Live Everyday as if It's ur Last Day: LAWS AND THEOREMS FOR CIRCUIT SOLVING (PART 2)

Continuing our discussion on circuit theorems, let's have a look on the remaining theorems.

SUPERPOSITION THEOREM -

It states that, the net output of a system due to multiple inputs acting through a circuit simultaneously is equal to the sum of individual outputs when each input is acting separately.

A tough statement to look at, but simple to understand.

In a multiple input (supply / source) circuit, first take a single source and short the other sources (depending on the type of source present, discussed later).
Calculate the required parameter using the single source.
Then select the next source and again short the remaining sources.
Then again calculate the required parameter.
Continue this process for the number of sources present.

Then add the values of the same parameter obtained by the various sources and that's the final answer. !!!!

Eg -

Let's find Vx across R3 using superposition theorem.

First, short circuit the voltage source (V2)

V1 = 12V
R1 = 10Ω
R3 = 30Ω || R2 = 20Ω

Ckt looks as follows -

Tot. resistance (Rt) = 10 + (30 || 20)
= 22Ω

Tot. Current (I) = 12/22 = 0.54A

Applying KVL,

» 12 - 10I - 30(Ia-Ib) = 0
» 30(Ia-Ib) = 6.6 ..... (1)

Similarly, KVL in next loop,

» 20(Ib) - 30(Ib-Ia) = 0 ...... (2)

Solving,
Ia = -0.11A , Ib = -033A

» Vx1 = 30*(-0.11+0.33) = 6.6 V

Similarly,

Vx obtained in this loop is

Vx2 = 1.636 V

Therefore, the total voltage across the resistance R3 is,

Vx = Vx1 + Vx2 = 6.6 + 1.636
= 8.182V

Multisim output of Vx

THEVENIN'S AND NORTON'S THEOREMS -

Thevenin's theorem states that any given circuit can be converted into a simple circuit consisting of an equivalent voltage source and a resistance in series with the load. The voltage source is known as Thevenin's voltage source and the Resistance in series is called the Thevenin's resistance.

Norton's theorem is just an extension which is - when the same circuit gets replaced by a current source and a resistance in parallel with it. Then the current source is called Norton's equivalent current source, and the resistance in parallel is the Norton's equivalent resistance.

Eg -

Consider the above circuit, Let the parameter needed to be found out is the voltage across the load. Then by Thevenin's theorem, the above circuit can be converted into -

Where,

Vth = the voltage across the load when it is disconnected.
Rth = Total resistance when voltage source is removed.

The same circuit when solved using Norton's theorem.

In = Norton's current flowing through the load when the load is removed.

Rn = Equivalent resistance when source is removed.

It is to be noted that, Rth = Rn.

Steps For solving using Thevenin's and Norton's theorems.

Finding the Thevenin's voltage (Vth) -

Redraw the given circuit by disconnecting the branch across which output is required.
Mark the disconnected terminals A and B.
Using the circuit analysis theorems discussed previously, determine the voltage across the terminals A and B.
This voltage is the Thevenin's voltage (Vth).

Finding the Norton's current (In) -

Redraw given circuit, disconnecting the branch.
Short the terminals A and B.
Determine the current flowing through he shorted branch.
The current obtained is Norton's current.

Calculation of Rth -

Rth = Rn = Vth / In

Findin the required parameter -

Draw the Thevenin's or Norton's equivalent circuit.
Reconnect the disconnected branch ie the load.
Calculate the parameter required.

Important point to be noted -

While redrawing the circuits, note the effects of short circuiting and open circuiting the discinnected branch on the entire circuit.

A resistance in parallel with a short circuit is redundant.
For a redundant component, current and voltage across it is zero.

TYPES OF SOURCES -

There are only two types of sources -

Voltage source - When the voltage output dominates over the current output.
Current source - When the output current dominates over the voltage output.

They can also be classified as -

Independent source - When output of the source is not dependent on any other parameter of the components connected in the circuit.
Dependent source - When the output of the source is dependent on other parameters of the components connected in the circuit.