## ELECTRICAL NETWORK THEOREMS

Electric circuit theorems are always beneficial to help find voltage and currents in multi-loop circuits. These theorems use fundamental rules or formulas and basic equations of mathematics to analyze basic components of electrical or electronic parameters such as voltages, currents, resistance, and so on. These fundamental theorems include basic theorems like the Superposition theorem, Nortonâs theorem, the Maximum power transfer theorem, and Theveninâs theorems. Another group of network theorems that are mostly used in the circuit analysis process includes the Reciprocity theorem and Millmanâs theorem.

#### 1) SUPERPOSITION THEOREM:

As applicable to AC networks, it states as follows:

In any network made up of linear impedances and containing more than one source of emf, the current flowing in any branch is the phasor sum of the currents that would flow in that branch if each source were considered separately, all other emf sources being replaced for the time being, by their respective internal impedances (if any).

Note. It may be noted that independent sources can be âkilledâ i.e. removed leaving behind their internal impedances (if any) but dependent sources should not be killed.

#### 2) THEVENINâS THEOREM:

As applicable to AC networks, this theorem may be stated as follows:

The current through a load impedance ZL connected across any two terminals A and B of a linear network is given by Vth/(Zth + ZL) where Vth is the open-circuit voltage across A and B and Zth is the internal impedance of the network as viewed from the open-circuited terminals A and B with all voltage sources replaced by their internal impedances (if any) and current sources by infinite impedance.

#### 3) RECIPROCITY THEOREM

This theorem applies to networks containing linear bilateral elements and a single voltage source or a single current source. This theorem may be stated as follows:

If a voltage source in branch A of a network causes a current of 1 branch B, then shifting the voltage source (but not its impedance) of branch B will cause the same current I in branch A.

It may be noted that currents in other branches will generally not remain the same. A simple way of stating the above theorem is that if an ideal voltage source and an ideal ammeter are inter-changed, the ammeter reading would remain the same. The ratio of the input voltage in branch A to the output current in branch B is called the transfer impedance.

Similarly, if a current source between nodes 1 and 2 causes a potential difference of V between nodes 3 and 4, shifting the current source (but not its admittance) to nodes 3 and 4 causes the same voltage V between nodes 1 and 2.

In other words, the interchange of an ideal current source and an ideal voltmeter in any linear bilateral network does not change the voltmeter reading.

However, the voltages between other nodes would generally not remain the same. The ratio of the input current between one set of nodes to the output voltage between another set of nodes is called the transfer admittance.Â

#### 4) NORTONâS THEOREM:

As applied to AC networks, this theorem can be stated as under:

Any two-terminal active linear network containing voltage sources and impedances when viewed from its output terminals is equivalent to a constant current source and a parallel impedance. The constant current is equal to the current which would flow in a short-circuit placed across the terminals and the parallel impedance is the impedance of the network when viewed from open-circuited terminals after voltage sources have been replaced by their internal impedances (if any) and current sources by infinite impedance.Â

#### 5) MAXIMUM POWER TRANSFER:

For any power source, the maximum power transferred from the power source to the load is when the resistance of the load R_{L} is equal to the equivalent or input resistance of the power source (R_{in} = R_{Th} or R_{N}). The process used to make R_{L} = R_{in} is called impedance matching.

This theorem is particularly useful for analyzing communication networks where the goals is transfer of maximum power between two circuits and not highest efficiency.

*Cases:*

1. When load is purely resistive and adjustable, MPT is achieved when R_{L} = | Z_{g} | = â ( R^{2}_{g} + X^{2}_{g }).

2. When both load and source impedances are purely resistive (i.e. X_{L}= X_{g}= 0), MPT is achieved when R_{L} = R_{g.}

3. When

R_{L} and X_{L} are both independently adjustable, MPT is achieved when X_{L}= -X_{g }and R_{L} = R_{g.}

4. When X_{L} is fixed and R_{L }is adjustable, MPT is achieved when R_{L }= â [R^{2}_{g} + (X_{g}+ X_{L})^{2}]

#### 6) MILLMANâS THEOREM

It permits any number of parallel branches consisting of voltage sources and impedances to be reduced to a single equivalent voltage source and equivalent impedance. Such multi-branch circuits are frequently encountered in both electronics and power applications.

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