Linearity theory and study

A function is said to be linear if it satisfies the properties of additivity and homogeneity. We can express these properties, mathematically, as:

₁

₂

Note that if we multiply the parameter x of the function by any constant, the result will be the same as multiplying the image of the function, at the point x, by the constant.

So, we can state that electrical circuits composed of resistors and voltage sources or current sources, whether independent or dependent (as long as they are linear), are systems of linear functions, where the mathematical properties of linear functions can be applied. It should be added that it is possible to apply these properties to circuits that have capacitors and inductors, as long as they have initial stored energy equal to zero. Thus, these circuits will also be governed by linear relationships, resulting in a circuit of linear functions.

2. Linearity

"For circuit analysis, the linearity technique involves setting a certain value for the quantity you want to calculate and then calculating the voltage or current at the source that produces the chosen value of the quantity."

In this way, we will obtain a certain value from the source that produces the arbitrated value of the quantity. Now, to find the true value of the quantity in the original circuit, simply apply a simple rule of three. Or, it is also possible to determine the multiplier constant of the function. For greater clarification on the use of this technique, let's look at an example practical.

Practical example

Let the circuit shown in Figure 17-01. We want to calculate the current value i₃.

Let's start solving the problem by arranging a value for i₃. To facilitate calculations, we chose integer values. If i₃ = 2 A, then e₁ = 30 i₃ = 30 x 2 = 60 V. As a consequence, i₂ = e₁/ 20 = 60 / 20 = 3 A. By the law of knots, we have to i₁ = i₂ + i₃ = 3 + 2 = 5 A . With this result we can calculate the value of V, since V = 8 i₁ + e₁ = 8 x 5 + 60 = 100 V. Therefore, we obtain the circuit shown in Figure 17-02.

Let's interpret our calculations. What we have just calculated means that when we have a voltage source equal to 100 V we obtain a current i₃ = 2 A. As we know that the circuit is linear, to know what the value of i₃ is when we change the value of the voltage source to 60 V, simply apply a rule of three simple.

Another way is to determine the constant multiplier of the function. So, let's call this constant k and define it as:

k = i₃ / V = 2 / 100 = 1 / 50

With this data we can write the function that determines the value of i₃ as a function of the value of the voltage source, that is:

i₃ = k V = V / 50

Logo, nesse exemplo temos:

i₃ = 60 / 50 = 1.2 A

As you can see, this technique is very simple and effective in solving problems. It can also be used in conjunction with other theorems, such as the superposition theorem. You can compare this result with the one shown in Problem 10-1.

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