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The main theorem of the circuit theory

Superposition theorem

Superposition theorem can be formulated as follows.

The reaction of the linear electric circuit on an arbitrary action to be representing a linear combination of the more simple actions, is equal to a linear combination of reactions caused by each of the actions taken separately.

The Proof:

When considering the method of loop currents loop current I of the i - th loop in the general case is defined by the expression (4.215)

(4.232)

 

where , - respectively the determinant of the matrix loop impedances system and the determinant obtained from replacing the i - th column by the column of loop EMF - MLE. Expending of the determinant on elements of the matrix-column - MLE, we get

(4.233)

 

where , , ... , , are algebraic adjunct of the determinant to the elements , , ... , , .

With that in the general case

 

(4.234)

 

Here - in a minor of determinant , obtained by deletion from the j - th row the i - th column.

From (4.233) it is shown the loop current I , considered as a circuit reaction, is equal to the sum of N components, each of which is a separate current, considered as a reaction to the loop EMF E , E , ... , E , considered as separate effects.

Similarly, when considering the method of nodal voltages node voltage U of i - th node in the general case, determined by the expression (4.231).

 

(4.235)

 

where , - respectively the determinant of the system of the matrix nodal admittances and the determinant obtained from by replacing the i - th column of the matrix - column node currents MNC

Expending the determinant on elements of the matrix column MNC, we get

 

(4.236)

 

where , , ... , , are algebraic adjunct of the determinant key to the elements , , ... , , . Here is determined by (4.234).

From (4.236) it is shown a single voltage U , considered as a circuit reaction, is the sum of N components, each of which represents a separate voltage, considered as a reaction to the nodal currents J , J , ... , J , considered as separate effects.

Superposition theorem does not apply to calculation of the powers through the currents or through voltages as the power is quadratic, that is nonlinear function of current or voltage.

 

Fig. 4.36

 

As an illustration of superposition theorem we consider its application to the calculation of circuits on the fig. 4.36. Here in the circuit there are two sources of energy (two effects): - current source J and voltage source E . The original circuit, based on the superposition principle is represented by two partial circuits, of the first of which there is only one source of energy - the current source J , and the second - another source of energy - voltage source E . From the action of the current source J in the first circuit currents I , I , I flow in the branches with the resistances . From the action of the voltage source E in the second circuit currents I , I , I flow in these branches . The full currents I , I , I , proceeding from the superposition principle, are defined as sums (4.237)



 

(4.237)

 

We calculate the first of partial circuits. The currents in the branches define the rule of alien resistance

 

(4.238)

 

(4.239)

 

(4.240)

 

Calculate the second of the partial circuit. Get

 

(4.241)

 

(4.242)

 

(4.243)

 

The full currents of (4.237)

(4.244)

 

(4.245)

 

(4.246)

 


Date: 2015-12-18; view: 834


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