Solving for currents and voltages in multi-loop electric circuits can be quite complicated, particularly for AC circuits. The voltage law and current law always apply, but using them may lead to long systems of equations. Certain theorems help with network analysis:
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| Fig. Basic Network |
The superposition theorem is applicable to linear and bilateral networks.
Statement: It states that in an active linear network containing several sources, the voltage across or current through any element of that network is equal to the sum of individual voltages or currents, produced separately across or in that element by each source acting independently with all the remaining sources replaced by their respective internal impedances.
Statement: It states that in an active linear network containing several sources, the voltage across or current through any element of that network is equal to the sum of individual voltages or currents, produced separately across or in that element by each source acting independently with all the remaining sources replaced by their respective internal impedances.
The Thevenin's theorem is used for simplification of complicated networks.
Statement: Any network containing active and passive elements and one or more dependent and independent voltage/current sources can be replaced by an equivalent network containing a voltage source ( Vth or Voc) and a series impedance (Zth). Vth is the Thevenin's equivalent voltage. It is the open circuit voltage measured at two terminals of interest with load impedance ZL removed. Zth is the Thevenin's equivalent impedance. It is the impedance with load impedance ZL removed when all the independent voltage/current sources are replaced by their internal impedances.
Statement: Any network containing active and passive elements and one or more dependent and independent voltage/current sources can be replaced by an equivalent network containing a voltage source ( Vth or Voc) and a series impedance (Zth). Vth is the Thevenin's equivalent voltage. It is the open circuit voltage measured at two terminals of interest with load impedance ZL removed. Zth is the Thevenin's equivalent impedance. It is the impedance with load impedance ZL removed when all the independent voltage/current sources are replaced by their internal impedances.
This theorem is called as the dual of the Thevenin's theorem. This is because the Thevenin's equivalent voltage source is converted to an equivalent current source. The obtained equivalent is called Norton's equivalent.
Statement: Any linear network containing of active sources and bilateral elements can be replaced by an equivalent circuit containing a current source called Norton's equivalent current (IN or ISC) and an equivalent impedance in parallel (Zth or Zeq) across the two terminals of load. IN is the short circuit current flowing through the short circuited path that is replaced of load impedance ZL. Hence it is also called short circuit current ISC or Norton's current IN. The equivalent impedance Zth is obtained when looking through load impedance, with load impedance ZL removed and all the independent voltage/current sources are replaced by their internal impedances.
Statement: Any linear network containing of active sources and bilateral elements can be replaced by an equivalent circuit containing a current source called Norton's equivalent current (IN or ISC) and an equivalent impedance in parallel (Zth or Zeq) across the two terminals of load. IN is the short circuit current flowing through the short circuited path that is replaced of load impedance ZL. Hence it is also called short circuit current ISC or Norton's current IN. The equivalent impedance Zth is obtained when looking through load impedance, with load impedance ZL removed and all the independent voltage/current sources are replaced by their internal impedances.
The maximum power transfer theorem is used in communication circuits for transferring maximum power from source to load.
Statement: In an active network, maximum power will be transferred from source to load if the load impedance is the complex conjugate of the internal impedance of the circuit as seen from terminals of the load.
Statement: In an active network, maximum power will be transferred from source to load if the load impedance is the complex conjugate of the internal impedance of the circuit as seen from terminals of the load.
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