we divide each term in row 2 by (5) to yield
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
We can then divide the terms of row 3 by (36/5) to produce
Interpretation of the third row tells us that the value for the third unknown (
I3 ) is 2 A. We can use the coefficients from the second row along with the value for I
3 to solve for I
2 .
which yields the result
Lastly, we may use the coefficients of the first row along with the previously determined values for
I2 and
I3 to produce the result for
I1 .
Insertion of the previously found unknowns yields
So we find the value for
I1 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
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
,
, and
. 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
and
. Use the matrix inversion method to find the two mesh current.
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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