Compensation theorem

Compensation theorem can be formulated as follows.

Currents and voltages in the electric circuit will not change, if any branch of this circuit replace an ideal voltage source , the EMF of which is equal to the voltage at the terminals of this branch, and is the opposite in the direction of it, or to the ideal current source, the current of which is equal to the current of the branch and coincides with it in the direction.

Proof:

Fig. 4.40

Let us consider Fig. 4.40. Here in the electric circuit is picked out branch c - d with impedance Z and current I (Fig. 4.40.a). Branch voltage

(4.262)

Include in the branch c - d two voltage sources, EMF of which are equal to the voltage U and directions of which are the opposite (Fig.4.40.b). Obviously, the current I will not change. As the voltage between points a - b and b - d are equal in magnitude and opposite in direction, the resulting voltage between points a - d is equal to zero and these points can be connected by a short-circuit jumper (the dotted line in Fig. 4.40.b). The result is equivalent circuit on Fig. 4.40.c, in which the branch c - d is included only EMF E = U and current which is equal to the original current I .

Thellegen theorem

Thellegen theorem is extremely general. It applies to any electric circuits with the concentrated parameters, which contain any elements: linear and non-linear, passive and active, changing in time or permanent. This unity is achieved due to the fact that Thellegen theorem is based on the Kirchhoff’s laws.

Thellegen theorem (published in 1952) can be formulated as follows: The sum of the voltages and currents in each of the branches of the electric circuit is equal to zero. That is

(4.263)

where u , i - instantaneous values of voltage and current in the k-th circuit branch.

Proof:

Fig. 4.41

Let us consider Fig. 4.41. Here and nodes of electric circuit are connected with k-branch by impedance Z, in which the current flows of indicated direction. Voltage nodes relative to the base node 0 are marked , . Voltage of the k-th branch is indicated by the u . Denote also

(4.264)

It Is obvious

(4.265)

Write down the product

(4.266)

or

(4.267)

where

(4.268)

Add (4.266) and (4.267). Get

(4.269)

Let us sum up (4.269) for all branches of the electric circuit. We get to b branches

(4.270)

Here the double summation is introduced, as work under the sign of the amount is performed for all possible combinations of branches. The n - the number of nodes of the circuit. If there is no branch connecting the node with the node , then write i = I in (4.270) . The right part of equation (4.270) can be divided as follows

(4.271)

In (4.271) for each value of the sum

(4.272)

since is the sum of the currents of all branches going out of the node . Also, for each value sum

(4.273)

so how is the sum of the currents of all branches going out of the node . It follows from the law of Kirchhoff for currents.

Thus, from (4.270) - (4.273), we obtain

(4.274)

The expressions in the left part of the the expression (4.274) is the sum of the instantaneous powers of all branches of the circuit, that is identical to the condition of the instantaneous powers balance (3.66, p). Hence, the condition of the power balance is a special case of Tellegen theorem. The theorem is proved.

Fig. 4.42

As an illustration of Tellegen theorem let us consider the network of electric circuit on Fig. 4.42. Here for each of the branches specified instantaneous currents and voltages. Let for some time

(4.275)

(4.276)

Tellegen theorem fair for electrical circuits, which satisfy the Kirchhoff laws. Here according to the Kirchhoff’s law for the currents:

for node 1:

(4.277)

for node 2:

(4.278)

for node 3:

(4.279)

to node 4:

(4.280)

According to the Kirchhoff’s law for the voltages:

for the loop e - L - C :

(4.281)

for the loop C - L - R :

(4.282)

Now, by Thellegen theorem

(4.283)

Date: 2015-12-18; view: 769