9k=Theorems of sources with Null-Action (Vaschy’s Theorems)

            Vaschy’s Theorem About Electromotive Sources

            Enounce: If we’ll add sources of electromotive voltage with equal values on all the 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 circuit we should 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 (Theorem of the Equivalent Voltage – Generator)

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

Enounce: The current Iab through the side (ab) will be equal with the ratio between the voltage of idle-running at the terminals of the side (ab), Uab0, at witch we add the value of the electromotive voltage of this side, Eab and the sum between the impedance of the external circuit at the terminals of the side (ab), Zab0 with passivised sources, at which we added the impedance of the side (ab), Zab :


Norton’s Theorem (or Theorem of Equivalent Generator of Current)

xf5dKWyuY1cpB1i2X4FmXLV1ZykWtDXOl2WSUSrj            Using Norton’s theorem we’ll be able to 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), Zab.

Enounce “The voltage between the terminals of a side (ab) of an electric circuit is equal with the report between the current of short – circuit of the side (ab) and the sum between the admittance of the side (ab) and the admittance of the circuit which is exterior to this side with passivised sources


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


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

iPn79AvwLOJBUEAA7 is the admittance of the side (ab).

RplW7RtTYt3DjyjVdgm7TOgkq8E1N8g4oPW+nFPL represents the equivalent admittance of the circuit which is exterior to the side (ab), with passivized sources (equal with the inverse of the equivalent impedance exterior to the side (ab)).

Theorem of Power Conservations in P S P R

According with Kirchhoff’s first theorem, for generally un-isolated network we have V7h9tVu1XWvfG0gYScNyUstJoVS0hBOW5MLESocA      

which can be expressed in complex using the theorems of complex representation:OTpbCmABCwwIhwioitnSzbxzkgJo4eIFhj6h1JTu          

Applying the properties of the conjugate at complex numbers we find:074OwQ7c3dN2LyvLTrSGS3sudOFDQhMAt1++8Ezk                                       DdxAmZGZhhridWHBscmTomYaBAA7 

then we’ll sum for all the nodes of the circuit (a=1, 2.. n), resulting: gR7W5Q6deVLjUAG0CidGGrFwLLdoBNQXA+VGmL9g  

Tthe theorem of powers conservation may be enounced as follows: “The sum of the complex apparent powers received by an electric network through the interconnecting terminals (MbpiXKoIjSmjCYrCsx1NnM2ydbswLV5R0cJAgA7)  is equal with the sum of the complex apparent powers received by all the sides of the interconnected network (+AgYJlCwGGh4iID4N9EIYREkoTFIuMfI4BFWNJDh).

Using Joubert’s theorem for a generally active side of circuit: JMYcCt5GRKO33apcXPcpgnMNOMkbIV7z4TgLTRCD 

we can deduce: geBfzVOEQQAOw==                                                                                              

            “The complex apparent power received by the passive elements of the circuit is due to the active elements of 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): 3eoSJCHCF0XBr8AKgxkIUCqY+zahVJmYqFFJSEAA                                                                                                    (5.332)

the relation (5.332) can also be written under the form: I2Y2EppRrR709JQlAEAA7                       

hence: S5+ConBRdE4OUUaGqlilVKRWq7QAlJ2Ku3WurwGx     OSIDPmjSObOmzwMMCRQgQIhZFVNHF4D4xVSBUyOm                                                                                                 

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


1) The active powers are always received by passive elements of circuit (resistors); in exchange the active powers given by the circuit sources and the active powers received by 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 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 (Qz) can be positive, negative or null.

xmtKhBEDX6QyyIn+MUdc8dE+q7jOzXLWpZ2TZABjFor Zk and Zkj we have:

4wuGASYu5zSoBzNpqrpi6iCmISkB47RVfOCiHjYb  lZNeyGbhGWhIQgKFA2TFQrw1CrJix49R5K7Fe3Li             

For mutual couplings: Zkj=Zjk and:

Zkj . Ij . Ik*+Zjk . Ik . Ij* = Zkj . (Ij . Ik*+Ik . Ij*) =

Zkj . [Ij . Ik*+(Ij . Ik*)*] =

LvvsnbNOBfvtWwQBADs=                                                                                           (5.340)

5yj+G6ZbPvOabvPXNBwEAOw== (5.341)

Then from (5.341) we have: PoK138DZtfO8tZIbtq7oeA9d6mF7uBWElsgHqCCc                                                                        B0pQkzkgmJArqEIH+rvcsQJ3C41oPAmxS2pK9KLb                            (5.343)

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

-for active powers: 8B42P9m576DjaOAArQdOwijgM9AQmtlceA9xuahi                                                    

-for reactive powers:q661rdeF4HrWB8S57rYx86UDDIMjhamOdnXznZ3p               


            WnBTNWBKPgPV6dQB0KKWyS3TGPNACjjH3IR4aNMd                (Fig. 5.70 (a))                                         

             dnhaB+gsSDWo8tVeikYmvaGaSKdfnrLhaJqIwCpR               (Fig. 5.70 (b))

            QIBwSCwaj8ikEikIOAfLqHRKrQoJBYN1y+0qBQev ,  then: l0AFsForiTiwIWQZHGZetB68CMMbBQEAOw==  

Theorem of the Maxim Transfer of Active Power in PSPR

JO54n3ve6753sSMjCAA7            A load-impedance Zs=Rs+jXs, connected at the terminals of an electric network (fig. 5.71) receives a maximum active power if: nSCLGUh4AIBSMusKOnhcSAEhBMDDiBIoI0hlo8BU                                                         (5.353)                                                                


Dem. We can equivalate the exterior Zs circuit at terminals with an equivalent generator of electromotive voltage, according to Thevenin’s theorem with the parameters E  and gif;base64,R0lGODlhHwAZAHcAMSH+GlNvZnR3Y: Zgif;base64,R0lGODlhEwAYAHcAMSH+GlNvZnR3Y= R+jX                                                                             

In this situation the current through Zs will be: u+dqgQwDJa53Pdurpls8yEu0cydp5WilgCjBLCCu                               

The active power received by Zs will be: fVAMiFWDbwvn08Q0VW60RVsfXBa3CiVXDattLpqo           

To find the extreme value of PRs we’ll impose condition of extreme with partial derivatives: GiLUfUmvzPUe0WK59qNlcBXM8K19tzBA4CHRzRRk                                                     (5.357)

PDNoCSfb+AZIWB5TQz4XxMoGRMTfO0VcQxVD9Fk0  hence:

d2DTUoxx1FiA9CuxiSdVXYPVEXO2EIAAAOw==           (5.360)

hence:  RS   =   R                                                                                                              (5.361)

            gg5kr5FQg9YtacV9US1nJMbSd3nEunXrRPHnmt8I                                                                             (5.362)

hence: oSZumvWKj1663IjErEdBSxKsXXQLpULNXrBaYRb3                                                                                   (5.363)

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

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

             vDRQEADs=                                                                                  (5.365)