Linearity and superposition theorem

In summary, the conversation discusses a network of constant current with known resistances and currents at different switch positions. The problem is solved using the linearity and superposition theorem, which allows for the calculation of constants x and y. By putting the parallel network of R, 3R, and 4R inside the active network box, the constants are adjusted and the equation for current Ip is solved. The conversation also suggests using the compensation theorem and the proportionality theorem to solve for the current Ip'' when the switch is in the third position.
  • #1
Ivan Antunovic
111
4

Homework Statement


For the network of constant current shown in Figure 4 it is known that R1 = 50 Ω and , R = 10 Ω. When the switch P is
in the 1-position , current I = 50 mA and Ip = 70 mA known i . When the switch P is in
the 2-position , current I' = 40 mA and Ip' = 90 mA are known . Determine the current Ip''
when the switch switches to the third position.
Linearity_theorem.png

Homework Equations

The Attempt at a Solution


By linearity and superposition theorem we have a linear network and if we have our active load network we can write I = xU + y , where current I is the response and voltage U represents excitation , constants x and y are determined by the resistance of the active network ,where x has dimension of A/V and y has dimension of A.

By solving equation 1 and equation 2 I get that,
x = 0.2 and y = -0.23,
for the 3rd equation I get:
Ip" = xU1 + y = 0.2 U1 - 0.23
I have two unknowns Ip" ,U1 and only 1 equation.
What am I doing wrong?
 

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  • #2
What if you put the parallel network of R, 3R and 4R inside the active network box? You know the value of R and currents I and I'. Is it possible to find I'' using the same method you used above? I am not sure, it's just a suggestion. I'm not familiar with your method.
 
  • #3
cnh1995 said:
What if you put the parallel network of R, 3R and 4R inside the active network box? You know the value of R and currents I and I'. Is it possible to find I'' using the same method you used above? I am not sure, it's just a suggestion. I'm not familiar with your method.
It would only change constants x and y since the resistance of the active network would be changed , I would have form Ip = xI + y , where x would be non dimensional constant and y would be constant with dimension of amperes , still I would end up with 1 equation with 2 unknowns.
 
  • #4
Hm if my logic is right
fad.png


Answer should be for R I"p = 30 mA ?
 
  • #5
@cnh1995
For example look at those simulations, imagine this network in the black box is the active network and you don't know what is in it ( active since it has active sources, if there were only resistors equation would have y = 0)
active_network_1.png

active_network_2.png

capture

Let the V1 be excertation and the voltage measured by the voltmeter be response V2 .
So for the first two pictures we have
V1 = 200 V , V2 = 14.3 V
V1 = 300 V , V2 = 28.6 V
V2 = xV1 + y , and solving this equation you get
x = 0.142 and y = -14

Now suppose you wanted to find the voltage for V1 = 400 V
you already know constants so it's
V2 = V1*0.142 -14 = 400 * 0.142 -14 = 42. 8 V .

Look at the picture below , we get 42.8 V the same result.
active_network_3.png

screen shot on windows
That is the whole idea about this theorem as I said if there were only resistors in the black box we would have constant y = 0 , so equation would look like V2 = x*V1 ,linear homonogeous equation.
 
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  • #6
Try writing Ip in terms of that switched resistance (I'll denote it R* for convenience, i.e.,
Ip = kR* + y

If you need to find I'' then determine I as a similar linear relationship to R*.
 
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  • #7
NascentOxygen said:
Try writing Ip in terms of that switched resistance (I'll denote it R* for convenience, i.e.,
Ip = kR* + y

If you need to find I'' then determine I as a similar linear relationship to R*.
70 m = x30 +y
90 m = x40 +y

x = 0.002 , y = 0.01

Ip" = xR + y = 0.002 * 10 + 0.001 = 30mA , may I ask you how did you come up with this?
I was thinking it could be since R becomes excitation and by compensation theorem
http://www.electrical4u.com/compensation-theorem/

if a current or voltage is known on the resistor it can be replaced by ideal voltage/current source and vice versa
 
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  • #8
The only independent variable (called "input") in this system is the switch position, i.e., the resistance R. So it seemed that you should be looking for the linear relationship between output Ip and input R*.
 
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  • #9
NascentOxygen said:
The only independent variable (called "input") in this system is the switch position, i.e., the resistance R. So it seemed that you should be looking for the linear relationship between output Ip and input R*.
By the way I think it's called proportionality theorem in english literature.
 

What is linearity?

Linearity is a mathematical property that describes the relationship between two variables. In a linear system, the output is directly proportional to the input. This means that if the input is doubled, the output will also double. Linearity can be applied to various fields of science, including physics, engineering, and economics.

What is the superposition theorem?

The superposition theorem is a fundamental principle in physics and engineering that states that the response of a linear system to multiple inputs is equal to the sum of the individual responses to each input. In other words, the total response is the sum of the individual responses. This theorem is useful in analyzing complex systems that can be broken down into simpler parts.

How do linearity and superposition theorem relate to each other?

Linearity and superposition theorem are closely related concepts. The superposition theorem can only be applied to linear systems, as it relies on the linearity property. This means that the response of a non-linear system cannot be determined using the superposition theorem. Additionally, the superposition theorem can be used to prove the linearity of a system, as it shows that the output is directly proportional to the input.

What are some real-world applications of linearity and superposition theorem?

Linearity and superposition theorem have numerous applications in various fields of science and engineering. In electrical circuits, the superposition theorem can be used to analyze complex circuits and determine the voltage or current in a specific component. In economics, linearity and superposition theorem are used to model supply and demand relationships. In physics, they are used to study the behavior of waves and other physical systems.

Are there any limitations to linearity and superposition theorem?

While linearity and superposition theorem are powerful tools, they do have some limitations. As mentioned earlier, the superposition theorem can only be applied to linear systems. Additionally, the linearity property may not hold true in some cases, such as when dealing with highly non-linear systems or at extreme input levels. It is important to carefully consider the assumptions and limitations when applying linearity and superposition theorem to real-world problems.

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