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**Network Theorems** The fundamental theory on which many branches of electrical engineerings, such as electric power, electric machines, control, electronics, computers, communications, and instrumentation are built is the Electric circuit theory. So here the network theorem helps us to solve any complex network for a given condition.**Note: **All the theorems are only applicable to Linear Network only, according to the theory of Linear Network they follow the condition of Homogeneity & Additivity.

Homogeneity Principle if input x(t) gives → response y(t) then it must follow ⇔k x(t) →k y(t)

Additivity Principle if two input x_{1}(t)+ x_{2} (t)⇔ y_{1}(t) + y_{2}(t) then k_{1}x_{1}(t) + k_{2}x_{2} (t) ⇔ k_{1} y_{1}(t) + k_{2} y_{2}(t)**1. SUPERPOSITION THEOREM **

In any linear bilateral network containing two or more independent sources (voltage or current sources or combination of voltage and current sources), the resultant current / voltage in any branch is the algebraic sum of currents / voltages caused by each independent sources acting alone,with all other independent sources being replaced meanwhile by their respective internal resistances.

**Procedure for using the superposition theorem **

Step-1: Retain one source at a time in the circuit and replace all other sources with their internal resistances.

Step-2: Determine the output (current or voltage) due to the single source acting alone.

Step-3: Repeat steps 1 and 2 for each of the other independent sources.

Step-4: Find the total contribution by adding algebraically all the contributions due to the independent sources.

So for above given circuit the total response or say current I thrrough register R_{2} will be equal to the sum of individual response obtained by each source.**Removing of Active Element in Superposition Theorem****Limitation** Superposition cannot be applied to power effects because the power is related to the square of the voltage across a resistor or the current through a resistor. The squared term results in a nonlinear (a curve, not a straight line) relationship between the power and the determining current or voltage.

Example:**2. THEVENIN'S & NORTON'S THEOREM **

Thevenin’s theorem states that any two output terminals of an active linear network containing independent sources (it includes voltage and current sources) can be replaced by a simple voltage source of magnitude V_{TH} in series with a single resistor R_{TH} where R_{TH} is the equivalent resistance of the network when looking from the output terminals A & B with all sources (voltage and current) removed and replaced by their internal resistances and the magnitude of V_{TH} is equal to the open circuit voltage across the A & B terminals.**The procedure for applying Thevenin’s theorem **

To find a current L I through the load resistance RL using Thevenin’s theorem, the following steps are followed:**Step-1: **Disconnect the load resistance (R_{L}) from the circuit,**Step-2: **Calculate the open-circuit voltage V_{TH} at the load terminals (A & B) after disconnecting the load resistance (R_{L}).**Step-3:** Redraw the circuit with each practical source replaced by its internal resistance. **Note,** voltage sources should be short-circuited remove them and replace with plain wire and current sources should be open-circuited just removed.**Step-4:** Look backward into the resulting circuit from the load terminals (A & B) Calculate the resistance that would exist between the load terminals

**Step-5:** Place R_{TH i}n series with V_{TH} to form the Thevenin’s equivalent circuit replacing the imaginary fencing portion or fixed part of the circuit with an equivalent practical voltage source.**Step-6:** Reconnect the original load to the Thevenin voltage circuit as shown in the load’s voltage, current and power may be calculated by a simple arithmetic operation only.

**Note:** (i) One great advantage of Thevenin’s theorem over the normal circuit reduction technique or any other technique is this: once the Thevenin equivalent circuit has been formed, it can be reused in calculating load current (I_{L}), load voltage (V_{L}) and load power (PL) for different loads.

(ii) Fortunately, with help of this theorem, one can find the choice of load resistance RL that results in the maximum power transfer to the load.**3. NORTON'S THEOREM**

Norton’s theorem states that any linear network containing can be replaced by a current source and a parallel resistor. R_{N} = R_{TH} is the equivalent resistance of the network when looking from the output terminals A & B with all sources (voltage and current) removed and replaced by their internal resistances and the magnitude of V_{TH} is equal to the open circuit voltage across the A & B terminals.

I_{N} is the Load current

Consider obtaining the equivalent circuit across the points AE. Short circuit AE as shown. Then I_{sc} = I_{1-sc} + I_{2-sc }and I_{1-sc }= 12/1 = 12 A, I_{2-sc}= 10/2 = 5 A, so that I_{sc} = 12 + 5 = 17 A

i.e. = I_{norton} = 17 A.

Total conductance of a parallel circuit is the addition of the individual conductances.

∴ G_{norton} = 1/2 + 1/1 = 1.5 S

∴ Norton's equivalent circuit is

The current I is given by **3. MAXIMUM POWER TRANSFER THEOREM**

The maximum power transfer theorem states that, to obtain maximum external power from a source with a finite Internal Impedance (Say Resistance) the resistance of the load must equal to the resistance of the source as viewed from its output terminals.

Results of Maximum Power Transfer:

- The maximum power transfer takes place when the load resistance R
_{L }= R_{th} - The maximum power transferred to the load Pmax = P
_{L}(R_{L }= R_{th}) = (V_{th})^{2}/ 4R_{th} - The efficiency of power transfer η = P
_{L}/P_{s} - where P
_{L}- Power delivered to the Load - P
_{s}- Power Generated by Source

**Note: **Maximum power transfer condition results in 50 percent efficiency in Thevenin equivalent, however much lower efficiency in the original circuit.

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