Circuit Theorems: Vaschy, Thevenin, Norton, and Power Transfer

Posted on Dec 25, 2024 in Electronics

Vaschy’s Theorems for Null-Action Sources

Vaschy’s Theorem About Electromotive Sources

Enounce: If we add electromotive voltage sources with equal values on all sides converging to a node by serial connection, the source having identical orientations reported to the common node, then the currents from the sides of the circuit remain unchanged.

Vaschy’s Theorem About Current Sources

Enounce: If along a loop of a circuit we add current sources connected in parallel with the sides of the respective loop, having identical orientation in report with the sense of closing along the respective loop of circuit, the currents through the circuit branches remain unchanged.

Theorems of Equivalent Generators

Thevenin’s Theorem (Equivalent Voltage Generator)

The theorem of the equivalent voltage generator allows us to determine the current through a side of an electric circuit without determining all the currents of the circuit.

Enounce: The current I_ab through the side (ab) will be equal to the ratio between the idle-running voltage at the terminals of the side (ab), U_ab0, to which we add the value of the electromotive voltage of this side, E_ab, and the sum between the impedance of the external circuit at the terminals of the side (ab), Z_ab0 with passivized sources, to which we added the impedance of the side (ab), Z_ab :

Norton’s Theorem (Equivalent Current Generator)

Using Norton’s theorem, we can determine the voltage between the terminals of a side (ab). To find the current through the side (ab), we should divide the value of the voltage determined with this theorem by the value of the impedance of the side (ab), Z_ab.

Enounce: The voltage between the terminals of a side (ab) of an electric circuit is equal to the ratio between the short-circuit current of the side (ab) and the sum between the admittance of the side (ab) and the admittance of the circuit that is exterior to this side with passivized sources.

To prove Norton’s theorem (or the theorem of the equivalent current generator), we’ll start from Thevenin’s theorem and calculate the voltage between the terminals of the side (ab):

– is the short-circuit current through the side (ab), due to the sources exterior to this side, with the independence of the side (ab) in short circuit;

– is the admittance of the side (ab).

– represents the equivalent admittance of the circuit that is exterior to the side (ab), with passivized sources (equal to the inverse of the equivalent impedance exterior to the side (ab)).

Theorem of Power Conservation in PSPR

According to Kirchhoff’s first theorem, for a generally un-isolated network, we have

which can be expressed in complex form using the theorems of complex representation:

Applying the properties of the conjugate at complex numbers, we find:

Then, we’ll sum for all the nodes of the circuit (a=1, 2.. n), resulting in:

The theorem of power conservation may be stated as follows: “The sum of the complex apparent powers received by an electric network through the interconnecting terminals ( ) is equal to the sum of the complex apparent powers received by all the sides of the interconnected network ( ).”

Using Joubert’s theorem for a generally active side of a circuit:

we can deduce:

“The complex apparent power received by the passive elements of the circuit is due to the active elements of the circuit and to the apparent complex powers considered from the interconnecting terminals.”

As generally any complex apparent power has a real part (active power) and an imaginary part (reactive power): (5.332)

the relation (5.332) can also be written in the form:

hence:

“The active (reactive) power received by the passive elements of the circuit is due to the active (reactive) powers given by the circuit sources and to the active (reactive) powers received by the circuit through the interconnection terminals.”

Observations

1) The active powers are always received by passive elements of the circuit (resistors); in exchange, the active powers given by the circuit sources and the active powers received by the circuit through the interconnection terminals are not necessarily positive.

But generally speaking, the total active power due to the circuit sources is positive, for an identical rule of association voltage – current both at the active elements and at the passive elements.

2) The reactive powers “received” by the passive elements of the circuit in fact mean a reactive power with sign:

• positive – if the reactive elements are coils;
• negative – if the reactive elements are capacitors.

The total sum of these powers (Q_z) can be positive, negative, or null.

For Z_k and Z_kj we have:

For mutual couplings: Z_kj=Z_jk and:

Z_{kj .} I_{j .} I_k*+Z_{jk .} I_{k .} I_j* = Z_{kj .} (I_{j .} I_k*+I_{k .} I_j*) =

Z_{kj .} [I_{j .} I_k*+(I_j . I_k*)*] =

(5.340)

(5.341)

Then from (5.341) we have: (5.343)

With the relations (5.342) and (5.343), we can make balance sheets at electric networks interconnected as follows:

-for active powers:

-for reactive powers:

(Fig. 5.70 (a))

(Fig. 5.70 (b))

, then:

Theorem of the Maximum Transfer of Active Power in PSPR

A load-impedance Z_s=R_s+jX_s, connected at the terminals of an electric network (fig. 5.71), receives a maximum active power if: (5.353)

Dem. We can equate the exterior Z_s circuit at terminals with an equivalent generator of electromotive voltage, according to Thevenin’s theorem with the parameters E and : Z = R+jX

In this situation, the current through Z_s will be:

The active power received by Z_s will be:

To find the extreme value of P_Rs, we’ll impose the condition of extreme with partial derivatives: (5.357)

hence:

(5.360)

hence: R_S = R (5.361)

(5.362)

hence: (5.363)

As the active power given by the equivalent source of the circuit is:

the efficiency of the active power transmission to the load under the conditions of a maximum active power absorbed by the charge is:

(5.365)