Maximum power Transfer Theorem

As you are probably aware, a normal car battery is rated at 12 V and generally has an open circuit voltage of around 13.5 V. Similarly, if we take 9 pen-torch batteries, they too will have a terminal voltage of 9×1.5 = 13.5 V. However, you would also be aware, that if your car battery is dead, you cannot go to the nearest shop, buy 9 pen-torch batteries and start your car.Why is that ?Because the pen-torch batteries, although having the same open circuit voltage does not have the necessary power (or current capacity) and hence the required current could not be given.Or if stated in different terms, it has too high an internal resistance so that the voltage would drop without giving the necessary current.

This means that a given battery (or any other energy supply, such as the mains) can only give a limited amount of power to a load.The maximum power transfer theorem defines this power, and tells us the condition at which this occurs. 

For example, if we consider the above battery, maximum voltage would be given when the current is zero, and maximum current would be given when the load is short-circuit (load voltage is zero). Under both these conditions, there is no power delivered to the load. Thus obviously in between these two extremes must be the point at which maximum power is delivered. 

The Maximum Power Transfer theorem states that for maximum active power to be delivered to the load, load impedance must correspond to the conjugate of the source impedance (or in the case of direct quantities, be equal to the source impedance).

Compensation Theorem

In many circuits, after the circuit is analysed, it is realized that only a small change need to be made to a component to get a desired result. In such a case we would normally have to recalculate.The compensation theorem allows us to compensate properly for such changes without sacrificing accuracy. 

In any linear bilateral active network, if any branch carrying a current I has its impedance Z changed by an amount ∆Z, the resulting changes that occur in the other branches are the same as those which would have been caused by the injection of a voltage source of (-) I . ∆Z in the modified branch. 

Consider the voltage drop across the modified branch. 

V +∆V = (Z +∆Z)( I +∆I) = Z . I + ∆Z . I + (Z +∆Z) . ∆I 

from the original network, V = Z . I 

∴  ∆V = ∆Z . I + (Z +∆Z) . ∆I

Since the value I is already known from the earlier analysis, and the change required in the 

impedance, ∆Z , is also known, I .∆Z is a known fixed value of voltage and may thus be represented by a source of emf I.∆Z . 

Using superposition theorem, we can easily see that the original sources in the active network give rise to the original current I, while the change corresponding to the emf I.∆Z must produce the remaining changes in the network. 

Reciprocality Theorem

The reciprocality theorem tells us that in a linear passive bilateral network an excitation and the corresponding response may be interchanged. 

In a two port network, if an excitation e(t) at port (1) produces a certain response r(t) at a port (2), then if the same excitation e(t) is applied instead to port (2), then the same response r(t) would occur at the other port (1).

Norton’s Theorem

Norton’s Theorem  states that any linear, active, bilateral network, considered across one of its ports, can be replaced by an equivalent current source (Norton’s current source) and an equivalent shunt admittance (Norton’s Admittance).Since the two sides are identical, they must be true for all conditions. Thus if we compare the current through the port in each case under short circuit conditions, and measure the input admittance of the network with the sources removed (voltage sources short-circuited and current sources open-circuited), then 

Inorton =  Isc

Ynorton =  Yin

Thevenin’s Theorem

The Thevenin’s theorem,basically gives the equivalent voltage source corresponding to an active network.If a linear, active, bilateral network is considered across one of its ports, then it can be replaced by an equivalent voltage source (Thevenin’s voltage source) and an equivalent series impedance (Thevenin’s impedance). 

        Since the two sides are identical, they must be true for all conditions. Thus if we compare the voltage across the port in each case under open circuit conditions, and measure the input impedance of the network with the sources removed (voltage sources short-circuited and current sources open-circuited), then 

Ethevenin=  Voc, and 
Zthevenin=  Zin

What is Kirchoff’s voltage law?

Kirchoff’s voltage law is based on the principle of conservation of energy. This requires that the total work done in taking a unit positive charge around a closed path and ending up at the original point is zero. 

This gives us our basic Kirchoff’s law as the algebraic sum of the potential differences taken round a closed loop is zero.

i.e. around a loop,  ΣVr= 0, where Vrare the voltages across the branches in the loop. 

va+ vb+ vc+ vd– ve= 0 

This is also sometimes stated as the sum of the emfs taken around a closed loop is equal to the sum of the voltage drops around the loop. 

What is Kirchoff’s current law?

Kirchoff’s current law is based on the principle of conservation of charge.This requires that the algebraic sum of the charges within a system cannot change. Thus the total rate of change of charge must add up to zero. Rate of change of charge is current.

This gives us our basic Kirchoff’s current law as the algebraic sum of the currents meeting at a point is zero. 
i.e. at a node, ΣIr= 0,  where Ir are the currents in the branches meeting at the node.

This is also sometimes stated as the sum of the currents entering a node is equal to the sum of the current leaving the node. The theorem is applicable not only to a node, but to a closed system.