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)
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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)
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(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)
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(4.242)
(4.243)
The full currents of (4.237)
(4.244)
(4.245)
(4.246)
Theorem on the equivalent generator
This theorem is also known under the name Тhеvenin - Norton theorem and consists of two parts.
Thevenin - theorem – theorem on the equivalent voltage source - can be formulated as follows.
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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,с),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
Reciprocity theorem
Reciprocity theorem can be formulated as follows.
If a voltage source with the EMF E or current source J is included in the branch a - b of the linear electric circuits, not containing other energy sources, and creates in the branch c - d current I , than the same voltage source E or current source J ,included in the branch c - d, creates in the branch a - b the same current I.
Proof:
Fig. 4.39
Let us consider Fig. 4.39. Here voltage source E . included in the branch a – b of the passive linear electric circuits, creates in the branch c - d with impedance Z current I (Fig. 4.39,a). Take this source in the branch c - d. Define the current in the branch a - b. Let branch a - b is included in the loop n, and the branch c - d - in the loop k of linear electric circuit. Let's calculate the circuit according to the method of loop currents.
1. Let voltage source E is included to the loop n (4.39.a). Then the current of the k – th loop
(4.259)
where: - the determinant of the system of loop impedance matrix;
- the determinant, resulting from by disclosure of the on the column of loop EMF. As EMF E is included only in the loop n, and the rest of the circuit is passive, then all determinants, except the determinant , equal to zero.
2. Let voltage source E is included in the loop k (Fig. 4.39.b). Then the current of the n th loop
(4.260)
where: - the determinant, resulting in the disclosure of the on the column of loop EMF. As EMF E is included only in the loop k, and the rest of the circuit is passive, then all determinants, except the determinant , equal to zero.
It is known, that the matrix of the loop impedances is symmetric about the main diagonal, that is = . Therefore
(4.261)
The theorem is proved.
As an illustration of reciprocity theorem define the current I in the circuit of fig.4.36, used when considering the principle of the superposition. Current I in Fig.4.36 was been previously defined by the expression (4.242). Take voltage source E into the branch with impedance Z . Choose the direction of E coinciding with the direction of current I . Then the current
(4.262)
that coincides with (4.242).
Reciprocity theorem is valid only for linear passive circuits. For nonlinear and active circuits in the general case, reciprocity theorem is not performed.
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)
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