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Theorem on the equivalent generator

 

This theorem is also known under the name Òhåvenin - Norton theorem and consists of two parts.

Thevenin - theorem – theorem on the equivalent voltage source - can be formulated as follows.

The current in any branch of the linear electric circuit do not change if the rest of the circuit replace by the equivalent voltage source, the EMF of which is equal to the voltage at the terminals of open branches, and the internal resistance - the resistance between the breaking points.

The Proof:

 

Fig. 4.37

 

Let us given electric circuit with voltage sources E , E , ... , E and resistances Z , Z , ... , Z (fig. 4.37,a). Select in this circuit the branch with impedance Z , connected between terminals k, l, with a current I . Include in the branch of k - l two equal in magnitude and opposite voltage source with the EMF E = E = U , where U is the voltage between the open terminals k, l (Fig. 4.37,b). Obviously, the current I in the circuit will not change. On the basis of the superposition theorems of, this circuit can be represented as the sum of the two circuits, the first of which includes EMF E , E , ... , E and E , and the direction of the EMF E is opposite to the direction of the current I (Fig. 4.37,ñ),and the second - only E (Fig. 4.37.d). Then the current

 

(4.247)

 

It is obvious the current I in the circuit of fig. 4.37,c is equal to zero, as the equivalent of EMF sources E , E , ... , E and E are equal and oppositely directed. Therefore the current

(4.248)

If we replace the impedance Z , Z , ... , Z equivalent impedance Z , then on fig. 4.37,d

 

(4.249)

 

Equivalent circuit with Z and E is shown in Fig. 4.37,e. It is visible, in this circuit when we break the k - l branch we get the open circuit voltage on these terminals

 

(4.250)

 

The theorem is proved.

Norton theorem - theorem on the equivalent current source can be formulated as follows.

The current in any branch of the linear electric circuits do not change if the rest of the circuit replace by the equivalent current source, the current of which is equal to the current short-circuit of this branch, and the internal conductance - conductance between the points of breaking this branch.

The proof of this theorem immediately follows from Fig. 4.37.e, if an equivalent voltage source with EMF E and the internal resistance Z replaced by an equivalent current source J and internal conductance Y (Fig.4.37.g).

 

Fig. 4.37.g

 

Here

(4.251)

 

From Fig. 4.37.g it is shown the current by the short-circuited k, l terminals

 

(4.252)

 

The open k, l terminals from Fig. 437,e,g give the conductance from the open terminals

 

(4.253)

 

The theorem is proved.

As an illustration of the equivalent generator theorem define the current I in the network of Fig. 4.36, with the application of this theorem.

Switch off the branch with Z , in which the current I flows and calculate the voltage between terminals 1, 2 of the circuit (Fig. 4.38). Convert the voltage source with the EMF E and the internal resistance Z into the equivalent current source



 

(4.254)

 

Then define equivalent current source by means of the algebraic summation of current source J and J

 

(4.255)

 

Internal admittance Y

 

(4.256)

 

Then, from (4.255), (4.256) we get

 

(4.257)

 

Now from (4.249), (4.256), (4.257) we get

 

(4.258)

 

 

that coincides with (4.244).

 

Fig. 4.38

 


Date: 2015-12-18; view: 1049


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